101098rsta20021080

Mantle-circulation models with sequential dataassimilation inferring present-day mantle

structure from plate-motion histories

By Hans-Peter Bunge1 M A Richards2 an d J R Baumgardner3

1Department of Geosciences Princeton University PrincetonNJ 08544 USA (bungeprincetonedu)

2Department of Earth and Planetary Sciences University of CaliforniaBerkeley Berkeley CA 94720 USA (markrseismoberkeleyedu)

3T-Division Los Alamos National Laboratory Los AlamosNM 87545 USA (johnrblanlgov)

Published online 27 September 2002

Data assimilation is an approach to studying geodynamic models consistent simul-taneously with observables and the governing equations of mantle regow Such anapproach is essential in mantle circulation models where we seek to constrain anunknown initial condition some time in the past and thus cannot hope to use shy rst-principles convection calculations to infer the regow history of the mantle One of themost important observables for mantle-regow history comes from models of Mesozoicand Cenozoic plate motion that provide constraints not only on the surface veloc-ity of the mantle but also on the evolution of internal mantle-buoyancy forces dueto subducted oceanic slabs Here we present shy ve mantle circulation models withan assimilated plate-motion history spanning the past 120 Myr a time period forwhich reliable plate-motion reconstructions are available All models agree well withupper- and mid-mantle heterogeneity imaged by seismic tomography A simple stan-dard model of whole-mantle convection including a factor 40 viscosity increase fromthe upper to the lower mantle and predominantly internal heat generation revealsdownwellings related to Farallon and Tethys subduction Adding 35 bottom heat-ing from the core has the predictable enotect of producing prominent high-temperatureanomalies and a strong thermal boundary layer at the base of the mantle Signif-icantly delaying mantle regow through the transition zone either by modelling thedynamic enotects of an endothermic phase reaction or by including a steep factor 100viscosity rise from the upper to the lower mantle results in substantial transition-zoneheterogeneity enhanced by the enotects of trench migration implicit in the assimilatedplate-motion history An expected result is the failure to account for heterogeneitystructure in the deepest mantle below 1500 km which is inreguenced by Jurassic platemotions and thus cannot be modelled from sequential assimilation of plate motionhistories limited in age to the Cretaceous This result implies that sequential assimi-lation of past plate-motion models is inenotective in studying the temporal evolution ofcoremantle-boundary heterogeneity and that a method for extrapolating present-day information backwards in time is required For short time periods (of the order

One contribution of 14 to a Discussion Meeting `Chemical reservoirs and convection in the Earthrsquosmantlersquo

Phil Trans R Soc Lond A (2002) 360 25452567

2545

cdeg 2002 The Royal Society

2546 H-P Bunge M A Richards and J R Baumgardner

of perhaps a few tens of Myr) such a method exists in the form of crude `backwardrsquoconvection calculations For longer time periods (of the order of a mantle overturn)a rigorous approach to extrapolating information back in time exists in the form ofiterative nonlinear optimization methods that carry assimilated information into thepast through the use of an adjoint mantle convection model

Keywords mantle convection Earthrsquos interior geodynamics data assimilation

1 Introduction

Since the advent of global seismic tomography more than two decades ago (Dziewon-ski et al 1977) seismic-imaging studies have provided powerful constraints on man-tle regow Starting with the pioneering studies of Masters et al (1982) Dziewon-ski (1984) and Hager amp Clayton (1989) seismologists have demonstrated that theEarthrsquos mantle is heterogeneous on the largest scales with a prominent pattern offast lower-mantle seismic-velocity anomalies concentrated into a ring-like structurethat circumscribes the Pacishy c Richards amp Engebretson (1992) showed that thispattern correlates with the Earthrsquos relatively recent Cenozoic and Mesozoic subduc-tion history ie with the location of cold mantle downwellings at convergent plateboundaries over the past 200 Myr Their shy ndings illustrated the relationship betweendeep-mantle structure and the history of plate motion and supported the view thattomographic images are `snapshotsrsquo of a convecting mantle The seismic studies havematured to the point where the largest features are becoming well resolved through-out the mantle and the long-wavelength mantle heterogeneity structure is now widelyagreed upon (Masters et al 1996 Li amp Romanowicz 1996 Ritsema amp van Heijst2000)

In a complementary development shy ner spatial scales of mantle heterogeneity havecome into our focus with the recent high-resolution seismic-imaging studies of Grandet al (1997) and van der Hilst et al (1997) In fact considerable attention has beendevoted to mapping the fate of oceanic plates subducted since the Cretaceous Asa result detailed images of the Farallon slab under North America (van der Leeamp Nolet 1997) and the numerous slabs in the western Pacishy c (van der Hilst 1995)are now available Attention has also turned to the Mesozoic slabs of Jurassic ageburied in the Asian mantle under Siberia These ancient slabs were left behind fromthe closure of the Mongol-Okhotsk Ocean (van der Voo et al 1999) The consensusemerging from the seismic studies is consistent with highly time-dependent regowwhere some slabs enter the lower mantle relatively unimpeded while others interactwith the transition zone (van der Hilst 1995)

While seismologists progressed in mapping mantle heterogeneity geodynamicistsadvanced in mapping the parameter space of three dimensional (3D) spherical man-tle convection (Baumgardner 1985 Glatzmaier 1988 Bercovici et al 1989) Owinglargely to the dramatic increase in computational power of modern parallel comput-ers convection models are now available with sumacr cient spatial resolution to approx-imate the dynamic regime of global mantle regow (Tackley et al 1993 Bunge et al 1996 Zhong et al 2000) Early on these models emphasized the general characterof mantle convection which involves a number of complicated physical enotects Themodels highlighted the enotects of phase transitions (Tackley et al 1994) heatingmode and radial viscosity variations on mantle convection with some calculations

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2547

run at Rayleigh numbers as high as 108 (Bunge et al 1997) In a series of recenttechnical breakthroughs geodynamicists have also begun to simulate some of themost complicated aspects of mantle convection involving the enotects of brittle failurealong plate margins They achieved this technical advance either through modellinga mobile weak zone along plate margins (Zhong amp Gurnis 1996) or by using rela-tively complicated laws of strain-weakening rheology (Bercovici 1998) As a resultthe generation of plate tectonics can now be studied self-consistently in some man-tle convection models (Trompert amp Hansen 1998 Tackley 2000 Zhong et al 2000Richards et al 2001)

Unfortunately the striking progress of simulating complicated mantle convectionprocesses on a computer is beginning to reveal a fundamental problem of ab ini-tio geodynamic simulations The problem arises because mantle convection is highlysensitive to initial conditions The initial condition for mantle convection some timein the past is of course unknown This lack of initial-condition information is of greatconcern It implies that mantle convection calculations cannot be used to model theregow structures mapped by seismologists Simply put we cannot hope to explainpresent-day mantle heterogeneity by running a mantle convection model forward intime from some well-deshy ned initial state even if we had a perfect mantle convec-tion model because such a well-deshy ned initial state is of course unknown Thisstrong sensitivity to initial conditions ultimately dictates that a predictive mantleconvection model capable of reproducing seismic mantle heterogeneity is in a senseinescapably doomed to failure

Hager amp OrsquoConnell (1979) showed the initial-condition problem can be at leastpartly overcome through data assimilation By including ie assimilating present-day plate motion into analytic regow calculations Hager amp OrsquoConnell (1979) computedglobal mantle regow consistent with their model and the assimilated data Ricard etal (1993) went one step further with the assimilation problem Using a history ofsubduction they estimated the Mesozoic and Cenozoic evolution of mantle-buoyancyforces and included these buoyancy forces into global mantle regow thus pioneeringdata assimilation in time-dependent geodynamic models Stated in simple terms thepurpose of data assimilation in mantle convection is as follows using all availableinformation determine as accurately as possible the state of mantle regow at anygiven time The available information consists shy rst of the observations proper platemotions in the case of Hager amp OrsquoConnell (1979) and Ricard et al (1993) Thesecond source of information is the dynamic model and more precisely the physicallaws governing the regow These physical laws are fundamentally the principles ofconservation of mass momentum and energy and a computational model is nothingother than a numerically usable statement of these principles Before we come tothe technical aspects we recall that the initial-condition problem is encounteredin many areas of geophysical regow Not surprisingly data assimilation is used toovercome it Numerical weather-prediction models for example use a range of dataassimilation techniques (Talagrand 1997) The approach is also emerging prominentlyin oceanography (Bennett 1992 Wunsch 1996) Most often data are assimilated intoa computational model through a process known as `sequential shy lteringrsquo (Wunsch1996) Here the model is integrated over the time period for which observations havebeen made Whenever the model reaches an instant where observations are availablethe model is `updatedrsquo or `correctedrsquo with the new observation The calculation is

Phil Trans R Soc Lond A (2002)

2548 H-P Bunge M A Richards and J R Baumgardner

then restarted from the updated state and the process repeated until all availableinformation has been used

Bunge et al (1998) used this approach to compute mantle circulation models(MCMs) Starting from an assumed initial condition for the Late Cretaceous man-tle (necessarily an approximation for `truersquo Cretaceous mantle heterogeneity whichis of course unknown) they assimilated a history of Mesozoic and Cenozoic platemotion compiled by Lithgow-Bertelloni amp Richards (1998) into 3D spherical man-tle convection The assimilation updated the surface velocities computed from theirhigh-resolution mantle convection simulation with the record of past plate motionand spanned the past 120 Myr a time period for which reliable plate reconstruc-tions are available Bunge et al (1998) studied a simple mantle convection model ofuniform chemical composition heated primarily from within by radioactivity witha modest 20 component of bottom heating The model also included a signishy cantradial viscosity increase by a factor of 40 in the lower mantle and the lithosphererelative to the asthenosphere as well as the dynamic enotects of two well-establishedmantle phase transitions at depths of 410 and 670 km

In this paper we extend these models to explore a larger range of modellingparameters and to probe the enotects of realistic variations in the heating mode theradial mantle-viscosity proshy le and the strength of the two phase transitions all ofwhich are known to anotect vigorous 3D spherical mantle convection (Tackley et al 1994 Bunge et al 1997) Our calculations deshy ne a range of simple whole-mantleconvection models consistent with the governing equations of mantle regow tectonicconstraints provided by past plate-motion models and seismic tomographic observa-tions of the large-scale mantle-heterogeneity structure In the following sections webrieregy describe our numerical solution strategy and deshy ne the modelling parametersand thermodynamic reference state of our simulations We then present the lateraland spectral heterogeneity content of shy ve MCMs all of which assimilate a plate-tectonic history inferred by Lithgow-Bertelloni amp Richards (1998) for Mesozoic andCenozoic global plate motion We compare the MCMs with the high-resolution shearbody wave study of Grand et al (1997) Successful models allow for mass exchangebetween the upper and the lower mantle account for a depthwise increase in man-tle viscosity and include a substantial amount of bottom heating from the core Infact the models presented here agree remarkably well with the general character ofmantle heterogeneity inferred from seismic tomography both in their overall spectralheterogeneity content and in their particular location of cold downwelling slabs inthe upper and mid-mantle

We discuss the potential of using MCMs with assimilated plate-motion histories toconstruct time-dependent mantle regow models Applications of these models includetesting competing plate-tectonic hypotheses with seismic tomography to explore theconsistency between tectonic reconstructions of past plate motion and seismic imagesof subducted slabs Applications also include the potential to extract synthetic seis-mic datasets from the geodynamic models to compare these models more directlywith seismic observables Finally we address the most serious shortcoming in all cur-rent MCMs with assimilated plate-motion histories The limitation arises becausewell-constrained plate reconstructions are restricted to the past 120 Myr a timeperiod almost certainly less than the time-scale for generating thermal heterogeneityin the mantle The problem is aggravated by the fact that geodynamicists assimi-late past plate-motion histories into their models through sequential shy ltering The

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2549

approach carries information forward in time from the past into the present Butno information is carried back in time from the present to improve our estimatesof mantle regow at some earlier time using observations made later on The resultis profound geodynamicists cannot reliably infer the heterogeneity structure of thedeepest mantle leaving our estimates of the deep-mantle heterogeneity sensitive tothe assumed initial condition Arguably this constraint precludes us from overcomingthe initial-condition problem with sequentially assimilated plate histories a dimacr cultythat is in fact evident from the Jurassic slabs buried under Siberia as it is indeednot obvious how plate-motion histories that are limited in age to the Cretaceouscan be applied to modelling mantle heterogeneity associated with subduction in theJurassic

We conclude by noting that the initial-condition problem is not necessarily insur-mountable A more powerful approach to data assimilation involves global smoothing(Bennett 1992 Wunsch 1996) which explicitly carries information back in time fromthe present The approach is thus well suited for data assimilation in geodynamicmodels where information on present-day mantle structure is inshy nitely more detailedthan our knowledge of the mantle at some earlier time Arguably the most importantinformation on mantle structure comes from seismic-imaging studies It would there-fore seem advantageous to attempt to assimilate seismic data into time-dependentmantle circulation models This can be done with an adjoint MCM which allowsus to smooth constraints from the assimilated seismic data globally over the entiretime-domain of the MCM by relating present-day mantle structure explicitly to regowat some earlier time The adjoint equations for mantle convection have been derivedrecently together with simple numerical modelling experiments to demonstrate thatmantle regow can in principle be inferred back in time for 100 Myr (Bunge et al 2002) We suggest that data assimilation using adjoint MCMs holds some promisefor overcoming the initial-condition problem in mantle regow

2 Fluid mechanics of mantle convection

We solve the mantle convection equations using the numerical modelling code Terra(Bunge amp Baumgardner 1995) Terra solves for the momentum and energy balanceof mantle convection at inshy nite Prandtl number (no inertial forces) in a spherical shellwith the inner radius being that of the outer core and the outer radius correspondingto the Earthrsquos surface In all the following calculations we use a numerical mesh withmore than 10 million shy nite elements allowing us to provide a grid-point resolutionof ca 50 km throughout the mantle With this resolution we are able to resolve acharacteristic thermal boundary thickness of the order of 200 km

We incorporate the dynamic enotects of mantle compressibility through the anelasticliquid approximation (Glatzmaier 1988) using a Murnaghan equation (Murnaghan1951) The reference state is identical to that of Bunge et al (1997) Heat capacityand thermal conductivity are held constant with the former being set to 1134 pound103 J kgiexcl1 Kiexcl1 and the latter to 60 W miexcl1 Kiexcl1 An internal heating rate of 60 pound10iexcl12 W kgiexcl1 is imposed roughly the chondritic value (Urey 1956) We choose anupper-mantle viscosity of 80pound1021 Pa s The value is somewhat larger than estimatesof Earthrsquos upper-mantle viscosity derived from studies of post-glacial rebound (egMitrovica 1996) due to computational limitations A substantial viscosity increasefrom the upper to the lower mantle is included as suggested by studies of the geoid

Phil Trans R Soc Lond A (2002)

2550 H-P Bunge M A Richards and J R Baumgardner

Table 1 Parameter values used in this study

parameter value

outer shell radius 6370 km

inner shell radius 3480 km

T s u rface 300 K

TCM B (for internally 1900 Kheated cases)

TCM B (for bottom-heated 3300 K 3500 Kcase with high viscosity)

sup2 upper mantle 80 pound 1021 Pa s

sup2 lower mantle 40sup2 upper mantle

sup2 lithosphere 100sup2 upper mantle

thermal conductivity k 60 W m iexcl 1 Kiexcl 1

internal heating rate Qi n t 6000 pound 10iexcl 12 W kg iexcl 1

heat capacity 1134 pound 103 J kgiexcl 1 K iexcl 1

RaH (based on sup2 10 pound 108

upper mantle)

phase-change parameters

reg 410 2 MPa K iexcl 1

raquo 410 =4raquo 410 3918 kg m iexcl 3 =230 kg m iexcl 3

not 410 3026 pound 10iexcl 5 K iexcl 1

P 0034

reg 670 iexcl3 MPa K iexcl 1

raquo 670 =4raquo 670 4117 kg m iexcl 3 =380 kg m iexcl 3

not 670 2671 pound 10iexcl 5 K iexcl 1

P iexcl0087

(Ricard et al 1993) and post-glacial rebound (Mitrovica 1996) In all calculationsthat follow mantle viscosity increases monotonically through the transition zone bya factor of 40 (unless specishy ed otherwise) while the viscosity of the lithosphere|theupper 100 km in our models|is always larger by factor of 100 relative to the uppermantle The thermal boundary conditions are always constant temperature at thesurface and the coremantle boundary (CMB) with the CMB temperature specishy edto result in a bottom heat regux of less than 1 of the global surface heat regux in ourmodels with predominantly internal heat production We also investigate two modelswith a substantial heat regux from the core (35 of the total surface value) In thesemodels we increase the CMB temperature to accommodate the larger bottom heatregux Mechanical boundary conditions are specishy ed (plate) velocities at the surfaceand free-slip (no shear-stress) at the CMB The parameter values of our calculationsare listed in table 1 With these modelling parameters the intensity of internallyheated convection in our models is characterized by a Rayleigh number RaH of108 based on the upper-mantle viscosity

Arguably the most important modelling assumption in our data-assimilation mod-els is that of the initial condition We adopt the following simplifying choice weapproximate the unknown Mid-Cretaceous mantle heterogeneity as a quasi-steady

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2551

(a) (d)

(b) (e)

(c) (f)

Figure 1 Plate boundary maps for Mesozoic and Cenozoic plate stages (a) 119100 Ma(b) 9484 Ma (c) 7474 Ma (d) 4843 Ma (e) 4325 Ma and (f ) present day (100 Ma) (fromLithgow-Bertelloni amp Richards 1998) with Cenozoic plate motion compiled from the globalreconstructions of Gordon amp Jurdy (1986) Arrows indicate the direction of plate motion Theirlength is proportional to the plate speed AF denotes Africa AN Antarctica AR ArabiaAU Australia CA Caribbean CO Cocos CR Chatham Rise EU Eurasia FA Farallon INIndia KU Kula NA North America NZ Nazca PA Pacimacrc PL Philippines PH Phoenixand SA South America In the Mesozoic the ancient Izanagi (IZA) Farallon (FA) Kula (KU)and Phoenix (PH) plates occupy most of the Pacimacrc Basin Today these plates have largelydisappeared In the Cenozoic the Pacimacrc (PA) plate is inferred to undergo a rapid change frompredominantly northward to westward motion after ca 43 Myr

state of mantle regow with the oldest available plate reconstruction corresponding toMid-Cretaceous plate motion (119 Ma see shy gure 1) and assimilate Mid-Cretaceousplate motion for all of the Earthrsquos history (45 Gyr) prior to that time We accountfor the reduced convective vigour of our models by scaling the root-mean-square(RMS) plate velocities so that convective motion is neither increased nor reducedby the assimilated velocities ie we scale the RMS plate velocity to match theRMS surface velocity of convection with no imposed plate motion thus keeping the

Phil Trans R Soc Lond A (2002)

2552 H-P Bunge M A Richards and J R Baumgardner

model Peclet number unchanged The smaller surface velocities of our calculationsimply a slower overall temporal evolution of our models compared with the EarthConsequently absolute times are not meaningful in our models Instead we expresstime in terms of transit times (Gurnis amp Davies 1986a) where we deshy ne one transittime as the ratio of mantle depth to model RMS surface velocity In all calcula-tions that follow model time is normalized to `Earth timersquo by multiplying by theratio of Earthrsquos plate RMS velocities (ca 5 cm yriexcl1) to the RMS surface velocityof convection with no imposed plate motion (ca 15 cm yriexcl1) Thus imposing Mid-Cretaceous plate motion for 45 billion years of `Earth timersquo corresponds to a modeltime of ca 135 billion years Mantle regow after 45 billion `Earth yearsrsquo of assimilatedMid-Cretaceous plate motion represents heterogeneity in quasi steady state with theimposed plate motion and in lieu of other information we may regard it as a crudeapproximation for large-scale thermal heterogeneity in the Mid-Cretaceous mantle

3 Comparison of deguid and seismic mantle models

(a) Lateral heterogeneity

Before proceeding we note a key result from our previous work on MCMs In Bungeet al (1998) we performed simple model experiments designed to demonstrate theenotective spin-up time-scale associated with changes in plate motions We probed thisquestion by shy rst running convection models for a long time up to quasi-equilibriumwith a single regime of imposed plate motions (the 119100 Ma stage) and thensuddenly imposing present-day plate motions The models thus involved only twoplate-motion stages and showed that (depending on the exact model parametersused) the approximate re-equilibration time for the adjustment of mantle convectionto the new regime of plate motions varied from ca 50 Myr for the upper and mid-mantle to ca 150200 Myr for the deep mantle These fundamental response times(applicable of course only to whole-MCMs) are important to keep in mind as weconsider the enotects of changing plate motions in MCMs constrained by plate-motionhistories For example they imply directly that assimilation of plate-motion datainto mantle regow through sequential shy ltering cannot be used to infer even the recentevolution of CMB structure because well-constrained plate motions are simply notavailable for a long enough period of time (Richards et al 2000)

With the above caveat in mind we now turn our attention to study the lat-eral heterogeneity of MCMs with assimilated plate-motion histories Our referencemodel is decidedly simple It includes pure internal heat generation and a viscos-ity increase of a factor of 40 from the upper to the lower mantle in what may beregarded roughly as a geodynamic `standard modelrsquo for whole-mantle regow (Daviesamp Richards 1992) The initial condition for our calculation is mantle regow in quasi-steady state with Mid-Cretaceous plate motions (119 Ma) as described above Start-ing from this initial condition we integrate convection forward in time for 100 Myrof Earth time and assimilate past plate motions deduced from the global reconstruc-tions of Lithgow-Bertelloni amp Richards (1998) (see shy gure 1) as a time-dependentsurface-velocity boundary condition RMS plate velocities are scaled to match thesurface RMS velocities of convection without assimilated plate motion as notedabove Figure 2b shows the lateral heterogeneity of our data-assimilation model inthe transition zone (at a depth of 670 km) We also present a tomographic modelof mantle heterogeneity (shy gure 2a) inferred from the shear body-wave study of

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2546 H-P Bunge M A Richards and J R Baumgardner

of perhaps a few tens of Myr) such a method exists in the form of crude `backwardrsquoconvection calculations For longer time periods (of the order of a mantle overturn)a rigorous approach to extrapolating information back in time exists in the form ofiterative nonlinear optimization methods that carry assimilated information into thepast through the use of an adjoint mantle convection model

Keywords mantle convection Earthrsquos interior geodynamics data assimilation

1 Introduction

Since the advent of global seismic tomography more than two decades ago (Dziewon-ski et al 1977) seismic-imaging studies have provided powerful constraints on man-tle regow Starting with the pioneering studies of Masters et al (1982) Dziewon-ski (1984) and Hager amp Clayton (1989) seismologists have demonstrated that theEarthrsquos mantle is heterogeneous on the largest scales with a prominent pattern offast lower-mantle seismic-velocity anomalies concentrated into a ring-like structurethat circumscribes the Pacishy c Richards amp Engebretson (1992) showed that thispattern correlates with the Earthrsquos relatively recent Cenozoic and Mesozoic subduc-tion history ie with the location of cold mantle downwellings at convergent plateboundaries over the past 200 Myr Their shy ndings illustrated the relationship betweendeep-mantle structure and the history of plate motion and supported the view thattomographic images are `snapshotsrsquo of a convecting mantle The seismic studies havematured to the point where the largest features are becoming well resolved through-out the mantle and the long-wavelength mantle heterogeneity structure is now widelyagreed upon (Masters et al 1996 Li amp Romanowicz 1996 Ritsema amp van Heijst2000)

In a complementary development shy ner spatial scales of mantle heterogeneity havecome into our focus with the recent high-resolution seismic-imaging studies of Grandet al (1997) and van der Hilst et al (1997) In fact considerable attention has beendevoted to mapping the fate of oceanic plates subducted since the Cretaceous Asa result detailed images of the Farallon slab under North America (van der Leeamp Nolet 1997) and the numerous slabs in the western Pacishy c (van der Hilst 1995)are now available Attention has also turned to the Mesozoic slabs of Jurassic ageburied in the Asian mantle under Siberia These ancient slabs were left behind fromthe closure of the Mongol-Okhotsk Ocean (van der Voo et al 1999) The consensusemerging from the seismic studies is consistent with highly time-dependent regowwhere some slabs enter the lower mantle relatively unimpeded while others interactwith the transition zone (van der Hilst 1995)

While seismologists progressed in mapping mantle heterogeneity geodynamicistsadvanced in mapping the parameter space of three dimensional (3D) spherical man-tle convection (Baumgardner 1985 Glatzmaier 1988 Bercovici et al 1989) Owinglargely to the dramatic increase in computational power of modern parallel comput-ers convection models are now available with sumacr cient spatial resolution to approx-imate the dynamic regime of global mantle regow (Tackley et al 1993 Bunge et al 1996 Zhong et al 2000) Early on these models emphasized the general characterof mantle convection which involves a number of complicated physical enotects Themodels highlighted the enotects of phase transitions (Tackley et al 1994) heatingmode and radial viscosity variations on mantle convection with some calculations

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2547

run at Rayleigh numbers as high as 108 (Bunge et al 1997) In a series of recenttechnical breakthroughs geodynamicists have also begun to simulate some of themost complicated aspects of mantle convection involving the enotects of brittle failurealong plate margins They achieved this technical advance either through modellinga mobile weak zone along plate margins (Zhong amp Gurnis 1996) or by using rela-tively complicated laws of strain-weakening rheology (Bercovici 1998) As a resultthe generation of plate tectonics can now be studied self-consistently in some man-tle convection models (Trompert amp Hansen 1998 Tackley 2000 Zhong et al 2000Richards et al 2001)

Unfortunately the striking progress of simulating complicated mantle convectionprocesses on a computer is beginning to reveal a fundamental problem of ab ini-tio geodynamic simulations The problem arises because mantle convection is highlysensitive to initial conditions The initial condition for mantle convection some timein the past is of course unknown This lack of initial-condition information is of greatconcern It implies that mantle convection calculations cannot be used to model theregow structures mapped by seismologists Simply put we cannot hope to explainpresent-day mantle heterogeneity by running a mantle convection model forward intime from some well-deshy ned initial state even if we had a perfect mantle convec-tion model because such a well-deshy ned initial state is of course unknown Thisstrong sensitivity to initial conditions ultimately dictates that a predictive mantleconvection model capable of reproducing seismic mantle heterogeneity is in a senseinescapably doomed to failure

Hager amp OrsquoConnell (1979) showed the initial-condition problem can be at leastpartly overcome through data assimilation By including ie assimilating present-day plate motion into analytic regow calculations Hager amp OrsquoConnell (1979) computedglobal mantle regow consistent with their model and the assimilated data Ricard etal (1993) went one step further with the assimilation problem Using a history ofsubduction they estimated the Mesozoic and Cenozoic evolution of mantle-buoyancyforces and included these buoyancy forces into global mantle regow thus pioneeringdata assimilation in time-dependent geodynamic models Stated in simple terms thepurpose of data assimilation in mantle convection is as follows using all availableinformation determine as accurately as possible the state of mantle regow at anygiven time The available information consists shy rst of the observations proper platemotions in the case of Hager amp OrsquoConnell (1979) and Ricard et al (1993) Thesecond source of information is the dynamic model and more precisely the physicallaws governing the regow These physical laws are fundamentally the principles ofconservation of mass momentum and energy and a computational model is nothingother than a numerically usable statement of these principles Before we come tothe technical aspects we recall that the initial-condition problem is encounteredin many areas of geophysical regow Not surprisingly data assimilation is used toovercome it Numerical weather-prediction models for example use a range of dataassimilation techniques (Talagrand 1997) The approach is also emerging prominentlyin oceanography (Bennett 1992 Wunsch 1996) Most often data are assimilated intoa computational model through a process known as `sequential shy lteringrsquo (Wunsch1996) Here the model is integrated over the time period for which observations havebeen made Whenever the model reaches an instant where observations are availablethe model is `updatedrsquo or `correctedrsquo with the new observation The calculation is

Phil Trans R Soc Lond A (2002)

2548 H-P Bunge M A Richards and J R Baumgardner

then restarted from the updated state and the process repeated until all availableinformation has been used

Bunge et al (1998) used this approach to compute mantle circulation models(MCMs) Starting from an assumed initial condition for the Late Cretaceous man-tle (necessarily an approximation for `truersquo Cretaceous mantle heterogeneity whichis of course unknown) they assimilated a history of Mesozoic and Cenozoic platemotion compiled by Lithgow-Bertelloni amp Richards (1998) into 3D spherical man-tle convection The assimilation updated the surface velocities computed from theirhigh-resolution mantle convection simulation with the record of past plate motionand spanned the past 120 Myr a time period for which reliable plate reconstruc-tions are available Bunge et al (1998) studied a simple mantle convection model ofuniform chemical composition heated primarily from within by radioactivity witha modest 20 component of bottom heating The model also included a signishy cantradial viscosity increase by a factor of 40 in the lower mantle and the lithosphererelative to the asthenosphere as well as the dynamic enotects of two well-establishedmantle phase transitions at depths of 410 and 670 km

In this paper we extend these models to explore a larger range of modellingparameters and to probe the enotects of realistic variations in the heating mode theradial mantle-viscosity proshy le and the strength of the two phase transitions all ofwhich are known to anotect vigorous 3D spherical mantle convection (Tackley et al 1994 Bunge et al 1997) Our calculations deshy ne a range of simple whole-mantleconvection models consistent with the governing equations of mantle regow tectonicconstraints provided by past plate-motion models and seismic tomographic observa-tions of the large-scale mantle-heterogeneity structure In the following sections webrieregy describe our numerical solution strategy and deshy ne the modelling parametersand thermodynamic reference state of our simulations We then present the lateraland spectral heterogeneity content of shy ve MCMs all of which assimilate a plate-tectonic history inferred by Lithgow-Bertelloni amp Richards (1998) for Mesozoic andCenozoic global plate motion We compare the MCMs with the high-resolution shearbody wave study of Grand et al (1997) Successful models allow for mass exchangebetween the upper and the lower mantle account for a depthwise increase in man-tle viscosity and include a substantial amount of bottom heating from the core Infact the models presented here agree remarkably well with the general character ofmantle heterogeneity inferred from seismic tomography both in their overall spectralheterogeneity content and in their particular location of cold downwelling slabs inthe upper and mid-mantle

We discuss the potential of using MCMs with assimilated plate-motion histories toconstruct time-dependent mantle regow models Applications of these models includetesting competing plate-tectonic hypotheses with seismic tomography to explore theconsistency between tectonic reconstructions of past plate motion and seismic imagesof subducted slabs Applications also include the potential to extract synthetic seis-mic datasets from the geodynamic models to compare these models more directlywith seismic observables Finally we address the most serious shortcoming in all cur-rent MCMs with assimilated plate-motion histories The limitation arises becausewell-constrained plate reconstructions are restricted to the past 120 Myr a timeperiod almost certainly less than the time-scale for generating thermal heterogeneityin the mantle The problem is aggravated by the fact that geodynamicists assimi-late past plate-motion histories into their models through sequential shy ltering The

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2549

approach carries information forward in time from the past into the present Butno information is carried back in time from the present to improve our estimatesof mantle regow at some earlier time using observations made later on The resultis profound geodynamicists cannot reliably infer the heterogeneity structure of thedeepest mantle leaving our estimates of the deep-mantle heterogeneity sensitive tothe assumed initial condition Arguably this constraint precludes us from overcomingthe initial-condition problem with sequentially assimilated plate histories a dimacr cultythat is in fact evident from the Jurassic slabs buried under Siberia as it is indeednot obvious how plate-motion histories that are limited in age to the Cretaceouscan be applied to modelling mantle heterogeneity associated with subduction in theJurassic

We conclude by noting that the initial-condition problem is not necessarily insur-mountable A more powerful approach to data assimilation involves global smoothing(Bennett 1992 Wunsch 1996) which explicitly carries information back in time fromthe present The approach is thus well suited for data assimilation in geodynamicmodels where information on present-day mantle structure is inshy nitely more detailedthan our knowledge of the mantle at some earlier time Arguably the most importantinformation on mantle structure comes from seismic-imaging studies It would there-fore seem advantageous to attempt to assimilate seismic data into time-dependentmantle circulation models This can be done with an adjoint MCM which allowsus to smooth constraints from the assimilated seismic data globally over the entiretime-domain of the MCM by relating present-day mantle structure explicitly to regowat some earlier time The adjoint equations for mantle convection have been derivedrecently together with simple numerical modelling experiments to demonstrate thatmantle regow can in principle be inferred back in time for 100 Myr (Bunge et al 2002) We suggest that data assimilation using adjoint MCMs holds some promisefor overcoming the initial-condition problem in mantle regow

2 Fluid mechanics of mantle convection

We solve the mantle convection equations using the numerical modelling code Terra(Bunge amp Baumgardner 1995) Terra solves for the momentum and energy balanceof mantle convection at inshy nite Prandtl number (no inertial forces) in a spherical shellwith the inner radius being that of the outer core and the outer radius correspondingto the Earthrsquos surface In all the following calculations we use a numerical mesh withmore than 10 million shy nite elements allowing us to provide a grid-point resolutionof ca 50 km throughout the mantle With this resolution we are able to resolve acharacteristic thermal boundary thickness of the order of 200 km

We incorporate the dynamic enotects of mantle compressibility through the anelasticliquid approximation (Glatzmaier 1988) using a Murnaghan equation (Murnaghan1951) The reference state is identical to that of Bunge et al (1997) Heat capacityand thermal conductivity are held constant with the former being set to 1134 pound103 J kgiexcl1 Kiexcl1 and the latter to 60 W miexcl1 Kiexcl1 An internal heating rate of 60 pound10iexcl12 W kgiexcl1 is imposed roughly the chondritic value (Urey 1956) We choose anupper-mantle viscosity of 80pound1021 Pa s The value is somewhat larger than estimatesof Earthrsquos upper-mantle viscosity derived from studies of post-glacial rebound (egMitrovica 1996) due to computational limitations A substantial viscosity increasefrom the upper to the lower mantle is included as suggested by studies of the geoid

Phil Trans R Soc Lond A (2002)

2550 H-P Bunge M A Richards and J R Baumgardner

Table 1 Parameter values used in this study

parameter value

outer shell radius 6370 km

inner shell radius 3480 km

T s u rface 300 K

TCM B (for internally 1900 Kheated cases)

TCM B (for bottom-heated 3300 K 3500 Kcase with high viscosity)

sup2 upper mantle 80 pound 1021 Pa s

sup2 lower mantle 40sup2 upper mantle

sup2 lithosphere 100sup2 upper mantle

thermal conductivity k 60 W m iexcl 1 Kiexcl 1

internal heating rate Qi n t 6000 pound 10iexcl 12 W kg iexcl 1

heat capacity 1134 pound 103 J kgiexcl 1 K iexcl 1

RaH (based on sup2 10 pound 108

upper mantle)

phase-change parameters

reg 410 2 MPa K iexcl 1

raquo 410 =4raquo 410 3918 kg m iexcl 3 =230 kg m iexcl 3

not 410 3026 pound 10iexcl 5 K iexcl 1

P 0034

reg 670 iexcl3 MPa K iexcl 1

raquo 670 =4raquo 670 4117 kg m iexcl 3 =380 kg m iexcl 3

not 670 2671 pound 10iexcl 5 K iexcl 1

P iexcl0087

(Ricard et al 1993) and post-glacial rebound (Mitrovica 1996) In all calculationsthat follow mantle viscosity increases monotonically through the transition zone bya factor of 40 (unless specishy ed otherwise) while the viscosity of the lithosphere|theupper 100 km in our models|is always larger by factor of 100 relative to the uppermantle The thermal boundary conditions are always constant temperature at thesurface and the coremantle boundary (CMB) with the CMB temperature specishy edto result in a bottom heat regux of less than 1 of the global surface heat regux in ourmodels with predominantly internal heat production We also investigate two modelswith a substantial heat regux from the core (35 of the total surface value) In thesemodels we increase the CMB temperature to accommodate the larger bottom heatregux Mechanical boundary conditions are specishy ed (plate) velocities at the surfaceand free-slip (no shear-stress) at the CMB The parameter values of our calculationsare listed in table 1 With these modelling parameters the intensity of internallyheated convection in our models is characterized by a Rayleigh number RaH of108 based on the upper-mantle viscosity

Arguably the most important modelling assumption in our data-assimilation mod-els is that of the initial condition We adopt the following simplifying choice weapproximate the unknown Mid-Cretaceous mantle heterogeneity as a quasi-steady

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2551

(a) (d)

(b) (e)

(c) (f)

Figure 1 Plate boundary maps for Mesozoic and Cenozoic plate stages (a) 119100 Ma(b) 9484 Ma (c) 7474 Ma (d) 4843 Ma (e) 4325 Ma and (f ) present day (100 Ma) (fromLithgow-Bertelloni amp Richards 1998) with Cenozoic plate motion compiled from the globalreconstructions of Gordon amp Jurdy (1986) Arrows indicate the direction of plate motion Theirlength is proportional to the plate speed AF denotes Africa AN Antarctica AR ArabiaAU Australia CA Caribbean CO Cocos CR Chatham Rise EU Eurasia FA Farallon INIndia KU Kula NA North America NZ Nazca PA Pacimacrc PL Philippines PH Phoenixand SA South America In the Mesozoic the ancient Izanagi (IZA) Farallon (FA) Kula (KU)and Phoenix (PH) plates occupy most of the Pacimacrc Basin Today these plates have largelydisappeared In the Cenozoic the Pacimacrc (PA) plate is inferred to undergo a rapid change frompredominantly northward to westward motion after ca 43 Myr

state of mantle regow with the oldest available plate reconstruction corresponding toMid-Cretaceous plate motion (119 Ma see shy gure 1) and assimilate Mid-Cretaceousplate motion for all of the Earthrsquos history (45 Gyr) prior to that time We accountfor the reduced convective vigour of our models by scaling the root-mean-square(RMS) plate velocities so that convective motion is neither increased nor reducedby the assimilated velocities ie we scale the RMS plate velocity to match theRMS surface velocity of convection with no imposed plate motion thus keeping the

Phil Trans R Soc Lond A (2002)

2552 H-P Bunge M A Richards and J R Baumgardner

model Peclet number unchanged The smaller surface velocities of our calculationsimply a slower overall temporal evolution of our models compared with the EarthConsequently absolute times are not meaningful in our models Instead we expresstime in terms of transit times (Gurnis amp Davies 1986a) where we deshy ne one transittime as the ratio of mantle depth to model RMS surface velocity In all calcula-tions that follow model time is normalized to `Earth timersquo by multiplying by theratio of Earthrsquos plate RMS velocities (ca 5 cm yriexcl1) to the RMS surface velocityof convection with no imposed plate motion (ca 15 cm yriexcl1) Thus imposing Mid-Cretaceous plate motion for 45 billion years of `Earth timersquo corresponds to a modeltime of ca 135 billion years Mantle regow after 45 billion `Earth yearsrsquo of assimilatedMid-Cretaceous plate motion represents heterogeneity in quasi steady state with theimposed plate motion and in lieu of other information we may regard it as a crudeapproximation for large-scale thermal heterogeneity in the Mid-Cretaceous mantle

3 Comparison of deguid and seismic mantle models

(a) Lateral heterogeneity

Before proceeding we note a key result from our previous work on MCMs In Bungeet al (1998) we performed simple model experiments designed to demonstrate theenotective spin-up time-scale associated with changes in plate motions We probed thisquestion by shy rst running convection models for a long time up to quasi-equilibriumwith a single regime of imposed plate motions (the 119100 Ma stage) and thensuddenly imposing present-day plate motions The models thus involved only twoplate-motion stages and showed that (depending on the exact model parametersused) the approximate re-equilibration time for the adjustment of mantle convectionto the new regime of plate motions varied from ca 50 Myr for the upper and mid-mantle to ca 150200 Myr for the deep mantle These fundamental response times(applicable of course only to whole-MCMs) are important to keep in mind as weconsider the enotects of changing plate motions in MCMs constrained by plate-motionhistories For example they imply directly that assimilation of plate-motion datainto mantle regow through sequential shy ltering cannot be used to infer even the recentevolution of CMB structure because well-constrained plate motions are simply notavailable for a long enough period of time (Richards et al 2000)

With the above caveat in mind we now turn our attention to study the lat-eral heterogeneity of MCMs with assimilated plate-motion histories Our referencemodel is decidedly simple It includes pure internal heat generation and a viscos-ity increase of a factor of 40 from the upper to the lower mantle in what may beregarded roughly as a geodynamic `standard modelrsquo for whole-mantle regow (Daviesamp Richards 1992) The initial condition for our calculation is mantle regow in quasi-steady state with Mid-Cretaceous plate motions (119 Ma) as described above Start-ing from this initial condition we integrate convection forward in time for 100 Myrof Earth time and assimilate past plate motions deduced from the global reconstruc-tions of Lithgow-Bertelloni amp Richards (1998) (see shy gure 1) as a time-dependentsurface-velocity boundary condition RMS plate velocities are scaled to match thesurface RMS velocities of convection without assimilated plate motion as notedabove Figure 2b shows the lateral heterogeneity of our data-assimilation model inthe transition zone (at a depth of 670 km) We also present a tomographic modelof mantle heterogeneity (shy gure 2a) inferred from the shear body-wave study of

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2547

run at Rayleigh numbers as high as 108 (Bunge et al 1997) In a series of recenttechnical breakthroughs geodynamicists have also begun to simulate some of themost complicated aspects of mantle convection involving the enotects of brittle failurealong plate margins They achieved this technical advance either through modellinga mobile weak zone along plate margins (Zhong amp Gurnis 1996) or by using rela-tively complicated laws of strain-weakening rheology (Bercovici 1998) As a resultthe generation of plate tectonics can now be studied self-consistently in some man-tle convection models (Trompert amp Hansen 1998 Tackley 2000 Zhong et al 2000Richards et al 2001)

Unfortunately the striking progress of simulating complicated mantle convectionprocesses on a computer is beginning to reveal a fundamental problem of ab ini-tio geodynamic simulations The problem arises because mantle convection is highlysensitive to initial conditions The initial condition for mantle convection some timein the past is of course unknown This lack of initial-condition information is of greatconcern It implies that mantle convection calculations cannot be used to model theregow structures mapped by seismologists Simply put we cannot hope to explainpresent-day mantle heterogeneity by running a mantle convection model forward intime from some well-deshy ned initial state even if we had a perfect mantle convec-tion model because such a well-deshy ned initial state is of course unknown Thisstrong sensitivity to initial conditions ultimately dictates that a predictive mantleconvection model capable of reproducing seismic mantle heterogeneity is in a senseinescapably doomed to failure

Hager amp OrsquoConnell (1979) showed the initial-condition problem can be at leastpartly overcome through data assimilation By including ie assimilating present-day plate motion into analytic regow calculations Hager amp OrsquoConnell (1979) computedglobal mantle regow consistent with their model and the assimilated data Ricard etal (1993) went one step further with the assimilation problem Using a history ofsubduction they estimated the Mesozoic and Cenozoic evolution of mantle-buoyancyforces and included these buoyancy forces into global mantle regow thus pioneeringdata assimilation in time-dependent geodynamic models Stated in simple terms thepurpose of data assimilation in mantle convection is as follows using all availableinformation determine as accurately as possible the state of mantle regow at anygiven time The available information consists shy rst of the observations proper platemotions in the case of Hager amp OrsquoConnell (1979) and Ricard et al (1993) Thesecond source of information is the dynamic model and more precisely the physicallaws governing the regow These physical laws are fundamentally the principles ofconservation of mass momentum and energy and a computational model is nothingother than a numerically usable statement of these principles Before we come tothe technical aspects we recall that the initial-condition problem is encounteredin many areas of geophysical regow Not surprisingly data assimilation is used toovercome it Numerical weather-prediction models for example use a range of dataassimilation techniques (Talagrand 1997) The approach is also emerging prominentlyin oceanography (Bennett 1992 Wunsch 1996) Most often data are assimilated intoa computational model through a process known as `sequential shy lteringrsquo (Wunsch1996) Here the model is integrated over the time period for which observations havebeen made Whenever the model reaches an instant where observations are availablethe model is `updatedrsquo or `correctedrsquo with the new observation The calculation is

Phil Trans R Soc Lond A (2002)

2548 H-P Bunge M A Richards and J R Baumgardner

then restarted from the updated state and the process repeated until all availableinformation has been used

Bunge et al (1998) used this approach to compute mantle circulation models(MCMs) Starting from an assumed initial condition for the Late Cretaceous man-tle (necessarily an approximation for `truersquo Cretaceous mantle heterogeneity whichis of course unknown) they assimilated a history of Mesozoic and Cenozoic platemotion compiled by Lithgow-Bertelloni amp Richards (1998) into 3D spherical man-tle convection The assimilation updated the surface velocities computed from theirhigh-resolution mantle convection simulation with the record of past plate motionand spanned the past 120 Myr a time period for which reliable plate reconstruc-tions are available Bunge et al (1998) studied a simple mantle convection model ofuniform chemical composition heated primarily from within by radioactivity witha modest 20 component of bottom heating The model also included a signishy cantradial viscosity increase by a factor of 40 in the lower mantle and the lithosphererelative to the asthenosphere as well as the dynamic enotects of two well-establishedmantle phase transitions at depths of 410 and 670 km

In this paper we extend these models to explore a larger range of modellingparameters and to probe the enotects of realistic variations in the heating mode theradial mantle-viscosity proshy le and the strength of the two phase transitions all ofwhich are known to anotect vigorous 3D spherical mantle convection (Tackley et al 1994 Bunge et al 1997) Our calculations deshy ne a range of simple whole-mantleconvection models consistent with the governing equations of mantle regow tectonicconstraints provided by past plate-motion models and seismic tomographic observa-tions of the large-scale mantle-heterogeneity structure In the following sections webrieregy describe our numerical solution strategy and deshy ne the modelling parametersand thermodynamic reference state of our simulations We then present the lateraland spectral heterogeneity content of shy ve MCMs all of which assimilate a plate-tectonic history inferred by Lithgow-Bertelloni amp Richards (1998) for Mesozoic andCenozoic global plate motion We compare the MCMs with the high-resolution shearbody wave study of Grand et al (1997) Successful models allow for mass exchangebetween the upper and the lower mantle account for a depthwise increase in man-tle viscosity and include a substantial amount of bottom heating from the core Infact the models presented here agree remarkably well with the general character ofmantle heterogeneity inferred from seismic tomography both in their overall spectralheterogeneity content and in their particular location of cold downwelling slabs inthe upper and mid-mantle

We discuss the potential of using MCMs with assimilated plate-motion histories toconstruct time-dependent mantle regow models Applications of these models includetesting competing plate-tectonic hypotheses with seismic tomography to explore theconsistency between tectonic reconstructions of past plate motion and seismic imagesof subducted slabs Applications also include the potential to extract synthetic seis-mic datasets from the geodynamic models to compare these models more directlywith seismic observables Finally we address the most serious shortcoming in all cur-rent MCMs with assimilated plate-motion histories The limitation arises becausewell-constrained plate reconstructions are restricted to the past 120 Myr a timeperiod almost certainly less than the time-scale for generating thermal heterogeneityin the mantle The problem is aggravated by the fact that geodynamicists assimi-late past plate-motion histories into their models through sequential shy ltering The

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2549

approach carries information forward in time from the past into the present Butno information is carried back in time from the present to improve our estimatesof mantle regow at some earlier time using observations made later on The resultis profound geodynamicists cannot reliably infer the heterogeneity structure of thedeepest mantle leaving our estimates of the deep-mantle heterogeneity sensitive tothe assumed initial condition Arguably this constraint precludes us from overcomingthe initial-condition problem with sequentially assimilated plate histories a dimacr cultythat is in fact evident from the Jurassic slabs buried under Siberia as it is indeednot obvious how plate-motion histories that are limited in age to the Cretaceouscan be applied to modelling mantle heterogeneity associated with subduction in theJurassic

We conclude by noting that the initial-condition problem is not necessarily insur-mountable A more powerful approach to data assimilation involves global smoothing(Bennett 1992 Wunsch 1996) which explicitly carries information back in time fromthe present The approach is thus well suited for data assimilation in geodynamicmodels where information on present-day mantle structure is inshy nitely more detailedthan our knowledge of the mantle at some earlier time Arguably the most importantinformation on mantle structure comes from seismic-imaging studies It would there-fore seem advantageous to attempt to assimilate seismic data into time-dependentmantle circulation models This can be done with an adjoint MCM which allowsus to smooth constraints from the assimilated seismic data globally over the entiretime-domain of the MCM by relating present-day mantle structure explicitly to regowat some earlier time The adjoint equations for mantle convection have been derivedrecently together with simple numerical modelling experiments to demonstrate thatmantle regow can in principle be inferred back in time for 100 Myr (Bunge et al 2002) We suggest that data assimilation using adjoint MCMs holds some promisefor overcoming the initial-condition problem in mantle regow

2 Fluid mechanics of mantle convection

We solve the mantle convection equations using the numerical modelling code Terra(Bunge amp Baumgardner 1995) Terra solves for the momentum and energy balanceof mantle convection at inshy nite Prandtl number (no inertial forces) in a spherical shellwith the inner radius being that of the outer core and the outer radius correspondingto the Earthrsquos surface In all the following calculations we use a numerical mesh withmore than 10 million shy nite elements allowing us to provide a grid-point resolutionof ca 50 km throughout the mantle With this resolution we are able to resolve acharacteristic thermal boundary thickness of the order of 200 km

We incorporate the dynamic enotects of mantle compressibility through the anelasticliquid approximation (Glatzmaier 1988) using a Murnaghan equation (Murnaghan1951) The reference state is identical to that of Bunge et al (1997) Heat capacityand thermal conductivity are held constant with the former being set to 1134 pound103 J kgiexcl1 Kiexcl1 and the latter to 60 W miexcl1 Kiexcl1 An internal heating rate of 60 pound10iexcl12 W kgiexcl1 is imposed roughly the chondritic value (Urey 1956) We choose anupper-mantle viscosity of 80pound1021 Pa s The value is somewhat larger than estimatesof Earthrsquos upper-mantle viscosity derived from studies of post-glacial rebound (egMitrovica 1996) due to computational limitations A substantial viscosity increasefrom the upper to the lower mantle is included as suggested by studies of the geoid

Phil Trans R Soc Lond A (2002)

2550 H-P Bunge M A Richards and J R Baumgardner

Table 1 Parameter values used in this study

parameter value

outer shell radius 6370 km

inner shell radius 3480 km

T s u rface 300 K

TCM B (for internally 1900 Kheated cases)

TCM B (for bottom-heated 3300 K 3500 Kcase with high viscosity)

sup2 upper mantle 80 pound 1021 Pa s

sup2 lower mantle 40sup2 upper mantle

sup2 lithosphere 100sup2 upper mantle

thermal conductivity k 60 W m iexcl 1 Kiexcl 1

internal heating rate Qi n t 6000 pound 10iexcl 12 W kg iexcl 1

heat capacity 1134 pound 103 J kgiexcl 1 K iexcl 1

RaH (based on sup2 10 pound 108

upper mantle)

phase-change parameters

reg 410 2 MPa K iexcl 1

raquo 410 =4raquo 410 3918 kg m iexcl 3 =230 kg m iexcl 3

not 410 3026 pound 10iexcl 5 K iexcl 1

P 0034

reg 670 iexcl3 MPa K iexcl 1

raquo 670 =4raquo 670 4117 kg m iexcl 3 =380 kg m iexcl 3

not 670 2671 pound 10iexcl 5 K iexcl 1

P iexcl0087

(Ricard et al 1993) and post-glacial rebound (Mitrovica 1996) In all calculationsthat follow mantle viscosity increases monotonically through the transition zone bya factor of 40 (unless specishy ed otherwise) while the viscosity of the lithosphere|theupper 100 km in our models|is always larger by factor of 100 relative to the uppermantle The thermal boundary conditions are always constant temperature at thesurface and the coremantle boundary (CMB) with the CMB temperature specishy edto result in a bottom heat regux of less than 1 of the global surface heat regux in ourmodels with predominantly internal heat production We also investigate two modelswith a substantial heat regux from the core (35 of the total surface value) In thesemodels we increase the CMB temperature to accommodate the larger bottom heatregux Mechanical boundary conditions are specishy ed (plate) velocities at the surfaceand free-slip (no shear-stress) at the CMB The parameter values of our calculationsare listed in table 1 With these modelling parameters the intensity of internallyheated convection in our models is characterized by a Rayleigh number RaH of108 based on the upper-mantle viscosity

Arguably the most important modelling assumption in our data-assimilation mod-els is that of the initial condition We adopt the following simplifying choice weapproximate the unknown Mid-Cretaceous mantle heterogeneity as a quasi-steady

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2551

(a) (d)

(b) (e)

(c) (f)

Figure 1 Plate boundary maps for Mesozoic and Cenozoic plate stages (a) 119100 Ma(b) 9484 Ma (c) 7474 Ma (d) 4843 Ma (e) 4325 Ma and (f ) present day (100 Ma) (fromLithgow-Bertelloni amp Richards 1998) with Cenozoic plate motion compiled from the globalreconstructions of Gordon amp Jurdy (1986) Arrows indicate the direction of plate motion Theirlength is proportional to the plate speed AF denotes Africa AN Antarctica AR ArabiaAU Australia CA Caribbean CO Cocos CR Chatham Rise EU Eurasia FA Farallon INIndia KU Kula NA North America NZ Nazca PA Pacimacrc PL Philippines PH Phoenixand SA South America In the Mesozoic the ancient Izanagi (IZA) Farallon (FA) Kula (KU)and Phoenix (PH) plates occupy most of the Pacimacrc Basin Today these plates have largelydisappeared In the Cenozoic the Pacimacrc (PA) plate is inferred to undergo a rapid change frompredominantly northward to westward motion after ca 43 Myr

state of mantle regow with the oldest available plate reconstruction corresponding toMid-Cretaceous plate motion (119 Ma see shy gure 1) and assimilate Mid-Cretaceousplate motion for all of the Earthrsquos history (45 Gyr) prior to that time We accountfor the reduced convective vigour of our models by scaling the root-mean-square(RMS) plate velocities so that convective motion is neither increased nor reducedby the assimilated velocities ie we scale the RMS plate velocity to match theRMS surface velocity of convection with no imposed plate motion thus keeping the

Phil Trans R Soc Lond A (2002)

2552 H-P Bunge M A Richards and J R Baumgardner

model Peclet number unchanged The smaller surface velocities of our calculationsimply a slower overall temporal evolution of our models compared with the EarthConsequently absolute times are not meaningful in our models Instead we expresstime in terms of transit times (Gurnis amp Davies 1986a) where we deshy ne one transittime as the ratio of mantle depth to model RMS surface velocity In all calcula-tions that follow model time is normalized to `Earth timersquo by multiplying by theratio of Earthrsquos plate RMS velocities (ca 5 cm yriexcl1) to the RMS surface velocityof convection with no imposed plate motion (ca 15 cm yriexcl1) Thus imposing Mid-Cretaceous plate motion for 45 billion years of `Earth timersquo corresponds to a modeltime of ca 135 billion years Mantle regow after 45 billion `Earth yearsrsquo of assimilatedMid-Cretaceous plate motion represents heterogeneity in quasi steady state with theimposed plate motion and in lieu of other information we may regard it as a crudeapproximation for large-scale thermal heterogeneity in the Mid-Cretaceous mantle

3 Comparison of deguid and seismic mantle models

(a) Lateral heterogeneity

Before proceeding we note a key result from our previous work on MCMs In Bungeet al (1998) we performed simple model experiments designed to demonstrate theenotective spin-up time-scale associated with changes in plate motions We probed thisquestion by shy rst running convection models for a long time up to quasi-equilibriumwith a single regime of imposed plate motions (the 119100 Ma stage) and thensuddenly imposing present-day plate motions The models thus involved only twoplate-motion stages and showed that (depending on the exact model parametersused) the approximate re-equilibration time for the adjustment of mantle convectionto the new regime of plate motions varied from ca 50 Myr for the upper and mid-mantle to ca 150200 Myr for the deep mantle These fundamental response times(applicable of course only to whole-MCMs) are important to keep in mind as weconsider the enotects of changing plate motions in MCMs constrained by plate-motionhistories For example they imply directly that assimilation of plate-motion datainto mantle regow through sequential shy ltering cannot be used to infer even the recentevolution of CMB structure because well-constrained plate motions are simply notavailable for a long enough period of time (Richards et al 2000)

With the above caveat in mind we now turn our attention to study the lat-eral heterogeneity of MCMs with assimilated plate-motion histories Our referencemodel is decidedly simple It includes pure internal heat generation and a viscos-ity increase of a factor of 40 from the upper to the lower mantle in what may beregarded roughly as a geodynamic `standard modelrsquo for whole-mantle regow (Daviesamp Richards 1992) The initial condition for our calculation is mantle regow in quasi-steady state with Mid-Cretaceous plate motions (119 Ma) as described above Start-ing from this initial condition we integrate convection forward in time for 100 Myrof Earth time and assimilate past plate motions deduced from the global reconstruc-tions of Lithgow-Bertelloni amp Richards (1998) (see shy gure 1) as a time-dependentsurface-velocity boundary condition RMS plate velocities are scaled to match thesurface RMS velocities of convection without assimilated plate motion as notedabove Figure 2b shows the lateral heterogeneity of our data-assimilation model inthe transition zone (at a depth of 670 km) We also present a tomographic modelof mantle heterogeneity (shy gure 2a) inferred from the shear body-wave study of

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2548 H-P Bunge M A Richards and J R Baumgardner

then restarted from the updated state and the process repeated until all availableinformation has been used

Bunge et al (1998) used this approach to compute mantle circulation models(MCMs) Starting from an assumed initial condition for the Late Cretaceous man-tle (necessarily an approximation for `truersquo Cretaceous mantle heterogeneity whichis of course unknown) they assimilated a history of Mesozoic and Cenozoic platemotion compiled by Lithgow-Bertelloni amp Richards (1998) into 3D spherical man-tle convection The assimilation updated the surface velocities computed from theirhigh-resolution mantle convection simulation with the record of past plate motionand spanned the past 120 Myr a time period for which reliable plate reconstruc-tions are available Bunge et al (1998) studied a simple mantle convection model ofuniform chemical composition heated primarily from within by radioactivity witha modest 20 component of bottom heating The model also included a signishy cantradial viscosity increase by a factor of 40 in the lower mantle and the lithosphererelative to the asthenosphere as well as the dynamic enotects of two well-establishedmantle phase transitions at depths of 410 and 670 km

In this paper we extend these models to explore a larger range of modellingparameters and to probe the enotects of realistic variations in the heating mode theradial mantle-viscosity proshy le and the strength of the two phase transitions all ofwhich are known to anotect vigorous 3D spherical mantle convection (Tackley et al 1994 Bunge et al 1997) Our calculations deshy ne a range of simple whole-mantleconvection models consistent with the governing equations of mantle regow tectonicconstraints provided by past plate-motion models and seismic tomographic observa-tions of the large-scale mantle-heterogeneity structure In the following sections webrieregy describe our numerical solution strategy and deshy ne the modelling parametersand thermodynamic reference state of our simulations We then present the lateraland spectral heterogeneity content of shy ve MCMs all of which assimilate a plate-tectonic history inferred by Lithgow-Bertelloni amp Richards (1998) for Mesozoic andCenozoic global plate motion We compare the MCMs with the high-resolution shearbody wave study of Grand et al (1997) Successful models allow for mass exchangebetween the upper and the lower mantle account for a depthwise increase in man-tle viscosity and include a substantial amount of bottom heating from the core Infact the models presented here agree remarkably well with the general character ofmantle heterogeneity inferred from seismic tomography both in their overall spectralheterogeneity content and in their particular location of cold downwelling slabs inthe upper and mid-mantle

We discuss the potential of using MCMs with assimilated plate-motion histories toconstruct time-dependent mantle regow models Applications of these models includetesting competing plate-tectonic hypotheses with seismic tomography to explore theconsistency between tectonic reconstructions of past plate motion and seismic imagesof subducted slabs Applications also include the potential to extract synthetic seis-mic datasets from the geodynamic models to compare these models more directlywith seismic observables Finally we address the most serious shortcoming in all cur-rent MCMs with assimilated plate-motion histories The limitation arises becausewell-constrained plate reconstructions are restricted to the past 120 Myr a timeperiod almost certainly less than the time-scale for generating thermal heterogeneityin the mantle The problem is aggravated by the fact that geodynamicists assimi-late past plate-motion histories into their models through sequential shy ltering The

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2549

approach carries information forward in time from the past into the present Butno information is carried back in time from the present to improve our estimatesof mantle regow at some earlier time using observations made later on The resultis profound geodynamicists cannot reliably infer the heterogeneity structure of thedeepest mantle leaving our estimates of the deep-mantle heterogeneity sensitive tothe assumed initial condition Arguably this constraint precludes us from overcomingthe initial-condition problem with sequentially assimilated plate histories a dimacr cultythat is in fact evident from the Jurassic slabs buried under Siberia as it is indeednot obvious how plate-motion histories that are limited in age to the Cretaceouscan be applied to modelling mantle heterogeneity associated with subduction in theJurassic

We conclude by noting that the initial-condition problem is not necessarily insur-mountable A more powerful approach to data assimilation involves global smoothing(Bennett 1992 Wunsch 1996) which explicitly carries information back in time fromthe present The approach is thus well suited for data assimilation in geodynamicmodels where information on present-day mantle structure is inshy nitely more detailedthan our knowledge of the mantle at some earlier time Arguably the most importantinformation on mantle structure comes from seismic-imaging studies It would there-fore seem advantageous to attempt to assimilate seismic data into time-dependentmantle circulation models This can be done with an adjoint MCM which allowsus to smooth constraints from the assimilated seismic data globally over the entiretime-domain of the MCM by relating present-day mantle structure explicitly to regowat some earlier time The adjoint equations for mantle convection have been derivedrecently together with simple numerical modelling experiments to demonstrate thatmantle regow can in principle be inferred back in time for 100 Myr (Bunge et al 2002) We suggest that data assimilation using adjoint MCMs holds some promisefor overcoming the initial-condition problem in mantle regow

2 Fluid mechanics of mantle convection

We solve the mantle convection equations using the numerical modelling code Terra(Bunge amp Baumgardner 1995) Terra solves for the momentum and energy balanceof mantle convection at inshy nite Prandtl number (no inertial forces) in a spherical shellwith the inner radius being that of the outer core and the outer radius correspondingto the Earthrsquos surface In all the following calculations we use a numerical mesh withmore than 10 million shy nite elements allowing us to provide a grid-point resolutionof ca 50 km throughout the mantle With this resolution we are able to resolve acharacteristic thermal boundary thickness of the order of 200 km

We incorporate the dynamic enotects of mantle compressibility through the anelasticliquid approximation (Glatzmaier 1988) using a Murnaghan equation (Murnaghan1951) The reference state is identical to that of Bunge et al (1997) Heat capacityand thermal conductivity are held constant with the former being set to 1134 pound103 J kgiexcl1 Kiexcl1 and the latter to 60 W miexcl1 Kiexcl1 An internal heating rate of 60 pound10iexcl12 W kgiexcl1 is imposed roughly the chondritic value (Urey 1956) We choose anupper-mantle viscosity of 80pound1021 Pa s The value is somewhat larger than estimatesof Earthrsquos upper-mantle viscosity derived from studies of post-glacial rebound (egMitrovica 1996) due to computational limitations A substantial viscosity increasefrom the upper to the lower mantle is included as suggested by studies of the geoid

Phil Trans R Soc Lond A (2002)

2550 H-P Bunge M A Richards and J R Baumgardner

Table 1 Parameter values used in this study

parameter value

outer shell radius 6370 km

inner shell radius 3480 km

T s u rface 300 K

TCM B (for internally 1900 Kheated cases)

TCM B (for bottom-heated 3300 K 3500 Kcase with high viscosity)

sup2 upper mantle 80 pound 1021 Pa s

sup2 lower mantle 40sup2 upper mantle

sup2 lithosphere 100sup2 upper mantle

thermal conductivity k 60 W m iexcl 1 Kiexcl 1

internal heating rate Qi n t 6000 pound 10iexcl 12 W kg iexcl 1

heat capacity 1134 pound 103 J kgiexcl 1 K iexcl 1

RaH (based on sup2 10 pound 108

upper mantle)

phase-change parameters

reg 410 2 MPa K iexcl 1

raquo 410 =4raquo 410 3918 kg m iexcl 3 =230 kg m iexcl 3

not 410 3026 pound 10iexcl 5 K iexcl 1

P 0034

reg 670 iexcl3 MPa K iexcl 1

raquo 670 =4raquo 670 4117 kg m iexcl 3 =380 kg m iexcl 3

not 670 2671 pound 10iexcl 5 K iexcl 1

P iexcl0087

(Ricard et al 1993) and post-glacial rebound (Mitrovica 1996) In all calculationsthat follow mantle viscosity increases monotonically through the transition zone bya factor of 40 (unless specishy ed otherwise) while the viscosity of the lithosphere|theupper 100 km in our models|is always larger by factor of 100 relative to the uppermantle The thermal boundary conditions are always constant temperature at thesurface and the coremantle boundary (CMB) with the CMB temperature specishy edto result in a bottom heat regux of less than 1 of the global surface heat regux in ourmodels with predominantly internal heat production We also investigate two modelswith a substantial heat regux from the core (35 of the total surface value) In thesemodels we increase the CMB temperature to accommodate the larger bottom heatregux Mechanical boundary conditions are specishy ed (plate) velocities at the surfaceand free-slip (no shear-stress) at the CMB The parameter values of our calculationsare listed in table 1 With these modelling parameters the intensity of internallyheated convection in our models is characterized by a Rayleigh number RaH of108 based on the upper-mantle viscosity

Arguably the most important modelling assumption in our data-assimilation mod-els is that of the initial condition We adopt the following simplifying choice weapproximate the unknown Mid-Cretaceous mantle heterogeneity as a quasi-steady

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2551

(a) (d)

(b) (e)

(c) (f)

Figure 1 Plate boundary maps for Mesozoic and Cenozoic plate stages (a) 119100 Ma(b) 9484 Ma (c) 7474 Ma (d) 4843 Ma (e) 4325 Ma and (f ) present day (100 Ma) (fromLithgow-Bertelloni amp Richards 1998) with Cenozoic plate motion compiled from the globalreconstructions of Gordon amp Jurdy (1986) Arrows indicate the direction of plate motion Theirlength is proportional to the plate speed AF denotes Africa AN Antarctica AR ArabiaAU Australia CA Caribbean CO Cocos CR Chatham Rise EU Eurasia FA Farallon INIndia KU Kula NA North America NZ Nazca PA Pacimacrc PL Philippines PH Phoenixand SA South America In the Mesozoic the ancient Izanagi (IZA) Farallon (FA) Kula (KU)and Phoenix (PH) plates occupy most of the Pacimacrc Basin Today these plates have largelydisappeared In the Cenozoic the Pacimacrc (PA) plate is inferred to undergo a rapid change frompredominantly northward to westward motion after ca 43 Myr

state of mantle regow with the oldest available plate reconstruction corresponding toMid-Cretaceous plate motion (119 Ma see shy gure 1) and assimilate Mid-Cretaceousplate motion for all of the Earthrsquos history (45 Gyr) prior to that time We accountfor the reduced convective vigour of our models by scaling the root-mean-square(RMS) plate velocities so that convective motion is neither increased nor reducedby the assimilated velocities ie we scale the RMS plate velocity to match theRMS surface velocity of convection with no imposed plate motion thus keeping the

Phil Trans R Soc Lond A (2002)

2552 H-P Bunge M A Richards and J R Baumgardner

model Peclet number unchanged The smaller surface velocities of our calculationsimply a slower overall temporal evolution of our models compared with the EarthConsequently absolute times are not meaningful in our models Instead we expresstime in terms of transit times (Gurnis amp Davies 1986a) where we deshy ne one transittime as the ratio of mantle depth to model RMS surface velocity In all calcula-tions that follow model time is normalized to `Earth timersquo by multiplying by theratio of Earthrsquos plate RMS velocities (ca 5 cm yriexcl1) to the RMS surface velocityof convection with no imposed plate motion (ca 15 cm yriexcl1) Thus imposing Mid-Cretaceous plate motion for 45 billion years of `Earth timersquo corresponds to a modeltime of ca 135 billion years Mantle regow after 45 billion `Earth yearsrsquo of assimilatedMid-Cretaceous plate motion represents heterogeneity in quasi steady state with theimposed plate motion and in lieu of other information we may regard it as a crudeapproximation for large-scale thermal heterogeneity in the Mid-Cretaceous mantle

3 Comparison of deguid and seismic mantle models

(a) Lateral heterogeneity

Before proceeding we note a key result from our previous work on MCMs In Bungeet al (1998) we performed simple model experiments designed to demonstrate theenotective spin-up time-scale associated with changes in plate motions We probed thisquestion by shy rst running convection models for a long time up to quasi-equilibriumwith a single regime of imposed plate motions (the 119100 Ma stage) and thensuddenly imposing present-day plate motions The models thus involved only twoplate-motion stages and showed that (depending on the exact model parametersused) the approximate re-equilibration time for the adjustment of mantle convectionto the new regime of plate motions varied from ca 50 Myr for the upper and mid-mantle to ca 150200 Myr for the deep mantle These fundamental response times(applicable of course only to whole-MCMs) are important to keep in mind as weconsider the enotects of changing plate motions in MCMs constrained by plate-motionhistories For example they imply directly that assimilation of plate-motion datainto mantle regow through sequential shy ltering cannot be used to infer even the recentevolution of CMB structure because well-constrained plate motions are simply notavailable for a long enough period of time (Richards et al 2000)

With the above caveat in mind we now turn our attention to study the lat-eral heterogeneity of MCMs with assimilated plate-motion histories Our referencemodel is decidedly simple It includes pure internal heat generation and a viscos-ity increase of a factor of 40 from the upper to the lower mantle in what may beregarded roughly as a geodynamic `standard modelrsquo for whole-mantle regow (Daviesamp Richards 1992) The initial condition for our calculation is mantle regow in quasi-steady state with Mid-Cretaceous plate motions (119 Ma) as described above Start-ing from this initial condition we integrate convection forward in time for 100 Myrof Earth time and assimilate past plate motions deduced from the global reconstruc-tions of Lithgow-Bertelloni amp Richards (1998) (see shy gure 1) as a time-dependentsurface-velocity boundary condition RMS plate velocities are scaled to match thesurface RMS velocities of convection without assimilated plate motion as notedabove Figure 2b shows the lateral heterogeneity of our data-assimilation model inthe transition zone (at a depth of 670 km) We also present a tomographic modelof mantle heterogeneity (shy gure 2a) inferred from the shear body-wave study of

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2549

approach carries information forward in time from the past into the present Butno information is carried back in time from the present to improve our estimatesof mantle regow at some earlier time using observations made later on The resultis profound geodynamicists cannot reliably infer the heterogeneity structure of thedeepest mantle leaving our estimates of the deep-mantle heterogeneity sensitive tothe assumed initial condition Arguably this constraint precludes us from overcomingthe initial-condition problem with sequentially assimilated plate histories a dimacr cultythat is in fact evident from the Jurassic slabs buried under Siberia as it is indeednot obvious how plate-motion histories that are limited in age to the Cretaceouscan be applied to modelling mantle heterogeneity associated with subduction in theJurassic

We conclude by noting that the initial-condition problem is not necessarily insur-mountable A more powerful approach to data assimilation involves global smoothing(Bennett 1992 Wunsch 1996) which explicitly carries information back in time fromthe present The approach is thus well suited for data assimilation in geodynamicmodels where information on present-day mantle structure is inshy nitely more detailedthan our knowledge of the mantle at some earlier time Arguably the most importantinformation on mantle structure comes from seismic-imaging studies It would there-fore seem advantageous to attempt to assimilate seismic data into time-dependentmantle circulation models This can be done with an adjoint MCM which allowsus to smooth constraints from the assimilated seismic data globally over the entiretime-domain of the MCM by relating present-day mantle structure explicitly to regowat some earlier time The adjoint equations for mantle convection have been derivedrecently together with simple numerical modelling experiments to demonstrate thatmantle regow can in principle be inferred back in time for 100 Myr (Bunge et al 2002) We suggest that data assimilation using adjoint MCMs holds some promisefor overcoming the initial-condition problem in mantle regow

2 Fluid mechanics of mantle convection

We solve the mantle convection equations using the numerical modelling code Terra(Bunge amp Baumgardner 1995) Terra solves for the momentum and energy balanceof mantle convection at inshy nite Prandtl number (no inertial forces) in a spherical shellwith the inner radius being that of the outer core and the outer radius correspondingto the Earthrsquos surface In all the following calculations we use a numerical mesh withmore than 10 million shy nite elements allowing us to provide a grid-point resolutionof ca 50 km throughout the mantle With this resolution we are able to resolve acharacteristic thermal boundary thickness of the order of 200 km

We incorporate the dynamic enotects of mantle compressibility through the anelasticliquid approximation (Glatzmaier 1988) using a Murnaghan equation (Murnaghan1951) The reference state is identical to that of Bunge et al (1997) Heat capacityand thermal conductivity are held constant with the former being set to 1134 pound103 J kgiexcl1 Kiexcl1 and the latter to 60 W miexcl1 Kiexcl1 An internal heating rate of 60 pound10iexcl12 W kgiexcl1 is imposed roughly the chondritic value (Urey 1956) We choose anupper-mantle viscosity of 80pound1021 Pa s The value is somewhat larger than estimatesof Earthrsquos upper-mantle viscosity derived from studies of post-glacial rebound (egMitrovica 1996) due to computational limitations A substantial viscosity increasefrom the upper to the lower mantle is included as suggested by studies of the geoid

Phil Trans R Soc Lond A (2002)

2550 H-P Bunge M A Richards and J R Baumgardner

Table 1 Parameter values used in this study

parameter value

outer shell radius 6370 km

inner shell radius 3480 km

T s u rface 300 K

TCM B (for internally 1900 Kheated cases)

TCM B (for bottom-heated 3300 K 3500 Kcase with high viscosity)

sup2 upper mantle 80 pound 1021 Pa s

sup2 lower mantle 40sup2 upper mantle

sup2 lithosphere 100sup2 upper mantle

thermal conductivity k 60 W m iexcl 1 Kiexcl 1

internal heating rate Qi n t 6000 pound 10iexcl 12 W kg iexcl 1

heat capacity 1134 pound 103 J kgiexcl 1 K iexcl 1

RaH (based on sup2 10 pound 108

upper mantle)

phase-change parameters

reg 410 2 MPa K iexcl 1

raquo 410 =4raquo 410 3918 kg m iexcl 3 =230 kg m iexcl 3

not 410 3026 pound 10iexcl 5 K iexcl 1

P 0034

reg 670 iexcl3 MPa K iexcl 1

raquo 670 =4raquo 670 4117 kg m iexcl 3 =380 kg m iexcl 3

not 670 2671 pound 10iexcl 5 K iexcl 1

P iexcl0087

(Ricard et al 1993) and post-glacial rebound (Mitrovica 1996) In all calculationsthat follow mantle viscosity increases monotonically through the transition zone bya factor of 40 (unless specishy ed otherwise) while the viscosity of the lithosphere|theupper 100 km in our models|is always larger by factor of 100 relative to the uppermantle The thermal boundary conditions are always constant temperature at thesurface and the coremantle boundary (CMB) with the CMB temperature specishy edto result in a bottom heat regux of less than 1 of the global surface heat regux in ourmodels with predominantly internal heat production We also investigate two modelswith a substantial heat regux from the core (35 of the total surface value) In thesemodels we increase the CMB temperature to accommodate the larger bottom heatregux Mechanical boundary conditions are specishy ed (plate) velocities at the surfaceand free-slip (no shear-stress) at the CMB The parameter values of our calculationsare listed in table 1 With these modelling parameters the intensity of internallyheated convection in our models is characterized by a Rayleigh number RaH of108 based on the upper-mantle viscosity

Arguably the most important modelling assumption in our data-assimilation mod-els is that of the initial condition We adopt the following simplifying choice weapproximate the unknown Mid-Cretaceous mantle heterogeneity as a quasi-steady

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2551

(a) (d)

(b) (e)

(c) (f)

Figure 1 Plate boundary maps for Mesozoic and Cenozoic plate stages (a) 119100 Ma(b) 9484 Ma (c) 7474 Ma (d) 4843 Ma (e) 4325 Ma and (f ) present day (100 Ma) (fromLithgow-Bertelloni amp Richards 1998) with Cenozoic plate motion compiled from the globalreconstructions of Gordon amp Jurdy (1986) Arrows indicate the direction of plate motion Theirlength is proportional to the plate speed AF denotes Africa AN Antarctica AR ArabiaAU Australia CA Caribbean CO Cocos CR Chatham Rise EU Eurasia FA Farallon INIndia KU Kula NA North America NZ Nazca PA Pacimacrc PL Philippines PH Phoenixand SA South America In the Mesozoic the ancient Izanagi (IZA) Farallon (FA) Kula (KU)and Phoenix (PH) plates occupy most of the Pacimacrc Basin Today these plates have largelydisappeared In the Cenozoic the Pacimacrc (PA) plate is inferred to undergo a rapid change frompredominantly northward to westward motion after ca 43 Myr

state of mantle regow with the oldest available plate reconstruction corresponding toMid-Cretaceous plate motion (119 Ma see shy gure 1) and assimilate Mid-Cretaceousplate motion for all of the Earthrsquos history (45 Gyr) prior to that time We accountfor the reduced convective vigour of our models by scaling the root-mean-square(RMS) plate velocities so that convective motion is neither increased nor reducedby the assimilated velocities ie we scale the RMS plate velocity to match theRMS surface velocity of convection with no imposed plate motion thus keeping the

Phil Trans R Soc Lond A (2002)

2552 H-P Bunge M A Richards and J R Baumgardner

model Peclet number unchanged The smaller surface velocities of our calculationsimply a slower overall temporal evolution of our models compared with the EarthConsequently absolute times are not meaningful in our models Instead we expresstime in terms of transit times (Gurnis amp Davies 1986a) where we deshy ne one transittime as the ratio of mantle depth to model RMS surface velocity In all calcula-tions that follow model time is normalized to `Earth timersquo by multiplying by theratio of Earthrsquos plate RMS velocities (ca 5 cm yriexcl1) to the RMS surface velocityof convection with no imposed plate motion (ca 15 cm yriexcl1) Thus imposing Mid-Cretaceous plate motion for 45 billion years of `Earth timersquo corresponds to a modeltime of ca 135 billion years Mantle regow after 45 billion `Earth yearsrsquo of assimilatedMid-Cretaceous plate motion represents heterogeneity in quasi steady state with theimposed plate motion and in lieu of other information we may regard it as a crudeapproximation for large-scale thermal heterogeneity in the Mid-Cretaceous mantle

3 Comparison of deguid and seismic mantle models

(a) Lateral heterogeneity

Before proceeding we note a key result from our previous work on MCMs In Bungeet al (1998) we performed simple model experiments designed to demonstrate theenotective spin-up time-scale associated with changes in plate motions We probed thisquestion by shy rst running convection models for a long time up to quasi-equilibriumwith a single regime of imposed plate motions (the 119100 Ma stage) and thensuddenly imposing present-day plate motions The models thus involved only twoplate-motion stages and showed that (depending on the exact model parametersused) the approximate re-equilibration time for the adjustment of mantle convectionto the new regime of plate motions varied from ca 50 Myr for the upper and mid-mantle to ca 150200 Myr for the deep mantle These fundamental response times(applicable of course only to whole-MCMs) are important to keep in mind as weconsider the enotects of changing plate motions in MCMs constrained by plate-motionhistories For example they imply directly that assimilation of plate-motion datainto mantle regow through sequential shy ltering cannot be used to infer even the recentevolution of CMB structure because well-constrained plate motions are simply notavailable for a long enough period of time (Richards et al 2000)

With the above caveat in mind we now turn our attention to study the lat-eral heterogeneity of MCMs with assimilated plate-motion histories Our referencemodel is decidedly simple It includes pure internal heat generation and a viscos-ity increase of a factor of 40 from the upper to the lower mantle in what may beregarded roughly as a geodynamic `standard modelrsquo for whole-mantle regow (Daviesamp Richards 1992) The initial condition for our calculation is mantle regow in quasi-steady state with Mid-Cretaceous plate motions (119 Ma) as described above Start-ing from this initial condition we integrate convection forward in time for 100 Myrof Earth time and assimilate past plate motions deduced from the global reconstruc-tions of Lithgow-Bertelloni amp Richards (1998) (see shy gure 1) as a time-dependentsurface-velocity boundary condition RMS plate velocities are scaled to match thesurface RMS velocities of convection without assimilated plate motion as notedabove Figure 2b shows the lateral heterogeneity of our data-assimilation model inthe transition zone (at a depth of 670 km) We also present a tomographic modelof mantle heterogeneity (shy gure 2a) inferred from the shear body-wave study of

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2550 H-P Bunge M A Richards and J R Baumgardner

Table 1 Parameter values used in this study

parameter value

outer shell radius 6370 km

inner shell radius 3480 km

T s u rface 300 K

TCM B (for internally 1900 Kheated cases)

TCM B (for bottom-heated 3300 K 3500 Kcase with high viscosity)

sup2 upper mantle 80 pound 1021 Pa s

sup2 lower mantle 40sup2 upper mantle

sup2 lithosphere 100sup2 upper mantle

thermal conductivity k 60 W m iexcl 1 Kiexcl 1

internal heating rate Qi n t 6000 pound 10iexcl 12 W kg iexcl 1

heat capacity 1134 pound 103 J kgiexcl 1 K iexcl 1

RaH (based on sup2 10 pound 108

upper mantle)

phase-change parameters

reg 410 2 MPa K iexcl 1

raquo 410 =4raquo 410 3918 kg m iexcl 3 =230 kg m iexcl 3

not 410 3026 pound 10iexcl 5 K iexcl 1

P 0034

reg 670 iexcl3 MPa K iexcl 1

raquo 670 =4raquo 670 4117 kg m iexcl 3 =380 kg m iexcl 3

not 670 2671 pound 10iexcl 5 K iexcl 1

P iexcl0087

(Ricard et al 1993) and post-glacial rebound (Mitrovica 1996) In all calculationsthat follow mantle viscosity increases monotonically through the transition zone bya factor of 40 (unless specishy ed otherwise) while the viscosity of the lithosphere|theupper 100 km in our models|is always larger by factor of 100 relative to the uppermantle The thermal boundary conditions are always constant temperature at thesurface and the coremantle boundary (CMB) with the CMB temperature specishy edto result in a bottom heat regux of less than 1 of the global surface heat regux in ourmodels with predominantly internal heat production We also investigate two modelswith a substantial heat regux from the core (35 of the total surface value) In thesemodels we increase the CMB temperature to accommodate the larger bottom heatregux Mechanical boundary conditions are specishy ed (plate) velocities at the surfaceand free-slip (no shear-stress) at the CMB The parameter values of our calculationsare listed in table 1 With these modelling parameters the intensity of internallyheated convection in our models is characterized by a Rayleigh number RaH of108 based on the upper-mantle viscosity

Arguably the most important modelling assumption in our data-assimilation mod-els is that of the initial condition We adopt the following simplifying choice weapproximate the unknown Mid-Cretaceous mantle heterogeneity as a quasi-steady

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2551

(a) (d)

(b) (e)

(c) (f)

Figure 1 Plate boundary maps for Mesozoic and Cenozoic plate stages (a) 119100 Ma(b) 9484 Ma (c) 7474 Ma (d) 4843 Ma (e) 4325 Ma and (f ) present day (100 Ma) (fromLithgow-Bertelloni amp Richards 1998) with Cenozoic plate motion compiled from the globalreconstructions of Gordon amp Jurdy (1986) Arrows indicate the direction of plate motion Theirlength is proportional to the plate speed AF denotes Africa AN Antarctica AR ArabiaAU Australia CA Caribbean CO Cocos CR Chatham Rise EU Eurasia FA Farallon INIndia KU Kula NA North America NZ Nazca PA Pacimacrc PL Philippines PH Phoenixand SA South America In the Mesozoic the ancient Izanagi (IZA) Farallon (FA) Kula (KU)and Phoenix (PH) plates occupy most of the Pacimacrc Basin Today these plates have largelydisappeared In the Cenozoic the Pacimacrc (PA) plate is inferred to undergo a rapid change frompredominantly northward to westward motion after ca 43 Myr

state of mantle regow with the oldest available plate reconstruction corresponding toMid-Cretaceous plate motion (119 Ma see shy gure 1) and assimilate Mid-Cretaceousplate motion for all of the Earthrsquos history (45 Gyr) prior to that time We accountfor the reduced convective vigour of our models by scaling the root-mean-square(RMS) plate velocities so that convective motion is neither increased nor reducedby the assimilated velocities ie we scale the RMS plate velocity to match theRMS surface velocity of convection with no imposed plate motion thus keeping the

Phil Trans R Soc Lond A (2002)

2552 H-P Bunge M A Richards and J R Baumgardner

model Peclet number unchanged The smaller surface velocities of our calculationsimply a slower overall temporal evolution of our models compared with the EarthConsequently absolute times are not meaningful in our models Instead we expresstime in terms of transit times (Gurnis amp Davies 1986a) where we deshy ne one transittime as the ratio of mantle depth to model RMS surface velocity In all calcula-tions that follow model time is normalized to `Earth timersquo by multiplying by theratio of Earthrsquos plate RMS velocities (ca 5 cm yriexcl1) to the RMS surface velocityof convection with no imposed plate motion (ca 15 cm yriexcl1) Thus imposing Mid-Cretaceous plate motion for 45 billion years of `Earth timersquo corresponds to a modeltime of ca 135 billion years Mantle regow after 45 billion `Earth yearsrsquo of assimilatedMid-Cretaceous plate motion represents heterogeneity in quasi steady state with theimposed plate motion and in lieu of other information we may regard it as a crudeapproximation for large-scale thermal heterogeneity in the Mid-Cretaceous mantle

3 Comparison of deguid and seismic mantle models

(a) Lateral heterogeneity

Before proceeding we note a key result from our previous work on MCMs In Bungeet al (1998) we performed simple model experiments designed to demonstrate theenotective spin-up time-scale associated with changes in plate motions We probed thisquestion by shy rst running convection models for a long time up to quasi-equilibriumwith a single regime of imposed plate motions (the 119100 Ma stage) and thensuddenly imposing present-day plate motions The models thus involved only twoplate-motion stages and showed that (depending on the exact model parametersused) the approximate re-equilibration time for the adjustment of mantle convectionto the new regime of plate motions varied from ca 50 Myr for the upper and mid-mantle to ca 150200 Myr for the deep mantle These fundamental response times(applicable of course only to whole-MCMs) are important to keep in mind as weconsider the enotects of changing plate motions in MCMs constrained by plate-motionhistories For example they imply directly that assimilation of plate-motion datainto mantle regow through sequential shy ltering cannot be used to infer even the recentevolution of CMB structure because well-constrained plate motions are simply notavailable for a long enough period of time (Richards et al 2000)

With the above caveat in mind we now turn our attention to study the lat-eral heterogeneity of MCMs with assimilated plate-motion histories Our referencemodel is decidedly simple It includes pure internal heat generation and a viscos-ity increase of a factor of 40 from the upper to the lower mantle in what may beregarded roughly as a geodynamic `standard modelrsquo for whole-mantle regow (Daviesamp Richards 1992) The initial condition for our calculation is mantle regow in quasi-steady state with Mid-Cretaceous plate motions (119 Ma) as described above Start-ing from this initial condition we integrate convection forward in time for 100 Myrof Earth time and assimilate past plate motions deduced from the global reconstruc-tions of Lithgow-Bertelloni amp Richards (1998) (see shy gure 1) as a time-dependentsurface-velocity boundary condition RMS plate velocities are scaled to match thesurface RMS velocities of convection without assimilated plate motion as notedabove Figure 2b shows the lateral heterogeneity of our data-assimilation model inthe transition zone (at a depth of 670 km) We also present a tomographic modelof mantle heterogeneity (shy gure 2a) inferred from the shear body-wave study of

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

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Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2551

(a) (d)

(b) (e)

(c) (f)

Figure 1 Plate boundary maps for Mesozoic and Cenozoic plate stages (a) 119100 Ma(b) 9484 Ma (c) 7474 Ma (d) 4843 Ma (e) 4325 Ma and (f ) present day (100 Ma) (fromLithgow-Bertelloni amp Richards 1998) with Cenozoic plate motion compiled from the globalreconstructions of Gordon amp Jurdy (1986) Arrows indicate the direction of plate motion Theirlength is proportional to the plate speed AF denotes Africa AN Antarctica AR ArabiaAU Australia CA Caribbean CO Cocos CR Chatham Rise EU Eurasia FA Farallon INIndia KU Kula NA North America NZ Nazca PA Pacimacrc PL Philippines PH Phoenixand SA South America In the Mesozoic the ancient Izanagi (IZA) Farallon (FA) Kula (KU)and Phoenix (PH) plates occupy most of the Pacimacrc Basin Today these plates have largelydisappeared In the Cenozoic the Pacimacrc (PA) plate is inferred to undergo a rapid change frompredominantly northward to westward motion after ca 43 Myr

state of mantle regow with the oldest available plate reconstruction corresponding toMid-Cretaceous plate motion (119 Ma see shy gure 1) and assimilate Mid-Cretaceousplate motion for all of the Earthrsquos history (45 Gyr) prior to that time We accountfor the reduced convective vigour of our models by scaling the root-mean-square(RMS) plate velocities so that convective motion is neither increased nor reducedby the assimilated velocities ie we scale the RMS plate velocity to match theRMS surface velocity of convection with no imposed plate motion thus keeping the

Phil Trans R Soc Lond A (2002)

2552 H-P Bunge M A Richards and J R Baumgardner

model Peclet number unchanged The smaller surface velocities of our calculationsimply a slower overall temporal evolution of our models compared with the EarthConsequently absolute times are not meaningful in our models Instead we expresstime in terms of transit times (Gurnis amp Davies 1986a) where we deshy ne one transittime as the ratio of mantle depth to model RMS surface velocity In all calcula-tions that follow model time is normalized to `Earth timersquo by multiplying by theratio of Earthrsquos plate RMS velocities (ca 5 cm yriexcl1) to the RMS surface velocityof convection with no imposed plate motion (ca 15 cm yriexcl1) Thus imposing Mid-Cretaceous plate motion for 45 billion years of `Earth timersquo corresponds to a modeltime of ca 135 billion years Mantle regow after 45 billion `Earth yearsrsquo of assimilatedMid-Cretaceous plate motion represents heterogeneity in quasi steady state with theimposed plate motion and in lieu of other information we may regard it as a crudeapproximation for large-scale thermal heterogeneity in the Mid-Cretaceous mantle

3 Comparison of deguid and seismic mantle models

(a) Lateral heterogeneity

Before proceeding we note a key result from our previous work on MCMs In Bungeet al (1998) we performed simple model experiments designed to demonstrate theenotective spin-up time-scale associated with changes in plate motions We probed thisquestion by shy rst running convection models for a long time up to quasi-equilibriumwith a single regime of imposed plate motions (the 119100 Ma stage) and thensuddenly imposing present-day plate motions The models thus involved only twoplate-motion stages and showed that (depending on the exact model parametersused) the approximate re-equilibration time for the adjustment of mantle convectionto the new regime of plate motions varied from ca 50 Myr for the upper and mid-mantle to ca 150200 Myr for the deep mantle These fundamental response times(applicable of course only to whole-MCMs) are important to keep in mind as weconsider the enotects of changing plate motions in MCMs constrained by plate-motionhistories For example they imply directly that assimilation of plate-motion datainto mantle regow through sequential shy ltering cannot be used to infer even the recentevolution of CMB structure because well-constrained plate motions are simply notavailable for a long enough period of time (Richards et al 2000)

With the above caveat in mind we now turn our attention to study the lat-eral heterogeneity of MCMs with assimilated plate-motion histories Our referencemodel is decidedly simple It includes pure internal heat generation and a viscos-ity increase of a factor of 40 from the upper to the lower mantle in what may beregarded roughly as a geodynamic `standard modelrsquo for whole-mantle regow (Daviesamp Richards 1992) The initial condition for our calculation is mantle regow in quasi-steady state with Mid-Cretaceous plate motions (119 Ma) as described above Start-ing from this initial condition we integrate convection forward in time for 100 Myrof Earth time and assimilate past plate motions deduced from the global reconstruc-tions of Lithgow-Bertelloni amp Richards (1998) (see shy gure 1) as a time-dependentsurface-velocity boundary condition RMS plate velocities are scaled to match thesurface RMS velocities of convection without assimilated plate motion as notedabove Figure 2b shows the lateral heterogeneity of our data-assimilation model inthe transition zone (at a depth of 670 km) We also present a tomographic modelof mantle heterogeneity (shy gure 2a) inferred from the shear body-wave study of

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2552 H-P Bunge M A Richards and J R Baumgardner

model Peclet number unchanged The smaller surface velocities of our calculationsimply a slower overall temporal evolution of our models compared with the EarthConsequently absolute times are not meaningful in our models Instead we expresstime in terms of transit times (Gurnis amp Davies 1986a) where we deshy ne one transittime as the ratio of mantle depth to model RMS surface velocity In all calcula-tions that follow model time is normalized to `Earth timersquo by multiplying by theratio of Earthrsquos plate RMS velocities (ca 5 cm yriexcl1) to the RMS surface velocityof convection with no imposed plate motion (ca 15 cm yriexcl1) Thus imposing Mid-Cretaceous plate motion for 45 billion years of `Earth timersquo corresponds to a modeltime of ca 135 billion years Mantle regow after 45 billion `Earth yearsrsquo of assimilatedMid-Cretaceous plate motion represents heterogeneity in quasi steady state with theimposed plate motion and in lieu of other information we may regard it as a crudeapproximation for large-scale thermal heterogeneity in the Mid-Cretaceous mantle

3 Comparison of deguid and seismic mantle models

(a) Lateral heterogeneity

Before proceeding we note a key result from our previous work on MCMs In Bungeet al (1998) we performed simple model experiments designed to demonstrate theenotective spin-up time-scale associated with changes in plate motions We probed thisquestion by shy rst running convection models for a long time up to quasi-equilibriumwith a single regime of imposed plate motions (the 119100 Ma stage) and thensuddenly imposing present-day plate motions The models thus involved only twoplate-motion stages and showed that (depending on the exact model parametersused) the approximate re-equilibration time for the adjustment of mantle convectionto the new regime of plate motions varied from ca 50 Myr for the upper and mid-mantle to ca 150200 Myr for the deep mantle These fundamental response times(applicable of course only to whole-MCMs) are important to keep in mind as weconsider the enotects of changing plate motions in MCMs constrained by plate-motionhistories For example they imply directly that assimilation of plate-motion datainto mantle regow through sequential shy ltering cannot be used to infer even the recentevolution of CMB structure because well-constrained plate motions are simply notavailable for a long enough period of time (Richards et al 2000)

With the above caveat in mind we now turn our attention to study the lat-eral heterogeneity of MCMs with assimilated plate-motion histories Our referencemodel is decidedly simple It includes pure internal heat generation and a viscos-ity increase of a factor of 40 from the upper to the lower mantle in what may beregarded roughly as a geodynamic `standard modelrsquo for whole-mantle regow (Daviesamp Richards 1992) The initial condition for our calculation is mantle regow in quasi-steady state with Mid-Cretaceous plate motions (119 Ma) as described above Start-ing from this initial condition we integrate convection forward in time for 100 Myrof Earth time and assimilate past plate motions deduced from the global reconstruc-tions of Lithgow-Bertelloni amp Richards (1998) (see shy gure 1) as a time-dependentsurface-velocity boundary condition RMS plate velocities are scaled to match thesurface RMS velocities of convection without assimilated plate motion as notedabove Figure 2b shows the lateral heterogeneity of our data-assimilation model inthe transition zone (at a depth of 670 km) We also present a tomographic modelof mantle heterogeneity (shy gure 2a) inferred from the shear body-wave study of

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2553

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 2 Demeaned lateral heterogeneity maps at a depth of 670 km for Grand et al rsquo s S-wavemodel (a) and the macrve MCMs discussed in the text (b) Reference MCM with pure internal heatgeneration and a factor-40 viscosity increase in the lower mantle (c) (d) Same as (b) but with(c) the ereg ects of two mantle phase reactions or (d) a viscosity contrast of factor 100 betweenthe upper and the lower mantle (e) (f) Same as (b) and (d) but with 35 added core heatingIn the MCMs blue denotes cold and red denotes hot (measured on a scale of sect400 K) For theseismic model blue is fast and red is slow (measured on a scale of sect05) The seismic studyshows slabs associated with subduction under the Americas India and the western Pacimacrc Thereference MCM part (b) is similar Parts (c) and (d) show strong transition-zone heterogeneitydue to slabs delayed on their way into the deep mantle Parts (e) and (f ) show additionalhigh-temperature anomalies due to hot rising mantle deg ow

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2554 H-P Bunge M A Richards and J R Baumgardner

Grand et al (1997) The secondary-wave (S-wave) model is remarkably similar tothe independent compressional primary-body-wave tomographic model of van derHilst et al (1997) Looking shy rst at the tomographic model shy gure 2a shows sheetsof fast seismic-velocity anomalies associated with subducted slabs beneath Northand South America and corresponding to the Farallon and Nazca plates Thereare also fast seismic-velocity anomalies under India and the western Pacishy c asso-ciated with Cenozoic subduction of the Indian and Pacishy c plates beneath Eura-sia Isolated point-like low-velocity anomalies exist for example under the Red SeaRift and the East African Rift The overall heterogeneity of the data-assimilationmodel (shy gure 2b) is characterized by a similar pattern with cold anomalies con-centrated under places of past plate convergence although there is a lack of hotactive mantle upwellings owing to the absence of core heating Away from mantledownwellings temperatures are nearly uniform as expected for internally heatedconvection

We proceed in shy gure 2c by adding the dynamic enotects of the 410 km and 670 kmphase transitions to our reference model while leaving all other modelling parame-ters unchanged We use a Clapeyron slope of reg 410 = 2 MPa Kiexcl1 to model the 410 kmolivine-to-spinel phase transformation and a value of reg 670 = iexcl 3 MPa Kiexcl1 for theshy -spinel-to-perovskite transition at a depth of 670 km in agreement with experi-mental data (eg Katsura amp Ito 1989 Akaogi amp Ito 1993) Our choices for reg 410

and reg 670 imply values of the dynamically signishy cant phase-change buoyancy param-eter (Christensen amp Yuen 1984) of P410 = 0034 and P670 = iexcl 0087 respectively(see table 1) The inhibiting buoyancy enotects of the 670 km phase reaction causeupwellings and downwellings to pause in the transition zone as expected There arealso some downwellings away from plate boundaries under slow moving plates suchas Africa arising from thermal boundary-layer instabilities The delay of mantleregow through the transition zone increases the overall amplitude of transition-zoneheterogeneity compared with the standard model (shy gure 2b) in agreement with ear-lier mantle convection studies (Solheim amp Peltier 1994 Tackley et al 1994 Bungeet al 1997) But there is little enotect otherwise The sensitivity of mantle circula-tion to increases in the radial mantle-viscosity proshy le is probed by examining con-vection involving a steep factor 100 viscosity jump from the upper to the lowermantle (shy gure 2d) All other modelling parameters of the high-viscosity case areheld unchanged from the standard model in shy gure 2b As expected there is sluggishmass transport from the upper to the lower mantle not unlike the reduced masstransport we observe in mantle regow involving the enotects of the endothermic phasereaction As a result downwellings are delayed with numerous slabs concentrated atthe upperlower-mantle boundary The sluggish regow of slabs into the deeper mantleis particularly evident in regions with substantial Cenozoic subduction An exampleis the mantle beneath the western Pacishy c where subduction rates increased dramat-ically after the Pacishy c plate-motion change from a predominantly northward to amore westward motion some 43 Ma (see shy gure 1) Not surprisingly we shy nd numer-ous Cenozoic slabs in this region a heterogeneity pattern that may be comparedwith the S-wave study of Grand et al (1997) We complement our internally heatedconvection models with two calculations that involve substantial heating from thecore to explore the sensitivity of mantle regow to bottom heating Adding 35 coreheat regux either to the standard model (shy gure 2b) or to the high-viscosity case (shy g-ure 2d) results in the mantle heterogeneity shown in shy gure 2e f The modelled core

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2555

heat regux is substantially larger than that which has been estimated for the mantlebased on observations of the dynamic topography over hotspots (eg Sleep 1990)The core-heated model in shy gure 2e shows additional hot upwellings located underthe Pacishy c and the Atlantic Otherwise the general heterogeneity character withcold mantle downwellings concentrated under plate-convergence zones is similar toour internally heated convection cases The hot anomalies are more pronounced inthe high-viscosity model with 35 bottom heating (shy gure 2f) An intensishy cation ofthe anomalies in this model is expected because upwellings require larger thermalbuoyancy while passing through the stinoter lower mantle compared with the modelin shy gure 2e

We next examine the mid-mantle heterogeneity at a depth of 1500 km (shy gure 3) inour models At this depth the S-wave study (shy gure 3a) is dominated by a prominentfast seismic-velocity anomaly located under eastern North America the Caribbeanand the northernmost part of South America due to Farallon subduction Anotherfast anomaly follows the AlpineHimalayan mountain belt and reregects subductionassociated with the closure of the ancient Tethys Ocean There are also two pro-nounced low-velocity anomalies under Africa and the Azores Heterogeneity in thestandard model (shy gure 3b) is similar with slabs buried under North and South Amer-ica and under the AlpineHimalayan suture although there is a lack of hot thermalanomalies as noted earlier due to the absence of core heating A similar hetero-geneity character is evident from the phase-change-modulated case in shy gure 3c andthe model in shy gure 3d with a high-viscosity lower mantle The major dinoterencebetween both models and the reference case is due to the slower overall sinkingvelocities of subducted slabs As a result mid-mantle heterogeneity is dominatedby downwellings corresponding to earlier stages of the assimilated plate-motion his-tory The bottom-heated cases in shy gure 3e f include substantial hot upwelling regowanomalies in addition to the downwelling slabs that characterize all MCMs Theseupwellings are concentrated into broadly similar regions in the mid-Atlantic andthe eastern Pacishy c with thermal anomalies comparable with the thermal pertur-bations associated with subducted slabs an observation that agrees well with theS-wave study where the prominent low-seismic-velocity anomaly under Africa iscomparable with seismic heterogeneity associated with subducted slabs The mostsubstantial dinoterence is the location of the MCM upwellings relative to the slow seis-mic anomalies in Grand et al rsquos (1997) model which is particularly striking for theslow seismic feature under Africa Our bottom-heated MCMs show hot upwellingsnot under Africa but located several thousand kilometres further west beneath themid-Atlantic

CMB heterogeneity in our MCMs is investigated in shy gure 4 Here the bottom-heated models (shy gure 4e f) are characterized by strong lateral heterogeneity vari-ations associated with a thermal boundary layer The spatial heterogeneity patternagrees remarkably well between the two bottom-heated models However as notedearlier in the mid-mantle panels in shy gure 3 there is a substantial dinoterence in thelocation of the hot thermal anomalies relative to the S-wave study especially for thepronounced low-seismic-velocity anomaly under Africa The great similarity of thebottom-heated cases is striking given that both models show small but noticeabledinoterences in upper and mid-mantle regow at depths of 670 and 1500 km We returnto the internally heated MCMs (shy gure 4bd) and verify a broadly similar spatial pat-tern of warm and cold temperature anomalies while noting that overall heterogene-

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2556 H-P Bunge M A Richards and J R Baumgardner

(a) (b)

(c) (d)

(e) (f)

ndash05+400K

+05ndash400K

Figure 3 Same maps as macrgure 2 but for the mid-mantle at a depth of 1500 km Grand etal rsquo s seismic study (a) is dominated by fast seismic anomalies under the Americas and Indiaarising from Farallon and Tethys subduction There is also a prominent low-seismic-velocityanomaly under Africa The reference MCM part (b) is similar but lacking hot thermal anoma-lies due to the absence of core heating Parts (c) and (d) show thermal anomalies associatedwith subduction but with slabs corresponding to earlier tectonic regimes due to the delayeddeg ux of slabs into the lower mantle Part (c) shows the ereg ects of two mantle phase reactionsand (d) shows the ereg ects with a viscosity contrast factor of 100 between the upper and thelower mantle as in macrgure 2 Parts (e) and (f ) (35 added core heating) show high-temperatureanomalies associated with hot rising mantle deg ow Note that all MCMs place the Farallon slabca 1500 km further west of its location in Grand et al rsquo s model which probably redeg ects the factthat we do not explicitly account for low-angle subduction under North America during theLate CretaceousEocene Laramide orogeny

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2557

(a) (b)

(c) (d)

(e) (f)

ndash2+400K

+2ndash400K

Figure 4 Same maps as macrgure 3 but for the CMB at a depth of 2800 km The seismic studyby Grand et al (1997) (a) is plotted on a scale of sect20 and dominated by a dramaticlow-seismic-velocity anomaly under Africa and the Pacimacrc with seismically faster mantle in theintervening regions The increase in heterogeneity amplitude by a factor of four relative tothe upper and mid-mantle is probably due to chemical heterogeneity at the mantle base Allinternally heated MCMs (b) (c) and (d) are characterized by the near absence of thermal het-erogeneity owing to the absence of heating from the core The bottom-heated models (e) and(f) show a near identical thermal heterogeneity pattern that correlates with the oldest assim-ilated plate-motion stage of 119100 Ma (macrgure 1) due to the artimacrcial nature of the assumedinitial condition (see text)

ity amplitudes are markedly subdued The small heterogeneity amplitude contrastssharply with the strong thermal heterogeneity observed in the bottom-heated mod-els and dinoters from the pronounced CMB heterogeneity amplitudes inferred fromGrand et al rsquos S-wave study Weak thermal heterogeneity is of course expected inour internally heated cases owing to the absence of a lower thermal boundary layerat the bottom of the mantle

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2558 H-P Bunge M A Richards and J R Baumgardner

(b) Spectral heterogeneity

A useful way to compare geodynamic and seismic Earth models is through spec-tral heterogeneity maps (SHMs) at least in a stochastic sense These maps contourthe spectral RMS amplitude as a function of mantle depth and spherical harmonicdegree (Tackley et al 1994) Figure 5b shows the SHM for our standard modelHeterogeneity is dominated by large-scale structure concentrated at low harmonicdegrees as expected for convection involving the enotects of plate motion and viscositystratishy cation (Bunge amp Richards 1996) We note that even without bottom heat-ing the strongest heterogeneity occurs at degree 2 near the bottom of the mantleAdding the two mantle phase transitions (shy gure 5c) or a high-viscosity lower mantle(shy gure 5d) adds an intermediate-wavelength heterogeneity near a depth of 670 kmwith little enotect otherwise Adding 35 core heating has the enotect of dramaticallystrengthening the amplitude of heterogeneity in the lower thermal boundary layerin our bottom-heated cases (shy gure 5e f ) These characteristics may be comparedwith Grand et al rsquos model (shy gure 5a) where strong heterogeneity amplitude at thelowest spherical harmonic degrees 24 is concentrated near the Earthrsquos surface andthe bottom of the mantle

In fact comparison of the SHM for Grand et al rsquos model with any of the MCMsshown in shy gure 5 reveals that overall the MCMs do not mimic the relative levels ofheterogeneity found in seismic tomography At the same time the actual shape ofmid-mantle heterogeneity is rather well-modelled as seen in shy gures 24 The explana-tion for the failure of MCMs to account for the strong concentration of heterogeneityamplitude near to the top and bottom of the mantle is clear Grand et al rsquos modellike all global tomography models is dominated in terms of heterogeneity amplitudeby the uppermost and lowermost mantle signatures both of which are probablydue to chemical boundary-layer heterogeneities not modelled in our MCM approachThe lithospheric signal (uppermost 300 km or so) strongly reregects the presence ofdeep continental roots (`tectospherersquo) which in turn are thought to reregect long-livedchemical heterogeneity carried beneath ancient continental areas (see for exampleJordan 1978) The lowermost mantle seismic-velocity anomalies are likewise believedto reregect chemical heterogeneity as evidenced by the negative correlation betweenbulk-sound and shear-wave speeds in this region (eg Kennett et al 1998) In otherwords whole-mantle MCMs may stand a chance of predicting thermal structure inthe mantle interior but are inherently incapable of predicting the seismic signaturesof the lithosphere and CMB region The larger CMB heterogeneity amplitude mod-elled from our MCMs with strong core heating goes some way towards explainingthe CMB signature observed in Grand et al rsquos model But it would be premature toconclude that the strong heterogeneity amplitude of the CMB region requires highcore heat regux The SHMs therefore remind us that the development of MCMs fromdata assimilation or from other techniques is at a stage of infancy even if the `shy xesrsquonecessary for greater realism are fairly obvious

4 Discussion

We have studied shy ve MCMs with assimilated plate-motion histories but we have byno means exhausted the relevant parameter space For example we have not inves-tigated the possible range of mantle viscosity proshy les and we have not studied theenotects of all reasonable variations in phase buoyancy parameters Our simple MCMs

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2559

(a) (b)

(c) (d)

(e) (f)

top

depth

CMB

0 2 4 6 8 10 12 0 2 4 6 8 10 12spherical harmonic degree spherical harmonic degree

Figure 5 Spectral heterogeneity maps (SHMs) for the macrve MCMs and the S-wave study of Grandet al (1997) shown in macrgures 24 (a) S-wave study Strong heterogeneity probably associatedwith chemical variations dominates the top and bottom of the mantle (b) Reference MCM (seetext) Heterogeneity is concentrated into the gravest spherical harmonic degrees 24 at the topand bottom of the mantle Parts (c) and (d) show the same maps as in (b) but with (c) theereg ects of two mantle phase reactions or (d) with a viscosity contrast factor of 100 betweenthe upper and the lower mantle An additional heterogeneity amplitude exists in the transitionzone Parts (e) (f ) show the same maps as in (b) and (d) but with 35 added core heatingStrong heterogeneity associated with the lower thermal boundary layer dominates the SHMsRMS spectral amplitude is contoured as a function of mantle depth (surface at the top CMB atthe bottom) and spherical harmonic degree (012) Each panel is normalized to the maximumamplitude for that panel and there are 10 contour intervals

do not include the enotects of lateral viscosity variations arising from lateral varia-tions in temperature and strain rate (eg Tackley 2000 Trompert amp Hansen 1998Zhong et al 2000) An even more important omission is that we have not considered

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2560 H-P Bunge M A Richards and J R Baumgardner

the enotects of compositional stratishy cation which are believed to be important in thedeep mantle (Kellogg et al 1999) The choice of a reduced convective vigour in ourmodels stems mainly from computational considerations Each model presented heresaturates the largest supercomputer available to us (a LINUX-based Beowulf PCcluster) However our results are sumacr cient to infer the general response of mantlecirculation with `Earth-likersquo viscosity stratishy cation core heating and phase changesat a Rayleigh number of 108 to the enotects of assimilated plate-motion histories Theresponse is in fact quite predictable Heterogeneity is dominated by subducted slabsdescending into the mantle at convergent plate boundaries The dominance of long-wavelength structure in our models agrees well with the prominent large-scale mantleheterogeneity structure imaged by seismologists (Dziewonski 1984) and with the redspectrum of the geoid which has a strong peak at spherical harmonic degrees 23Our models are similar in this respect to the data-assimilation models of Ricard etal (1993) which yield excellent shy ts to the observed geoid

There are however fundamental limitations Arguably the most profound resultis our inability to resolve deep-mantle structure The result is easily anticipatedfrom the prediction that deep-mantle advection velocities scale roughly with thelogarithm of the upperlower-mantle viscosity contrast (Gurnis amp Davies 1986b)The signishy cant depthwise viscosity increase in the mantle implies an enotective tran-sit time of the order of 150200 Myr as slabs sink through the mantle a periodconsiderably longer than the time spanned by models of past plate motion Thuswe should not hope to infer the heterogeneity structure of the deepest mantle fromassimilated plate-motion histories spanning only ca 100 Myr The di culty is obvi-ous in our bottom-heated models In fact both models show nearly identical regowstructure at the CMB despite their dinoterent radial viscosity proshy le The dimacr cultyis equally apparent from the near-stationary pattern of hot upwelling mantle regow inthese models evidenced for example by comparing hot mantle regions at the CMBwith the locations of high-temperature anomalies in the mid-Atlantic at a depth of1500 km The high-temperature anomalies agree well with patches of hot mantle atthe CMB but show little resemblance to the location of the African low-seismic-velocity anomaly in Grand et al rsquos S-wave study We can understand the spatial pat-tern of deep-mantle heterogeneity in our data-assimilation models to be an artefactof the model initialization This is evident when we compare it with the 119100 MaMid-Cretaceous plate reconstruction (shy gure 1) and note that it corresponds closelyto the location of spreading and subduction plate margins in the reconstructionsof Lithgow-Bertelloni amp Richards (1998) In other words deep-mantle heterogene-ity in our data-assimilation models is entirely controlled by the oldest assimilatedplate-motion stage used to initialize regow in the Mid-Cretaceous mantle We couldof course make other assumptions to approximate Mid-Cretaceous mantle hetero-geneity rather than assuming it can be inferred from regow in a quasi-steady statewith Mid-Cretaceous plate motion But regardless of whatever we may assume forheterogeneity in the Mid-Cretacous our results demonstrate clearly that sequentialassimilation of plate-motion histories into mantle regow is insumacr cient to overcome theinitial-condition problem

Our shy nding has important implications for regedgling studies of the temporal evolu-tion of mantle heterogeneity at the CMB There are a number of attractive reasonsfor trying to understand time variations in the large-scale structure of the CMBFor example recent magnetohydrodynamic simulations of the core indicate that the

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2561

geodynamo is sensitive to changes in CMB heterogeneity (Glatzmaier et al 1999Bloxham 2000) Palaeomagnetists have long speculated about possible causes forthe Cretaceous Normal Superchron (CNS) a remarkable time period lasting fromca 120 to 85 Ma when the geodynamo occupied a single magnetic polarity It appearsentirely plausible that the great stability of the geodynamo in the Mid-Cretaceousoccurred in response to large-scale temperature variations at the CMB associatedwith Mesozoic mantle convection Our results demonstrate that we cannot test thishypothesis in MCMs with sequentially assimilated plate-motion histories (see a moredetailed analysis of this problem in Richards et al (2000))

Our results for heterogeneity in the mid- and upper mantle are more encourag-ing Here the MCMs show small but noticeable dinoterences aside from the shy rst-orderobservation that cold downwellings agree well with subducted slabs in Grand etal rsquos S-wave study implying that plate-motion histories can be used to improve ourunderstanding of regow in the upper and mid-mantle There is however an importantadditional caveat When assimilating past plate-motion models we must realize thattectonic reconstructions particularly for those plates that have completely disap-peared (Izanagi Phoenix Kula) or are in the process of disappearing (Farallon) areonly approximate especially the locations of subduction zones and to a lesser extentthe poles of rotation The great uncertainties associated with plate-motion histo-ries imply equally signishy cant uncertainties in the location of mantle downwellingsin our models We take the mid-mantle location of the Farallon plate under NorthAmerica as an example All shy ve MCMs show the ancient Farallon slab further westof its location in Grand et al rsquos seismic study by ca 1500 km The displacement iswell outside the inherent spatial resolution both in the S-wave study and in our regowcalculations which employ a grid-point resolution of ca 50 km throughout the man-tle It arises probably from an unusual period during the Late CretaceousEoceneLaramide orogeny (from 70 to 40 Ma) when the Farallon plate subducted at a shal-low angle under North America (Dickenson amp Snyder 1978) We can test this plate-tectonic hypothesis explicitly by simulating the enotects of low-angle subduction inour models Assuming a shallow dip angle of the Farallon slab under North Americaduring the Laramide orogeny we shy nd that the MCMs match Grand et al rsquos imageof the Farallon slab rather well (Bunge amp Grand 2000) There are numerous otherplate-tectonic uncertainties related especially to the Mesozoic plate boundaries inthe western Pacishy c For example the ancient Izanagi subduction zone in the north-western Pacishy c is poorly known implying signishy cant uncertainties in mid-mantleMCM heterogeneity structure under Eurasia The MCMs thus onoter an intriguingmechanism to test competing plate-tectonic hypotheses with tomographic studies toimprove past plate-reconstruction models

The strong local enotect of the shy -spinel-to-perovskite phase reaction in the transitionzone may seem at shy rst surprising The associated MCM heterogeneity with numer-ous slabs located at a depth of 670 km is quite unlike the heterogeneity imagedin Grand et al rsquos S-wave study The dramatic phase-reaction enotect is not easilyanticipated given our modest choice for the value of the Clapeyron slope withreg 670 = iexcl 3 MPa Kiexcl1 However there are other factors besides the Clapeyron slopethat inreguence the ability of downwellings to penetrate the lower mantle such astrench migration ie the lateral movement of subduction zones (Christensen 1996)Considerable trench migration occurred over the past 120 Myr in the IndoEurasianplate-convergence zone and the subduction zones surrounding the Pacishy c evidenced

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2562 H-P Bunge M A Richards and J R Baumgardner

for example by the westward motion of subduction onot-shore of North and SouthAmerica due to the opening of the Atlantic Basin Trench migration enters our mod-els through the assimilated plate-motion history although we note that the strongoverall enotect of the phase reaction would be onotset in part if we modelled the enhancedstinotness of cold subducted lithosphere which promotes slab regow into the lower man-tle by acting as a stress guide (Zhong amp Gurnis 1994) However it is clear fromour calculations that assimilated plate-motion histories onoter important additionalconstraints on the development of mantle heterogeneity in the transition zone

We must make another qualifying remark our bottom-heated models draw 35 oftheir energy budget from the core an amount substantially larger than that whichhas been estimated for the mantle based on observations of dynamic topographyover hotspots (eg Sleep 1990) Analog mantle convection experiments indicate thatdeep-mantle plumes may be captured by large-scale mantle circulation suggestingthat there could be a signishy cant amount of `hiddenrsquo core heat regux in the mantleaside from the heat regux readily associated with plumes (Jellinek et al 2002) Inour bottom-heated models we specify a CMB temperature of 3300 K (3500 K for thecase with high viscosity see table 1) arguably a conservative value of that whichhas been estimated for the core based on the melting temperature of iron (Boehler2000) Using this thermal boundary condition the bottom-heated MCMs yield atemperature drop of more than 1000 K across the lower thermal boundary layer inagreement with independent mineral-physics estimates for the temperature contrastacross D00 (Williams 1997) The sharp temperature contrast across the bottom of themantle is promoted further because the mantle geotherm is likely to be subadiabaticoutside of thermal boundary layers (Bunge et al 2001) so a high core heat reguxwould seem plausible A signishy cant amount of core heating is suggested by the energyrequirements of numerical dynamo models (Glatzmaier amp Roberts 1995 Kuang ampBloxham 1997) and would be consistent with thermal history calculations of the core(Bunotett et al 1996) although we add that a chemically distinct layer at the mantlebase would lower the core heat regux and would help to reduce the excess temperatureof mantle plumes (Farnetani 1997) implied by our calculations The existence of thislayer appears likely from geochemical considerations (Hofmann 1997) and has beenexplored in dynamic models (Kellogg et al 1999)

There is a last remark we must make Geodynamicists tend to compare theirmantle regow models directly with images of mantle heterogeneity derived from tomo-graphic inversions of seismic data The approach is problematic because the unevendistribution of seismic sources and receivers and the non-uniqueness of the inverseproblem impose a complex tomographic shy lter on mantle structure that should beconsidered prior to any comparison of geodynamic and seismic mantle models Wecould include the enotects of tomographic shy ltering explicitly in our models by comput-ing synthetic travel-time anomalies from the MCMs and inverting these synthetictravel-time residuals for the mantle structure (eg Johnson et al 1993) In MCMsthis approach reveals that tomographic shy ltering is probably minor in areas of highray density (Bunge amp Davies 2001) However a far more interesting observation canbe drawn from such synthetic seismic datasets The travel-time anomalies obtainedfrom MCMs reveal that the co-location of earthquake sources and mantle down-wellings acts to bias the arrival times of the synthetics in a systematic way Thebias is evident from histograms of the synthetic travel-time residuals which demon-strate that the arrival-time residuals associated with all shy ve MCMs presented here

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2563

are characterized by a negative mean In other words synthetic arrival-time residualsin our MCMs reveal that the Earthrsquos seismic velocity structure as sampled by body-waves is fast compared with an average reference Earth as measured for examplefrom free oscillations (Davies amp Bunge 2001) Extracting synthetic seismic observ-ables from MCMs is an attractive way to advance our understanding of deep Earthstructure especially as seismologists make rapid progress in their abilities to modelcomplex theoretical phenomena of seismic wave propagation (Komatitsch amp Tromp1999 Dahlen et al 2000) and in their enotorts to collect new high-resolution datasetson a continental scale (eg Meltzer et al 1999)

5 Conclusion

We have studied mantle circulation by sequentially assimilating past plate-motionhistories into reguid-dynamic models of Earthrsquos mantle We observe excellent cor-relation of cold downwellings in our models with the location of subducted slabsinferred from seismic images of the mantle Our calculations deshy ne a range of sim-ple whole-MCMs consistent with the governing equations of mantle regow tectonicconstraints from past plate-motion models and seismic tomographic observations ofthe large-scale-mantle heterogeneity structure We shy nd that upper- and mid-mantleheterogeneity structure is sensitive to variations in the amount of core heating theradial mantle viscosity proshy le the strength of phase reactions and the uncertaintiesin the assimilated plate-motion histories implying that sensitivity studies in MCMscan be applied to improve our estimates of the radial mantle-viscosity proshy le theenotects of phase reactions the ratio of internal to external heating and reconstructionsof past plate movement

In contrast deep-mantle heterogeneity is insensitive to the constraints providedby the available record of Mesozoic and Cenozoic plate motion but is sensitive tothe assumed initial condition for heterogeneity in the Mid-Cretaceous mantle Ourresults suggest that sequential shy ltering of plate-motion histories (where the assim-ilated data inreguence mantle regow only at later times but no information is carriedback in time from the present) is insu cient to infer the temporal evolution of deep-mantle heterogeneity which could be modelled from a variational approach to assim-ilation by incorporating constraints for example from seismic data into mantle regowthrough adjoint MCMs Over short time periods (perhaps a few tens of Myr) a crudeapproach to assimilating seismic data into mantle regow can be derived from simplebackwards advection of the `knownrsquo present-day state of the mantle (eg Steinbergeramp OrsquoConnell 1997 Bunge amp Richards 1992) to study the most recent evolution ofCMB structure However for longer time-scales over which thermal dinotusion acrossboundary layers becomes important (ca 50 Myr or more) constraints from seismictomography must be assimilated into mantle regow through a more powerful approachto assimilation involving adjoint methods which are clearly superior in recoveringthe unknown initial conditions of the mantle (Bunge et al 2002) Data assimilationpromises to be a rich shy eld for advancing our theoretical understanding of mantledynamics in the coming decade

Computations for this work were carried out on a Beowulf cluster at Princeton Universityrsquo sDepartment of Geosciences and funded by NSF grant EAR-9814635 HPB acknowledges sup-port from NSF grants EAR-0106651 and EAR-9980457

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

2564 H-P Bunge M A Richards and J R Baumgardner

References

Akaogi M amp Ito E 1993 Remacrnement of enthalpy measurement of MgSiO3 perovskite andnegative pressuretemperature slopes for perovskite-forming reactions Geophys Res Lett20 18391842

Baumgardner J R 1985 Three dimensional treatment of convective deg ow in the Earthrsquo s mantleJ Stat Phys 39 501511

Bennett A F 1992 Inverse methods in physical oceanography Cambridge University Press

Bercovici D 1998 Generation of plate tectonics from lithospheremantle deg ow and void-volatileself-lubrication Earth Planet Sci Lett 154 139151

Bercovici D Schubert G amp Glatzmaier G A 1989 Three-dimensional spherical models ofconvection in the Earthrsquo s mantle Science 244 950955

Bloxham J 2000 Sensitivity of the geomagnetic axial dipole to thermal coremantle interactionsNature 405 6365

Boehler R 2000 High-pressure experiments and the phase diagram of lower mantle and corematerials Rev Geophys 38 221245

Bureg ett B A Huppert H E Lister J R amp Woods A W 1996 On the thermal evolution ofthe Earthrsquo s core J Geophys Res 101 79898006

Bunge H-P amp Baumgardner J R 1995 Mantle convection modeling on parallel virtualmachines Comput Phys 9 207215

Bunge H-P amp Davies J H 2001 Tomographic images of a mantle circulation model GeophysRes Lett 28 7780

Bunge H-P amp Grand S P 2000 Mesozoic plate-motion history below the northeast PacimacrcOcean from seismic images of the subducted Farallon slab Nature 405 337340

Bunge H-P amp Richards M A 1992 The backward-problem of plate tectonics and mantleconvection (abstract) Eos 73 281 (Spring Meeting Suppl)

Bunge H-P amp Richards M A 1996 The origin of large scale structure in mantle convectionereg ects of plate motion and viscosity stratimacrcation Geophys Res Lett 23 29872990

Bunge H-P Richards M A amp Baumgardner J R 1996 Ereg ect of depth-dependent viscosityon the planform of mantle convection Nature 376 436438

Bunge H-P Richards M A amp Baumgardner J R 1997 A sensitivity study of three-dimensional spherical mantle convection at 108 Rayleigh number ereg ects of depth-dependentviscosity heating mode and an endothermic phase change J Geophys Res 102 11 99112 007

Bunge H-P Richards M A Lithgow-Bertelloni C Baumgardner J R Grand S P ampRomanowicz B 1998 Time scales and heterogeneous structure in geodynamic Earth modelsScience 280 9195

Bunge H-P Ricard Y amp Matas J 2001 Non-adiabaticity in mantle convection Geophys ResLett 28 879882

Bunge H-P Hagelberg C amp Travis B 2002 Mantle circulation models with variational dataassimilation inferring past mantle deg ow and structure from plate motion histories and seismictomography (In the press)

Christensen U R 1996 The indeg uence of trench migration on slab penetration into the lowermantle Earth Planet Sci Lett 140 2739

Christensen U R amp Yuen D A 1984 The interaction of a subducting lithospheric slab with achemical or phase boundary J Geophys Res 89 43894402

Dahlen F A Hung S H amp Nolet G 2000 Frechet kernels for macrnite-frequency traveltimes ITheory Geophys J Int 141 157174

Davies J H amp Bunge H-P 2001 Seismically fastrsquo geodynamic mantle models Geophys ResLett 28 7376

Davies G F amp Richards M A 1992 Mantle convection J Geol 100 151206

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2565

Dickenson W R amp Snyder W S 1978 Plate tectonics of the Laramide orogeny GeophysicalSociety of America Memoir Series vol 151 pp 355366

Dziewonski A M 1984 Mapping the lower mantle determination of lateral heterogeneity in Pvelocity up to degree and order 6 J Geophys Res 89 59295952

Dziewonski A M Hager B H amp Orsquo Connell R J 1977 Large-scale heterogeneities in the lowermantle J Geophys Res 82 239255

Farnetani C G 1997 Excess temperature of mantle plumes the role of chemical stratimacrcationacross D 0 0 Geophys Res Lett 24 15831586

Glatzmaier G A 1988 Numerical simulations of mantle convection time-dependent three-dimensional compressible spherical shell Geophys Astrophys Fluid Dynam 43 223264

Glatzmaier G A amp Roberts P H 1995 A three-dimensional self-consistent computer simulationof a geomagnetic macreld reversal Nature 377 203209

Glatzmaier G A Coe R Hongre L amp Roberts P 1999 The role of the Earthrsquo s mantle incontrolling the frequency of geomagnetic reversals Nature 401 885890

Gordon R G amp Jurdy D M 1986 Cenozoic global plate motions J Geophys Res 91 12 38912 406

Grand S P van der Hilst R D amp Widiyantoro S 1997 Global seismic tomography a snapshotof convection in the Earth GSA Today 7 16

Gurnis M amp Davies G 1986a Mixing in numerical models of mantle convection incorporatingplate kinematics J Geophys Res 91 63756395

Gurnis M amp Davies G F 1986b Numerical study of high Rayleigh number convection in amedium with depth-dependent viscosity Geophys J R Astr Soc 85 523541

Hager B H amp Clayton R W 1989 Constraints on the structure of mantle convection usingseismic observations deg ow models and the geoid In Mantle convection plate tectonics andglobal dynamics (ed W R Peltier) pp 657763 New York Gordon amp Breach

Hager B H amp Orsquo Connell R J 1979 Kinematic models of large-scale deg ow in the Earthrsquo s mantleJ Geophys Res 84 10311048

Hofmann A W 1997 Mantle geochemistry the message from oceanic volcanism Nature 385219229

Jellinek A M Gunnerman H M amp Richards M A 2002 Plume capture by divergentplate motions implications for the distribution of hotspots geochemistry of mid-ocean ridgebasalts and heat deg ux at the coremantle boundary (In preparation)

Johnson S Masters T G Tackley P J amp Glatzmaier G A 1993 How well can we resolvea convecting Earth with seismic data (abstract) Eos 74(43) 80 (Fall Meeting Suppl)

Jordan T H 1978 Composition and development of continental tectosphere Nature 274 544548

Katsura T amp Ito E 1989 The system Mg2SiO4Fe2SiO4 at high pressures and temperaturesprecise determination of stabilities of olivine modimacred spinel and spinel J Geophys Res 9415 66315 670

Kellogg L H Hager B H amp van der Hilst R D 1999 Compositional stratimacrcation in thedeep mantle Science 283 18811884

Kennett B L N Widiyantoro S amp van der Hilst R D 1998 Joint seismic tomography forbulk sound and shear wave speed in the Earthrsquo s mantle J Geophys Res 103 12 46912 493

Komatitsch D amp Tromp J 1999 Introduction to the spectral element method for three-dimensional seismic wave propagation Geophys J Int 139 806822

Kuang W L amp Bloxham J 1997 An Earth-like numerical dynamo model Nature 389 371374

Li X D amp Romanowicz B 1996 Global mantle shear velocity model developed using nonlinearasymptotic coupling theory J Geophys Res 101 22 24522 272

Lithgow-Bertelloni C amp Richards M A 1998 The dynamics of Cenozoic and Mesozoic platemotions Rev Geophys 36 2778

Phil Trans R Soc Lond A (2002)

2566 H-P Bunge M A Richards and J R Baumgardner

Masters G Jordan T H Silver P G amp Gilbert F 1982 Aspherical Earth structure fromfundamental spheroidal-mode data Nature 298 609613

Masters G Johnson S Laske G amp Bolton H 1996 A shear-velocity model of the mantlePhil Trans R Soc Lond A 354 13851410

Meltzer A (and 11 others) 1999 USArray initiative GSA Today 9 810

Mitrovica J X 1996 Haskell (1935) revisited J Geophys Res 101 555569

Murnaghan F D 1951 Finite deformation of an elastic solid Wiley

Ricard Y Richards M A Lithgow-Bertelloni C amp Le Stunreg Y 1993 A geodynamic modelof mantle density heterogeneity J Geophys Res 98 21 89521 909

Richards M A amp Engebretson D C 1992 Large-scale mantle convection and the history ofsubduction Nature 355 437440

Richards M A Bunge H-P amp Lithgow-Bertelloni C 2000 Mantle convection and platemotion history toward general circulation models In The history and dynamics of globalplate motions (ed M A Richards et al ) Geophysical Monograph Series vol 121 pp 289307 American Geophysical Union

Richards M A Yang W-S Baumgardner J R amp Bunge H-P 2001 The role of a low vis-cosity zone in stabilizing plate tectonics implications for comparative terrestrial planetologyGeochem Geophys Geosyst 2 116

Ritsema J amp van Heijst H J 2000 Seismic imaging of structural heterogeneity in Earthrsquo smantle evidence for large-scale mantle deg ow Sci Prog 83 243259

Sleep N H 1990 Hotspots and mantle plumes some phenomenology J Geophys Res 9567156736

Solheim L P amp Peltier W R 1994 Avalanche ereg ects in phase transition modulated thermalconvection a model of the Earthrsquo s mantle J Geophys Res 99 69977018

Steinberger B amp Orsquo Connell R J 1997 Changes of the Earthrsquo s rotation axis owing to advectionof mantle density heterogeneities Nature 387 169173

Tackley P J 2000 Mantle convection and plate tectonics toward an integrated physical andchemical theory Science 288 20022007

Tackley P J Stevenson D J Glatzmaier G A amp Schubert G 1993 Ereg ects of an endothermicphase transition at 670 km depth on a spherical model of convection in Earthrsquo s mantle Nature361 699704

Tackley P J Stevenson D J Glatzmaier G A amp Schubert G 1994 Ereg ects of multiplephase transitions in a three-dimensional spherical model of convection in Earthrsquo s mantle JGeophys Res 99 15 87715 901

Talagrand O 1997 Assimilation of observations an introduction J Meteorol Soc Jpn 75191209

Trompert R amp Hansen U 1998 Mantle convection simulations with rheologies that generateplate-like behaviour Nature 395 686689

Urey H C 1956 The cosmic abundance of potassium uranium and thorium and the heatbalances of the Earth the Moon and Mars Proc Natl Acad Sci USA 42 889891

van der Hilst R D 1995 Complex morphology of subducted lithosphere in the mantle beneaththe Tonga trench Nature 374 154157

van der Hilst R D Widiyantoro S amp Engdahl E R 1997 Evidence for deep mantle circulationfrom global tomography Nature 386 578584

van der Lee S amp Nolet G 1997 Seismic image of the subducted trailing fragments of theFarallon plate Nature 386 266269

van der Voo R Spakman W amp Bijwaard H 1999 Mesozoic subducted slabs under SiberiaNature 397 246249

Williams Q 1997 The temperature contrast across D 0 0 In The coremantle boundary region(ed M Gurnis et al ) Geodynamic Series vol 28 pp 7382 American Geophysical Union

Phil Trans R Soc Lond A (2002)

Mantle-circulation models with sequential data assimilation 2567

Wunsch C 1996 The ocean circulation inverse problem Cambridge University Press

Zhong S amp Gurnis M 1994 Role of plates and temperature dependent viscosity in phase changedynamics J Geophys Res 99 15 90315 917

Zhong S amp Gurnis M 1996 Interaction of weak faults and non-newtonian rheology producesplate tectonics in a 3D model of mantle deg ow Nature 383 245247

Zhong S Zuber M T Moresi L amp Gurnis M 2000 Role of temperature-dependent viscosityand surface plates in spherical shell models of mantle convection J Geophys Res 10511 06311 082

Phil Trans R Soc Lond A (2002)