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Capabilitiesandlimitationsofnumericalicesheetmodels:AdiscussionforEarth-scientistsandmodelers

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Quaternary Science Reviews 30 (2011) 3691e3704

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Capabilities and limitations of numerical ice sheet models: a discussion forEarth-scientists and modelers

Nina Kirchnera,*, Kolumban Hutterb, Martin Jakobssonc, Richard Gyllencreutzc

aDepartment of Physical Geography and Quaternary Geology, Stockholm University, 10691 Stockholm, Swedenb c/o Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology, 8092 Zürich, SwitzerlandcDepartment of Geological Sciences, Stockholm University, 10691 Stockholm, Sweden

a r t i c l e i n f o

Article history:Received 12 April 2011Received in revised form13 September 2011Accepted 15 September 2011Available online 22 October 2011

Keywords:Spatial reconstructionsIce shelvesIce streamsIce sheet modelingShallow IceApproximationHigher Order modelsSvalbard

* Corresponding author. Tel.: þ46 8 162988/þ46 70E-mail addresses: nina.kirchner@natgeo.su.se (N.

ethz.ch (K. Hutter), martin.jakobsson@geo.su.se (M. Jakgeo.su.se (R. Gyllencreutz).

0277-3791/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.quascirev.2011.09.012

a b s t r a c t

The simulation of dynamically coupled ice sheet, ice stream, and ice shelf-systems poses a challenge tomost numerical ice sheet models. Here we review present ice sheet model limitations targeting a broaderaudience within Earth Sciences, also those with no specific background in numerical modeling, in orderto facilitate cross-disciplinary communication between especially paleoglaciologists, marine andterrestrial geologists, and numerical modelers. The ‘zero order’ (Shallow Ice Approximation, SIA)-, ‘higherorder’-, and ‘full Stokes’ ice sheet models are described conceptually and complemented by an outline oftheir derivations. We demonstrate that higher order models are required to simulate coupled ice sheet-ice shelf and ice sheet-ice stream systems, in particular if the results are aimed to complement spatial iceflow reconstructions based on higher resolution geological and geophysical data. The zero order SIAmodel limitations in capturing ice stream behavior are here illustrated by conceptual simulations ofa glaciation on Svalbard. The limitations are obvious from the equations comprising a zero order model.However, under certain circumstances, simulation results may falsely give the impression that icestreams indeed are simulated with a zero order SIA model.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Ice shelves are ice masses floating in the ocean while beingattached to and fed by grounded glaciers or an ice sheet (Figs. 1, 2).Large ice shelves are today encountered almost exclusively alongthe Antarctic coastline, although smaller ice shelves also existaround Greenland, northern Ellesmere Island, the Franz Josef LandArchipelago and Severnya Zemlya in the high Russian Arctic. Iceshelves are highly dynamic as illustrated by the recent partialcollapses of the Larsen andWilkins ice shelves (Antarctica) and thebreak ups of Petermann Glacier and the Ward Hunt Ice Shelf inNorthwest Greenland (Rott et al., 1996; Mueller et al., 2003; Glasserand Scambos 2008; Humbert and Braun 2008, Falkner et al., 2011).Oceanic forcing, rapid variation in intrinsic tributary glacier andice-stream dynamics, as well as changes in configuration of thecoupled ice sheet-ice shelf complex are the major processes related

6090588; fax: þ46 8 164818.Kirchner), hutter@vaw.baug.obsson), richard.gyllencreutz@

All rights reserved.

to ice shelf disintegration (Rignot et al., 2004; Shepherd et al., 2004;Rignot and Steffen 2008; Goldberg et al., 2009; Pritchard et al.,2009). Once rapidly retreating, ice shelves and floating glaciertongues affect the mass balances of their feeding ice sheets andglaciers considerably through accelerated drainage. Since thisultimately contributes to changes in sea level, changes in ice-shelfconfigurations are carefully observed and recorded by space-borne missions such as CryoSat, ICESat and IceBridge. The datafrom these missions serve as input in prognostic numerical icemodeling experiments focusing on the Greenland as well as theAntarctic ice sheets and their fringing ice shelves (e.g. Fricker andPadman 2006; Wingham et al., 2006; Khazendar et al., 2007; Ray2008; Fricker et al., 2009).

Paleo ice-shelf configurations are mainly understood throughspatial reconstructions based on interpretations of mapped, andideally also dated, marine geological evidence (Polyak et al., 2001;Spielhagen et al., 2004; Svendsen et al., 2004; Engels et al., 2007;Graham et al., 2008, 2010; Jakobsson et al., 2010a) (Fig. 3).However, most recently, results from a coupled ice sheet-ice shelf-climate model, run to investigate ice sheet-ice shelf reaction tooceanic changes on timescales relevant for paleosimulations havebeen presented to the scientific community where they are still

Fig. 1. Sketch of a coupled ice sheet-ice shelf system. In sufficient upstream (down-stream) distance from the grounding line, pure ice sheet flow (ice shelf flow) isdescribed by the zero order Shallow Ice Approximation SIA (zero order Shallow ShelfApproximation SSA). In a transition region across the grounding line, both sheet flowand shelf flow features prevail. There, SIA and SSA have to be extended before they canbe coupled using the second order SO-SIA and SO-SSA modeling approaches.

Fig. 2. Filchner Ice Shelf region, Weddell Sea sector, West Antarctica. Different flowregimes: A) Bulk ice sheet flow of shearing type over frictional bedrock topographyinduces undulations occasionally propagating to the ice sheet surface. B) Thinning anddiverging ice shelf flow (Filchner Ice Shelf) exhibiting an essentially flat and undis-turbed upper ice surface due to virtually frictionless motion over the ocean surface. C)Rapid flow of tributary glaciers (Slessor Glacier) and ice streams (Recovery Ice stream)interlinking sheet and shelf flow regimes. The white solid and stippled lines denote thegrounding line and the calving front of Filchner Ice Shelf, respectively, derived fromICESat and MOA data and provided by J. Robbins and H. J. Zwally, NASA GSFC. The cubesA) B) and C) are detailed in Fig. 4. Background picture from http://earthobservatory.nasa.gov/IOTD/view.php?id¼7620.

2 In the mid 19th century, ice shelves were effective barriers to the conquest of

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e37043692

open to discussion (Alvarez-Solas et al., 2011). Overall, paleo-iceshelf evolution from inception to decay is yet to a much lesserextent complemented by numerical modeling than investigationsof future ice shelf behavior where modeling is central. The mainreason for this is the weak mutual exchange between the disci-plines investigating paleo ice shelves and the disciplines addressingprognostic future ice shelf behavior. This highlights the generalneed for increased cross-disciplinary communication betweenpaleoglaciologists, marine and terrestrial geologists/geophysicistsand numerical modelers sharing a common interest in ice shelfcomplexes.1

Here, we suggest a common basis of departure for such cross-disciplinary communication: we aim to provide readers from theEarth Sciences with no specific background in modeling witha conceptual understanding of different types of numerical models(‘zero order’-, ‘higher order’-, and ‘Full Stokes’ ice shelf and/or icesheet models). As definition and benefit of each particular type ofmodel often remain obscure to non-modelers, we address the clas-sification, limitations and merits of these models (Section 2.2,2.3,3).Focusing on ice streams we demonstrate that zero order models areinsufficient to capture their dynamics correctly (Section 3.1,4.1). Thisis well-known to modelers and obvious from the equations under-lying any zero-order model. Yet, visualizations can veil these limita-tions and may even suggest that ice stream dynamics are accountedfor. Presumably, confusion resulting fromsuchpeculiarities has in thepast hindered cross-disciplinary communication. We thereforepresent conceptual simulations of a glaciation over Svalbard whichillustrate the mentioned limitations and pitfalls of zero order modelresults (Section 4). Also, we advocate, to the wider community ofEarth Science researchers interested in ice shelf complexes, thenecessity and use of higher order ice models.

1 We refer to one or more ice shelves being fed by an unspecified number of icestreams draining an attached inland ice mass as an ‘ice shelf complex’.

2. Diversity of ice shelf research

The identification of ice shelves as individual components of thecryosphere worthwhile modern2 scientific investigation dates backto the late 1950’s. Ice shelf specific research conducted during thepast six decades can be roughly divided into two periods: in theearly years, ice shelves were mainly investigated in isolation fromthe feeding ice sheets, the adjacent ocean and the atmosphere.From the mid 1970’ies, the dynamics and stability of ice shelfcomplexes, including grounding line dynamics, became a key issue(Weertman 1974, Thomas 1979, Bindschadler et al., 1987). Sincethen, an increasing number of disciplines included ice shelves intheir research agenda once their critical role in the global climatesystemwas realized (Polyak et al., 2001; Fricker and Padman 2006;Khazendar et al., 2007; Jenkins and Jacobs 2008). Despite overallsimilar scientific objectives, the disparity of the research fieldshindered cross-disciplinary collaboration with some few excep-tions. On the other hand, reconstructions of Quaternary ice-sheetconfigurations began in the late 1990’s taking cross-disciplinaryapproaches by combining numerical modeling and spatial recon-structions, though treating ice-shelf components in the numericalmodeling in a simplified manner only (e.g. Siegert and Dowdeswell1999; Siegert et al., 2001; Kleman et al., 2002; Tarasov and Peltier2004; Napieralski et al., 2007; Stokes and Tarasov 2010). Cross-disciplinary studies involving simulations with combined ice

the South Pole; when James Clark Ross, on January 9, 1841, first sighted the ice shelfnamed after him he called the ice mass the ‘Great Ice Barrier’. Otto Nordenskjöld,who crossed the site of the Larsen A ice shelf with his field party in 1902, was thefirst to introduce the terminology ‘ice shelf’ as a substitute for the term ‘barrier’suggested by Ross (Nordenskjöld 1909).

Fig. 3. Spatial reconstructions of Pan-Arctic paleoglacial configurations. Map showing the Arctic Ocean bathymetry revealing several large bathymetric troughs, in the continentalmargins, sculptured by large ice streams that were active during past glacial periods. These ice streams most likely fed ice shelves at several places. The inferred hypothesized iceshelf extension during Marine Isotope Stage 6 (MIS 6) by Jakobsson et al. (2010a) is shown by a gray stippled line. The extensions of the Barents and Kara Sea Ice Sheet during MIS6(Saalian maximum, white stippled line) and MIS 2 (LGM, black stippled line) is from Svendsen et al. (2004). The blue line represents the North American LGM ice sheet extension(Dyke et al. 2002). Bathymetry from IBCAO (Jakobsson et al. 2008a). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version ofthis article.)

3 Ice shelves “produce rather flat slabs of floating ice which, for the theoretician,are the simplest of all ice masses” (Thomas, 1979).

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 3693

sheet/ice shelf models and comparison with geological records areless common and occur in the scientific literature first morerecently (e.g. Pollard and DeConto 2007, 2009).

2.1. Early ice shelf specific research

If considered in isolation, the behavior of ice shelves as well asice sheets is well understood andmodeled since the late 1950’s. Thetiming of this advance in knowledge is linked to the Norwe-gianeBritisheSwedish 1949e1952 expedition to the Maudheim IceShelf as well as the First International Geophysical Year (the ThirdInternational Polar Year) 1957e1958 expeditions to the Filchnerand the Ross ice shelves (Roberts 1950; Giaever and Schytt 1952;Robin 1953; Odisha and Ruttenberg 1958; Zumberge et al., 1960;Crary et al., 1962). Those expeditions focused, for the first timesince the days of the early explorers, specifically on ice shelves.

From physiographic analogies between West Antarctica and theArctic Ocean, Mercer (1970) proposed the hypothesis of extensive

ice-shelves covering at least parts of the central Arctic Oceansometime during the Quaternary. Mercer’s Arctic Ocean ice shelveshave since been a recurrent topic among paleoglaciologists. Recentgeological and geophysical data show that there indeed existedextensive ice shelves, at least in the Amerasian Arctic Ocean, duringMarine Isotope Stage (MIS) 6 (Jakobsson et al., 2010a); a topicfurther discussed below. Since themid 1970’s, the importance of iceshelves for the persistence of marine ice sheets, such as the WestAntarctic Ice Sheet (WAIS), was recognized and related to both themigration of the grounding line and the intermittent mass loss atthe calving front in response to oceanic forcing (Weertman1974,1976; Thomas and Bentley 1978). Furthermore, it was recog-nized that the bulk3 of the ice shelf played a less critical role thanboth the sheet-shelf transition region across the grounding line,

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e37043694

and the calving front. Ice shelf complexes have been investigatedusing different techniques and with various foci by Weertman(1974), Robin (1975), Thomas and Bentley (1978), Chugunov andWilchinsky (1996), Hulbe and MacAyeal (1999) and Hulbe andFahnestock (2004). Attempts to e.g. access the cavity beneath anice shelf by drilling through it were made from the mid 1970swithin the multifaceted Ross Ice Shelf Project (RISP) (Thomas et al.,1984). In 1984, the Filchner-Ronne-Ice-Shelf-Programme (FRISP)was started as a subcommittee of the Scientific Committee onAntarctic Research (SCAR)Working Group of Glaciology; it has nowwidened its scope with the acronym FRISP instead denoting ‘Forumfor Research into Ice Shelf Processes’. The ongoing Amery Ice ShelfOcean Research (AMISOR) project aims to quantify ice shelf/oceaninteraction and succeeds the earlier Amery Ice Shelf Project (Buddet al., 1982). In the mid 1990’s, the state of numerical ice-shelfmodeling at that time was assessed in a model intercomparisonproject (MacAyeal et al., 1996). However, ice sheet modelingexperiments dominated because they supported deep drilling icecore projects such as GRIP, GISP and NGRIP in Greenland, and EPICAin Antarctica. The European Ice Sheet Modeling INiTiative (EIS-MINT) intercomparison and benchmark project accompanied thesedrilling efforts (Huybrechts et al., 1996; Payne et al., 2000).

2.2. Why separate ice shelf and ice sheet models?

From a viewpoint of basic physics, ice shelves and ice sheets arenot necessary to treat apart. They are both large ice masses underextreme slow creeping motion due to external action of gravity andclimate. The classical laws of physics, conservation of mass,Newton’s second law (force balance) and conservation of energy,essentially suffice to describe the evolution of their motion. Specificice shelf and ice sheet behavior is due to differences in the types ofboundaries and the actions of the environment across theseboundaries. For ice sheets, the latter are the contact surfaces withthe atmosphere, the solid ground, and the ocean (calving in theabsence of ice shelves). Internal boundaries delineate transitionregions with ice shelves or shear margins distinguishing fast flowfeatures such as ice streams. For ice shelves, the boundaries are thecontact surfaces with the atmosphere and the ocean. In regions ofice rises and ice rumples ice shelves are also in contact with solidground. Further boundaries comprise the shore lines. If for a givenice shelf or sheet, the processes describing how mass, momentumand heat are supplied from the outside at these boundaries areprescribed, then the thermo-physical processes inside the ice shelfor ice sheet are in principle determined. A distinction between iceshelves and ice sheets is thus as such not compulsory from thepoint of view of general physics.

However, there is the significant difference in that ice shelvesfloat in water while ice sheets are grounded on land or shallowseafloor. In all existing ice shelf and ice sheetmodels, modelers startfrom the classical laws of physics as stated above. Yet, to simplify themathematics and, later, the numerics, the force balance is typicallyreplaced byan approximate force balance inwhich certain terms areneglected. Thedegreeof simplification of the resultingmodel is thenmeasured by the number of neglected terms. The simplest modelsneglect a comparatively large number of terms and retain only thosethat capture the most typical behavior of ice sheets or ice shelves,respectively. By successivelyamending formerlyneglected terms, anentire hierarchy of ice shelf and ice sheet models can be developed.The underlying principle of the so-called zero order model-class isthe Shallow Shelf Approximation (SSA) and the Shallow IceApproximation (SIA) for ice shelf and ice sheetmodels, respectively,discussed later and detailed in Appendix A.

Within their class, SSA and SIA have the largest degree ofnegligence for which yet typical ice shelf and ice sheet behavior,

respectively, is reproduced. Further negligence would yielduntypical physical behavior. Disappointingly, SSA and SIA modelscan not be patched together because combined shelf-sheetbehavior can not be described properly using zero order models.However, within the class of ‘second order (SO)models’, the SO-SSAand SO-SIA models have the right degree of negligence to allow fora proper coupling of ice shelves and ice sheets (Fig. 1). Practically,the classes in the model hierarchy are distinguished based ona shallowness parameter

ε :¼ typical vertical ice mass extenttypical horizontal ice mass extent

:

This parameter measures the shallowness of an ice shelf or anice sheet, and enters the theory as a perturbation parameter(Appendix A). An nth order model contains contributions ofequations involving all powers of ε from ε

0 to εn ; SO-SSA and SO-

SIA thus contain ε0, ε1 and ε

2. A complete model would accountfor all non-negative integer powers of ε, implying that no shal-lowness at all is assumed for the ice mass under question. Acomplete model is simultaneously valid for ice shelves and icesheets and corresponds to what is know as the full Stokes model influid dynamics (e.g. Batchelor, 1970).

2.3. Numerical model challenges of coupled ice shelf/ice sheetsystems

Since themid 1990’s, the refinement of numerical codes capableof treating coupled ice sheet/ice shelf systems is a topic of intenseresearch. More recently, the ice shelf-ocean interface and circula-tion in ice shelf cavities has received increased attention (e.g.Shepherd et al., 2004; Holland, D.M. et al., 2008a; Holland, P.R.et al., 2008b; Jenkins and Jacobs 2008). Considerable progress hasbeen achieved, but severe theoretical and numerical challenges stillremain to be addressed and overcome before simulations of, forexample, the ice shelf/ice sheet complexes that appear to havecovered large parts of the Arctic Ocean and its surrounding land-masses are expected to yield satisfying results (IPCC 2007; van derVeen and ISMASS members, 2007; Jakobsson et al., 2010a; SeaRISE;Ice2Sea).

The challenges associated with coupled ice shelf/ice stream/icesheet complexes are due to the components’ distinct spatialextensions, response time to external forcings, material inhomo-geneities, thermal regimes and mechanical stresses dominatingtheir flow behavior, and the generally poorly known subglacialbasal conditions (Figs. 2,4, Appendix A). The different regimesseparating sheet- from shelf- from stream flow have neither staticnor sharp boundaries, and their mutual interaction can not becaptured by patching together interacting regimes along commoninterfaces. Coupling of ice sheet/ice shelf/ice stream dynamics takesplace in a space-time varying transition zone extending up- anddownstream of the grounding line. There, a mixture of shelf andsheet flow prevails in varying portions depending on the distanceto the grounding line.

Four major challenges are: First, to properly couple the differentice dynamical flow regimes. Second, to properly account for espe-cially processes at the ice-bed interface. Third, to numericallysimulate dynamically coupled ice sheet/ice stream/ice shelfsystems at high resolution through entire glacial cycles. Fourth, tofurther develop existing models of grounding line migration,requiring thermodynamical investigation of the merging continua‘ice’, ‘water’, and ‘sediment’.

Fig. 4. Velocity profiles and stress states. Schematic visualization of characteristic velocity profiles (top row) and stress components (bottom row) arising in ice sheet (A), ice shelf(B) and ice stream (C) flow, cf. cubes A, B and C in Fig. 2. A) Velocity profile in the SIA (ice sheet flow), resulting from sliding at the base and gliding due to internal deformation. Flowis dominated by (xy) -parallel shearing stresses Txz and Tyz. The element dxdydz has z -dependent cryostatic pressure pð,; zÞ and shear stress distributions Txzð,; zÞ; Tyzð,; zÞ. Theindicated stress components are the only ones retained in a zero order model. B) Vertically uniform horizontal velocity field in the SSA (ice shelf flow). This velocity field is inducedby the depth-integrated (xy) -parallel normal and shear stresses TExx , T

Eyy , T

Exy . The integrated stress components are often referred to as membrane forces as illustrated for an element

dxdyH (H ¼ ice shelf thickness, as in Fig. 1). C) Velocity profile (ice stream flow) with pronounced contributions from sliding, resembling a combined sheet- and shelf-flow profile.Employing all six stress components of the Cauchy stress tensor corresponds to a full Stokes approach. A proper scaling analysis for ice streams would reveal the relative importanceof normal stresses Txx, Tyy, Tzz and shear stresses Txy, Txz, Tyz ; i.e. which ones could be assigned weights ε

n, n�1 without losing characteristic ice stream features.

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 3695

2.4. Spatial reconstructions of ice shelf/ice sheet complexes

Advances in high-latitude seafloor mapping in the 1990’s led tonew insights into the glacial histories of the Antarctic and theArctic. In particular, the data showed that disintegration of iceshelves and grounding line retreat is not unique to the 21st century.For example, glacial landforms on the Antarctic continental shelfrevealed that the extent of theWAIS varied considerably during thelast glacial cycle (e.g. Anderson 1999; Conway et al., 1999; Shippet al., 1999; Evans et al., 2006). Geological records of ice-shelfdisintegration, grounding line retreat and enhanced drainage ofinland ice facilitated by ice streams indicate that the processescurrently ongoing at West Antarctica’s marine interface indeedhave a history (Canals et. al., 2000; Dowdeswell et al., 2004;Graham et al., 2010; Jakobsson et al., 2011).

Spatial reconstructions of the Quaternary ice sheet extents in theArctic regionwere in focus for the previous programs PONAM (POlarNorth Atlantic Margins; Elverhøi et al., 1998) and QUEEN (QUater-nary Environments of the Eurasian North; Svendsen et al., 2004), aswell as for the active ‘Arctic Paleoclimates and its Extremes’ (APEX)program(Jakobssonet al., 2008c, 2010b) (Fig. 3). The reconstructionsproduced within these programs portray a scenario involvingmultiple episodes of glacial advances and retreats on the Arcticcontinental margins, likely associated with ice shelves fringing thewaxing and waning continental Eurasian and Amerasian ice sheets.Glacigenic features on the Morris Jesup Rise, Chukchi Borderlandand Yermak Plateau, three extensions of the Arctic Ocean conti-nental shelf, as well as on the submarine Lomonosov Ridge suggestthat ice shelves, perhaps similar in size to the Ross Ice Shelf and

Filchner Ice Shelf of present-day Antarctica, existed during theglacial maximum of MIS 6 (Polyak et al., 2001; Jakobsson et al.,2008b; Dowdeswell et al., 2010; Jakobsson et al., 2010a). Thedeepest groundings of MIS 6 ice shelf fragments occurred at ca 1000mwater depth on the Lomonosov Ridge crest and the Morris JesupRise. However, thinner ice shelves floating above these deep phys-iographic features may have existed during other glacial periodsthan MIS 6. Large glacial troughs formed in the shallow continentalshelf surrounding the central Arctic Ocean basin seem like a strongindication of that ice streams interlinked the continental ice sheetswith Arctic Ocean ice shelves (Stokes et al., 2005, 2006; Engels et al.,2007; Kleman and Glasser 2007; England et al., 2009). This isprecisely what is observed currently in Antarctica. In summary, it isnow understood from geophysical data and mapped marine glaci-genic landforms that Arctic Ocean ice-shelf complexes existed informer glacial periods and decayed in ensuing warmer climates(Vogtet al.,1994, Polyaket al., 2001,Dykeet al., 2002; Svendsenet al.,2004; Jakobsson et al., 2010a; O’Regan et al., 2010) (Fig. 3), toeventually be virtually absent at present.

3. A systematization of numerical ice shelf and ice sheetmodels

The dynamical behavior of ice shelves and ice sheets in isolationis commonly modeled using an asymptotic perturbation expansionapproach from thefield of appliedmathematics (e.g. vanDyke 1964;Kevorkian andCole 1981). In this approach, the parameter εdefiningthe shallowness of an ice shelf or an ice sheet enters the equationsdescribing iceflow. As indicated in Section 2.2, a hierarchy ofmodels

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e37043696

based on powers of ε can be derived. The perturbation expansionapproach can ultimately be used to couple ice sheet and ice-shelfdynamics in the transition zone where matching instead of patch-ing is needed to correctly resolve the dynamic interaction of icesheets, shelves and ice streams. Central to the description of icedynamics is the force balance (Newton’s second law), viz:

rdyðx; tÞ

dt¼ �grad pþ div TEðx; tÞ þ rg: (1)

Eq. (1) balances the product of time rate of change of velocity y

and ice density r with surface forces (stresses TE, pressure p) andvolume forces (gravity g), cf. Appendix A. From Eq. (1), ice shelf andice sheet models of different order, zero (SSA/SIA), first (FO-SSA/FO-SIA), and second (SO-SSA/SO-SIA), are derived.

3.1. Zero order models

Describing ice dynamics using the SSA and the SIA is thecoarsest possible approximation where the only terms left arethose that derive from zero-order powers of the shallownessparameter, ε0. Written as three scalar equations in x -, y - and zdirection, respectively, the final non-dimensionalized zero orderSSA/SIA equations corresponding to Eq. (1) are (for a derivation andnotation cf. Appendix A):

ðSSAÞ

0 ¼ �vpð0Þvx

þvTExxð0Þvx

þvTE

xyð0Þvy

þvTE

xzð0Þvz

0 ¼ �vpð0Þvy

þvTE

xyð0Þvx

þvTE

yyð0Þvy

þvTEyzð0Þvz

rswrsw � r

¼ �vpð0Þvz

þvTE

zzð0Þvz

(2)

ðSIAÞ

0 ¼ �vpð0Þvx

þvTE

xzð0Þvz

0 ¼ �vpð0Þvy

þvTEyzð0Þvz

1 ¼ �vpð0Þvz

:

(3)

Reducing the full force balance (1) to (Eq. 2), (Eq. 3) results invirtually unconstrained computability of ice shelves and ice sheetsin isolation. Therefore, numerical SSA/SIA-codes for ice shelves andice sheets can be run separately over several hundreds of thousandsof years. However, a critical limitation from retaining only zero-order terms is that the SSA and the SIA are valid only in the iceshelf and sheet, respectively.

Furthermore, the SIA is incapable of describing ice streambehavior. Ice streams interlink terrestrial and marine glacial envi-ronments hosting ice sheets and ice shelves, respectively, withevidence of paleo ice streams found in both environments (e.g.Stokes and Clarke 2001;Mosola and Anderson 2005; DeAngelis andKleman 2008; Dowdeswell et al., 2008). For example, a network ofice streams in the Canadian Arctic Archipelago drained the Lau-rentide ice sheet and fed large ice shelves in the Amerasian basin ofthe Arctic Ocean (Jakobsson et al., 2010a, Stokes and Tarasov 2010).At present, the Siple Coast of West Antarctica is a typical regionwith coupled ice sheet/ice stream/ice shelf flow (Bamber et al.,2000). Ice streams are rapid flow features which are embedded inthe surrounding slow moving or even close to stagnant ice, withpronounced zones of lateral shearing along the sides. As basalsliding often is enhanced in ice streams (Alley et al., 2004;Winsborrow et al., 2010), vertical shearing (Txz,Tyz) is no longerdominating ice dynamics and xy -parallel shear stresses (Txy) and

normal stresses (Txx,Tyy,Tzz) become increasingly important. Thus,ice streams exhibit many features that are also typical for ice shelfbehavior (Fig. 4), and can hence not be modeled appropriatelywithin the SIA.

Also, it is impossible to patch the SSA and SIAmodels together asstress components in the SSA force balance do not necessarily havecounterparts in the corresponding SIA force balance upstream ofthe grounding line, cf. Eq. 2. In the SSA equations, gradients ofnormal and xy -parallel ice shelf stresses (Txx, Tyy, Tzz and Txy) areretained because their impact is comparable to the one of ice shelfvertical stresses (Txz, Tyz). Moreover, seawater density rsw enters theSSA equations. For ice sheet flow, however, the vertical stressesdominate to such an extent that the other stress components areneglected in the zero order approximation.

A consistent coupling between shelf and sheet flow can only beperformed if each stress component accounted for in ice shelf flowhas a counterpart in ice sheet flow. Second order models have theleast order for which this is the case, allowing coupling between iceflowacross the grounding line (Figs.1, 4). This is also the frameworkfor ice sheet-ice stream coupling as modeling embedded icestreams is conceptually equivalent to the matching of ice sheet flowand ice shelf flow: For the latter, the transition from one flow regionto another takes place mainly in the along-flow direction whencrossing the grounding line. Characteristic flow behavior of icestream/ice sheet complexes is howevermost easily identified in thedirection perpendicular to the ice flow. The lateral shear marginsrepresent boundary layers separating regions of sheet flow fromregions of stream flow, each with different ice dynamics. Therefore,coupled modeling of ice stream-ice sheet flow transition ought tobe achieved with a compound of SO-SIA and SO-SSA models. Themodeling of Whillans Ice Stream based on a perturbation expan-sion approach has been discussed by MacAyeal (1989).

3.2. Second order models

In the Second Order Shallow Shelf- and Second Order ShallowIce Approximation equations (SO-SSA and SO-SIA), terms weightedby powers ε

n with n ¼ 0,1,2 are retained. The equations corre-sponding to (2) read (Appendix A)

ðSO� SSAÞ

0 ¼ �vpð2Þvx

þvTExxð2Þvx

þvTExyð2Þvy

þvTExzð2Þvz

;

0 ¼ �vpð2Þvy

þvTExyð2Þvx

þvTEyyð2Þvy

þvTE

yzð2Þvz

;

0 ¼ �vpð2Þvz

þvTExzð0Þvx

þvTE

yzð0Þvy

þvTE

zzð2Þvz

;

(4)

ðSO� SIAÞ

0 ¼ �vpð2Þvx

þvTExxð0Þvx

þvTE

xyð0Þvy

þvTExzð2Þvz

;

0 ¼ �vpð2Þvy

þvTExyð0Þvx

þvTEyyð0Þvy

þvTE

yzð2Þvz

;

0 ¼ �vpð2Þvz

þvTExzð0Þvx

þvTE

yzð0Þvy

þvTE

zzð0Þvz

;

(5)

Eqs. (4) and (5) show that stress components can now be tracedacross the grounding line: terms ½vðTijðkÞÞ=vðx; y; zÞ�shelf can bematched with terms ½vðTijðkÞÞ=vðx; y; zÞ�sheet, (k¼0,2). Still, thismatching requires careful treatment within a delicate procedureknown as ‘matched asymptotic expansion technique’ (van Dyke,1964; Kevorkian and Cole 1981; Schoof and Hindmarsh, 2010).This is necessary since some gradients appear with different orderson both sides of the grounding line; e.g. vTE

xyð2Þ=vx (shelf) vs.vTE

xyð0Þ=vx (sheet). The matched solution is valid in the transition

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 3697

zone and approaches the SSA solution on the ice shelf side and theSIA solution on ice sheet side (Fig. 1).

To our knowledge, no numerical SO-SSA code exists, while twosimplified versions of the SO-SIA have recently been implementedfor the purpose of simulating glaciation and landscape evolution(Egholm et al., 2011) and ice stream dynamics (Ahlkrona 2011).

3.3. Full Stokes models

The perturbation expansion permits infinite expansion in theshallowness parameter ε, see Eq. (A.2). Retaining termsweighted bypowers εnwith n/N, the equations for ice dynamics become exactand applicable for ice shelf and sheet flow simultaneously:matching across the grounding line becomes obsolete. Thus, forn/N, we obtain a unified set of equations, the so-called Full Stokes(FS) equations:

0¼�vpðx;y;z;tÞvx

þvTExxðx;y;z;tÞvx

þvTExyðx;y;z;tÞvy

þvTExzðx;y;z;tÞvz

;

0¼�vpðx;y;z;tÞvx

þvTExyðx;y;z;tÞvx

þvTEyyðx;y;z;tÞvy

þvTEyzðx;y;z;tÞ

vz;

0¼�vpðx;y;z;tÞvz

þvTExzðx;y;z;tÞvx

þvTEyzðx;y;z;tÞ

vyþvTE

zzðx;y;z;tÞvz

þrg:

(6)

The FS equations coincide with (1) if the acceleration terms inthe latter are neglected. FS models are computationally costly andare therefore restricted to spatio-temporal domains coveringhundreds of square kilometers and hundreds of years only. Routinesimulations of coupled paleo ice shelf/ice sheet systems are to datenot possible with FS models.

4. The need for higher order models in integrated modeling

Because a rigorous second order model capable of modeling iceshelf/ice stream/ice sheet dynamics at spatial resolutions and timescales matching those employed in spatial reconstructions based ongeological and geophysical data is not yet available, we can notevaluate its performance by comparing simulation results with fielddata. The need for such a model is however obvious, as we willillustrate below with a simulation experiment of the Svalbard icesheet. However, it should be noted that the higher order hybridmodelofPollardandDeConto (2009)hassuccessfullybeenapplied tomodel the West Antarctic Ice Sheet growth and decay through thepast 5 million years. Their simulations show good agreements withthe geological records of ANDRILL-AND1B. However, a relativelycoarse grid of 40 km resolution for continental experiments and 10km resolution for nested experiments in the Ross embayment andtheAND-1Bdrill sitewasused.This implies that comparisonbetweenmodel results andmappedglacigenic featureswith scales less thanoreven slightly more than 10km is difficult or even impossible.

4.1. Svalbards ice streams

Modeling coupled ice stream/ice sheet complexes may bea strategic intermediate step to be further developed beforemodeling the entire ice shelf/ice stream/ice sheet system. Apartfrom having to deal with only two instead of three components, icestream/ice sheet complexes are conceptually much simpler sincethe marine dimension is absent: to large extents, coupled icestream/ ice sheet systems are land based.

In contrast tofloating ice shelves, ice streams leave distinct traceson the continental shelf because they often are grounded on theseafloor during peak glacials. Ice streams are therefore particularly

attractive to investigate with combined use of spatial reconstruc-tions and numerical modeling. From submarine glacial landformsderived from high-resolution sea-floor imagery, Ottesen et al.,(2007) reconstructed the Late Weichselian ice sheet flow regimeincluding ice streams on thewestern and northern Svalbardmargin(Fig. 5A). Using the numerical SIA ice sheet code SICOPOLIS (Greve,1997; 2010), we performed conceptual simulations for the Sval-bard region, with the setup chosen to conform to existing spatialreconstructions at both regional and continental scales. Results arepresented in Fig. 5, panels C toF. Thepurposeof ourexperiments is 1/to show that despite unmodified ice flow dynamics, the paleo-icestreams mapped by Ottesen et al., (2007) are not reproduced well,while flow patterns over the contemporary topography of Svalbardindicate ice streams, 2/ to explain this inconsistency and 3/ to relateit to the limitations of zero order SIA ice sheet model. Details of themodeling setup are provided in Appendix B.

In summary, two simulations were carried out, run TOPO-LGMwith topographic boundary conditions conforming to the LastGlacial Maximum (LGM), and run TOPO-PRES with present daytopographic boundary conditions. Both runs start from assumed icefree conditions at arbitrary model time t ¼ 0, and evolve for 18 kyron a 10 km � 10 km grid. Simplified time-independent ‘glacialclimates’ are employed to force the ice sheet model through time.For run TOPO-LGM, air surface temperature T(x,y,z) in the entiremodeling domain is set to T(zsurf)¼�10 �C, while T(zsurf)¼�15 �C forrun TOPO-PRES. Variations of temperature with altitude, T(z), areaccomplished through an atmospheric lapse rate of �4.5 �C per1000m increase in elevation. RunTOPO-LGM is forced by a spatiallyvariable precipitation field P(x,y) derived from the data set ofLegates and Willmott (1990). For run TOPO-PRES, a constantprecipitation field P has been employed. The elevation-desertification feedback has for simplicity not been taken intoaccount, P does thus not vary with altitude. Since our simulationsaim at highlighting Shallow Ice Approximation (SIA) limitationswith regard to modeling ice streams, the use of such simplifiedclimate forcings seems justified. More realistic in-depth paleo-simulations of the Svalbard region would necessitate extensiveanalysis of climate data prior to employing it to force the ice sheetmodel (Pollard et al., 2000; Kirchner et al., 2011).

A general inspection of the simulation results (Fig. 5), based onthe absolute value of the horizontal ice velocities at the ice sheet

surface, yð0Þ :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2xð0Þ þ y2yð0Þ

qas an indicator of fast streaming ice,

suggests that for TOPO-LGM, there is no clear evidence for anystream-like ice flow (Fig. 5A, D). For TOPO-PRES, the simulatedpattern of surface velocities apparently indicate ice streams inplausible places, judging from the locations of the heads of mappedpaleo ice streams (Fig. 5A, F). A closer inspection of Fig. 5D, F revealsthat y(0) is generally highest at the ice margins following the conti-nental shelf break (TOPO-LGM) and Svalbards present day coastline(TOPO-PRES), respectively. Locally, high velocities are modeled tooccuralso in the interiorof the ice sheet: For TOPO-PRES, thesezonesof high velocity extend from the interior of the ice cap to its marinemargin, suggesting realistic ice stream flowwith mouths and headsin reasonable locations. For TOPO-LGM, high surface flow velocitiesdo, in contrast, not continuously stretch from the interior of the icesheet to the icemargin at the continental shelf break. Away from themargin, regions of high ice surface velocities instead coincide withregions of pronounced changes in ice thickness (Fig. 5C). Note thatalso for TOPO-PRES, changes in ice thickness can be related to highflow velocities (Fig. 5E).

These observations conform to the general result that thehorizontal ice velocities in an SIA model are proportional to thegradient of the ice sheet surface, where the degree of proportion-ality partly depends on local ice thickness (Greve and Blatter 2009):

Fig. 5. Modeling results. A: Reconstruction of Svalbard’s paleo-ice streams (Ottesen et al. 2007) and the LGM-ice margin in this region (Svendsen et al. 2004). B: Employed modelingdomain and basal topography/bathymetry as derived from the ETOPO1-dataset (Amante & Eakins 2009). C-F:Model results for twodifferent runs TOPO-LGMand TOPO-PRES using theSIA ice sheet codeSICOPOLIS (cf. AppendixB fordetails of the simulation set-up). C: 2D-viewof isolines of ice thickness for runTOPO-LGM.D: Planviewof ice surface velocityplottedoverthe ice sheet surface topography for run TOPO-LGM, cf. C. The SIA model is unable to reproduce paleo-ice streams that continuously extend from the interior of the ice sheet to theirtermination at the ice margin. E: 2D-view of isolines of ice thickness for run TOPO-PRES. F: Absolute velocity for run TOPO-PRES in E. Ice surface velocity is generally highest at the icemargin, which for TOPO-PRES is essentially dictated by the current land/oceanmask. Likewise, high surface velocity ismodeled to occur in the interiorwhereever ice surface gradientsare steep. Though it appears as if ice streams are captured with sufficient accuracy in this simulation, the modeled features do not reflect true ice stream behavior.

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e37043698

yxð0ÞðzÞ; yyð0ÞðzÞ fvzsvx

;vzsvy

A ðTÞðzs � z�Þ3dz�: (7)

� � � �Zz

bHere, zs ¼ zs(x,y,t) is the ice sheet surface, and A ðTÞ the

temperature dependent rate factor in Glen’s flow law. The lower

limit of integration is the ice base b, where a no slip condition isassumed for simplicity (cf. Sect. 2.3). The upper limit of integrationis an arbitrary position z in the ice sheet (z < zs) or at its surface(z ¼ zs). We remark that Eq. (7) is obtained from integrating the

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 3699

stress-strain relation (Glen’s flow law, linking spatial velocitygradients to stresses) and in which the zero order vertical shearstresses Txz, Tyz can be replaced by vzs=vx$ðzs � z�Þ andvzs=vy$ðzs � z�Þ. To relate Eq. (7) to the simulation results presentedin Fig. 5, let us start with a simple experiment of thought:

For a slab-type ice mass of constant thickness, resting on a flatbed with constant angle of inclination, the gradient of the icesurface (vzs/vx,vzs/vy) is constant. Neglecting variations in thermaleffects entering through A ðTÞ, Eq. (7) thus renders ice surfacevelocities that are constant since all terms on the right hand side ofEq. (7) are constant. In the schematic visualization given in Fig. 1,this implies that fast streaming ice (depicted by the arrow) cannotbe detected as ice surface velocities for both the ice stream and thesurrounding bulk ice are the same according to Eq. (7).

Over real basal topographies, however, ice thickness is typicallynot constant, with greater thickness over topographic lows thanover highs. Variations in ice thickness influence the ice surfacevelocities through the term (zs�z*)3 in the integral in Eq. (7), and incombination with the surface gradient thus determine the icevelocities at the surface (again neglecting variations in A ðTÞ forsimplicity). The dependence of the ice surface velocities on the icethickness is the reason why so-called topographic ice streams(those flowing in a trough) may roughly be captured even by SIAmodels. In the schematic in Fig. 1, this implies that fast streamingice (indicated by the ice sheet arrow), is likely to be detected if theice stream is flowing in a topographic trough, while the adjacentbulk ice does not. As ice thickness will be larger over the bathy-meytric trough hosting the ice stream, Eq. (7) renders higher icesurface velocities over the trough than over the adjacent, thinnerparts of the ice sheet. This effect is observed also in Fig. 5D, F.

However, pure ice streams as such on the Siple Coast, Antarctica,are not topographically controlled (Winsborrow et al., 2010).Therefore, ice thicknesses in the regions of ice stream flow andadjacent ice sheetflowwill be similar, implying that gradients of theice surface will not vary drastically (the latter assumption might nolonger be true at small scales e.g. resolving crevassed lateral shearzones). As in the experiment of thought above, Eq. (7) will result, inthe SIA frameworkwithinwhich it is derived, in ice surface velocitiesthat are essentially the same for the ice streamand the ice sheet. Thisis observed in Fig. 5D, where the ice thickness and ice surfacegradient over the continental shelf appear to be too homogenous toresult in notable changes in ice surface velocities.

Only higher order ice sheet models, inwhich other stresses thanthe shear stresses Txz and Tyz enter, have the capability of rectifyingthis shortcoming of the SIA: in the second order SO-SIA, the shearstresses Txz and Tyz contain additional contributions beyond thosestemmimg from the surface gradient times the local ice thickness:instead of Txz¼vzs/vx(zs�z*) as in the SIA, the corresponding secondorder SO-SIA relation can be qualitatively summarized as (fordetails, see Ahlkrona 2011)

Txz ¼ vzsvx

ðzs�z�Þ þ v

vx

ZðTzz�TxxÞdz� � v

vy

ZTxydz�þ. (8)

and where higher order surface topography related terms havebeen omitted for simplicity. According to Glens flow law, thestresses are linked to spatial velocity gradients. Hence, if stressesare known, velocities can be derived through integration. It isobvious from even the qualitative representation of the Txz as in Eq.(8) that the velocities computed in a SO-SIA thus will be differentfrom those in the SIA, and in particular contain contributions notrepresented at all in the SIA (sheet-parallel and normal stresses). Inthe schematic in Fig. 1, this implies that fast streaming ice (indi-cated by the ice sheet arrow) can be captured by the SO-SIA even iftopographic forcing is small: for bulk ice sheet flow, where verticalshear stresses dominate, the ice surface velocities will essentially

be the surface-gradient dominated ones derived from the SIA.However, over the ice stream flow region, where longitudinalstresses become important, the ice surface velocities will bedifferent from the surrounding SIA-ones if computed in the SO-SIAframework, which by construction takes those in-plane horizontalstress into account, see Eq. (8). Thus, improved modeling of icestream flow is possible only with second (or higher) order ice sheetmodels. We expect a SO-SIA model or any other higher order modelto be able to correctly simulate ice streams in the set-up chosen forTOPO-LGM (Fig. 5D), cf. also Hindmarsh (2009) for a discussion ofice stream dynamics and the here neglected thermo-mechanicalcoupling and its feedback mechanisms.

4.2. Coupled ice shelf e ice sheet systems around ice rises

Above, modeling of ice stream/ice sheet systems has been sug-gested as an intermediate target to be achieved prior to modelingcoupled ice sheet/ice shelf systems. However, the latter do notnecessarily involve ice streams. This is because ice shelf/ice sheetsystems may exist also in the vicinity of ice rises. At bathymetricheights, ice shelvesmay locally run aground. If the grounded area issmall, the ice shelf may slide over the seafloor giving rise to smalland sometimes fractured ice surface undulations referred to as icerumples. They rise typically less than 50 meters above the flat iceshelf surface. Ice rumples exhibit flow features which are domi-nated by the dynamical behavior of the surrounding ice shelf flow.For large grounding areas, the ice likely develops flow patternswhich mimic the dynamics of miniature grounded ice sheetsembedded in the ice shelf. Such groundings are called ice rises andcan reach heights up to several hundred meters above the ice shelflevel. By exerting a restraining force (‘backstress’) on the upstreamice shelf, ice rumples and ice rises impact ice shelf dynamics overlarge spatial areas. Ice rises and rumples have long been recognizedto play key roles in the maintenance or destruction of ice sheet/iceshelf systems (Thomas 1979; Martin and Sanderson 1980; Smith1986; Bindschadler et al., 1987; Doake 1987; MacAyeal 1987;Hindmarsh 1996, 2006; Hindmarsh and LeMeur 2001).

While present ice shelf groundings mainly are detected usingremote sensing techniques of the ice surface, past groundings arerecorded through glacigenic features preserved in the seafloor. Theprocesses leading to their formation and hence the nature of thegrounding events causing the glacigenic features remain discussed.Apart from local grounding of ice shelves causing ice rises, theseprocesses include the interaction of the seafloor with (i) armadas ofice bergs, (ii) large singular icebergs, and (iii) grounded ice sheets.Recently, late-Quaternary grounding events at the Yermak Plateauand the Lomonosov Ridge in the Arctic Ocean have been inter-preted based on geotechnical analyzes of ice-overridden sedimentsfrom beneath these groundings (O’Regan et al., 2010).

Once numerical simulations with coupled ice shelf/ice sheetmodels become feasible on timescales relevant for paleo-simulations, their results can be contrasted to field evidence toeither support or dismiss hypotheses concerning the nature ofgrounding events. To our knowledge, no existing numerical codeaccounts for the coupling of ice shelf/sheet dynamics around icerises in ice shelves yet. Such a model would have to be at leastsecond order, involving SO-SSA and SO-SIA. Moreover, heavilycrevassed ice margins around ice rises indicate that damagemechanics might have to be incorporated into the modeling. Suchmodels have not yet been proposed.

5. Conclusions

In this paper, we present a review of the fundamentals ofcontemporary ice sheet modeling for a broad Earth Science

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e37043700

audience intended to be accessible also to those with no specificbackground in theoretical glaciology and numerical modeling. Themain goal is to provide a basis for increased communicationbetween especially non-modelers and modelers sharing a commoninterest in the dynamics of ice shelves and ice streams. Hence, tobridge the gap, the degree of mathematical detail and topicalinclusiveness has deliberately been limited: E.g., grounding linedynamics, marine instability and the role of sedimentary bedsbeneath fast flowing ice are excluded from our review because weconsider these topics too specialized to be included in a reviewintended to be cross-disciplinary

We discuss different levels of basic model assumptions; fromsimplified Shallow Ice Approximation (SIA) schemes (of zeroth,first, and second order) to the (exact) Full-Stokes equations. Thelimitations of a zero order ice sheet model are shown withconceptual simulations of a hypothetical glaciation over Svalbard,focusing on ice stream dynamics.

We compare those conceptual simulation results to Ottesenet al., (2007) spatial reconstruction of Svalbard paleo-icestreams, and explain the discrepancies by discussing the equa-tion governing ice surface velocities (naturally indicative of fastflowing ice) in any SIA model. The mathematics involved are few,and tightly interlinked with the examples presented. We showthat in a zero order model, where the ice surface velocities areproportional to the ice surface gradient and the ice thickness, pureice streams can hardly be modeled, while topographically con-troled ones may occasionally appear to be modeled correctly,simply because higher local ice thickness (as e.g. over troughs)give rise to higher surface velocities. We emphasize that these SIA-limitations in capturing ice stream dynamics are related to theneglection of other than vertical shear stresses (Txz,Tyz) in any zeroorder model. We further sketch why higher order models, takinginto account sheet-parallel stresses, can improve simulations offast streaming ice.

Acknowledgments

Financial support from the Bert Bolin Center for ClimateResearch enabled KH to visit Stockholm University and is gratefullyacknowledged. J. Robbins and H.J. Zwally kindly provided part ofthe data used in Fig. 2. This is a contribution from the Bert BolinCenter for Climate Research at Stockholm University.

Appendix A. Derivation of zero order-, higher order and FullStokes models

Zero order models

The derivation of zero order models relies on a perturbationexpansion approach. The procedure is illustrated for themomentum balance. Higher order models are obtained in the sameframework, as we will show later. In the literature, ice sheet modeldescriptions often start from the evolution equation of ice thick-ness. The latter stems from the mass balance and is different fromEq. (A.1).

rdyðx; tÞ

dt¼ �gradpþ divTEðx; tÞ þ rg: (A.1)

The momentum balance Eq. (A.1) does not distinguishbetween ice-shelf flow and ice-sheet flow. In Eq. (A.1), r is themass density, y the velocity, p the pressure, TE the symmetricextra Cauchy stress tensor, and g the gravitational acceleration.The material time derivative is represented by d/dt with tdenoting time. The operators grad and div take the gradient of

the pressure field and the divergence of the stress field TE atx¼(x,y,z) and time t.

Separating hydrostatic pressure contributions from the totalstresses is common in glaciology. The splitting of the total stresstensor T is typically written as T¼�pIþTE where pI is the hydro-static pressure tensor (Paterson 1994). Imaged as 3 � 3 matrices, pIhas pressure p on the diagonal entries, and zeros elsewhere. Theextra Cauchy stress TE has the normal stresses TE

xx ¼ Txx þ p, TEyy ¼

Tyy þ p and TEzz ¼ Tzz þ p on the diagonal entries, and the vertical

shear stresses TExz ¼ Txz ¼ Tzx, TE

yz ¼ Tyz ¼ Tzy and the xy -parallelshear stress TExy ¼ Txy ¼ Tyx on the off-diagonal entries. Theorientation of these stresses is depicted in Fig. 4C.

Central to the perturbation expansion approach is

� a parameter ε, accounting for the shallowness of ice shelves andsheets,

� a suitable scaling of the ice dynamic equations in terms of ε,� the representation of the functions describing e.g. the stressdistribution in the ice as infinite series in ε.

Discussing these ingredients outlines the derivation of theShallow Shelf Approximation (SSA) and the Shallow Ice Approxi-mation (SIA) equations, respectively, see Eqs. (2),(3).

First, ε is introduced to reflect that the vertical dimensions ofice sheets and shelves are much smaller than their horizontalones: they are shallow. For ice shelves, typical vertical andhorizontal dimensions are [H] ¼ 500 m, [L] ¼ 500 km, whiletypical horizontal and vertical velocities are given by[VH] ¼ 1 ma�1 and [VL] ¼ 500 ma�1. For ice sheets, typicalvertical and horizontal dimensions are [H] ¼ 1 km and[L] ¼ 1000 km, with typical horizontal and vertical velocitiesgiven by [VH] ¼ 0.1 ma�1 and [VL] ¼ 100 ma�1. Definingε:¼[H]/[L]¼[VH]/[VL] yields εz10�3 for both ice shelves and icesheets.

Second, the shallowness parameter is used to scale the variablesin the ice flow equations after non-dimensionalization. Forinstance, stress components are made dimensionless by intro-ducing a common stress scale rgH, where r and H are ice densityand ice thickness, and g the gravity constant. Further, a validassumption for the bulk of an ice sheet is that vertical shear stressesTExz; T

Eyz dominate over sheet-parallel shear stresses TE

xy. This can beintroduced into the ice flow equations by e.g. multiplying (’scaling’)the different stress components with different powers of ε : thehigher the power of ε in e.g. εnTE

xy with n ¼ 1,2,3. is, the less‘weight’ is assigned to TE

xy. An important result of the scalingprocedure is that the acceleration terms on the left hand side of(A.1) can be neglected.

Third, a perturbation scheme is employed according to whichquantities arising in the force balance of ice shelf and ice-sheetflow are written as infinite power series in the aspect ratio ε. Forthe hydrostatic pressure p this infinite series reads (note thatε0¼1)

p ¼ pð0Þ þ εpð1Þ þ ε2pð2Þ þ ε

3pð3Þ þ. ¼XNi¼0

εipðiÞ: (A.2)

Eq. (A.2) is called a perturbation expansion of p in the pertur-bation parameter ε.

The further procedure is as follows: All functions arising in theforce balance of ice dynamics are expressed as infinite power seriesin ε. Those functions which were assigned ‘weight’ ε0 in the scalingprocess appear in the form given by Eq. (A.2). Quantities ‘weighted’with ε

n,n�1 enter the force balance with corresponding higherpowers of ε. Specifically, for ice sheet flow the verticalshear stresses enter as εTExz ¼ εTExzð0Þ þ ε

2TExzð1Þ þ. and

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 3701

εTEyz ¼ εTEyzð0Þ þ ε

2TEyzð1Þ þ.. The sheet parallel shear stresses are

scaled with ε2 in the SIA: ε2TE

xy ¼ ε2TE

xyð0Þ þ ε3TE

xyð1Þ þ.. Likewise,the normal stress components enter through ε

2TExx ¼

ε2TE

xxð0Þ þ ε3TE

xxð1Þ þ.; ε2TEyy ¼ ε

2TEyyð0Þ þ ε3TE

yyð1Þ þ., and ε2TE

zz ¼ε2TE

zzð0Þ þ ε3TEzzð1Þ þ.. Conceptually, Eq. (A.1) then takes on the

form A¼0, or rather, A0 þ εA1 þ ε2A2 þ. ¼ 0 ¼ 0,ð1þ ε

þε2 þ.Þ. This equation is requested to be valid for all powers of ε

individually, implying that terms of equal powers of ε are equated:for ε

0, comparison of coefficients yields A0¼0. For ε1, A1¼0 is ob-

tained, etc. For the ice flow equation, equating terms which are oforder ε0 leads to the Shallow Ice Approximation (SIA) for ice sheets(Fowler and Larson 1978; Morland and Johnson 1980; Hutter 1983;Greve 1997; Baral 1999; Baral et al., 2001; Greve and Blatter 2009).

A different scaling is appropriate for ice shelves. The dimen-sionless normal and in-plane stresses p, TE

xx, TEyy, T

Ezz and TExy are

assumed to be of order unity ( ε0), while the vertical shear stressesTExz, T

Eyz are of order ε1. The vertical shear stresses are assigned less

weight because they are negligibly small at the free surface and atthe ice-ocean interface; therefore they must be small inside the iceshelf. The derivatives vTExz=vz, vT

Eyz=vz are, however, of order ε

0.Then, the perturbation expansions for the stress quantities aresubstituted into the force balances. Equating terms of order ε0 thenyields Shallow Shelf Approximation (SSA) equations (Morland andShoemaker 1982; Morland 1987; Weis et al., 1999; Baral 1999).

Both equation sets, the SIA and the SSA, are given in Eqs (2) and(3), (3.1) in the main text (Section 3.1). In a subsequent step notdiscussed here, the stress components are linked to the strain ratecomponents, expressing the material response of ice undercreeping motion: this is typically achieved by employing Glen’sflow law (Paterson 1994) or one of its generalizations (Hutter 1983;Greve and Blatter, 2009).

Discussion of SSA and SIA models

� Obvious differences in the SSA and SIA in (2), (3) stem from thedifferent scaling of the stresses: the stress components TExx, T

Eyy,

TEzz and TExy are O(ε2) in the SIA, but O(ε) in the SSA. A consistent

merging of ice shelf/ice sheet dynamics must thereforenecessarily be at least of O(ε2). A detailed description of thescaling procedure is given in Hutter (1983), Baral (1999) andWeis et al., (1999).

� The SSA model satisfies the floatation condition for ice shelvesexpressed as

hboard ¼�1� r

rsw

�H;hdraft ¼ r

rswH:

Here, hboard and hdraft are the ice shelf freeboard height anddraft, related by the local ice thickness H as H¼hboardþhdraft (Fig. 1).rH ¼ RH

0 rð,; zÞdz is the average density in the ice column. Thefloatation condition is derived from vertical integration of the SSA-Eq. (2).3 During integration, zero atmospheric pressure conditionsat the free ice shelf surface are exploited. At the ice-ocean interface,pressure equals the weight of the overburden water column.

� The shear stresses TExz and TEyz at the base of an ice sheet are

given by the overburden pressure times the surface slope of theice sheet. This result is frequently used as an a priori fact inmore phenomenological ice modeling approaches (Paterson1994). Here, it is derived from vertical integration of SIA-Eq.(3).1,2

� Horizontal velocities vx, vy are depth-independent in the SSA,but depth-dependent in the SIA. Indeed, vertical profiles ofhorizontal velocities in the SIA at a fixed position (x,y) and fixedtime t generally have a sliding and a gliding contribution. The

latter reflects the influence of material shearing, the former isdue to viscous and frictional basal slip (Fig. 4A). The relativeimportance of gliding and sliding varies through the depth ofthe ice sheet. Consequently, the velocities in the x - and y-directions satisfy vx¼vx(x,y,z,t) and vy¼vy(x,y,z,t), respectively.By contrast, SSA horizontal velocity components are uniformlydistributed over depth (Fig. 4B). However, they vary in thehorizontal flow directions according to the material action ofthe membrane forces

RH0 ðTExx; TE

yy; TEzzÞdz. Consequently,

vx¼vx(x,y,t),vy¼vy(x,y,t). In summary, SIA behavior is reminis-cent of z -dependent simple shearing, subject to z -dependentcryostatic pressure (Fig. 4A). SSA behavior is akin to membranebehavior, subject to normal and shearingmembrane stresses, ofwhich the strength is governed by the horizontal thicknessgradient (Fig. 4B). Clearly, the SIA formulation must be amen-ded by adding membrane behavior to the simple shearing, andthe SSA formulation must be extended by adding some verticalshearing if a coupled shelf/sheet model is to be adequatelydescribed. The same is expected to be required to model icestreams: a typical ice stream velocity profile approaches the icesheet velocity profile if sliding is reduced, and the ice shelfvelocity profile if sliding is dominating over gliding (Fig. 4C).

First order models

The framework of perturbation expansions allows also thederivation of higher order models. Equating terms of order ε1 leadsto the ‘First Order Shallow Shelf Approximation (FO-SSA)’ and ‘FirstOrder Shallow Ice Approximation (FO-SIA)’. As FO-SIA and FO-SSAare formally no different from the SSA- and SIA approximations,their value has been discussed controversially in the literature(Johanneson 1992; Blatter 1995; Mangeney and Califano 1998;Baral et al., 2001). Of relevance in the present context is thatthese equations, just as their SSA and SIA counterparts, do neitherallow for a consistent coupling of ice shelves and ice sheets withinthe class of first order models.

Higher order and second order models

Equating terms of order ε2 leads to the ‘Second Order ShallowShelf Approximation (SO-SSA)’, and ‘Second Order Shallow IceApproximation (SO-SIA)’. These equations are given in (4) and (5)and (3.2) in the main text.

Previously, some zero order ice sheet models have been amen-ded by selected first and/or second order contributions. However,the former are not full second order models as not all terms thatshould be present have been included. Often, such models aresimply called ‘higher ordermodels’. Despite the lack of rigor in theirderivation, they work well for the applications they were designedfor. Examples include glacier and ice sheet flow (Blatter 1995;Blatter et al., 1998; Colinge and Blatter 1998; Hubbard 2000;Pattyn 2003), simplified continental-scale ice-age ice sheet/shelfdynamics (Pollard and DeConto 2009), and modeling of sliding atthe base of the ice sheet (Schoof and Hindmarsh, 2010). Higherorder and full Stokes models have recently been benchmarked inthe Ice Sheet Modeling IntercomParison e Higher Order Models(ISMIP-HOM) project (Gagliardini and Zwinger 2008; Pattyn et al.,2008). A review of different higher order ice approximationmethods from the viewpoint of mathematical geophysics is givenby Hindmarsh (2004).

It is remarked that the term ‘hybrid models’ is becomingincreasing popular in the context of higher order coupled ice shelf/ice sheet flow (Pollard and DeConto 2009). It is also used inconnection with inverse modeling strategies in which e.g. ice

N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e37043702

stream basal parameters are optimized until modeled and observedice surface velocities show minimal discrepancy (MacAyeal 1992;Khazendar et al., 2007; Goldberg and Sergienko 2010).

Full Stokes models for coupled ice shelf/ice sheet systems

Full Stokes models solve Eq. (1) under the assumption ofnegligible accelerations. The resulting set of equations is presentedas Eq. (6) in the main text. As no other approximations are made,Eq. (6) are most exact, but also most costly to solve. Due to thecomplexity of Eq. (6), full Stokes simulations require substantialcomputational resources. When simulating glacier flow, a typicalmodeling domain size is 10 km� 10 km, resolved by a 50 m� 50 mgrid in the horizontal. A typical time period covered by sucha simulation is 200 years. If computed serially, the effective timeneeded to complete such a simulation is in the order of days (Jouvetet al., 2008, 2009). Using a zero order ice sheet model allows, incontrast, to simulate e in the same amount of time (days) e theevolution of ice sheets over domains as large as an entire hemi-sphere, and over hundreds of thousands of years, with horizontalspatial resolutions ranging typically from 10 km � 10 km to40 km � 40 km. Full Stokes models 4 are typically discretized usingFinite Element technique, and thus allow in principle for the use ofunstructured and adaptive meshes. This implies that from a theo-retical point of view, there are virtually no limitations to the grid-size employed. In glaciological applications, the mesh size willhowever be dictated by the problem to be investigated: to date,meshes typically do not employ horizontal resolutions smaller than5 m (LeMeur et al., 2004; Zwinger et al., 2007; Zwinger and Moore2009). While data used to initially constrain (e.g. bedrock topog-raphy) or to subsequently force (e.g. precipitation and air surfacetemperature) the ice model are typically not available at such highresolution, the situation is different when numerical modelingresults are to be compared with field evidence. For example,mapping of glacigenic features in the glacial troughs occasionallyreaching depths greater than 1000 m on glaciated continentalmargins, have generally been carried out using deep water multi-beam systemswith horizontal resolutions limited to approximately15e20 m and vertical of about 0.5% of the water depth. However,recent mapping of Pine Island Trough, West Antarctica, with thenew generation deep water system with higher resolution capa-bilities revealed smaller scale features of critical importance for theglacial dynamics (Jakobsson et al., 2011). These were corrugationridges, regular features the length of which ranges between 60 mand 200m, while their height varies between 1m and 2m only. Thecorrugation ridges are indicative of an ice shelf break up in PineIsland Bay at about 12 kyr. Numerical reconstructions of this iceshelf break up will crucially depend on models that can be run athigh spatial resolution if model output is to be compared with fieldevidence, here e among others e in the form of the small scale andspatially sparsely distributed corrugation moraines.

Appendix B. Ice sheet modeling set up

We employ SICOPOLIS (SImulation COde for POLythermal IceSheets), which is a 3D, thermodynamically coupled ice sheet modelbased on the Shallow Ice Approximation (SIA). SICOPOLIS isapplicable to polythermal ice masses, but has been run in cold-icemode for the Svalbard modeling experiments. SICOPOLIS providestime-dependent extent, thickness, velocity, temperature, water

4 The most frequently used Full Stokes model in glaciological applications is‘Elmer’, which is freely available from the finnish CSC-IT center for Science (http://www.csc.fi).

content and age for grounded ice sheets in response to externalforcing. The latter is given by precipitation, surface temperatureand geothermal heat flux. SICOPOLIS employs Glen’s law todescribe the flow of ice. The thermo-mechanical coupling isdescribed by the rate factor A which is temperature-dependent forcold ice, and water-content dependent for temperate ice. Isostaticeffects due to ice load are modeled by the Elastic-Lithosphere-Relaxing-Asthenosphere approach (Greve and Blatter 2009).Effects of heat conduction through the lithosphere are accountedfor through the geothermal heat flux, which is assumed constant ata value of 40 � 10�3 Wm�2 for the Svalbard modeling experiment.More detailed information concerning SICOPOLIS can be found inGreve (1997; 2010).

The Svalbard modeling domain is shown in Fig. 5A. A10 km � 10 km discretization in the horizontal is employed whichresults in 91 � 81 in-plane gridpoints. In the vertical, the cold icecolumn and the lithosphere are discretized by 81 and 11 gridpoints, respectively. Simulations start with assumed initial ice freeconditions at arbitrary model time t¼0 kyr and evolve for 18 kyr.Topography/bathymetry is derived from the ETOPO1 data set andinterpolated to the SICOPOLIS-grid (Amante and Eakins, 2009).

The different topographic boundary conditions employed inruns TOPO-LGM and TOPO-PRES are derived from different land/ocean masks. For TOPO-LGM, gridpoints with elevations less than500 m below current sea level are assigned ‘ocean’, while theremaining ones are treated as ‘land’. Thus, land available forglaciation extends to the continental shelf break in TOPO-LGM. Forrun TOPO-PRES, the threshold value distinguishing ‘land’ from‘ocean’ gridpoints is 0 m; the land/ocean mask corresponds to thepresent situation. A standard PDD model is used to model accu-mulation and ablation at the surface of the ice sheet. At its base,a Weertman-type sliding law is employed with uniform moderatesliding coefficient as suggested in SICOPOLIS’ ‘Austfonna-module’(Greve 2010). The effects of different and varying basal boundaryconditions and their implications for the dynamics of the Austfonnaice cap is discussed in Dunse et al. (2011).

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