Earth and Planetary Science Letters 284 (2009) 409–416

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Earth and Planetary Science Letters

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From plutons to magma chambers: Thermal constraints on the accumulation oferuptible silicic magma in the upper crust

C. Annen ⁎Section des Sciences de la Terre, Université de Genève, 13 rue des Maraîchers, 1205 Genève, Switzerland

⁎ Tel.: +41 22 379 66 23; fax: +41 22 379 32 10.E-mail address: Catherine.annen@unige.ch.

0012-821X/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.epsl.2009.05.006

a b s t r a c t

a r t i c l e i n f o

Article history:Received 17 June 2008Received in revised form 6 April 2009Accepted 6 May 2009Available online 7 June 2009

Editor: C.P. Jaupart

Keywords:plutonmagma chambersuper-eruptionmagma fluxsillheat transfer

In order to provide new insights into the relationship between plutonism and volcanism, numericalsimulations involving heat transfer computation were used to estimate the conditions required for theformation of large magma chambers within plutons that grow by vertical stacking of sills. Large magmachambers can develop within plutons if sill accretion rates exceed 10−2 m/yr. For 10 km thick plutons, thevolumes of eruptible magma are large enough to feed the most voluminous silicic explosive eruptions only ifmagma fluxes exceed 10−2 km3/yr. Emplacement rates required for the formation of a crystal mush fromwhich a melt layer could be extracted by compaction are only slightly lower than the emplacement ratesrequired to directly forming reservoirs of magma that are hot enough to be eruptible.The long-term average pluton emplacement rates inferred from the geochronological data (10−3 m/yr) aretoo low to allow for the formation of large magma chambers. However, some shallow laccoliths wereemplaced much more rapidly and super-eruptions of 103 km3 of ignimbrites associated with caldera collapseare evidence of the existence of large shallow magma chambers. Taken together, magma fluxes estimated onthe basis of geochronological data on plutons and laccoliths, and on the basis of current large-scaledeformation in magmatic provinces, the occurrence of super-eruptions, and the results of numericalsimulation suggest that the growth of plutons is a multi-timescale process with large magma chambersdeveloping during the episodes of highest magma fluxes.

© 2009 Elsevier B.V. All rights reserved.

1. Background

The formation of large silicic magma chambers and the relation-ship between plutonism and volcanism remain a topic of livelydiscussion (e.g. Glazner et al., 2004; Lipman, 2007; Miller, 2008).Recently there has been a change of paradigm regarding the wayplutons are emplaced and grown in the upper crust based on thefollowing observations:

1. Many plutons are low aspect ratio tabular bodies (Vigneresse,1995;Cruden, 1998).

2. Many plutons and batholiths form by agglomeration of discretepulses that commonly take the shape of horizontal sheets (sills)(de Saint-Blanquat et al., 2006; Pasquarè and Tibaldi, 2007;Menand, 2008; Michel et al., 2008).

3. Geochronological data indicate that plutons and batholiths that areseveral kilometres thick were emplaced over several million years(Coleman et al., 2004; Matzel et al., 2006).

Protracted emplacement durations imply that the successivepulses that build up plutons were separated by long time intervalsso that each magma pulse may have time to solidify before the arrival

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of the next pulse. In this case, a pluton cannot represent a fossilisedmagma tank of comparable volume, which led Glazner (2004) topropose that large magma chambers able to feed super-eruptions ofthousands of cubic kilometres are rare and short-lived. Large-volumeignimbrite eruptions associated with caldera collapses and underlaidby batholith-size intrusions are evidence of the existence of largeshallow magma chambers (Lipman, 2007) and it appears fromgeochronological and petrological data that large bodies of silicicmagma can accumulate quite rapidly in the upper crust prior to largecaldera-forming eruptions (e.g. Wilson et al., 2006).

The objective of the present paper is to use numerical models ofpluton growth and compute the temperature evolution of plutons andcountry rocks in order to provide further insight into the pluton–magma chamber relationship.

2. Modelling

The conditions needed for a pluton to evolve in a large magmachamber are evaluated by computing with a finite difference schemetemperatures and melt fractions in a crust section where a pluton isgrowing by accretion of sills. In the context of incremental plutongrowth, a large persistent magma chamber is defined as a volume ofmagma that exceeds the size of one pulse and that stays molten anderuptible between emplacements of two pulses. Incubation times are

Fig. 1. Melt fraction–temperature relationship as taken from Annen et al. (2008).Thecurve used in the numerical simulation is interpolated from experimental data byMartel et al. (1999) on a silicic andesite (circles) and by Piwinskii and Wyllie (1968) ona compositionally close tonalite (triangles). The equations relating melt fraction andtemperature are given in Appendix A.

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the times between emplacement of the first sill and the onset ofaccumulation of large volumes of crystal mush or mobile magma. Amagma is mobile if the crystal fraction is less than a critical fraction of40–60% (Lejeune and Richet,1995). At higher crystal fractions, crystalsform a rigid network and the magma is a crystal mush. Extraction of alayer of melt by compaction of a crystal mush (Jackson et al., 2003;Bachmann and Bergantz, 2004) explains the occurrence of crystal-poor rhyolites (Bachmann and Bergantz, 2008). Crystal-rich inter-mediate eruptions (Hildreth, 1981) require either that the magma ishot enough and molten enough to be mobile or that seismicitydestabilises a meta-stable crystal mush and triggers eruption(Gottsmann et al., 2009). Both mechanisms that involve crystalmushes (compaction and destabilisation) require melt fraction of atleast 40% (Bachmann and Bergantz, 2004; Gottsmann et al., 2009).

In the numerical simulations, the initial temperature of the crust isa geothermal gradient of 30 °C/km.Melt fractions are calculated on thebase of the melt fraction–temperature curves for a tonalitic composi-tion obtained through experimental petrology (Fig. 1) (Piwinskii andWyllie, 1968; Martel et al., 1999). The magma injection temperature isits liquidus at 990 °C. The solidus temperature is 760 °C. The criticaltemperature that corresponds to a melt fraction of 60%, and abovewhich the magma is considered mobile, is 875 °C (Martel et al., 1999).The temperature for a crystal mushwith 40%melt is 840 °C. For amoresilicic magma, the solidus temperature and the temperature for acrystal mush are lower than for a tonalitic magma, i.e. less heat isneeded for a liquid body to develop. But the liquidus temperature andthus the sensible heat of an injection are also lower. So, first-orderresults should not be affected by slightly different compositions. Thecountry rock is assumed to be of the same composition and is allowedtomelt and to absorb latent heat if its temperature exceeds the solidus.The heat balance equation as well as the governing equations for thetemperature dependence of the melt fraction is given in Appendix A.The physical parameters are reported in Table 1.

Table 1Model input parameters.

Parameters Values Units

ρ Magma and crust density 2500 kg m−3

cp Magma and crust specific heat capacity 1000 J kg−1 K−1

k Magma and crust conductivity 2.5 W m−1 K−1

L Magma and crust latent heat 3×105 J kg−1

TL Magma and crust liquidus temperature 990 °CTs Magma and crust solidus temperature 760 °C

The successive sills that build up the pluton are assumed to bewafer-shaped (Fig. 2). The sills are vertically stacked to build up acylindrical pluton that grows vertically. The diameter of the pluton isequivalent to the diameter of the sills. This growth dynamics isjustified by the physics of sill emplacement in a layered environment(Menand, 2008) and by the observation that the horizontal dimensionof laccoliths and plutons is only weakly dependent on their thickness(Cruden, 1998). The simulations are run with sill diameters of 10 and20 km, which correspond to realistic extensions for plutons (Petfordet al., 2000) and for large silicic calderas (Roche and Druitt, 2001). Sillvolumes are accommodated by downward displacement of under-lying country rocks in agreement with field observations andmechanical models (Paterson et al., 1996; Cruden, 1998; Grocott et al.,2009). The modelled system is axisymetric (Fig. 2) and all simulationsare run in 2D using cylindrical coordinates (quasi 3D), which allowscomputationof volumes and volumetricfluxes.As longas temperaturesare too low for magma to accumulate, successive sills are emplaced atthe top of the sill pile as the solidified sill-country rock boundary formsa rigidity contrast favourable to sill emplacement (Kavanagh et al.,2006; Menand, 2008). As soon as mobile magma is accumulatingwithin the pluton, further injections are trapped by the mobile magmalayer so that new sills are emplaced at the interface between the crystalmush and the mobile magma in agreement with observations onplutons (Wiebe andCollins,1998). Thefirst sill is emplaced at a depth of5 km where the initial crust temperature is 150 °C. The pluton growswith successive sill emplacement until it is 10 km thick. Thefinal plutonvertical extension is from 5 to 15 km depth.

Themodel does not aim to reproduce the full complexity of nature.Rather it looks at first-order requirements for forming a magmachamber in the upper crust by magma advection of heat. It providesfirst-order estimations of minimum sill accretion rates (in m/yr) andmagma fluxes (in km3/yr) that are needed for the accumulation oflarge volumes of crystal mush and mobile magma. It does notintegrate deformation that may put limit on possible emplacementrates. It does not take into account the possibility of delayed latentheat release related to quenching of thin sills (Michaut and Jaupart,2006).

In the model, heat transfer is conductive and thermal conductivityis constant. Heat transfer due to convection within the magma andhydrothermal circulation within the country rock is neglected.

Fig. 2. The modeled pluton is growing by accretion of wafer-shaped sill. The first sill isemplaced at 5 km depth and the final thickness of the pluton is 10 km. Drawing is not toscale. The final number of sills that build up the modeled pluton is 50.

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Neglecting magma convection does not significantly affect the resultsin terms of durations of incubation periods and of minimal emplace-ment rates required for the formation of large volumes of liquid in theform of crystal mush, because the thermal evolution of the system iscontrolled by the amount of heat that can be conducted away from themagma through the crust and towards the colder shallower crustallevels. Annen et al. (2008) show that magma convection and gassparging (Bachmann and Bergantz, 2006) do not change the order ofmagnitude of results. Convection is limited to magma layers wheremelt fraction exceeds 60%. As long as successive sills cool down below875 °C between two injections, convection is limited to the last sill anddoes not influence significantly heat transfer in the system. Aftermobile magma has started to accumulate on thicknesses larger than asingle sill, magma convection should result in a faster cooling of themobile magma than conduction only (as assumed here). But coolingby convection would proceed down to the critical temperature for amobile magma, and a crystal mush would remain. Moreover, in athermally mature igneous body that grew incrementally, the wallrocks have been preheated during the incubation period and lowtemperature gradients within the mobile magma and between themobile magma and the chamber walls (Fig. 3) limit the efficiency ofmagma convection. Thus the cooling effect of convection is much lessimportant than in the case of a large magma body instantaneouslyemplaced in a cold crust (e.g. Huppert and Sparks, 1988). In contrast,hydrothermal circulation can significantly enhance cooling of thepluton and may result in much longer incubation times limitingsignificantly the ability of a pluton to develop in a large magmachamber. A recent study by Whittington et al. (2009) indicates thatthermal diffusivity decreases with increasing temperature, an effect

Fig. 3. Cross section in a modelled half-pluton. In this example, the pluton is 2.5 kmthick and formed by accretion of sill 20 km in diameter emplaced at a rate of 0.1 m/y(3×10−2 km3/yr) over 25,000 yr. This snapshot shows the state of themagma chamber1000 yr after the intrusion of the last 200 m thick sill. Contours of temperatures (50 °Cintervals) and melt fractions (0.2 intervals) are shown. The 0.6 melt fraction contourmarks the limit of the mobile magma chamber and the 0.4 melt fraction contour marksthe limit of the mush that could lead to eruption if subject to compaction (Bachmannand Bergantz, 2004) or if destabilised by seismicity (Gottsmann et al., 2009). Thedashed line shows the extension of the pluton. The maximum temperature is 910 °C.

that was not taken into account in the modelling. In contrast withconvection that increases heat transfer and accelerates cooling, adecrease in diffusivity reduces heat transfer and magma cooling.

3. Results

3.1. Incubation times

In a system where an intrusive body grows by accretion of sills inan initially cold crust, the first sills are emplaced in a cold environmentand rapidly cool and solidify. With time, as successive sills transfertheir sensible and latent heat to the crust, the temperatures of thewhole system (intrusions + country rocks) progressively increase.When the solidus temperature is reached at the locus of injection, avolume of highly crystalline magma composed of interconnectedcrystals and interstitial melts starts to grow. When the melt fractionreaches about 40%, melt can be extracted from the mush bycompaction (Bachmann and Bergantz, 2004) or the mush can bedestabilised by seismic activity and erupts (Gottsmann et al., 2009).Eventually the temperature corresponding to the critical melt fractionis exceeded and a reservoir of mobile magma begins to grow. Becausethe pulses are low aspect ratio sills, heat transfer between theaccumulating sills and the country rock is mostly through the walls ofthe sills and the cooling of the sills is controlled by the rate of sillaccretion in m/yr (or pluton average emplacement rate, c.f. Annenet al., 2008) and not by the volumetric magma flux (in m3/yr). Thusduration of the incubation period between injection of the first sill andthe formation of amush and eventually of a mobilemagma chamber iscontrolled by the sill accretion rate q (sill thickness divided by timeinterval between sills emplacement) and is independent of sillsdiameter (Fig. 4). The model was tested for different time intervalsbetween intrusions corresponding to different emplacement rates.The thickness of the sills is fixed to 200 m to optimize computationtimes. In nature, sills are probably thinner, but because the long-termthermal evolution of the system is controlled by the averageemplacement rate and not by the exact values of sill thicknesses andtime intervals, the computed incubation times are independent of thechosen thickness. This would not be true however if the release oflatent heat would be delayed by quenching as modelled by Michautand Jaupart (2006).

Incubation times are weakly dependent on the initial geothermunless the upper crust has already been heated by previousconsolidated plutons. Incubation times for the growth of a crystallinemush (melt fraction N40%) and for accumulation of mobile magma(melt fraction N60%) vary as q−2 (Fig. 4a) in agreement with theanalysis ofMichaut and Jaupart (2006). Note that the incubation timesrequired to form a crystal mush able to feed an eruption bycompaction or by destabilisation related to seismicity are not muchlower than the incubation times needed to accumulate mobile andeasily eruptible magma.

These results can also be expressed in terms of critical thickness(Fig. 4b), which is the cumulated thickness of sills that are emplacedduring the incubation period. The critical thickness is equal to theincubation time multiplied by the accretion rate. A large magmachamber can develop within a pluton emplaced at a steady rate only ifthe pluton thickness is larger than the critical thickness. For accretionrates that are less than a fewcentimetres per year, the critical thicknessexceeds 10 km (Fig. 4b), i.e. the size of most plutons (Petford et al.,2000). For accretion rates that are more than about 3×10−2 m/yr alarge magma chamber can develop (see also Hanson and Glazner,1995; Yoshinobu et al., 1998; Annen et al., 2008). This result contrastswith results obtained from simulations applied to the High Himalayanleucogranites that showed that emplacement rates of a few milli-metres per year were sufficient for a large convectivemagma chamberto develop (Annen et al., 2006b). In the case of High Himalayanleucogranites significantly less heat needs to be advected to the system

Fig. 5. Evolution over time of (a) the intruded and mobile volumes and (b) the portionof the pluton that is mobile. In this example, the pluton is 20 km in diameter and isemplaced at a rate of 5×10−2 m/yr (1.6×10−2 km3/yr).

Fig. 4. Incubation times (a) and critical thicknesses (b) for different sill accretion rates.Incubation time is the time interval between emplacement of the first sill andaccumulation of a thickness of crystal mush (melt fraction N40%) or mobile magma(melt fraction N60%) larger than one sill. Critical thickness is the thickness intrudedduring the incubation time and before accumulation of mush or mobile magma. Thecritical thickness is equal to the sill accretion rate multiplied by the incubation time. Fora given sill accretion rate, incubations times and critical thicknesses are the same for silldiameters of 10 and 20 km. For plutons that are 10 km thick or less, no crystal mush ormobile magma accumulate if accretion rates are less than 3×10−2 m/yr (dashed line),which corresponds to fluxes of 2.4×10−3 km3/yr for a pluton diameter of 10 km or9.4×10−3 km3/yr for a pluton diameter of 20 km.

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to form a magma chamber in comparison with the general modelpresented here because large emplacement depths (10–12 km) resultin larger initial crust temperatures (more than 300 °C) and also thecritical melt fraction of the highly differentiated magma is reached atmuch lower temperatures (about 640 °C instead of 875 °C).

3.2. Shape of the magma reservoir

The magma chamber starts to be sill-like and, if emplacement ratesare high enough, it eventually evolves towards a bell shaped chamber(Fig. 3). Because new injectedmagma is trapped by the high-melt layer,the magma chamber grows from top to bottom so that the highesttemperatures are at the bottom of themagma body. The crystal mush isat the top and on the side of the mobile magma chamber (Fig. 3).However, the simulation does not take into account exchanges of mass.

In nature, crystals in suspension in the magma chamber may settle andaccumulate to formamushon the chamberfloor although this process islimited bymagma convection (Sparks et al.,1984).Moreover, part of thecrystal mush on the magma chamber walls and roof may collapse intothe chamber resulting in redistribution of the mush and the mobilemagma. Convection is expected to result in increased heating of thechamber roof resulting in re-melting (Huppert and Sparks, 1988) andenhanced cooling of the mobile magma so that, depending on theeffectiveness of convection, the magma reservoir may be larger andmore crystalline than represented in Fig. 3.However, in an incrementallygrowing igneous body, the crust must heat up to high temperaturesbefore themobilemagma can accumulate, which results in temperaturecontrasts between themagma reservoir and thewall rock that aremuchlower than for an instantaneously emplaced body. The highesttemperature gradients are located in the solid rocks and the temperaturegradient in the mobile part of the magma body is low which limits theeffect of magma convection on temperature distributions.

Note that emplacement rates resulting in the magma chamberillustrated by Fig. 3, if sustained on the whole emplacement durationof a 10 km thick pluton, would result in the pluton being emplaced

Fig. 6. For different magma fluxes, maximum portion of a pluton that is a magmareservoir (a) and volumes of eruptible magma within a 10 km thick pluton (b). Themodelled pluton diameters d are10 and 20 km. The plain circles and crosses are portionsand volumes of magma with a melt fraction at or above 60%, the open circles areportions and volumes of magma with a melt fraction at or above 40% (for d=10 km).On (b) the horizontal grey linemarks the volume thatmust be exceeded to feed a super-eruption (450 km3). Arrows point to the estimated long-term average magma fluxes forplutons (Crisp, 1984; Coleman et al., 2004; Matzel et al., 2006), the episodic higherfluxes recorded by Mt Stuart intrusion (Matzel et al., 2006), the fluxes estimated forTorres del Paine laccolith (Michel et al., 2008) and for the assembly of the magma thatfed the Oranui eruption of Taupo volcano (Wilson et al., 2006).

413C. Annen / Earth and Planetary Science Letters 284 (2009) 409–416

over 100,000 yr which is contradicted by geochronological data(Coleman et al., 2004; Matzel et al., 2006).

3.3. Eruptible volumes and eruptible portion of the pluton

The incubation time needed before eruptible magma starts toaccumulate depends only on sill accretion rates (see Section 3.1) whereasthe portion of the pluton that is a crystal mush or that is amobilemagma,and the volumes of crystal mush andmobile magma, depend both on theaccretion rate and on the pluton diameter, i.e. on the injected volumetricmagma flux. Fig. 5 shows the evolution in time of themobile volumes andof theportionof pluton that ismobile (mobile volumedividedby intrudedvolume) for a pluton 20 km in diameter that grows at a high rate of 5×10−2 m/yr (flux is 1.6×10−2 km3/yr). During the incubation period, themobile volume drops to zero between sill intrusions (Fig. 5a). Eventually,when the temperatures arehighenough formobilemagmatoaccumulate,the mobile volume steadily increases with time and with each newinjection so that, after a sharp increase, the portion of the pluton that ismobile tends toward a constant value (Fig. 5b).

If the magma body final thickness is 10 km or less, sill diametersmust be more than 7.5 km for the magma body volume to exceed thevolume of super-eruptions (N450 km3) (Mason et al., 2004; Self,2005; Sparks et al., 2005). For a pluton of 10 km in diameter and 10 kmin thickness to be able to feed a super-eruption, more than 50% of thepluton volume must erupt. This portion drops to 14% if the plutondiameter is 20 km.

Fig. 6 shows the final mush and mobile portions (Fig. 6a) andvolumes (Fig. 6b) for modelled plutons 10 km thick that grew byaccretion of sills 10 and 20 km in diameter. For fluxes more than about7.5×10−2 km3/yr (i.e. accretion rates of about 1 m/yr for sillsdiameter of 10 km and accretion rates of about 0.25 m/yr for silldiameters of 20 km) the eruptible portion reaches 80%. With suchfluxes the pluton is a large magma tank, but it implies that a 10 kmthick pluton is emplaced over a few tens of thousand years only. Forfluxes less than a few 10−2 km3/yr, the eruptible portion drops to lessthan 50% and for fluxes less than a few 10−3 km3/yr, no large magmachamber grows in plutons that are 10 km thick or less. Fig. 6b showsthat magma fluxes of at least 10−2 km3/yr are required for theaccumulation of volumes of magma large enough to feed the largestexplosive caldera-forming eruptions. This minimum flux is limited bythe sill diameters and by the sill accretion rate. For small sills, notenough volume is available to feed super-eruptions whatever theaccretion rate. For low emplacement rates, low aspect ratio sillssolidify between injections whatever their diameters.

4. Discussion

4.1. Volcanic output

Long-term volcanic output rates for silicic (from andesitesto rhyolites) individual volcanoes or small volcanic fields ascompiled by White et al. (2006) cover a wide range from 2.3×10−6

to 1.2×10−2 km3/yr. If those volcanic products are fed by long-livedshallow crustal magma chambers, their volumes cannot exceed theeruptible portion of themagma body. Maximum erupted fluxes can beestimated on the basis of the data represented on Fig. 6 bymultiplyingthe intruded magma flux by the eruptible portion. Calculated eruptedfluxes for intrusion horizontal extensions between 10 and 20 km fitvolcanic output fluxes compiled by White et al. (2006) if intrudedfluxes are in the range of 2.2×10−3 to 2.7×10−2 km3/yr.

4.2. High-volume explosive eruptions

The occurrence of large magnitude explosive eruptions thatproduce hundreds to thousands of km3 of magma and are associatedwith caldera collapse shows that large magma chambers can develop

in the upper crust. Super-eruptions involve magma volumes of morethan 450 km3 (Mason et al., 2004; Self, 2005; Sparks et al., 2005) andthe largest silicic eruption recorded (Fish Canyon) reaches over5000 km3 (Lipman, 2007). The horizontal extension of a caldera likelycorresponds approximately to the horizontal extension of the drainedmagma chamber (Roche and Druitt, 2001). Dividing erupted volumesby the corresponding caldera surface (as reported in Lipman, 1984,2007) gives magma chamber thickness of more than 500 m up toseveral kilometres, which is more than the thickness of the thickestobserved sills and thus must represent incremental accumulation aswell as rates of magma accumulation in the upper crust able togenerate large volumes of eruptible magma. Results of numericalsimulation suggest that magma fluxes of at least 10−2 km3/yr in thecrust are required to accumulate the volumes of eruptible magmaneeded to feed the largest explosive eruptions.

4.3. Plutons geochronology

Geochronological data indicate that some plutons are constructedover several million years (Coleman et al., 2004; Matzel et al., 2006)

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corresponding to long-term average emplacement rate of 10−3 m/yrand average magma flux of 10−4 km3/yr. Crisp (1984) estimatedsimilar intrusive fluxes of 10−3–10−4 km3/yr for batholiths andbatholithic belts by dividing batholiths approximated volumes byintervals between minimum and maximum ages of plutons withinbatholiths. Heat transfer computation shows that at such lowemplacement rates, each magma pulse crystallizes before the nextone is injected and the magma chamber size is not larger than onesingle pulse, as also inferred by Glazner et al. (2004). Even the processof Runaway Devitrification that would involve the rapid generation ofmelt in partly glassy intrusions cannot explain the formation of largemagma chambers if emplacement is over a million years (Michaut andJaupart, 2006, Fig. 8). Similarly, an emplacement rate of 10−3 m/yr istoo low for the long-term accumulation of large volume crystal mushand formation of a melt layer by compaction (Jackson et al., 2003;Bachmann and Bergantz, 2004). However, U/Pb geochronologicalstudy of Mt Stuart batholith by Matzel et al. (2006) indicates that thebatholith emplacement was punctuated by four periods of high fluxesinterrupted by a repose period of 1–2My. During the period of highestmagma flux, ~520 km3 of magma was emplaced over ~170±90 kycorresponding to a magma flux of about 3.1×10−3 km3/yr (Matzelet al., 2006). My results indicate that even at such relatively high rate,the high-melt mobile magma chamber would represent less than 10%of the volume injected and the volume of mobile magma and mushwould represent less than 25% of the total volume (Fig. 6). Thusthe volumes that could be erupted are lower than those of super-eruptions, although in the specific case of Mt Stuart batholith levels ofemplacement are deeper than modelled here, which would result in aslightly larger magma chamber. A recent study on Torres del Painelaccolith showed that the granitewas intruded as a series of sills over aperiod of 90±40 ky (Michel et al., 2008) corresponding to emplace-ment rates of about 10−2 m/yr. The laccolith emplacement depth wasshallow (2–3 km) and at this level the emplacement ratewas probablytoo low for the formation of a large magma chamber.

4.4. Magma fluxes

In summary, geochronological data suggest overall emplacement ofseveral kilometres thick plutons and batholiths at a rate of 10−4 km3/yr(Crisp, 1984; Coleman et al., 2004; Matzel et al., 2006), more detailedgeochronology on the emplacement history of some plutons indicatetransient higher fluxes of the order of 10−3 km3/yr (Matzel et al., 2006;Burgess and Miller, 2008). The shallow Torres del Paine laccolith wasemplacedata rate of10−2m/yr (Michel et al., 2008) andaccording todeSaint-Blanquat et al. (2006), the 150 m–250 m thick Black Mesa plutonmay have been incrementally assembled over a few hundred yearsonly, corresponding to an accretion rate of 2 m/yr. Modellingindicates that fluxes of at least 10−2 km3/yr and emplacement rates ofat least 10−2 m/yr are required to accumulate enough magma to feedlarge caldera-forming eruptions (This study, Hanson and Glazner, 1995;Yoshinobu et al., 1998). If runaway devitrification occurs, slightly loweremplacement rates can lead to magma accumulation (Michaut andJaupart, 2006). In the Altiplano of the central Andes, time-averagedmagmaaccumulation rates of around 10−3 km3/yr are estimated for thelarge calderas and associated ignimbrite systems but major eruptionscorrespond to higher fluxes of 4×10−3 to 1.2×10−2 km3/yr (de Silvaand Gosnold, 2007). Large-scale ground uplifts of 1–2×10−2 m/ yr(1992–2008) are recorded at Uturuncu volcano in SWBolivia (Pritchardand Simons, 2002) suggesting that magma is currently being emplacedat rates of at least 10−2 km3/yr at depths of about 18 km (Sparks et al.,2008). These observations suggest that mid-crustal magma intrusionrates can exceed time-averaged rates by over an order of magnitude.

Other examples of ground uplifts of a few 10−2 m/yr associatedwith large magmatic systems include Yellowstone caldera (USA)(Chang et al., 2007), Long Valley caldera (USA) (Tizzani et al., 2009)and Lazufre volcanic area (Chile) (Ruch et al., 2008). In contrast, the

uplift associated with the Socorro magma body, New Mexico, is afew 10−3 m/yr only and is interpreted as being related to magmavolume increases of 6–8×10−3 km3/yr (Fialko and Simons, 2001).The numerical results presented in this paper support Fialko andSimons' (2001) conclusion that the freezing rate of the magma ex-ceeds its emplacement rate preventing accumulation of large volumesof magma.

In the Taupo volcanic zone of New Zealand, detailed stratigraphyand use of zircon geochronology indicate that the 530 km3 of rhyoliticmagma erupted in the Orunanui eruption was developed over lessthan 40 ky (Charlier et al., 2005) corresponding to a flux of more than1.3×10−2 km3/yr.

Taken together observations on various plutonic and volcanicsystems and the results of numerical simulation suggest that growthof plutons in the upper crust is a multi-timescales process. Timeevolution of intrusive magma fluxes in the upper crust may becomparable to the fractal behaviour process and power law distribu-tion of recent eruption sequences that have been identified on severalvolcanoes (Telesca et al., 2002). The development of large magmachambers would be possible during periods of high magma fluxes inexcess of 10−2–10−1 km3/yr. Thermal maturation of a pluton allowsthe accumulation of larger volumes of eruptible magmas and largereruptions during episodes of high magma fluxes as observed forexample in the Altiplano–Puna volcanic complex (de Silva andGosnold, 2007). The origin of the sequencing of magma input mayresult from complex interaction between processes of magmageneration, extraction and emplacement occurring on differenttimescales at the source level (De Bremond d'Ars et al., 1995; Jacksonet al., 2003; Annen et al., 2006a).

At low emplacement rates (a few 10−3 m/yr), if the sills that formthe building blocks of an igneous body are a few tens of meters thick,the time interval between two injections is a few thousand years. Thisleaves enough time for visco-elastic relaxation of the stresses inducedin the crust by successive injections. However, the necessity toemplace magma at high rates to generate magma chambers largeenough to feed voluminous ignimbrites poses the problem of thedeformation that can be accommodated by the crust. Analyticalmodels involving a spherical magma chamber suggest that in a coldcrust high magma fluxes result in dyke opening and eruptions thatdrain the magma chamber (Jellinek and DePaolo, 2003). For a sill-likemagma chamber, high magma flux can also result in sill horizontalpropagation. Thus in case of a cold crust, thickening of a magmachambermay be limited and accumulation of large volumes of magmamay be possible only at a later stage in the magma body emplacementwhen the crust is hot enough to respond viscously to the deformation(Jellinek and DePaolo, 2003; Miller et al., 2006). However, accordingto Cruden (1998) and Cruden and McCaffrey (2001), high rates (morethan 1 m/yr) of magma can be accommodated in the upper crust by amechanism of mass exchange between a shallow and a deep reservoir.Room is created in the shallow reservoir by floor depressionassociated with brittle faulting and/or elastic deformation whereasductile flow probably occurs in the source reservoir which is located inthe mid-lower crust.

5. Conclusions

A range of observations on plutonic and volcanic systems,including high precision geochronology, detailed stratigraphy anddeformation measurements, suggests that the growth of magmabodies is episodic and that magma fluxes vary with time by orders ofmagnitude. Numerical models of plutons growth by accretion ofdiscrete sills indicate that the accumulation of several hundreds toseveral thousands of km3 of eruptible magma needed to feed thelargest explosive eruptions requires sill accretion rates of at least a few10−2 m/yr and fluxes of several 10−2 km3/yr, corresponding toperiods of the highest fluxes estimated on magmatic systems. With

415C. Annen / Earth and Planetary Science Letters 284 (2009) 409–416

maturation of the system over time, larger volumes of eruptiblemagma can be produced and accommodated during periods of highflux, but episodic growth of plutons and low long-term averagemagma fluxes imply that magma chambers present at any timerepresent only small portions of the final size of plutons.

Acknowledgements

The data for the melt–temperature relationship were compiled byMichel Pichavant. The manuscript benefited from the constructivecomments and suggestions of Steve Sparks. I thank Catherine Ginibrefor her help in clarifying the manuscript, and Sandy Cruden andThierry Menand for sharing their knowledge on the emplacement ofmagmas and on the rheology of the crust. The comments of twoanonymous referees helped to significantly improve the manuscript.

Appendix A

The model is conductive and includes the latent heat generated bycrystallization of the magma and by melting of previously emplacedmagma or country rock so that:

ρcpATAt

+ ρLAXAt

= kj2T ðA1Þ

where ρ is density, cp is magma heat capacity, T is temperature, t istime, k is thermal conductivity, L is latent heat and X is melt fraction.

The system is discretized over a squared grid and temperatures arecomputed in 2D at the nodes of the grid with a finite difference explicitschemeusing cylindrical coordinates (quasi 3D) (c.f. Annen et al., 2008).

The melt fraction–temperature relationship is:

X Tð Þ =

X 876ð Þ876− Ts

T − Tsð Þ : 876BC N T N Ts

−4:668 × 10−5T2 + 8:636 × 10−2T − 39:244 : 930BCzTz876BC1− X 930ð ÞTL − 930

T − TLð Þ + 1 : TL N T N 930BC

8>>>>>><>>>>>>:

ðA2Þ

X(876) and X(930) are melt fractions at temperatures of 876 and930 °C respectively and are calculated with the equation valid in thistemperature interval. TL andTs are the liquidus and solidus temperatures.

Because of thenon linearityof theX–T relationship, Eq. (A1) is solvediteratively between solidus and liquidus with a bisection method.

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