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Capabilities and limitations of numerical ice sheet models: A discussion for Earth-scientists and modelers

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

Nina Kirchnera,*, Kolumban Hutterb, Martin Jakobssonc, Richard Gyllencreutzc a Department of Physical Geography and Quaternary Geology, Stockholm University, 10691 Stockholm, Sweden b c/o Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology, 8092 Zürich, Switzerland c Department of Geological Sciences, Stockholm University, 10691 Stockholm, Sweden article info abstract

Article history: The simulation of dynamically coupled ice sheet, ice stream, and ice shelf-systems poses a challenge to Received 12 April 2011 most numerical ice sheet models. Here we review present ice sheet model limitations targeting a broader Received in revised form audience within Earth Sciences, also those with no speciﬁc background in numerical modeling, in order 13 September 2011 to facilitate cross-disciplinary communication between especially paleoglaciologists, marine and Accepted 15 September 2011 terrestrial geologists, and numerical modelers. The ‘zero order’ (Shallow Ice Approximation, SIA)-, ‘higher Available online 22 October 2011 order’-, and ‘full Stokes’ ice sheet models are described conceptually and complemented by an outline of their derivations. We demonstrate that higher order models are required to simulate coupled ice sheet- Keywords: Spatial reconstructions ice shelf and ice sheet-ice stream systems, in particular if the results are aimed to complement spatial ice ﬂ Ice shelves ow reconstructions based on higher resolution geological and geophysical data. The zero order SIA Ice streams model limitations in capturing ice stream behavior are here illustrated by conceptual simulations of Ice sheet modeling a glaciation on Svalbard. The limitations are obvious from the equations comprising a zero order model. Shallow Ice However, under certain circumstances, simulation results may falsely give the impression that ice Approximation streams indeed are simulated with a zero order SIA model. Higher Order models Ó 2011 Elsevier Ltd. All rights reserved. Svalbard

1. Introduction to ice shelf disintegration (Rignot et al., 2004; Shepherd et al., 2004; Rignot and Steffen 2008; Goldberg et al., 2009; Pritchard et al., Ice shelves are ice masses floating in the ocean while being 2009). Once rapidly retreating, ice shelves and floating glacier attached to and fed by grounded glaciers or an ice sheet (Figs. 1, 2). tongues affect the mass balances of their feeding ice sheets and Large ice shelves are today encountered almost exclusively along glaciers considerably through accelerated drainage. Since this the Antarctic coastline, although smaller ice shelves also exist ultimately contributes to changes in sea level, changes in ice-shelf around Greenland, northern Ellesmere Island, the Franz Josef Land configurations are carefully observed and recorded by space- Archipelago and Severnya Zemlya in the high Russian Arctic. Ice borne missions such as CryoSat, ICESat and IceBridge. The data shelves are highly dynamic as illustrated by the recent partial from these missions serve as input in prognostic numerical ice collapses of the Larsen and Wilkins ice shelves (Antarctica) and the modeling experiments focusing on the Greenland as well as the break ups of Petermann Glacier and the Ward Hunt Ice Shelf in Antarctic ice sheets and their fringing ice shelves (e.g. Fricker and Northwest Greenland (Rott et al., 1996; Mueller et al., 2003; Glasser Padman 2006; Wingham et al., 2006; Khazendar et al., 2007; Ray and Scambos 2008; Humbert and Braun 2008, Falkner et al., 2011). 2008; Fricker et al., 2009). Oceanic forcing, rapid variation in intrinsic tributary glacier and Paleo ice-shelf configurations are mainly understood through ice-stream dynamics, as well as changes in configuration of the spatial reconstructions based on interpretations of mapped, and coupled ice sheet-ice shelf complex are the major processes related ideally 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- * Corresponding author. Tel.: þ46 8 162988/þ46 70 6090588; fax: þ46 8 164818. climate model, run to investigate ice sheet-ice shelf reaction to E-mail addresses: [email protected] (N. Kirchner), [email protected]. ethz.ch (K. Hutter), [email protected] (M. Jakobsson), richard.gyllencreutz@ oceanic changes on timescales relevant for paleosimulations have geo.su.se (R. Gyllencreutz). been presented to the scientific community where they are still

Fig. 2. Filchner Ice Shelf region, Weddell Sea sector, West Antarctica. Different flow regimes: A) Bulk ice sheet flow of shearing type over frictional bedrock topography Fig. 1. Sketch of a coupled ice sheet-ice shelf system. In sufficient upstream (down- induces undulations occasionally propagating to the ice sheet surface. B) Thinning and fl fl stream) distance from the grounding line, pure ice sheet flow (ice shelf flow) is diverging ice shelf ow (Filchner Ice Shelf) exhibiting an essentially at and undis- described by the zero order Shallow Ice Approximation SIA (zero order Shallow Shelf turbed upper ice surface due to virtually frictionless motion over the ocean surface. C) fl Approximation SSA). In a transition region across the grounding line, both sheet flow Rapid ow of tributary glaciers (Slessor Glacier) and ice streams (Recovery Ice stream) fl and shelf flow features prevail. There, SIA and SSA have to be extended before they can interlinking sheet and shelf ow regimes. The white solid and stippled lines denote the be coupled using the second order SO-SIA and SO-SSA modeling approaches. grounding line and the calving front of Filchner Ice Shelf, respectively, derived from ICESat and MOA data and provided by J. Robbins and H. J. Zwally, NASA GSFC. The cubes A) B) and C) are detailed in Fig. 4. Background picture from http://earthobservatory. open to discussion (Alvarez-Solas et al., 2011). Overall, paleo-ice nasa.gov/IOTD/view.php?id¼7620. shelf evolution from inception to decay is yet to a much lesser extent complemented by numerical modeling than investigations of future ice shelf behavior where modeling is central. The main 2. Diversity of ice shelf research reason for this is the weak mutual exchange between the disciplines investigating paleo ice shelves and the disciplines addressing The identification of ice shelves as individual components of the 2 prognostic future ice shelf behavior. This highlights the general cryosphere worthwhile modern scientific investigation dates back need for increased cross-disciplinary communication between to the late 1950’s. Ice shelf specific research conducted during the paleoglaciologists, marine and terrestrial geologists/geophysicists past six decades can be roughly divided into two periods: in the and numerical modelers sharing a common interest in ice shelf early years, ice shelves were mainly investigated in isolation from complexes.1 the feeding ice sheets, the adjacent ocean and the atmosphere. Here, we suggest a common basis of departure for such cross- From the mid 1970’ies, the dynamics and stability of ice shelf disciplinary communication: we aim to provide readers from the complexes, including grounding line dynamics, became a key issue Earth Sciences with no specific background in modeling with (Weertman 1974, Thomas 1979, Bindschadler et al., 1987). Since a conceptual understanding of different types of numerical models then, an increasing number of disciplines included ice shelves in (‘zero order’-, ‘higher order’-, and ‘Full Stokes’ ice shelf and/or ice their research agenda once their critical role in the global climate sheet models). As definition and benefit of each particular type of system was realized (Polyak et al., 2001; Fricker and Padman 2006; model often remain obscure to non-modelers, we address the clas- Khazendar et al., 2007; Jenkins and Jacobs 2008). Despite overall sification, limitations and merits of these models (Section 2.2,2.3,3). similar scientific objectives, the disparity of the research fields Focusing on ice streams we demonstrate that zero order models are hindered cross-disciplinary collaboration with some few excep- insufficient to capture their dynamics correctly (Section 3.1,4.1). This tions. On the other hand, reconstructions of Quaternary ice-sheet is well-known to modelers and obvious from the equations under- configurations began in the late 1990’s taking cross-disciplinary lying any zero-order model. Yet, visualizations can veil these limita- approaches by combining numerical modeling and spatial recon- tions and may even suggest that ice stream dynamics are accounted structions, though treating ice-shelf components in the numerical for. Presumably, confusion resulting from such peculiarities has in the modeling in a simplified manner only (e.g. Siegert and Dowdeswell past hindered cross-disciplinary communication. We therefore 1999; Siegert et al., 2001; Kleman et al., 2002; Tarasov and Peltier present conceptual simulations of a glaciation over Svalbard which 2004; Napieralski et al., 2007; Stokes and Tarasov 2010). Cross- illustrate the mentioned limitations and pitfalls of zero order model disciplinary studies involving simulations with combined ice results (Section 4). Also, we advocate, to the wider community of Earth Science researchers interested in ice shelf complexes, the necessity and use of higher order ice models. 2 In the mid 19th century, ice shelves were effective barriers to the conquest of the South Pole; when James Clark Ross, on January 9, 1841, first sighted the ice shelf named 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 the 1 We refer to one or more ice shelves being fed by an unspecified number of ice first to introduce the terminology ‘ice shelf’ as a substitute for the term ‘barrier’ streams draining an attached inland ice mass as an ‘ice shelf complex’. suggested by Ross (Nordenskjöld 1909). N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 3693

Fig. 3. Spatial reconstructions of Pan-Arctic paleoglacial configurations. Map showing the Arctic Ocean bathymetry revealing several large bathymetric troughs, in the continental margins, 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 ice shelf 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 of this article.) sheet/ice shelf models and comparison with geological records are ice-shelves covering at least parts of the central Arctic Ocean less common and occur in the scientific literature first more sometime during the Quaternary. Mercer’s Arctic Ocean ice shelves recently (e.g. Pollard and DeConto 2007, 2009). have since been a recurrent topic among paleoglaciologists. Recent geological and geophysical data show that there indeed existed 2.1. Early ice shelf specific research extensive ice shelves, at least in the Amerasian Arctic Ocean, during Marine Isotope Stage (MIS) 6 (Jakobsson et al., 2010a); a topic If considered in isolation, the behavior of ice shelves as well as further discussed below. Since the mid 1970’s, the importance of ice ice sheets is well understood and modeled since the late 1950’s. The shelves for the persistence of marine ice sheets, such as the West timing of this advance in knowledge is linked to the Norwe- Antarctic Ice Sheet (WAIS), was recognized and related to both the gianeBritisheSwedish 1949e1952 expedition to the Maudheim Ice migration of the grounding line and the intermittent mass loss at Shelf as well as the First International Geophysical Year (the Third the calving front in response to oceanic forcing (Weertman International Polar Year) 1957e1958 expeditions to the Filchner 1974,1976; Thomas and Bentley 1978). Furthermore, it was recog- and the Ross ice shelves (Roberts 1950; Giaever and Schytt 1952; nized that the bulk3 of the ice shelf played a less critical role than Robin 1953; Odisha and Ruttenberg 1958; Zumberge et al., 1960; both the sheet-shelf transition region across the grounding line, Crary et al., 1962). Those expeditions focused, for the first time since the days of the early explorers, specifically on ice shelves. From physiographic analogies between West Antarctica and the 3 Ice shelves “produce rather flat slabs of floating ice which, for the theoretician, Arctic Ocean, Mercer (1970) proposed the hypothesis of extensive are the simplest of all ice masses” (Thomas, 1979). 3694 N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 and the calving front. Ice shelf complexes have been investigated respectively, is reproduced. Further negligence would yield using different techniques and with various foci by Weertman untypical physical behavior. Disappointingly, SSA and SIA models (1974), Robin (1975), Thomas and Bentley (1978), Chugunov and can not be patched together because combined shelf-sheet Wilchinsky (1996), Hulbe and MacAyeal (1999) and Hulbe and behavior can not be described properly using zero order models. Fahnestock (2004). Attempts to e.g. access the cavity beneath an However, within the class of ‘second order (SO) models’, the SO-SSA ice shelf by drilling through it were made from the mid 1970s and SO-SIA models have the right degree of negligence to allow for within the multifaceted Ross Ice Shelf Project (RISP) (Thomas et al., a proper coupling of ice shelves and ice sheets (Fig. 1). Practically, 1984). In 1984, the Filchner-Ronne-Ice-Shelf-Programme (FRISP) the classes in the model hierarchy are distinguished based on was started as a subcommittee of the Scientific Committee on a shallowness parameter Antarctic Research (SCAR) Working Group of Glaciology; it has now widened its scope with the acronym FRISP instead denoting ‘Forum ε : ¼ typical vertical ice mass extent : for Research into Ice Shelf Processes’. The ongoing Amery Ice Shelf typical horizontal ice mass extent Ocean Research (AMISOR) project aims to quantify ice shelf/ocean This parameter measures the shallowness of an ice shelf or an interaction and succeeds the earlier Amery Ice Shelf Project (Budd ice sheet, and enters the theory as a perturbation parameter et al., 1982). In the mid 1990’s, the state of numerical ice-shelf (Appendix A). An nth order model contains contributions of modeling at that time was assessed in a model intercomparison equations involving all powers of ε from ε0 to εn ; SO-SSA and SO- project (MacAyeal et al., 1996). However, ice sheet modeling SIA thus contain ε0, ε1 and ε2. A complete model would account experiments dominated because they supported deep drilling ice for all non-negative integer powers of ε, implying that no shal- core projects such as GRIP, GISP and NGRIP in Greenland, and EPICA lowness at all is assumed for the ice mass under question. A in Antarctica. The European Ice Sheet Modeling INiTiative (EIS- complete model is simultaneously valid for ice shelves and ice MINT) intercomparison and benchmark project accompanied these sheets and corresponds to what is know as the full Stokes model in drilling efforts (Huybrechts et al., 1996; Payne et al., 2000). fluid dynamics (e.g. Batchelor, 1970).

2.2. Why separate ice shelf and ice sheet models?

From a viewpoint of basic physics, ice shelves and ice sheets are not necessary to treat apart. They are both large ice masses under extreme slow creeping motion due to external action of gravity and 2.3. Numerical model challenges of coupled ice shelf/ice sheet climate. The classical laws of physics, conservation of mass, systems Newton’s second law (force balance) and conservation of energy, essentially suffice to describe the evolution of their motion. Specific Since the mid 1990’s, the refinement of numerical codes capable ice shelf and ice sheet behavior is due to differences in the types of of treating coupled ice sheet/ice shelf systems is a topic of intense boundaries and the actions of the environment across these research. More recently, the ice shelf-ocean interface and circula- boundaries. For ice sheets, the latter are the contact surfaces with tion in ice shelf cavities has received increased attention (e.g. the atmosphere, the solid ground, and the ocean (calving in the Shepherd et al., 2004; Holland, D.M. et al., 2008a; Holland, P.R. absence of ice shelves). Internal boundaries delineate transition et al., 2008b; Jenkins and Jacobs 2008). Considerable progress has regions with ice shelves or shear margins distinguishing fast flow been achieved, but severe theoretical and numerical challenges still features such as ice streams. For ice shelves, the boundaries are the remain to be addressed and overcome before simulations of, for contact surfaces with the atmosphere and the ocean. In regions of example, the ice shelf/ice sheet complexes that appear to have ice rises and ice rumples ice shelves are also in contact with solid covered large parts of the Arctic Ocean and its surrounding land- ground. Further boundaries comprise the shore lines. If for a given masses are expected to yield satisfying results (IPCC 2007; van der ice shelf or sheet, the processes describing how mass, momentum Veen and ISMASS members, 2007; Jakobsson et al., 2010a; SeaRISE; and heat are supplied from the outside at these boundaries are Ice2Sea). prescribed, then the thermo-physical processes inside the ice shelf The challenges associated with coupled ice shelf/ice stream/ice or ice sheet are in principle determined. A distinction between ice sheet complexes are due to the components’ distinct spatial shelves and ice sheets is thus as such not compulsory from the extensions, response time to external forcings, material inhomo- point of view of general physics. geneities, thermal regimes and mechanical stresses dominating However, there is the significant difference in that ice shelves their flow behavior, and the generally poorly known subglacial float in water while ice sheets are grounded on land or shallow basal conditions (Figs. 2,4, Appendix A). The different regimes seafloor. In all existing ice shelf and ice sheet models, modelers start separating sheet- from shelf- from stream flow have neither static from the classical laws of physics as stated above. Yet, to simplify the nor sharp boundaries, and their mutual interaction can not be mathematics and, later, the numerics, the force balance is typically captured by patching together interacting regimes along common replaced by an approximate force balance in which certain terms are interfaces. Coupling of ice sheet/ice shelf/ice stream dynamics takes neglected. The degree of simplification of the resulting model is then place in a space-time varying transition zone extending up- and measured by the number of neglected terms. The simplest models downstream of the grounding line. There, a mixture of shelf and neglect a comparatively large number of terms and retain only those sheet flow prevails in varying portions depending on the distance that capture the most typical behavior of ice sheets or ice shelves, to the grounding line. respectively. By successively amending formerly neglected terms, an Four major challenges are: First, to properly couple the different entire hierarchy of ice shelf and ice sheet models can be developed. ice dynamical flow regimes. Second, to properly account for espe- The underlying principle of the so-called zero order model-class is cially processes at the ice-bed interface. Third, to numerically the Shallow Shelf Approximation (SSA) and the Shallow Ice simulate dynamically coupled ice sheet/ice stream/ice shelf Approximation (SIA) for ice shelf and ice sheet models, respectively, systems at high resolution through entire glacial cycles. Fourth, to discussed later and detailed in Appendix A. further develop existing models of grounding line migration, Within their class, SSA and SIA have the largest degree of requiring thermodynamical investigation of the merging continua negligence for which yet typical ice shelf and ice sheet behavior, ‘ice’, ‘water’, and ‘sediment’. N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 3695

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. Flow is 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. The indicated 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 induced E E E by the depth-integrated (xy) -parallel normal and shear stresses Txx, Tyy, Txy. 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 importance n of normal stresses Txx, Tyy, Tzz and shear stresses Txy, Txz, Tyz ; i.e. which ones could be assigned weights ε , n1 without losing characteristic ice stream features.

2.4. Spatial reconstructions of ice shelf/ice sheet complexes Filchner Ice Shelf of present-day Antarctica, existed during the glacial maximum of MIS 6 (Polyak et al., 2001; Jakobsson et al., Advances in high-latitude seafloor mapping in the 1990’s led to 2008b; Dowdeswell et al., 2010; Jakobsson et al., 2010a). The new insights into the glacial histories of the Antarctic and the deepest groundings of MIS 6 ice shelf fragments occurred at ca 1000 Arctic. In particular, the data showed that disintegration of ice m water depth on the Lomonosov Ridge crest and the Morris Jesup shelves and grounding line retreat is not unique to the 21st century. Rise. However, thinner ice shelves floating above these deep phys- For example, glacial landforms on the Antarctic continental shelf iographic features may have existed during other glacial periods revealed that the extent of the WAIS varied considerably during the than MIS 6. Large glacial troughs formed in the shallow continental last glacial cycle (e.g. Anderson 1999; Conway et al., 1999; Shipp shelf surrounding the central Arctic Ocean basin seem like a strong et al., 1999; Evans et al., 2006). Geological records of ice-shelf indication of that ice streams interlinked the continental ice sheets disintegration, grounding line retreat and enhanced drainage of with Arctic Ocean ice shelves (Stokes et al., 2005, 2006; Engels et al., inland ice facilitated by ice streams indicate that the processes 2007; Kleman and Glasser 2007; England et al., 2009). This is currently ongoing at West Antarctica’s marine interface indeed precisely what is observed currently in Antarctica. In summary, it is have a history (Canals et. al., 2000; Dowdeswell et al., 2004; now understood from geophysical data and mapped marine glaci- Graham et al., 2010; Jakobsson et al., 2011). genic landforms that Arctic Ocean ice-shelf complexes existed in Spatial reconstructions of the Quaternary ice sheet extents in the former glacial periods and decayed in ensuing warmer climates Arctic region were in focus for the previous programs PONAM (POlar (Vogt et al.,1994, Polyak et al., 2001, Dyke et al., 2002; Svendsen et al., North Atlantic Margins; Elverhøi et al., 1998) and QUEEN (QUater- 2004; Jakobsson et al., 2010a; O’Regan et al., 2010)(Fig. 3), to nary Environments of the Eurasian North; Svendsen et al., 2004), as eventually be virtually absent at present. well as for the active ‘Arctic Paleoclimates and its Extremes’ (APEX) program (Jakobsson et al., 2008c, 2010b)(Fig. 3). The reconstructions 3. A systematization of numerical ice shelf and ice sheet produced within these programs portray a scenario involving models multiple episodes of glacial advances and retreats on the Arctic continental margins, likely associated with ice shelves fringing the The dynamical behavior of ice shelves and ice sheets in isolation waxing and waning continental Eurasian and Amerasian ice sheets. is commonly modeled using an asymptotic perturbation expansion Glacigenic features on the Morris Jesup Rise, Chukchi Borderland approach from the field of applied mathematics (e.g. van Dyke 1964; and Yermak Plateau, three extensions of the Arctic Ocean conti- Kevorkian and Cole 1981). In this approach, the parameter ε defining nental shelf, as well as on the submarine Lomonosov Ridge suggest the shallowness of an ice shelf or an ice sheet enters the equations that ice shelves, perhaps similar in size to the Ross Ice Shelf and describing ice flow. As indicated in Section 2.2, a hierarchy of models 3696 N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 based on powers of ε can be derived. The perturbation expansion normal stresses (Txx,Tyy,Tzz) become increasingly important. Thus, approach can ultimately be used to couple ice sheet and ice-shelf ice streams exhibit many features that are also typical for ice shelf dynamics in the transition zone where matching instead of patch- behavior (Fig. 4), and can hence not be modeled appropriately ing is needed to correctly resolve the dynamic interaction of ice within the SIA. sheets, shelves and ice streams. Central to the description of ice Also, it is impossible to patch the SSA and SIA models together as dynamics is the force balance (Newton’s second law), viz: stress components in the SSA force balance do not necessarily have counterparts in the corresponding SIA force balance upstream of dyðx; tÞ r ¼grad p þ div TEðx; tÞþrg: (1) the grounding line, cf. Eq. 2. In the SSA equations, gradients of dt normal and xy -parallel ice shelf stresses (Txx, Tyy, Tzz and Txy)are Eq. (1) balances the product of time rate of change of velocity y retained because their impact is comparable to the one of ice shelf r and ice density r with surface forces (stresses TE, pressure p) and vertical stresses (Txz, Tyz). Moreover, seawater density sw enters the fl volume forces (gravity g), cf. Appendix A. From Eq. (1), ice shelf and SSA equations. For ice sheet ow, however, the vertical stresses ice sheet models of different order, zero (SSA/SIA), first (FO-SSA/FO- dominate to such an extent that the other stress components are SIA), and second (SO-SSA/SO-SIA), are derived. neglected in the zero order approximation. A consistent coupling between shelf and sheet flow can only be performed if each stress component accounted for in ice shelf flow 3.1. Zero order models has a counterpart in ice sheet flow. Second order models have the least order for which this is the case, allowing coupling between ice Describing ice dynamics using the SSA and the SIA is the flow across the grounding line (Figs. 1, 4). This is also the framework coarsest possible approximation where the only terms left are for ice sheet-ice stream coupling as modeling embedded ice those that derive from zero-order powers of the shallowness streams is conceptually equivalent to the matching of ice sheet flow parameter, ε0. Written as three scalar equations in x -, y - and z and ice shelf flow: For the latter, the transition from one flow region direction, respectively, the final non-dimensionalized zero order to another takes place mainly in the along-flow direction when SSA/SIA equations corresponding to Eq. (1) are (for a derivation and crossing the grounding line. Characteristic flow behavior of ice notation cf. Appendix A): stream/ice sheet complexes is however most easily identified in the direction perpendicular to the ice flow. The lateral shear margins v v E v E v E fl pð0Þ Txxð0Þ Txyð0Þ Txzð0Þ represent boundary layers separating regions of sheet ow from 0 ¼ þ þ þ fl vx vx vy vz regions of stream ow, each with different ice dynamics. Therefore, v E v E v E coupled modeling of ice stream-ice sheet flow transition ought to vpð0Þ Txyð0Þ Tyyð0Þ Tyzð0Þ ðSSAÞ 0 ¼ þ þ þ (2) be achieved with a compound of SO-SIA and SO-SSA models. The vy vx vy vz modeling of Whillans Ice Stream based on a perturbation expan- r vp vTE sw ¼ ð0Þ þ zzð0Þ sion approach has been discussed by MacAyeal (1989). r r v v sw z z 3.2. Second order models v E vpð0Þ Txzð0Þ 0 ¼ þ vx vz In the Second Order Shallow Shelf- and Second Order Shallow v v E ð Þ pð0Þ Tyzð0Þ Ice Approximation equations (SO-SSA and SO-SIA), terms weighted SIA 0 ¼ þ (3) vy vz by powers εn with n ¼ 0,1,2 are retained. The equations corre- vpð0Þ sponding to (2) read (Appendix A) 1 ¼ : vz E E E vpð Þ vT ð Þ vT ð Þ vT ð Þ Reducing the full force balance (1) to (Eq. 2), (Eq. 3) results in ¼ 2 þ xx 2 þ xy 2 þ xz 2 ; 0 v v v v virtually unconstrained computability of ice shelves and ice sheets x x y z v v E v E v E in isolation. Therefore, numerical SSA/SIA-codes for ice shelves and pð2Þ Txyð2Þ Tyyð2Þ Tyzð2Þ ðSO SSAÞ 0 ¼ þ þ þ ; (4) ice sheets can be run separately over several hundreds of thousands vy vx vy vz v E v E v E of years. However, a critical limitation from retaining only zero- vpð2Þ Txzð0Þ Tyzð0Þ Tzzð2Þ 0 ¼ þ þ þ ; order terms is that the SSA and the SIA are valid only in the ice vz vx vy vz shelf and sheet, respectively. Furthermore, the SIA is incapable of describing ice stream v E v E v E vpð2Þ Txxð0Þ Txyð0Þ Txzð2Þ behavior. Ice streams interlink terrestrial and marine glacial envi- 0 ¼ þ þ þ ; ronments hosting ice sheets and ice shelves, respectively, with vx vx vy vz v E v E v E evidence of paleo ice streams found in both environments (e.g. vpð2Þ Txyð0Þ Tyyð0Þ Tyzð2Þ ðSO SIAÞ 0 ¼ þ þ þ ; (5) Stokes and Clarke 2001; Mosola and Anderson 2005; DeAngelis and vy vx vy vz Kleman 2008; Dowdeswell et al., 2008). For example, a network of v E v E v E vpð2Þ Txzð0Þ Tyzð0Þ Tzzð0Þ ice streams in the Canadian Arctic Archipelago drained the Lau- 0 ¼ þ þ þ ; vz vx vy vz rentide ice sheet and fed large ice shelves in the Amerasian basin of the Arctic Ocean (Jakobsson et al., 2010a, Stokes and Tarasov 2010). Eqs. (4) and (5) show that stress components can now be traced ½vð Þ=vð ; ; Þ At present, the Siple Coast of West Antarctica is a typical region across the grounding line: terms TijðkÞ x y z shelf can be fl ½vð Þ=vð ; ; Þ ¼ with coupled ice sheet/ice stream/ice shelf ow (Bamber et al., matched with terms TijðkÞ x y z sheet,(k 0,2). Still, this 2000). Ice streams are rapid flow features which are embedded in matching requires careful treatment within a delicate procedure the surrounding slow moving or even close to stagnant ice, with known as ‘matched asymptotic expansion technique’ (van Dyke, pronounced zones of lateral shearing along the sides. As basal 1964; Kevorkian and Cole 1981; Schoof and Hindmarsh, 2010). sliding often is enhanced in ice streams (Alley et al., 2004; This is necessary since some gradients appear with different orders v E =v Winsborrow et al., 2010), vertical shearing (Txz,Tyz) is no longer on both sides of the grounding line; e.g. Txyð2Þ x (shelf) vs. v E =v dominating ice dynamics and xy -parallel shear stresses (Txy) and Txyð0Þ x (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 the attractive to investigate with combined use of spatial reconstruc- SIA solution on ice sheet side (Fig. 1). tions and numerical modeling. From submarine glacial landforms To our knowledge, no numerical SO-SSA code exists, while two derived from high-resolution sea-floor imagery, Ottesen et al., simplified versions of the SO-SIA have recently been implemented (2007) reconstructed the Late Weichselian ice sheet flow regime for the purpose of simulating glaciation and landscape evolution including ice streams on the western and northern Svalbard margin (Egholm et al., 2011) and ice stream dynamics (Ahlkrona 2011). (Fig. 5A). Using the numerical SIA ice sheet code SICOPOLIS (Greve, 1997; 2010), we performed conceptual simulations for the Sval- 3.3. Full Stokes models bard region, with the setup chosen to conform to existing spatial reconstructions at both regional and continental scales. Results are The perturbation expansion permits infinite expansion in the presented in Fig. 5, panels C to F. The purpose of our experiments is 1/ fi fl shallowness parameter ε, see Eq. (A.2). Retaining terms weighted by to show that despite unmodi ed ice ow dynamics, the paleo-ice powers εn with n/N, the equations for ice dynamics become exact streams mapped by Ottesen et al., (2007) are not reproduced well, fl and applicable for ice shelf and sheet flow simultaneously: while ow patterns over the contemporary topography of Svalbard matching across the grounding line becomes obsolete. Thus, for indicate ice streams, 2/ to explain this inconsistency and 3/ to relate n/N, we obtain a unified set of equations, the so-called Full Stokes it to the limitations of zero order SIA ice sheet model. Details of the (FS) equations: modeling setup are provided in Appendix B. In summary, two simulations were carried out, run TOPO-LGM vpðx;y;z;tÞ vTE ðx;y;z;tÞ vTE ðx;y;z;tÞ vTE ðx;y;z;tÞ with topographic boundary conditions conforming to the Last 0 ¼ þ xx þ xy þ xz ; vx vx vy vz Glacial Maximum (LGM), and run TOPO-PRES with present day v E ð ; ; ; Þ v E ð ; ; ; Þ v E ð ; ; ; Þ topographic boundary conditions. Both runs start from assumed ice vpðx;y;z;tÞ Txy x y z t Tyy x y z t Tyz x y z t 0 ¼ þ þ þ ; free conditions at arbitrary model time t ¼ 0, and evolve for 18 kyr vx vx vy vz on a 10 km 10 km grid. Simplified time-independent ‘glacial v ð ; ; ; Þ v E ð ; ; ; Þ vTE ðx;y;z;tÞ v E ð ; ; ; Þ ¼ p x y z t þ Txz x y z t þ yz þ Tzz x y z t þr : climates ’ are employed to force the ice sheet model through time. 0 v v v v g z x y z For run TOPO-LGM, air surface temperature T(x,y,z) in the entire (6) modeling domain is set to T(zsurf)¼10 C, while T(zsurf)¼15 C for run TOPO-PRES. Variations of temperature with altitude, T(z), are The FS equations coincide with (1) if the acceleration terms in accomplished through an atmospheric lapse rate of 4.5 C per the latter are neglected. FS models are computationally costly and 1000 m increase in elevation. Run TOPO-LGM is forced by a spatially are therefore restricted to spatio-temporal domains covering variable precipitation field P(x,y) derived from the data set of hundreds of square kilometers and hundreds of years only. Routine Legates and Willmott (1990). For run TOPO-PRES, a constant simulations of coupled paleo ice shelf/ice sheet systems are to date precipitation field P has been employed. The elevation- not possible with FS models. desertification feedback has for simplicity not been taken into account, P does thus not vary with altitude. Since our simulations 4. The need for higher order models in integrated modeling aim at highlighting Shallow Ice Approximation (SIA) limitations with regard to modeling ice streams, the use of such simplified Because a rigorous second order model capable of modeling ice climate forcings seems justified. More realistic in-depth paleo- shelf/ice stream/ice sheet dynamics at spatial resolutions and time simulations of the Svalbard region would necessitate extensive scales matching those employed in spatial reconstructions based on analysis of climate data prior to employing it to force the ice sheet geological and geophysical data is not yet available, we can not model (Pollard et al., 2000; Kirchner et al., 2011). fi evaluate its performance by comparing simulation results with eld A general inspection of the simulation results (Fig. 5), based on data. The need for such a model is however obvious, as we will the absolute valueqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the horizontal ice velocities at the ice sheet illustrate below with a simulation experiment of the Svalbard ice surface, y : ¼ y2 þ y2 as an indicator of fast streaming ice, sheet. However, it should be noted that the higher order hybrid ð0Þ xð0Þ yð0Þ model of Pollard and DeConto (2009) has successfully been applied to suggests that for TOPO-LGM, there is no clear evidence for any model the West Antarctic Ice Sheet growth and decay through the stream-like ice flow (Fig. 5A, D). For TOPO-PRES, the simulated past 5 million years. Their simulations show good agreements with pattern of surface velocities apparently indicate ice streams in the geological records of ANDRILL-AND1B. However, a relatively plausible places, judging from the locations of the heads of mapped coarse grid of 40 km resolution for continental experiments and 10 paleo ice streams (Fig. 5A, F). A closer inspection of Fig. 5D, F reveals km resolution for nested experiments in the Ross embayment and that y(0) is generally highest at the ice margins following the conti- the AND-1B drill site was used. This implies that comparison between nental shelf break (TOPO-LGM) and Svalbards present day coastline model results and mapped glacigenic features with scales less than or (TOPO-PRES), respectively. Locally, high velocities are modeled to even slightly more than 10km is difficult or even impossible. occur also in the interior of the ice sheet: For TOPO-PRES, these zones of high velocity extend from the interior of the ice cap to its marine fl 4.1. Svalbards ice streams margin, suggesting realistic ice stream ow with mouths and heads in reasonable locations. For TOPO-LGM, high surface flow velocities Modeling coupled ice stream/ice sheet complexes may be do, in contrast, not continuously stretch from the interior of the ice a strategic intermediate step to be further developed before sheet to the ice margin at the continental shelf break. Away from the modeling the entire ice shelf/ice stream/ice sheet system. Apart margin, regions of high ice surface velocities instead coincide with from having to deal with only two instead of three components, ice regions of pronounced changes in ice thickness (Fig. 5C). Note that stream/ice sheet complexes are conceptually much simpler since also for TOPO-PRES, changes in ice thickness can be related to high fl the marine dimension is absent: to large extents, coupled ice ow velocities (Fig. 5E). stream/ ice sheet systems are land based. These observations conform to the general result that the In contrast to floating ice shelves, ice streams leave distinct traces horizontal ice velocities in an SIA model are proportional to the on the continental shelf because they often are grounded on the gradient of the ice sheet surface, where the degree of proportion- seafloor during peak glacials. Ice streams are therefore particularly ality partly depends on local ice thickness (Greve and Blatter 2009): 3698 N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704

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 modeling domain and basal topography/bathymetry as derived from the ETOPO1-dataset (Amante & Eakins 2009). C-F: Model results for two different runs TOPO-LGM and TOPO-PRES using the SIA ice sheet code SICOPOLIS (cf. Appendix B for details of the simulation set-up). C: 2D-view of isolines of ice thickness for runTOPO-LGM. D: Plan view of ice surface velocity plotted over the 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 their termination 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 ice margin, which for TOPO-PRES is essentially dictated by the current land/ocean mask. Likewise, high surface velocity is modeled to occur in the interior whereever ice surface gradients are 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.

Zz vz vz y ð Þ; y ð Þ f s; s A ð Þð Þ3 : limit of integration is the ice base b, where a no slip condition is xð0Þ z yð0Þ z v v T zs z dz (7) x y assumed for simplicity (cf. Sect. 2.3). The upper limit of integration b < Here, zs ¼ zs(x,y,t) is the ice sheet surface, and A ðTÞ the is an arbitrary position z in the ice sheet (z zs) or at its surface ¼ temperature dependent rate factor in Glen’s flow law. The lower (z 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 velocity be the surface-gradient dominated ones derived from the SIA. gradients to stresses) and in which the zero order vertical shear However, over the ice stream flow region, where longitudinal stresses Txz, Tyz can be replaced by vzs=vx$ðzs z Þ and stresses become important, the ice surface velocities will be vzs=vy$ðzs z Þ. To relate Eq. (7) to the simulation results presented different from the surrounding SIA-ones if computed in the SO-SIA in Fig. 5, let us start with a simple experiment of thought: framework, which by construction takes those in-plane horizontal For a slab-type ice mass of constant thickness, resting on a flat stress into account, see Eq. (8). Thus, improved modeling of ice bed with constant angle of inclination, the gradient of the ice stream flow is possible only with second (or higher) order ice sheet surface (vzs/vx,vzs/vy) is constant. Neglecting variations in thermal models. We expect a SO-SIA model or any other higher order model effects entering through A ðTÞ, Eq. (7) thus renders ice surface to be able to correctly simulate ice streams in the set-up chosen for velocities that are constant since all terms on the right hand side of TOPO-LGM (Fig. 5D), cf. also Hindmarsh (2009) for a discussion of Eq. (7) are constant. In the schematic visualization given in Fig. 1, ice stream dynamics and the here neglected thermo-mechanical this implies that fast streaming ice (depicted by the arrow) cannot coupling and its feedback mechanisms. be detected as ice surface velocities for both the ice stream and the surrounding bulk ice are the same according to Eq. (7). 4.2. Coupled ice shelf e ice sheet systems around ice rises Over real basal topographies, however, ice thickness is typically not constant, with greater thickness over topographic lows than Above, modeling of ice stream/ice sheet systems has been sug- over highs. Variations in ice thickness influence the ice surface gested as an intermediate target to be achieved prior to modeling * 3 velocities through the term (zsz ) in the integral in Eq. (7), and in coupled ice sheet/ice shelf systems. However, the latter do not combination with the surface gradient thus determine the ice necessarily involve ice streams. This is because ice shelf/ice sheet velocities at the surface (again neglecting variations in A ðTÞ for systems may exist also in the vicinity of ice rises. At bathymetric simplicity). The dependence of the ice surface velocities on the ice heights, ice shelves may locally run aground. If the grounded area is thickness is the reason why so-called topographic ice streams small, the ice shelf may slide over the seafloor giving rise to small (those flowing in a trough) may roughly be captured even by SIA and sometimes fractured ice surface undulations referred to as ice models. In the schematic in Fig. 1, this implies that fast streaming rumples. They rise typically less than 50 meters above the flat ice ice (indicated by the ice sheet arrow), is likely to be detected if the shelf surface. Ice rumples exhibit flow features which are domi- ice stream is flowing in a topographic trough, while the adjacent nated by the dynamical behavior of the surrounding ice shelf flow. bulk ice does not. As ice thickness will be larger over the bathy- For large grounding areas, the ice likely develops flow patterns meytric trough hosting the ice stream, Eq. (7) renders higher ice which mimic the dynamics of miniature grounded ice sheets surface velocities over the trough than over the adjacent, thinner embedded in the ice shelf. Such groundings are called ice rises and parts of the ice sheet. This effect is observed also in Fig. 5D, F. can reach heights up to several hundred meters above the ice shelf However, pure ice streams as such on the Siple Coast, Antarctica, level. By exerting a restraining force (‘backstress’) on the upstream are not topographically controlled (Winsborrow et al., 2010). ice shelf, ice rumples and ice rises impact ice shelf dynamics over Therefore, ice thicknesses in the regions of ice stream flow and large spatial areas. Ice rises and rumples have long been recognized adjacent ice sheet flow will be similar, implying that gradients of the to play key roles in the maintenance or destruction of ice sheet/ice ice surface will not vary drastically (the latter assumption might no shelf systems (Thomas 1979; Martin and Sanderson 1980; Smith longer be true at small scales e.g. resolving crevassed lateral shear 1986; Bindschadler et al., 1987; Doake 1987; MacAyeal 1987; zones). As in the experiment of thought above, Eq. (7) will result, in Hindmarsh 1996, 2006; Hindmarsh and LeMeur 2001). the SIA framework within which it is derived, in ice surface velocities While present ice shelf groundings mainly are detected using that are essentially the same for the ice stream and the ice sheet. This remote sensing techniques of the ice surface, past groundings are is observed in Fig. 5D, where the ice thickness and ice surface recorded through glacigenic features preserved in the seafloor. The gradient over the continental shelf appear to be too homogenous to processes leading to their formation and hence the nature of the result in notable changes in ice surface velocities. grounding events causing the glacigenic features remain discussed. Only higher order ice sheet models, in which other stresses than Apart from local grounding of ice shelves causing ice rises, these the shear stresses Txz and Tyz enter, have the capability of rectifying processes include the interaction of the seafloor with (i) armadas of this shortcoming of the SIA: in the second order SO-SIA, the shear ice bergs, (ii) large singular icebergs, and (iii) grounded ice sheets. stresses Txz and Tyz contain additional contributions beyond those Recently, late-Quaternary grounding events at the Yermak Plateau stemmimg from the surface gradient times the local ice thickness: and the Lomonosov Ridge in the Arctic Ocean have been inter- * instead of Txz¼vzs/vx(zsz ) as in the SIA, the corresponding second preted based on geotechnical analyzes of ice-overridden sediments order SO-SIA relation can be qualitatively summarized as (for from beneath these groundings (O’Regan et al., 2010). details, see Ahlkrona 2011) Once numerical simulations with coupled ice shelf/ice sheet Z Z models become feasible on timescales relevant for paleo- vzs v v Txz ¼ ðzs z Þ þ ðTzz TxxÞdz Txydz þ. (8) simulations, their results can be contrasted to field evidence to vx vx vy either support or dismiss hypotheses concerning the nature of and where higher order surface topography related terms have grounding events. To our knowledge, no existing numerical code been omitted for simplicity. According to Glens flow law, the accounts for the coupling of ice shelf/sheet dynamics around ice stresses are linked to spatial velocity gradients. Hence, if stresses rises in ice shelves yet. Such a model would have to be at least are known, velocities can be derived through integration. It is second order, involving SO-SSA and SO-SIA. Moreover, heavily obvious from even the qualitative representation of the Txz as in Eq. crevassed ice margins around ice rises indicate that damage (8) that the velocities computed in a SO-SIA thus will be different mechanics might have to be incorporated into the modeling. Such from those in the SIA, and in particular contain contributions not models have not yet been proposed. represented at all in the SIA (sheet-parallel and normal stresses). In the schematic in Fig. 1, this implies that fast streaming ice (indi- 5. Conclusions cated by the ice sheet arrow) can be captured by the SO-SIA even if topographic forcing is small: for bulk ice sheet flow, where vertical In this paper, we present a review of the fundamentals of shear stresses dominate, the ice surface velocities will essentially contemporary ice sheet modeling for a broad Earth Science 3700 N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 audience intended to be accessible also to those with no specific the pressure field and the divergence of the stress field TE at background in theoretical glaciology and numerical modeling. The x¼(x,y,z)andtimet. main goal is to provide a basis for increased communication Separating hydrostatic pressure contributions from the total between especially non-modelers and modelers sharing a common stresses is common in glaciology. The splitting of the total stress interest in the dynamics of ice shelves and ice streams. Hence, to tensor T is typically written as T¼pIþTE where pI is the hydro- bridge the gap, the degree of mathematical detail and topical static pressure tensor (Paterson 1994). Imaged as 3 3 matrices, pI inclusiveness has deliberately been limited: E.g., grounding line has pressure p on the diagonal entries, and zeros elsewhere. The E E ¼ þ E ¼ dynamics, marine instability and the role of sedimentary beds extra Cauchy stress T has the normal stresses Txx Txx p, Tyy fl þ E ¼ þ beneath fast owing ice are excluded from our review because we Tyy p and Tzz Tzz p on the diagonal entries, and the vertical E ¼ ¼ E ¼ ¼ consider these topics too specialized to be included in a review shear stresses Txz Txz Tzx, Tyz Tyz Tzy and the xy -parallel E ¼ ¼ intended to be cross-disciplinary shear stress Txy Txy Tyx on the off-diagonal entries. The We discuss different levels of basic model assumptions; from orientation of these stresses is depicted in Fig. 4C. simplified Shallow Ice Approximation (SIA) schemes (of zeroth, Central to the perturbation expansion approach is first, and second order) to the (exact) Full-Stokes equations. The limitations of a zero order ice sheet model are shown with a parameter ε, accounting for the shallowness of ice shelves and conceptual simulations of a hypothetical glaciation over Svalbard, sheets, focusing on ice stream dynamics. a suitable scaling of the ice dynamic equations in terms of ε, We compare those conceptual simulation results to Ottesen the representation of the functions describing e.g. the stress et al., (2007) spatial reconstruction of Svalbard paleo-ice distribution in the ice as infinite series in ε. streams, and explain the discrepancies by discussing the equation governing ice surface velocities (naturally indicative of fast Discussing these ingredients outlines the derivation of the flowing ice) in any SIA model. The mathematics involved are few, Shallow Shelf Approximation (SSA) and the Shallow Ice Approxi- and tightly interlinked with the examples presented. We show mation (SIA) equations, respectively, see Eqs. (2),(3). that in a zero order model, where the ice surface velocities are First, ε is introduced to reflect that the vertical dimensions of proportional to the ice surface gradient and the ice thickness, pure ice sheets and shelves are much smaller than their horizontal ice streams can hardly be modeled, while topographically con- ones: they are shallow. For ice shelves, typical vertical and troled ones may occasionally appear to be modeled correctly, horizontal dimensions are [H] ¼ 500 m, [L] ¼ 500 km, while simply because higher local ice thickness (as e.g. over troughs) typical horizontal and vertical velocities are given by 1 1 give rise to higher surface velocities. We emphasize that these SIA- [VH] ¼ 1ma and [VL] ¼ 500 ma . For ice sheets, typical limitations in capturing ice stream dynamics are related to the vertical and horizontal dimensions are [H] ¼ 1kmand neglection of other than vertical shear stresses (Txz,Tyz) in any zero [L] ¼ 1000 km, with typical horizontal and vertical velocities 1 1 order model. We further sketch why higher order models, taking given by [VH] ¼ 0.1 ma and [VL] ¼ 100 ma .Defining 3 into account sheet-parallel stresses, can improve simulations of ε:¼[H]/[L]¼[VH]/[VL] yields εz10 for both ice shelves and ice fast streaming ice. sheets. Second, the shallowness parameter is used to scale the variables fl Acknowledgments in the ice ow equations after non-dimensionalization. For instance, stress components are made dimensionless by intro- r r Financial support from the Bert Bolin Center for Climate ducing a common stress scale gH, where and H are ice density Research enabled KH to visit Stockholm University and is gratefully and ice thickness, and g the gravity constant. Further, a valid acknowledged. J. Robbins and H.J. Zwally kindly provided part of assumption for the bulk of an ice sheet is that vertical shear stresses E ; E E the data used in Fig. 2. This is a contribution from the Bert Bolin Txz Tyz dominate over sheet-parallel shear stresses Txy. This can be fl ’ ’ Center for Climate Research at Stockholm University. introduced into the ice ow equations by e.g. multiplying ( scaling ) the different stress components with different powers of ε : the ε εn E ¼ . higher the power of in e.g. Txy with n 1,2,3 is, the less Appendix A. Derivation of zero order-, higher order and Full ‘ ’ E weight is assigned to Txy. An important result of the scaling Stokes models procedure is that the acceleration terms on the left hand side of (A.1) can be neglected. Zero order models Third, a perturbation scheme is employed according to which quantities arising in the force balance of ice shelf and ice-sheet The derivation of zero order models relies on a perturbation flow are written as infinite power series in the aspect ratio ε.For expansion approach. The procedure is illustrated for the the hydrostatic pressure p this infinite series reads (note that momentum balance. Higher order models are obtained in the same ε0¼1) framework, as we will show later. In the literature, ice sheet model descriptions often start from the evolution equation of ice thick- XN ¼ þ ε þ ε2 þ ε3 þ . ¼ εi : ness. The latter stems from the mass balance and is different from p pð0Þ pð1Þ pð2Þ pð3Þ pðiÞ (A.2) ¼ Eq. (A.1). i 0 Eq. (A.2) is called a perturbation expansion of p in the pertur- dyðx; tÞ r ¼gradp þ divTEðx; tÞþrg: (A.1) bation parameter ε. dt The further procedure is as follows: All functions arising in the The momentum balance Eq. (A.1) does not distinguish force balance of ice dynamics are expressed as infinite power series 0 between ice-shelf flow and ice-sheet flow. In Eq. (A.1), r is the in ε. Those functions which were assigned ‘weight’ ε in the scaling mass density, y the velocity, p the pressure, TE the symmetric process appear in the form given by Eq. (A.2). Quantities ‘weighted’ n extra Cauchy stress tensor, and g the gravitational acceleration. with ε ,n1 enter the force balance with corresponding higher Thematerialtimederivativeisrepresentedbyd/dt with t powers of ε. Specifically, for ice sheet flow the vertical ε E ¼ ε E þ ε2 E þ . denoting time. The operators grad and div take the gradient of shear stresses enter as Txz Txzð0Þ Txzð1Þ and N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 3701

ε E ¼ ε E þ ε2 E þ . fl fl Tyz Tyzð0Þ Tyzð1Þ . The sheet parallel shear stresses are latter re ects the in uence of material shearing, the former is ε2 ε2 E ¼ ε2 E þ ε3 E þ . scaled with in the SIA: Txy Txyð0Þ Txyð1Þ . Likewise, due to viscous and frictional basal slip (Fig. 4A). The relative ε2 E ¼ the normal stress components enter through Txx importance of gliding and sliding varies through the depth of ε2 E þ ε3 E þ .; ε2 E ¼ ε2 E þ ε3 E þ . ε2 E ¼ Txxð0Þ Txxð1Þ Tyy Tyyð0Þ Tyyð1Þ , and Tzz the ice sheet. Consequently, the velocities in the x - and y ε2 E þ ε3 E þ . ¼ ¼ Tzzð0Þ Tzzð1Þ . Conceptually, Eq. (A.1) then takes on the -directions satisfy vx vx(x,y,z,t) and vy vy(x,y,z,t), respectively. 2 form A¼0, or rather, A0 þ εA1 þ ε A2 þ . ¼ 0 ¼ 0,ð1 þ ε By contrast, SSA horizontal velocity components are uniformly þε2 þ .Þ. This equation is requested to be valid for all powers of ε distributed over depth (Fig. 4B). However, they vary in the fl individually, implying that terms of equal powers of ε are equated: horizontal ow directions accordingR to the material action of ε0 fi ¼ ε1 ¼ Hð E ; E ; E Þ for , comparison of coef cients yields A0 0. For , A1 0 is ob- the membrane forces 0 Txx Tyy Tzz dz. Consequently, tained, etc. For the ice flow equation, equating terms which are of vx¼vx(x,y,t),vy¼vy(x,y,t). In summary, SIA behavior is reminis- order ε0 leads to the Shallow Ice Approximation (SIA) for ice sheets cent of z -dependent simple shearing, subject to z -dependent (Fowler and Larson 1978; Morland and Johnson 1980; Hutter 1983; cryostatic pressure (Fig. 4A). SSA behavior is akin to membrane Greve 1997; Baral 1999; Baral et al., 2001; Greve and Blatter 2009). behavior, subject to normal and shearing membrane stresses, of A different scaling is appropriate for ice shelves. The dimen- which the strength is governed by the horizontal thickness E E E E sionless normal and in-plane stresses p, Txx, Tyy, Tzz and Txy are gradient (Fig. 4B). Clearly, the SIA formulation must be amen- assumed to be of order unity ( ε0), while the vertical shear stresses ded by adding membrane behavior to the simple shearing, and E E ε1 Txz, Tyz are of order . The vertical shear stresses are assigned less the SSA formulation must be extended by adding some vertical weight because they are negligibly small at the free surface and at shearing if a coupled shelf/sheet model is to be adequately the ice-ocean interface; therefore they must be small inside the ice described. The same is expected to be required to model ice v E =v v E =v ε0 fi shelf. The derivatives Txz z, Tyz z are, however, of order . streams: a typical ice stream velocity pro le approaches the ice Then, the perturbation expansions for the stress quantities are sheet velocity profile if sliding is reduced, and the ice shelf substituted into the force balances. Equating terms of order ε0 then velocity profile if sliding is dominating over gliding (Fig. 4C). yields Shallow Shelf Approximation (SSA) equations (Morland and Shoemaker 1982; Morland 1987; Weis et al., 1999; Baral 1999). Both equation sets, the SIA and the SSA, are given in Eqs (2) and First order models (3), (3.1) in the main text (Section 3.1). In a subsequent step not discussed here, the stress components are linked to the strain rate The framework of perturbation expansions allows also the components, expressing the material response of ice under derivation of higher order models. Equating terms of order ε1 leads creeping motion: this is typically achieved by employing Glen’s to the ‘First Order Shallow Shelf Approximation (FO-SSA)’ and ‘First flow law (Paterson 1994) or one of its generalizations (Hutter 1983; Order Shallow Ice Approximation (FO-SIA)’. As FO-SIA and FO-SSA Greve and Blatter, 2009). are 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; Discussion of SSA and SIA models Baral et al., 2001). Of relevance in the present context is that these equations, just as their SSA and SIA counterparts, do neither Obvious differences in the SSA and SIA in (2), (3) stem from the allow for a consistent coupling of ice shelves and ice sheets within E E different scaling of the stresses: the stress components Txx, Tyy, the class of first order models. E E ε2 ε Tzz and Txy are O( ) in the SIA, but O( ) in the SSA. A consistent merging of ice shelf/ice sheet dynamics must therefore Higher order and second order models necessarily be at least of O(ε2). A detailed description of the scaling procedure is given in Hutter (1983), Baral (1999) and Equating terms of order ε2 leads to the ‘Second Order Shallow Weis et al., (1999). Shelf Approximation (SO-SSA)’, and ‘Second Order Shallow Ice fi fl The SSA model satis es the oatation condition for ice shelves Approximation (SO-SIA)’. These equations are given in (4) and (5) expressed as and (3.2) in the main text. r r Previously, some zero order ice sheet models have been amen- ¼ ; ¼ : hboard 1 r H hdraft r H ded by selected first and/or second order contributions. However, sw sw the former are not full second order models as not all terms that Here, hboard and hdraft are the ice shelf freeboard height and should be present have been included. Often, such models are ¼ þ ‘ ’ draft, relatedR by the local ice thickness H as H hboard hdraft (Fig. 1). simply called higher order models . Despite the lack of rigor in their r ¼ H rð,; Þ H 0 z dz is the average density in the ice column. The derivation, they work well for the applications they were designed floatation condition is derived from vertical integration of the SSA- for. Examples include glacier and ice sheet flow (Blatter 1995; Eq. (2).3 During integration, zero atmospheric pressure conditions Blatter et al., 1998; Colinge and Blatter 1998; Hubbard 2000; at the free ice shelf surface are exploited. At the ice-ocean interface, Pattyn 2003), simplified continental-scale ice-age ice sheet/shelf pressure equals the weight of the overburden water column. dynamics (Pollard and DeConto 2009), and modeling of sliding at the base of the ice sheet (Schoof and Hindmarsh, 2010). Higher E E The shear stresses Txz and Tyz at the base of an ice sheet are order and full Stokes models have recently been benchmarked in given by the overburden pressure times the surface slope of the the Ice Sheet Modeling IntercomParison e Higher Order Models ice sheet. This result is frequently used as an a priori fact in (ISMIP-HOM) project (Gagliardini and Zwinger 2008; Pattyn et al., more phenomenological ice modeling approaches (Paterson 2008). A review of different higher order ice approximation 1994). Here, it is derived from vertical integration of SIA-Eq. methods from the viewpoint of mathematical geophysics is given (3).1,2 by Hindmarsh (2004). Horizontal velocities vx, vy are depth-independent in the SSA, It is remarked that the term ‘hybrid models’ is becoming but depth-dependent in the SIA. Indeed, vertical profiles of increasing popular in the context of higher order coupled ice shelf/ horizontal velocities in the SIA at a fixed position (x,y) and fixed ice sheet flow (Pollard and DeConto 2009). It is also used in time t generally have a sliding and a gliding contribution. The connection with inverse modeling strategies in which e.g. ice 3702 N. Kirchner et al. / Quaternary Science Reviews 30 (2011) 3691e3704 stream basal parameters are optimized until modeled and observed content and age for grounded ice sheets in response to external ice surface velocities show minimal discrepancy (MacAyeal 1992; forcing. The latter is given by precipitation, surface temperature Khazendar et al., 2007; Goldberg and Sergienko 2010). and geothermal heat flux. SICOPOLIS employs Glen’slawto describe the flow of ice. The thermo-mechanical coupling is Full Stokes models for coupled ice shelf/ice sheet systems described by the rate factor A which is temperature-dependent for cold ice, and water-content dependent for temperate ice. Isostatic Full Stokes models solve Eq. (1) under the assumption of effects due to ice load are modeled by the Elastic-Lithosphere- negligible accelerations. The resulting set of equations is presented Relaxing-Asthenosphere approach (Greve and Blatter 2009). as Eq. (6) in the main text. As no other approximations are made, Effects of heat conduction through the lithosphere are accounted Eq. (6) are most exact, but also most costly to solve. Due to the for through the geothermal heat flux, which is assumed constant at 3 2 complexity of Eq. (6), full Stokes simulations require substantial a value of 40 10 Wm for the Svalbard modeling experiment. computational resources. When simulating glacier flow, a typical More detailed information concerning SICOPOLIS can be found in modeling domain size is 10 km 10 km, resolved by a 50 m 50 m Greve (1997; 2010). grid in the horizontal. A typical time period covered by such The Svalbard modeling domain is shown in Fig. 5A. A a simulation is 200 years. If computed serially, the effective time 10 km 10 km discretization in the horizontal is employed which needed to complete such a simulation is in the order of days (Jouvet results in 91 81 in-plane gridpoints. In the vertical, the cold ice et al., 2008, 2009). Using a zero order ice sheet model allows, in column and the lithosphere are discretized by 81 and 11 grid contrast, to simulate e in the same amount of time (days) e the points, respectively. Simulations start with assumed initial ice free evolution of ice sheets over domains as large as an entire hemi- conditions at arbitrary model time t¼0 kyr and evolve for 18 kyr. sphere, and over hundreds of thousands of years, with horizontal Topography/bathymetry is derived from the ETOPO1 data set and spatial resolutions ranging typically from 10 km 10 km to interpolated to the SICOPOLIS-grid (Amante and Eakins, 2009). 40 km 40 km. Full Stokes models 4 are typically discretized using The different topographic boundary conditions employed in Finite Element technique, and thus allow in principle for the use of runs TOPO-LGM and TOPO-PRES are derived from different land/ unstructured and adaptive meshes. This implies that from a theo- ocean masks. For TOPO-LGM, gridpoints with elevations less than retical point of view, there are virtually no limitations to the grid- 500 m below current sea level are assigned ‘ocean’, while the size employed. In glaciological applications, the mesh size will remaining ones are treated as ‘land’. Thus, land available for however be dictated by the problem to be investigated: to date, glaciation extends to the continental shelf break in TOPO-LGM. For meshes typically do not employ horizontal resolutions smaller than run TOPO-PRES, the threshold value distinguishing ‘land’ from 5m(LeMeur et al., 2004; Zwinger et al., 2007; Zwinger and Moore ‘ocean’ gridpoints is 0 m; the land/ocean mask corresponds to the 2009). While data used to initially constrain (e.g. bedrock topog- present situation. A standard PDD model is used to model accu- raphy) or to subsequently force (e.g. precipitation and air surface mulation and ablation at the surface of the ice sheet. At its base, temperature) the ice model are typically not available at such high a Weertman-type sliding law is employed with uniform moderate resolution, the situation is different when numerical modeling sliding coefficient as suggested in SICOPOLIS’‘Austfonna-module’ results are to be compared with field evidence. For example, (Greve 2010). The effects of different and varying basal boundary mapping of glacigenic features in the glacial troughs occasionally conditions and their implications for the dynamics of the Austfonna reaching depths greater than 1000 m on glaciated continental ice cap is discussed in Dunse et al. (2011). margins, have generally been carried out using deep water multi- beam systems with horizontal resolutions limited to approximately References 15e20 m and vertical of about 0.5% of the water depth. However, recent mapping of Pine Island Trough, West Antarctica, with the Ahlkrona, J., 2011. Implementing Higher Order Dynamics into the Ice Sheet Model SICOPOLIS. Master's Thesis, Uppsala University. new generation deep water system with higher resolution capa- Alley, R.B., Anandakrishnan, S., Dupont, T.K., Parizek, B.R., 2004. Ice streams e fast, bilities revealed smaller scale features of critical importance for the and faster? C. R. Physiques 5, 723e734. glacial dynamics (Jakobsson et al., 2011). These were corrugation Alvarez-Solas, J., Montoya, M., Ritz, C., Ramstein, G., Charbit, S., Dumas, C., Nisancioglu, K., Dokken, T., Ganopolski, A., 2011. Heinrich event 1: an example ridges, regular features the length of which ranges between 60 m of dynamical ice-sheet reaction to oceanic changes. Climate of the Past and 200 m, while their height varies between 1 m and 2 m only. The Discussions 7 (3), 1567e1583. corrugation ridges are indicative of an ice shelf break up in Pine Amante, C., Eakins, B.W., 2009. ETOPO1 1 Arc-Minute Global Relief Model: Proce- dures, Data Sources and Analysis. 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