A New Three-Dimensional Higher-Order Thermomechanical Ice

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B8, 2382, doi:10.1029/2002JB002329, 2003

A new three-dimensional higher-order thermomechanical ice sheet model: Basic sensitivity, ice stream development, and ice flow across subglacial lakes Frank Pattyn Department of Geography, Vrije Universiteit Brussel, Brussels, Belgium Received 29 November 2002; revised 27 March 2003; accepted 17 April 2003; published 16 August 2003.

[1] A new three-dimensional thermomechanically coupled ice sheet model is developed. Contrary to the majority of three-dimensional ice sheet models (shallow ice approximation), higher-order stress gradients (longitudinal and transverse) are accounted for in the force balance equations. The horizontal velocity field is determined from the force balance equations in their ‘‘derivative form’’ (elliptical equation) using the method presented by Pattyn [2000, 2002a]. The model is solved on a regular grid in the horizontal and an irregular grid in the vertical and is numerically stable. Basic experiments include the European Ice Sheet Modeling Initiative (EISMINT) benchmarks for large- scale ice sheet models and a comparison with the Saito-Blatter ice sheet model including higher-order stress gradients [Saito et al., 2003]. Detailed calculations of ice flow over three-dimensional bedrock perturbations showed the validity of the higher-order solution. The model is capable of simulating the evolution of an ice stream within the ice sheet and shows important aspects of observed ice stream features, such as the surface flattening and the importance of side drag. The simulation of the ice flow over a subglacial lake results in a flattening of the surface, a local velocity increase over the lake, and a deviation of the ice flow from the main flow direction, features which are also observed at Lake Vostok, Antarctica. INDEX TERMS: 1827 Hydrology: Glaciology (1863); 1863 Hydrology: Snow and ice (1827); 3230 Mathematical Geophysics: Numerical solutions; KEYWORDS: higher-order, ice sheet model, thermomechanical, ice stream, subglacial lake Citation: Pattyn, F., A new three-dimensional higher-order thermomechanical ice sheet model: Basic sensitivity, ice stream development, and ice flow across subglacial lakes, J. Geophys. Res., 108(B8), 2382, doi:10.1029/2002JB002329, 2003.

1. Introduction tance to derive the vertical velocity, which determines age versus depth, thinning of the stratigraphic layers by [2] Most ice sheet models are based on the so-called vertical strain, and vertical advection of heat [Dansgaard ‘‘shallow ice approximation’’ (SIA) [Hutter, 1983], which and Johnsen, 1969]. Numerical calculations have shown is valid for an ice mass with a small aspect ratio (H L), that the profiles of velocity versus depth have different where flow is dominated by internal shear deformation: ice shapes at divides compared to flank positions [Raymond, flow is driven by gravity, and vertical shearing is concen- 1983; Schøtt Hvidberg, 1996]. trated close to the bedrock. Almost no shearing exists [3] Near the margin of ice sheets (which comprises close to the surface. However, the SIA is not valid at all grounding lines, transition zones, outlet glaciers and ice places in an ice sheet, such as at the ice divide or near the streams), all forces in the force balance become equally ice sheet margin [Baral et al., 2001]. Below the ice divide important, especially when basal sliding comes into play or there is theoretically no shearing present. This would mean even dominates the horizontal flow field. Longitudinal that according to the SIA and for a Glen-type rheology, the stretching and lateral shearing play an essential role in the effective stress becomes zero, implying an infinitely large dynamics of large outlet glaciers and major Antarctic ice viscosity so that the free surface is stiff against shearing. streams. The governing processes near the margin involve a This is not realistic as longitudinal stresses must develop reduction of the basal drag, leading to significant basal which enhance the effective stress and lower the viscosity. motion (due to sliding and/or basal sediment deformation), This effect goes beyond the lowest order of the shallow ice which increases the complexity of the problem. Furthermore, approximation [Baral et al., 2001]. A precise knowledge thermomechanical control amplifies this complexity which on the stress field underneath an ice divide is of impor- might even lead to channeling of ice flow [Hindmarsh, 2001]. Copyright 2003 by the American Geophysical Union. [4] However, all these feedback processes are not clearly 0148-0227/03/2002JB002329$09.00 understood as thermomechanical models that solve the

EPM 4 - 1 EPM 4 - 2 PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL

Stokes problem in three dimensions are not widely devel- Table 1. Constants Used in the Numerical Model oped. Present-day three-dimensional numerical ice sheet Symbol Constant Value Units models including full thermomechanical coupling are those b dependence of melting 9.8 Â 10À8 KPaÀ1 developed by Huybrechts [1990], Ritz et al. [1997], Greve on pressure [1997], or Marshall and Clarke [1997], to name a few. All r ice density 910 kg mÀ3 1/n B0 flow rate factor 2.207 Pa a these models have been successfully applied to simulate K major present and paleo-ice sheet behavior on the globe. C flow rate factor 0.16612 K G geothermal heat flux 4.2 Â 10À2 WmÀ2 They are all based on the SIA and do not include longitu- K flow rate exponent 1.17 dinal nor transverse stress gradients in a systematic way. L specific latent heat of fusion 3.35 Â 105 JkgÀ1 The transition zone between different flow types (e.g., Q activation energy for creep 7.88 Â 104 J molÀ1 À1 À1 between ice sheet and ice shelf flow, or the onset regions R universal gas constant 8.31 J mol K qr limit temperature in 273.39 K of ice streams) is often not properly accounted for, even flow rate factor À1 À1 though all stresses and stress gradients are of equal impor- cp heat capacity of ice 2009 J kg K tance in the force balance. Three-dimensional thermome- g gravitational constant 9.81 m sÀ2 À7 À1 À1 À1 chanically coupled ice sheet models including higher-order ki thermal conductivity of ice 6.62 Â 10 Jm K yr m enhancement factor in flow law 1 stress gradients are sparsely seeded. Among them are the n exponent in Glen’s flow law 3 models of Mayer [1996], Albrecht [2000] and Saito [2002], the latter two based on the model by Blatter [1995] and Colinge and Blatter [1998]. The reason for this sparsity must be sought in the complexity of the model description, where r is the ice density, g is gravitational acceleration, v is the difficulty to obtain a numerically stable result, and the the velocity vector, [s] is the stress tensor, q is the ice high computational cost, despite the exponential increase temperature, cp and ki are the heat capacity and thermal over time of computer power. conductivity of the ice, respectively, and is internal [5] This paper presents the mathematical and numerical frictional heating due to deformation. Values for constants details of a new three-dimensional higher-order model (HO used in this paper are given in Table 1. Considering a model; including longitudinal and transverse stress gra- Cartesian coordinate system (x, y, z) with the z axis dients) with full thermomechanical coupling. It is based vertically pointing upward (z = 0 at sea level), and denoting on a two-dimensional version developed earlier [Pattyn, the velocity components in either direction as u, v, and w, 2000, 2002a]. Although transverse stresses and their gra- the conservation of mass equation (1) can be rewritten as dients are properly accounted for, some assumptions are made, such as neglecting the variational stress term. An @u @v @w þ þ ¼ 0: ð4Þ approach different from that of Blatter [1995] was taken to @x @y @z obtain a solution to the force balance equations. Basic experiments with the model include the standard European Neglecting acceleration terms in equation (2) and consider- Ice Sheet Modeling Initiative (EISMINT) I benchmark ing the gravitational acceleration only important in the experiments [Huybrechts et al., 1996] as well the EISMINT vertical direction, the linear momentum becomes II experiments with thermomechanical coupling [Payne et al., 2000]. Special attention is paid to the flow field underneath the ice divide and a comparison is made with @s @s @s xx þ xy þ xz ¼ 0; ð5Þ a similar 3-D model according to the SIA. Both experiments @x @y @z with and without (isotherm) thermomechanical coupling are considered. Furthermore, the flow over small-scale bedrock perturbations is considered in a diagnostic model experi- @s @s @s yx þ yy þ yz ¼ 0; ð6Þ ment. The evolution of an ice stream is investigated as well @x @y @z as the ice flow over a subglacial lake, such as Lake Vostok, Antarctica. @s @s @s zx þ zy þ zz ¼ rg: ð7Þ 2. Field Equations @x @y @z

[6] The model approach is based on continuum thermo- As a complete solution to the stress equilibrium equations dynamic modeling, and encompasses balance laws of mass, (5)–(7) is quite complicated, these equations are solved in a momentum and energy, extended with a constitutive equa- somewhat reduced form. The major assumption to the force tion. Treating ice as an incompressible fluid with constant balance equations is to apply the hydrostatic approximation density, the equations for conservation of mass, momentum in the vertical, so that equation (7) reduces to and energy are written as

@szz r v ¼ 0; ð1Þ ﬃ rg: ð8Þ @z dv r ¼r s þ rg; ð2Þ dt This implies that the variational stress is neglected, i.e., the resistance to a varying stress gradient in the direction of d cpq motion (the so-called T term) in the vertically integrated r ¼rðÞþk rq ; ð3Þ dt i force budget [Budd, 1970a, 1970b, 1971]. This term is also PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL EPM 4 - 3 known as the vertical resistive stress [Van der Veen and Here, another assumption is made, i.e., that horizontal Whillans, 1989]. Numerical calculations have shown that gradients of the vertical velocity are small compared to the vertical resistive stress occurs in regions where the flow vertical gradient of the horizontal velocity, or @w/@x @u/@z regime changes, such as near the ice divide or near the and @w/@y @v/@z. This is a common assumption in margin, but that this stress component is almost 2 orders of ice sheet modeling, and is valid for most of the ice sheet magnitude smaller than other normal and shear stress domain. Making use of the principle of mass conservation components [Pattyn, 2000]. Normal stress deviators are not equation (4), the effective strain rate is written as neglected and accounted for properly. Neglecting atmo- 1 spheric pressure, an expression for s is obtained by 2 zz e_ e_2 e_2 e_ e_ e_2 e_2 e_2 : 14 integrating equation (8) from the surface s to a height z in ¼ xx þ yy þ xx yy þ xy þ xz þ yz ð Þ the ice body. Since the flow relation (see below) is unaffected by hydrostatic pressure it is convenient to work Combining the flow law equation (11) and the horizontal with stress deviators. The stress deviator components are stress field equations (9) and (10), and replacing the strain obtained by subtractingP the amplitude of the hydrostatic rate components by velocity gradients using equations (13) 0 1 pressure, sij ¼ sij À 3 dij k skk ; where dij is the Kronecker and (14), one obtains delta. Inserting the stress deviator components and assuming a zero normal stress at the surface, the horizontal force @ @u @v @ @u @v @ @u @s 4h þ 2h þ h þ h þ h ¼ rg ; balance equations (5) and (6) become after some algebraic @x @x @y @y @y @x @z @z @x manipulation ð15Þ @s0 0 @ 0 0 xy @sxz @s 2sxx þ syy þ þ ¼ rg ; ð9Þ @ @v @u @ @u @v @ @v @s @x @y @z @x 4h þ 2h þ h þ h þ h ¼ rg ; @y @y @x @x @y @x @z @z @y ð16Þ @s0 @s0 @ 0 0 xy yz @s 2syy þ sxx þ þ ¼ rg : ð10Þ @y @x @z @y where ( The constitutive equation governing the creep of poly- 1 @u 2 @v 2 @u @v crystalline ice and relating the deviatoric stresses to the h ¼ AðÞq* À1=n þ þ strain rates is taken as a Glen-type flow law with exponent 2 @x @y @x @y ) n =3[Paterson, 1994] 2 2 2 ðÞ1Àn =2n 1 @u @v 1 @u 1 @v 2 þ þ þ þe_0 : ð17Þ 1 4 @y @x 4 @z 4 @z s0 ¼ 2he_ ; h ¼ AðÞq* À1=nðÞe_ þ e_ ðÞ1Àn =n; ð11Þ ij ij 2 0 Expanding equations (15) and (16) and rearranging terms, where e_ is the secondP invariant of the strain rate tensor, leads to 2 1 defined by e_ ¼ ij 2 e_ije_ij and h is the effective viscosity. À30 2 2 2 e_0 is a small number (10 ) to make the viscosity finite. @h @u @h @u @h @u @ u @ u @ u 4 þ þ þ h 4 þ þ When using a power law rheology, as is the case with Glen’s @x @x @y @y @z @z @x2 @y2 @z2 flow law, a singularity might occur at the bed under a divide @s @h @v @h @v @2v (no shearing) when no slip boundary conditions hold. The ¼ rg À 2 À À 3h ; ð18Þ @ @ @ @ @ @ @ use of such a small number does not influence the numerical x x y y x x y outcome of the model. The flow law rate factor A(q*) is a function of temperature, where q* is the ice temperature (K) @h @v @h @v @h @v @2v @2v @2v corrected for pressure melting, i.e., q*=q + bP, and where P is 4 þ þ þ h 4 þ þ 0 0 @y @y @x @x @z @z @y2 @x2 @z2 the ice pressure (P = s xx + s yy À rg (s À z)). Following Hooke [1981], A(q*) is parameterized as an Arrhenius @s @h @u @h @u @2u ¼ rg À 2 À À 3h ; ð19Þ relationship: @y @y @x @x @y @x@y ! 1 n 3C Q An expression for the vertical velocity w is obtained through AðÞ¼q* m exp K À : ð12Þ vertical integration of the incompressibility condition (4) B0 q q* Rq* ðÞr À from the base of the glacier to a height z

Here Q is the activation energy for ice creep, R is the Zz @u @v universal gas constant, and m is an enhancement factor (or wzðÞÀwbðÞ¼À þ dz: ð20Þ tuning parameter). By definition, strain rates are related to @x @y velocity gradients, so that b 0 1 Neglecting horizontal diffusion, the energy balance 0 1 @u 1 @u þ @v 1 @u equation (3) is written as e_xx e_xy e_xz B @x 2 @y @x 2 @z C @ A B C e_yx e_yy e_yz ¼ @ 1 @u @v @v 1 @v A: ð13Þ 2 2 @y þ @x @y 2 @z @q @ q @q @q @q e_zx e_zy e_zz _ 1 @u 1 @v @w rcp ¼ ki 2 À rcp u þ v þ w þ 2es; ð21Þ 2 @z 2 @z @z @t @z @x @y @z EPM 4 - 4 PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL where s is the effective stress or the second invariant of resistive forces and written as [Van der Veen and Whillans, the stress tensor. The heat transfer is thus a result of 1989] vertical diffusion, horizontal and vertical advection, and internal deformational heating. @b @b t ¼ s0 ðÞÀb 2s0 ðÞþb s0 ðÞb À s0 ðÞb ; ð25Þ [7] Using a kinematic boundary condition at the upper and bx xz xx yy @x xy @y lower surface of the ice mass (see below), the mass balance equation (1) is integrated along the vertical in order to obtain an expression for the change of local ice thickness in space 0 0 0 @b 0 @b tby ¼ s ðÞÀb 2s ðÞþb s ðÞb À s ðÞb : ð26Þ Z yz yy xx @y xy @x @H s ¼Àr udz þ Ms À Mb ¼ÀrðÞþuH Ms À Mb; ð22Þ @t b The stress-free surface implies that [Van der Veen and Whillans, 1989] where u is the horizontal velocity vector (m yrÀ1), H is the ice thickness (m), Ms is the local accumulation-ablation 0 0 @s 0 @s À1 2s ðÞþs s ðÞs þ s À sxzðÞ¼s 0 ð27Þ function (m yr ice equivalent), and Mb is the melting xx yy @x xy @y rate at the base of the glacier (m yrÀ1). Negative values of Mb imply refreezing, which is not considered here. Local ice thickness variations are thus in balance with the 0 0 @s 0 @s 2s ðÞþs s ðÞs þ s À syzðÞ¼s 0: ð28Þ divergence of the ice flux and the net input of mass at the yy xx @y xy @x upper and lower surface of the ice mass. [8] In order to verify the results of the HO model, a Written in terms of velocity gradients, this results in ‘‘shallow ice’’ model was implemented as well. According to the SIA the horizontal velocity field is defined as [e.g., @u @v @s @u @v @s @u 4 þ 2 þ þ À ¼ 0; ð29Þ Huybrechts et al., 1996; Payne et al., 2000] @x @y @x @y @x @y @z

Zz n nÀ1 n @v @u @s @u @v @s @v uðÞ¼Àz 2ðÞrg jjrs rs AðÞq* ðÞs À z dz þ ub: ð23Þ 4 þ 2 þ þ À ¼ 0: ð30Þ b @y @x @y @y @x @x @z

Note that according to the SIA the velocity is obtained A similar expression can be obtained for a stress-free base, through vertical integration over the ice column and is only as is the case in an ice shelf or an ice stream, simply by a function of the local geometry of the ice mass. replacing s in equations (29) and (30) by b. Boundary conditions to the thermodynamic equation (21) follow from the mean annual air temperature at the surface. At the base, 3. Boundary Conditions the temperature gradient is defined as [9] Boundary conditions to the ice mass are zero ice thickness (H = 0) at the edges of the model domain, a @q G þ t u b ¼À b b ; ð31Þ stress-free surface and no-slip boundary condition at the base @z ki (ub = vb = 0). In some of the experiments below a stress-free boundary condition was applied at some grid points of the where G is the geothermal heat flux. However, a constant numerical domain (free-slip condition or zero traction). geothermal heat flux is not strictly correct; it might be more These two treatments of basal conditions represent only realistic to consider heat conduction in the bedrock, so that the most extreme end-members of reality and are not G = Àkr (@T/@z), where kr is the thermal conductivity in the applicable to many natural situations (including ice streams, underlying bedrock. However, a constant value of G is a where basal drag is small but not zero). However, such common boundary condition in ice sheet and glacier conditions make it easier to compare the model with bench- models, and facilitates model intercomparison according mark tests. Using extreme boundary conditions pushes the to the EISMINT benchmarks. The basal temperature in the model to its limits of application and demonstrate its ice mass is kept at the pressure melting point whenever it is performance in terms of numerical stability under such reached, and the basal melt rate Mb is calculated as extreme conditions. Experiments carried out with varying basal traction (according to a sliding law) resulted in more 1 @q M ¼ k b j þ G þ u t ; ð32Þ realistic simulations, but did not alter the general perform- b rL i @z c b b ance of the model as shown in the experiments below. [10] The kinematic boundary conditions at the lower ice where L is the specific latent heat of fusion and the subscript surface reads c stands for the temperature gradient after correction for pressure melting. @b wbðÞ¼ þ ubrb À Mb: ð24Þ @t 4. Coordinate Transformation

Starting from the vertical velocity at the base (according to [11] For numerical convenience a dimensionless vertical equation (24)), equation (20) is numerically integrated to the coordinate is introduced to account for ice thickness surface. The basal drag is defined as the sum of all basal variations, and which is defined as z (s À z)/H, so that PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL EPM 4 - 5 z = 0 at the upper surface and z = 1 at the base of the ice Making use of equations (34) to (42), equation (18) then mass. The coordinate transformation maps (x, y, z) ! becomes after coordinate transformation (equation (19) is (x0, y0, z), so that the function derivatives transform to transformed accordingly): [Lliboutry, 1987] @u @u @2u 4h 0 þ h 0 þ 4h @f @f @x0 @f @y0 @f @z x @x 0 y @y 0 @x02 þ þ : ð33Þ @x @x0 @x @y0 @x @z @x 1 @h @u þ 4a h 0 þ h 4b þ b þ a h 0 þ x x x y y y H2 @z @z Similar expressions are obtained for @f /@y and @f/@z.A 0 0 0 2 2 2 further assumption is made that @x /@x, @y /@y ﬃ 1, and @x / @ u 2 2 1 @ u @ u þ h þ h 4a þ a þ þ 8axh @y, @x0/@z, @y0/@x, @y0/@z ﬃ 0, which is valid as long as @y02 x y H2 @z2 @x 0@z surface and bedrock gradients are not too large. The function derivatives can thus be rewritten as @2u @s @v @v þ 2a h ¼ rg À h 0 À 2h 0 y @y 0@z @x 0 y @x 0 x @y 0

@f @f @f 2 þ a ; ð34Þ @v @ v 0 x À 2ayh 0 þ axh 0 þ 3cxyh À 3axayh @x @x @z x y @z @z2

@2v @2v @2v 3h 3a h 3a h ; 44 2 2 2 2 À 0 0 À y 0 À x 0 ð Þ @ f @ f @f 2 @ f @ f @x @y @x @z @y @z þ bx þ a þ 2ax ; ð35Þ @x2 @x02 @z x @z2 @x0@z where

@f 1 @f @h @h h 0 a : 45 À ; ð36Þ x ¼ 0 þ x ð Þ @z H @z @x @z

[12] Similarly, the upper boundary condition (29) becomes @2f 1 @2f ; ð37Þ @z2 H2 @z2 @s @u @s @u @s @s 1 @u 4 þ þ 4a þ a þ @x 0 @x 0 @y 0 @y 0 x @x 0 y @y 0 H @z @2f 1 1 @H @f @2f @2f a ; 38 @s @v @s @v @s @s @v 0 À 0 À x 2 ð Þ @x@z H H @x @z @x @z @z ¼À À 2 À 2ay þ ax : ð46Þ @y 0 @x 0 @x 0 @y 0 @x 0 @y 0 @z

[13] Similar transformations are carried out for @2f @2f @f @2f @2f @2f þ cxy þ ay þ ax þ axay ; ð39Þ equation (30), as well as for the strain rate definition (13). @x@y @x0@y0 @z @x 0@z @y 0@z @z2 Finally, the thermodynamic equation (21) is transformed as follows: where @q @q 1 @s @H þ a u þ a v þ À z À w @t @z x y H @t @t 1 @s @H ax ¼ À z ; ð40Þ H @x 0 @x 0 @q @q k @2q 2 u v es_ : 47 þ 0 þ 0 À 2 2 ¼ ð Þ @x @y H rcp @z rcp 2 2 @ax @ax 1 @ s @ H @H bx ¼ þ ax ¼ À z À 2ax ; ð41Þ [14] Equations (46) to (47) are solved on a regular grid in @x 0 @z H @x02 @x02 @x 0 x0 and y0, and an irregular grid in z. First and second central difference approximations on a regular and irregular grid @a @a @a @a are given by Pattyn [2002a]. At the surface boundaries (e.g., c ¼ y þ a y x þ a x : ð42Þ k = 1 and k = N ), first and second derivatives are computed xy @ 0 x @ @ 0 y @ z x z y z using three-point upstream differences. With the exception of the surface boundaries, central differences are used Similar expressions can be obtained for @f/@y, @2f/@y2, @2f/@y everywhere. Using central differences leads to numerically @z, ay, and by. The incompressibility condition (4) thus stable solutions as long as the ice flow is relatively slow becomes (e.g. ice sheet flow, ice flow at transition zones or grounding lines). Whenever the ice moves considerably faster, i.e., for model experiments where velocities of >1000 m yrÀ1 are @w @u @u @v @v ¼ H þ ax þ þ ay : ð43Þ encountered, three-point upstream differences are applied @z @x0 @z @y 0 @z over the whole model domain for determining horizontal EPM 4 - 6 PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL velocity gradients in the nonlinear viscosity term h, to yield conjugate gradient method. Although the coding of sparse a stable solution. matrices is rather complicated, they are far more efficient in terms of computation time compared to point relaxation 5. Numerical Solution algorithms on the full or even parts of the matrix. A ‘‘good’’ convergence is obtained when the quantity jA.u À bj/jbj is 5.1. Velocity Field 4 less than 10À . [15] The basic problem is the determination of the veloc- 5.1.2. Iterating the Nonlinear System ity field v in an ice mass. The problem is reduced to [18] Because of the nonlinear nature of equation (44) and determining the velocity in the horizontal direction u(u, v), its transverse counterpart, A and C contain a parameter that is as the vertical velocity is obtained from the incompress- still a function of u and v, i.e., the viscosity term h, which has ibility condition (4), using the kinematic boundary condi- to be determined in an iterative fashion. The successive tion at the lower surface (equation (24)). Although the substitution method or Picard iteration might be used for this problem is in itself a nonlinear one, equation (44) can be purpose, but is not favored as values for h vary a few orders of treated as a pair of coupled linear equations with u (and v)as magnitude over the whole ice sheet domain. This puts high unknowns, if h is approximated from the previous iteration. demands on the convergence scheme, and may eventually The nonlinear part is given by the term h in above lead to a divergence of the solution. In order to optimize the equations, which is in itself a function of u and v through rate of convergence a relaxation formula was used based on the effective strain rate in the nonlinear flow law. The the unstable manifold correction [Hindmarsh and Payne, problem is thus twofold: (1) solving a linear set of equations 1996; Pattyn, 2002a]. Therefore equation (48) is written as using an estimate of the horizontal velocity field and (a similar operation is carried out for equation (49)): (2) iterating the nonlinear part by updating h with new estimates of u and v. Au‘ Á u* ¼ bu‘ ; ð50Þ 5.1.1. Solving Sparse Linear Systems [16] The finite difference notation of the transformed differential equations for u and v (equation (44)) form a set where u* is the velocity estimate obtained with the conjugate of linear equations with u(x0, y0, z) and v(x0, y0, z) as unknowns. gradient method. Consider an iterative solution of a nonlinear equation which generates a series of approximate solutions There are two ways to proceed. The easiest way is to solve ‘+1 ‘ equation (44) for u, based on estimates for h and v. Subse- u , u ,..., being updated by a series of correction vectors c‘+1, c‘,..., such that u‘+1 = u‘ + c‘. Since the correction quently, the related equation for the velocity in the transverse ‘ ‘ direction is solved for v, based on previous estimates for h and vector is defined as c = u* À u , the Picard iteration would simply update the velocity u‘+1 with this correction vector, so u. A more complex way of solving this set of equations is by ‘+1 ‘+1 ‘ solving both equations at once for u and v. The coefficient that u = u*. If e , e ,...is taken as the error in the solution vector u‘+1, u‘,..., then we can state that (e‘+1, e‘,...)=a matrix will now be 2N Â 2N instead of N Â N, where N is the (c‘+1, c‘,...). Assuming that the decay is on a straight line in number of velocity estimates (N = Nx Â Ny Â Nz). However, for tests carried out in this study, there were no additional the correction space, we obtain [Hindmarsh and Payne, benefits in terms of a more stable solution and it only 1996] increased computational time. Therefore the two-step ap- k c‘À1 k proach is favored. In matrix notation, the above problem is a ¼ ; ð51Þ written as k c‘ À c‘À1 k Au‘ Á u‘þ1 ¼ bu‘ ; ð48Þ and a new update of the velocity vector is obtained by

u‘þ1 ¼ au* þ ðÞ1 À a u‘; ð52Þ ‘ ‘þ1 ‘ Cv Á v ¼ dv ; ð49Þ ‘ ‘ where the modified correction vector becomes c* a c . The where ‘ is the iteration number, A and C are the coefficient direction w between successive correction vectors is matrices (A is formed by the left-hand side of equation (44)), computed as and b and d are the solution vectors (b is formed by the c‘ Á c‘À1 right-hand side of equation (44)). w ¼ arccos ; ð53Þ k c‘ kÁkc‘À1 k [17] Starting from a zeroth-order estimate of the horizon- 0 1 tal velocity field u , a new estimation u of the velocity is where the norms refer to the L2 norm. Whenever this angle is obtained by solving the set of linear equations. A quick close to 0 or p, the subspace iteration is applied. Using this analysis of the coefficient matrix A shows that this matrix subspace relaxation algorithm the solution vector converges is highly sparse: for a horizontal grid of 61 Â 61 nodes and properly. 41 layers in the vertical (N =61Â 61 Â 41 = 152561), the matrix A contains N Â N or more than 23 billion elements, 5.2. Ice Sheet Evolution and Thermodynamics of which at most 2 898 659 elements are strictly nonzero, [19] The continuity equation (22) is reformulated as a clustered around the diagonal, which is 0.012% of A. diffusion equation for ice thickness H. Defining the diffu- Although A resembles a band-diagonal matrix, it is more sivities as convenient to regard it as a sparse matrix and to solve the À1 À1 linear system of equation (48) using the sparse matrix @s @s Dx ¼ uH ; Dy ¼ vH ; ð54Þ algorithms of Press et al. [1992], which are based on the @x @y PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL EPM 4 - 7 the local change in ice thickness is written as

@H ÀrðÞ¼rDrH ðÞþDrb M À M ; ð55Þ @t s b where D =(Dx, Dy) is the diffusion vector. Diffusivities are calculated on a staggered grid, where Diþ1 is calculated as the mean of two diffusivities defined on the2 neighboring grid points i and i + 1, and corresponds to the type II diffusivity defined by Huybrechts et al. [1996]. Such a scheme is not mass conserving near the edge of the model domain, but is generally more stable than the proper calculation method between grid points (type I). Equation (55) results in a sparse system of Nx Â Ny equations, and is solved using the conjugate gradient method given by Press et al. [1992]. [20] The thermodynamic equation (47) is solved implic- itly in the vertical, giving rise to a tridiagonal system of Nz equations and which is solved using the tridiagonal algorithm of Press et al. [1992]. A two-point upstream difference notation was employed for the horizontal, while central differences were used in the vertical. The horizontal implicit terms are found by iteration of this scheme. Figure 1. Predicted surface topography of the steady state Only a few iterations are necessary to obtain a good ice sheet after 200,000 years of integration according to convergence. experiment A1. The thick black line shows the position of the ‘‘ice stream’’ for experiment D1. 6. Experiments and Results

[21] Basic experiments were carried out following the summit, according to the benchmark experiments by Payne EISMINT benchmark for large-scale ice sheet models, et al. [2000], more specifically the ‘‘moving margin’’ benchmark of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Huybrechts et al. [1996]. The model domain is 1500 Â 2 2 q ðÞ¼x; y q þ S ðÞx À x þðÞy À y ; ð57Þ 1500 km in the horizontal. Following the EISMINT II s min q sum sum experiments [Payne et al., 2000], a grid resolution of 25 km in both horizontal directions was taken, which where qmin is the minimum surface air temperature and Sq corresponds to a numerical grid of 61 Â 61 grid points. is the gradient of air-temperature change with distance form the summit. A time step of 20 years was taken for all The accumulation/ablation rate Ms is a function of the distance from the ice divide, i.e., experiments for the evolution of the continuity equation (55) as well as for the thermodynamic part equation (47). The qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi model was run from zero ice thickness until a steady state 2 2 MsðÞ¼x; y min Mmax; Sb Rel À ðÞx À xsum þðÞy À ysum ; was reached (after 200,000 years of calculation). Two types of experiments were carried out with both the HO model ð56Þ (subscript 1) and the SIA model (subscript 0), similar to the experiments described by Saito et al. [2003]. Experiment A relax where Mmax is the maximum accumulation rate and Sb is the isothermal experiment. Its numerical parameters are the gradient of accumulation rate change with horizontal listed in Table 2. Experiment B is similar to experiment A, distance. The accumulation rate is zero at a radial distance but with thermomechanical coupling, i.e. A(q*) is determined Rel from the summit (xsum, ysum), where x and y are distance from equation (12). Experiment C is a diagnostic experiment in km. At a distance larger than Rel from the summit, surface of ice flow over bedrock undulations. Experiment D mass balance becomes negative (ablation). The ice surface demonstrates the development of an ice stream within an temperature is also made a function of distance from the ice sheet, by considering a frictionless zone (tb =0)atthe base of an ice sheet. Another example of a stress-free (lubricated) spot at the base of an ice sheet is the ice flow over a subglacial lake, simulated in experiment E. Table 2. List of Parameters Used for Experiment A Parameter Value Units 6.1. Isotherm Ice Sheet (Experiment A) À1 ub, vb 0myr[22] Figure 1 displays the surface topography of the ice A(q*) 10À16 PaÀ1 yrÀ1 sheet in steady state for A . This ice sheet configuration is À1 1 Mmax 0.5 m yr À2 À1 À1 similar to the SIA solution A0, as is expected, since the SIA Sb 10 myr km is valid for large ice sheets resting on a flat bed. The Rel 450 km qmin 238.15 K difference between A1 and A0 amounts to 10 m (Figure 2). À2 À1 Sq 1.67 Â 10 Kkm A1 results in a somewhat smaller ice sheet and has a flatter xsum, ysum 750 km ice divide. The differences at the divide are traced back to EPM 4 - 8 PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL

Figure 2. Difference in ice thickness between A1 and A0 (solid circles) and between B1 and B0 (open circles) after 200,000 years of integration in time. the effect of the inclusion of longitudinal stress gradients in the HO model, which is the only nonzero stress component Figure 4. Predicted horizontal velocity at a distance of at the symmetrical ice divide. 25 km from the ice divide for A1 (solid circles) and A0 (open [23] The vertical velocity field demonstrates the major circles). difference between both models (Figure 3). A lower vertical velocity under the ice divide for A1 results in a ‘‘bump’’ in the vertical velocity field, confined to the immediate region of compensate for the higher surface velocity: as the surface the ice divide (Figure 3a). This corresponds to the so-called mass balance is only a function of distance from the summit, ‘‘Raymond bump’’ [Raymond, 1983; Schøtt Hvidberg, the steady state ice flux at any point should be the same for 1996], due to the fact that ice deep under an ice divide should both A0 and A1. Since ice thickness is similar in both be very hard and slow to flow, so that upper layers would experiments (a difference of only ±10 m), the vertical tend to drape themselves over it. This anomaly in vertical integrated horizontal velocity should be the same for both velocity is a result of differences in the horizontal velocity models. Further away from the ice divide, the difference in field: close to the ice divide (at 25 km form the divide, horizontal velocity between both models disappears. The Figure 4) surface horizontal velocity is higher for A1 divide anomaly is also observed in the basal temperature: a compared to A0. However, horizontal velocity at depth is ‘‘hot spot’’ arises underneath the ice divide according to A1 lower than for A0. This lower velocity is necessary to (Figure 5), due to the difference in vertical advection.

Figure 3. Predicted vertical velocity: (a) near the ice divide for A1; (b) at the ice divide for A1 (solid circles) and A0 (open circles). Note the ‘‘bump’’ in vertical velocity at the ice divide, which is not found in A0. PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL EPM 4 - 9

Figure 5. Predicted basal temperature qb for A1 (solid circles) and A0 (open circles). Note the ‘‘hot spot’’ under the ice divide.

[24] All these results, anomalies in vertical and horizontal profile also occur in B1. The divide anomaly between both flow field and temperature near the ice divide, are corrob- models is more amplified and further generalized within the orated by the experiments carried out by Saito et al. [2003]. central part of the ice sheet, as the temperature (and the hot However, their model results show a slightly thicker A1 ice spot) influences the velocity field in a direct way. The sheet, while the present study shows a thinner steady state difference in ice thickness between B1 and B0 is almost ice sheet compared to A0. This difference is probably related 40 m at the divide, and gradually decreases from approx- to the type of numerical scheme used for the ice sheet imately 300 km from the divide (Figure 2). The smaller ice evolution equation (55). The hot spot under the divide was sheet for B1 is a direct result of the hot spot at the divide, determined previously by Dahl-Jensen [1989]. which leads to softer ice, a higher deformational velocity compared to B0 and thus a smaller ice sheet. The difference 6.2. Thermomechanical Coupling (Experiment B) between both models is therefore most pronounced in the [25] Results of the thermomechanical coupling (B)are interior part of the ice sheet. similar to the results of A, i.e., an anomaly in vertical [26] The stress field near the ice divide is shown in velocity at the divide and a hot spot in the basal temperature Figure 6. Shear stress is slightly higher than driving stress,

0 À1 Figure 6. (a) Predicted longitudinal stress gradient @s xx/@x (Pa m ); (b) shear stress sxz (z) (thick solid 0 line), driving stress ztd (thin solid line) and longitudinal stress deviator s xx (z) (circles) at a distance of 25 km from the ice divide according to B1. EPM 4 - 10 PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL

Figure 7. Horizontal surface velocity anomaly us =(us (C1) À us(C0))/us (C1) Â100 in the vicinity of the ice divide (x, y = 150,150). Contour lines show basal topography (m).

which means that the longitudinal stress gradient must be B1 are in agreement with the results by Saito et al. [2003], negative, although the longitudinal stress is positive as albeit that some minor differences remain, probably attrib- stretching prevails near the ice divide (Figure 6b). The uted to differences in numerical schemes. negative stress gradient is a result of the relatively flat ice divide in B1 and is only present at depth at a horizontal 6.3. Ice Flow Over an Undulating Bedrock distance of 25 to 100 km away from the ice divide. Further (Experiment C) away from the ice divide (100 km), the longitudinal stress [27] In this diagnostic experiment a similar grid of 61 Â gradient becomes positive (Figure 6). The overall results of 61 grid points was considered, but the horizontal grid size was

Figure 8. Evolution of an ice stream (D1) starting from the steady state configuration of A1. Predicted surface topography at t = 0 (solid line), at t = 50 year (dashed line) and at t = 100 year (dotted line) is displayed, after invoking a stress-free basal condition tb = 0 along y = 750 km and between x = 350 km from the divide and the edge of the model domain. Predicted surface velocity at t = 50 year is given in solid circles. PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL EPM 4 - 11 increased to 5 km, thus covering a surface of 300 Â 300 km. The surface of the small ice cap is taken from the steady state outcome of experiment A1. The bedrock is determined from a sine wave function with a double amplitude of 1000 m and a wavelength of 37.5 km. The ice was made stiffer by setting A(q*) equal to 10À18 PaÀ1 yrÀ1. In this way a high-resolution experiment is carried out in a region of slow ice flow over a highly undulated bedrock. The model geometry was kept constant. [28] Since the velocity field for C0 (SIA) is locally determined, it will be solely a function of the local ice thickness and surface gradient (see equation (23)). This implies that the velocity will be high for large ice thickness and/or high surface slopes. The C1 horizontal velocity field is a smoothed version of C0, i.e., high velocities in C0 attain lower values in C1, and low velocities in C0 become higher in C1. These anomalies are associated with the bedrock topography (Figure 7). The maximum (minimum) velocity in C1 still coincides with the maximum (minimum) velocity in C0, but longitudinal pushes and pulls, due to the bedrock perturbation, influence the velocity field in a global way and not locally as is the case with C0. Such smoothing of velocity and stress fields is typical for higher-order models. Similar results were obtained with other higher-order models [Blatter et al., 1998; Pattyn, 2002b]. 6.4. Ice Stream Evolution (Experiment D) [29] Starting from the steady state ice sheet conditions and temperature field from experiment A1, an ice stream was generated by adjusting the basal boundary condition to a stress-free surface (equations (29) and (30)) for the following coordinates: i = 31 and j 45. The ice stream thus starts at a distance of 350 km from the ice divide and stretches to the edge of the model domain (Figure 1). The width of the ice stream is one grid point, or 25 km. [30] Figure 8 shows the evolution of the ice stream for the first 100 year after invoking the stress-free boundary conditions. Rather rapidly the ice velocity increases with almost 2 orders of magnitude, to reach maximum values of Figure 9. Experiment D1. (a) Predicted surface topogra- À1 2 500 km yr , a typical value for an Antarctic ice stream. phy (m), (b) predicted basal shear stress sxz (b) (kPa), 0 Also, the surface topography changes dramatically to (c) predicted surface longitudinal stress deviator s xx (s) (kPa), become almost flat over the whole length of the ice (d) predicted surface transverse shear stress sxy (s) (kPa) at stream. Similar characteristics of ice stream behavior are t = 50 year. The ice stream is situated at y = 750 km and shown in early flow line models of Van der Veen [1987]. stretches between x = 1100 km and the edge of the model The major difference here is that both longitudinal stresses domain x = 1500 km. and transverse shear stresses are properly accounted and solved for in three dimensions. [31] The model result shows a concave inflection point at [32] Transverse shear stress shows a pattern that is related the onset of the ice stream, similar to the transition between to the existence of shear margins and is characterized by an ice sheet and an ice shelf (Figure 8). It should be noted high positive values at the southern shear margin of the ice here that the basal boundary condition applied to our ice stream and high negative values at the northern shear stream is similar to that of an ice shelf. Associated with the margin. This results in an important lateral drag over the sudden increase in mass flux is a forward migration of the central part of the ice stream (not shown), which will, due to edge of the ice sheet, forming a frontal ‘‘bulge’’ (Figures 8 and the absence of the basal shear as well as longitudinal stress 9a). Basal shear stress is zero along the whole length of gradients, balance the driving stress in the central part of the the ice stream (Figure 9b). As a result, the surface velocity ice stream. The shear margins are a consequence of the equals the basal velocity for the whole length of the ice rapid ice flow embedded in the slow moving ice sheet. Both stream (not shown). The longitudinal stress deviator reaches patterns of longitudinal and transverse shear stress are a maximum (extension flow) at the onset of the ice stream typical features for existing Antarctic ice streams, such as and a minimum (compressive flow) at the edge of the ice the Siple Coast ice streams [Whillans and Van der Veen, 0 sheet (Figure 9c), but remains fairly low (s xx 0) along the 1997] albeit that the magnitude of the marginal shear stress whole length of the ice stream. (20–80 kPa in Figure 9) is relatively small compared to the EPM 4 - 12 PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL

À1 3 Figure 10. Experiment E1. (a) Predicted surface velocity (m yr ), (b) predicted surface slope (Â10 ), 0 (c) predicted surface longitudinal stress deviator s xx (s) (kPa), (d) predicted surface longitudinal stress 0 deviator s yy (s) (kPa), (e) predicted surface transverse shear stress sxy (s) (kPa), (f ) predicted basal 2 2 1/2 shear stress (sxz (b)+syz (b)) . The ice divide is situated at (x, y) = (750, 750) km. The lake is situated at (x, y) = (875, 875) km and is 50 km long in x and 25 km wide in y. magnitude of marginal shear stresses calculated for modern a prominent morphological surface feature within the Ant- Antarctic ice streams. arctic ice sheet, as the ice sheet surface is relatively flat and featureless, consistent with the surface of an ice shelf. Ice 6.5. Ice Flow Over a Subglacial Lake (Experiment E) flow over a large subglacial lake should be analogous to the [33] Radio echo sounding in East Antarctica has revealed flow of an ice shelf, where the lack of basal shear stress the existence of numerous subglacial lakes. The largest one prevents deformation of internal layers [Siegert et al., is Lake Vostok (14,000 km2), near the homonymous Rus- 2000]. Other distinct features of the ice flow over Lake sian Station and drilling site. Lake Vostok is associated with Vostok are revealed by the surface velocity field determined PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL EPM 4 - 13

Figure 11. Evolution of a subglacial lake in the central region of the ice sheet (E1) starting from the steady state configuration of A1. Predicted surface topography (solid circles) without lake (dashed line) and over a lake (solid line). Surface velocities (open circles) without (dashed line) and over a lake (solid line). The lake is situated at (x, y) = (875, 875) km and is 50 km long in x and 25 km wide in y. from radar interferometry [Kwok et al., 2000]: (1) an low basal shear stress (Figure 10f ). The zone of influence of increase in ice velocity over the lake (extension flow) other stress components (longitudinal and transverse followed by a decrease in ice speed passed the lake stresses, Figures 10c, 10d, and 10e) is much larger than (compression flow), and (2) a slight turning of the ice flow the effective area occupied by the lake. These stress over the lake, most visible over the central part of Lake components show a distinct ‘‘butterfly’’ pattern that extents Vostok where surface slopes are small. Model simulations for more than 100 km in either horizontal direction. of the ice flow at Lake Vostok demands the use of a HO model. Mayer and Siegert [2000] investigated the ice flow along a longitudinal transect with a 2-D HO flow line 7. Discussion and Conclusions model. Their results demonstrate that ice dynamics across [36] The numerical model developed and presented here the inflow grounding line are similar to an ice sheet/ice shelf is a three dimensional thermomechanical ice sheet model transition. including higher-order (HO) stress gradients. Similar HO [34] Experiment E1 thus consists of simulating the ice models exist, most of them solving for the stress field, using flow over a subglacial lake with the 3-D HO model by the method proposed by Van der Veen and Whillans [1989] considering a stress-free basal surface (lubricated spot) and Van der Veen [1989]. The uniqueness of this model lies stretching over two grid points and situated approximately in the fact that it solves the force balance equations in their 100 km from the ice divide. The orientation of the so-called ‘derivative’ form, by formulating them in terms of velocity subglacial lake with respect to the major ice flow is similar gradients (using Glen’s flow law) and solving second to the situation of Lake Vostok. The initial conditions are derivatives analytically instead of numerically. The result- given by experiment A1 and the ice sheet was allowed to ing equations look rather cumbersome, but assure a high react to the change in basal boundary conditions until the numerical stability. Stability of the numerical scheme can ice sheet surface reached a steady state. Results of E1 be further improved by adapting upstream differences for indicate that the surface velocity over the subglacial lake velocity gradients in the effective viscosity. increases to reach a maximum near the center of the lake, [37] Basic experiments with the model include the stan- and decreases again at the downstream edge of the lake dard EISMINT I benchmark experiments [Huybrechts et (Figure 10a). Moreover, the ice sheet flow field seems only al., 1996] as well the EISMINT II experiments with locally influenced by the presence of the lake, as the thermomechanical coupling [Payne et al., 2000]. A com- velocities downstream from the lake are similar to those if parison is made with a similar 3-D model according to the no lake were present (Figure 11). These findings are SIA, and both isothermal and thermomechanical experi- corroborated by Siegert and Ridley [1998] and by the ments are considered. Results are in agreement with results measurements of Kwok et al. [2000]. Also the turning of from Saito et al. [2003], and the solution under the ice the ice flow is visible in experiment E1, as the surface slope divide (Raymond bump and hot spot) is confirmed by of the lake depends on the orientation of the lake toward the earlier flow line studies [Raymond, 1983; Dahl-Jensen, surrounding ice flow. The maximum slope is therefore 1989]. Model experiments by Schøtt Hvidberg [1996] were found in the longitudinal direction of the lake. carried out at much higher detail and show the rise of [35] A flattening of the surface topography above the lake isochrons (due to reduced vertical velocity with a HO is observed in Figures 10b and 11, conform observations in model) 1 to 2 ice thicknesses from the ice divide. The Antarctica [Siegert et al., 2000], which is associated with a model resolution of the present study (25 km) does not EPM 4 - 14 PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL allow to identify the span of the Raymond bump, as the grid Baral, D., K. Hutter, and R. Greve, Asymptotic theories of large-scale motion, temperature, and moisture distribution in land-based polythermal point closest to the divide is at least 8 ice thicknesses away. ice sheets: a critical review and new developments, Appl. Mech. Rev., 54, However, the solution right under the ice divide remains 215–256, 2001. correct, irrespective of horizontal model resolution. Similar Blatter, H., Velocity and stress fields in grounded glaciers: A simple algo- remarks can be made for the rising basal temperatures under rithm for including deviatoric stress gradients, J. Glaciol., 41, 333–344, 1995. the divide, associated with the reduced vertical advection. Blatter, H., G. Clarke, and J. Colinge, Stress and velocity fields in glaciers: [38] With the exception of the ice-flow-over-a-bumpy- Part II. sliding and basal stress distribution, J. Glaciol., 44, 457–466, bed experiment, all model runs are prognostic, which keeps 1998. Budd, W., The longitudinal stress and strain-rate gradients in ice masses, an internal consistency between the conservation equations. J. Glaciol., 9, 19–27, 1970a. Two examples of prognostic model runs that can only be Budd, W., Ice flow over bedrock perturbations, J. Glaciol., 9,29–48, solved with a HO model are the ice stream development 1970b. and the ice flow over a subglacial lake, both simulated by Budd, W., Stress variations with ice flow over undulations, J. Glaciol., 10, 177–195, 1971. considering locally a stress-free basal surface. Changing Colinge, J., and H. Blatter, Stress and velocity fields in glaciers: part I. from a basal stress dominated regime to a basal stress-free finite-difference schemes for higher-order glacier models, J. Glaciol., 44, regime is similar to the ice sheet/ice shelf/grounding line 448–456, 1998. Dahl-Jensen, D., Steady thermomechanical flow along two-dimensional problem, which remains a hot topic amidst the ice sheet flow lines in large grounded ice sheets, J. Geophys. Res., 94, 10,355– modeling community. For instance, Marshall and Clarke 10,362, 1989. [1997] solved this problem by treating the ice sheet area as Dansgaard, W., and S. Johnsen, A flow model and a time scale for the ice core for Camp Century, Greenland, J. Glaciol., 8, 215–223, 1969. a binary mixture of sheet ice (shallow ice approximation) Greve, R., Application of a polythermal three-dimensional ice sheet model and stream ice (vertically integrated force balance equa- to the Greenland ice sheet: Response to steady-state and transient climate tions), which reduces the model complexity. In the present scenarios, J. Clim., 10, 901–918, 1997. model a proper treatment of the transition zone is given, as Hindmarsh, R., Influence of channelling on heating in ice-sheet flows, Geophys. Res. Lett., 28, 3681–3684, 2001. the ice mass is considered as one single continuum. In Hindmarsh, R., and A. Payne, Time-step limits for stable solutions of the reality, the reduction of basal drag in an ice stream is a ice-sheet equation, Ann. Glaciol., 23, 74–85, 1996. complex process involving basal hydrology, sediment de- Hooke, R., Flow law for polycrystalline ice in glaciers: Comparison of theoretical predictions, laboratory data, and field measurements, Rev. formation, basal melting and sliding, which lies beyond the Geophys., 19, 664–672, 1981. scope of this paper. The simple basal boundary switch Hutter, K., Theoretical Glaciology, Kluwer Acad., Norwell, Mass., 1983. seems sufficient to explain major processes that are ob- Huybrechts, P., A 3-D model for the Antarctic ice sheet: A sensitivity study on the glacial-interglacial contrast, Clim. Dyn., 5, 79–92, 1990. served in real ice masses. For simulations of existing Huybrechts, P., and T. Payne, The EISMINT benchmarks for testing ice- glaciers and ice sheets, more complex basal boundary sheet models, Ann. Glaciol., 23, 1–12, 1996. conditions will become necessary. Kwok, R., M. Siegert, and F. Carsey, Ice motion over Lake Vostok, Ant- arctica: Constraints on inferences regarding the accreted ice, J. Glaciol., [39] A major deficiency of the model runs remains the 46, 689–694, 2000. low grid resolution used (with the exception of the ice-flow- Lliboutry, L. A., Very Slow Flows of Solids, Martinus Nijhoff, Zoetermeer, over-a-bumpy-bed experiment). For instance, the model Netherlands, 1987. lake occupies only two grid points, which is insufficient Marshall, S., and G. Clarke, A continuum mixture model of ice stream thermomechanics in the Laurentide ice sheet: 1. Theory, J. Geophys. to draw solid conclusions. Nevertheless, the simulation is Res., 102, 20,599–20,614, 1997. successful, since major characteristics of the ice flow over a Mayer, C., Numerische Modellierung der U¨ bergangszone zwischen Eiss- subglacial lake, such as Lake Vostok, are represented. The child und Shelfeis (numerical modelling of the transition zone between an ice sheet and an ice shelf), Ber. Polarforsch., 214, 1–150, 1996. situation at Lake Vostok is also slightly different from the Mayer, C., and M. Siegert, Numerical modelling of ice-sheet dynamics model lake over a flat bedrock, as Lake Vostok lies in a across the Vostok subglacial lake, central East Antarctica, J. Glaciol., subglacial trench and the bedrock slope near the grounding 46, 197–205, 2000. line dips toward the lake. Since major components of the Paterson, W., The Physics of Glaciers, 3rd ed., Oxford, Pergamon, Tarry- town, N.J., 1994. flow field could be simulated, this indicates that the stress- Pattyn, F., Ice-sheet modelling at different spatial resolutions: Focus on the free basal boundary condition is a decisive factor in the ice grounding line, Ann. Glaciol., 31, 211–216, 2000. dynamics over a subglacial lake, and not the geometric Pattyn, F., Transient glacier response with a higher-order numerical ice-flow model, J. Glaciol., 48, 467–477, 2002a. configuration. The fact that the ‘real’ lake lies in a sub- Pattyn, F., Ice-flow characteristics over a rough bedrock: Implications for glacial trench overlain by thick ice adds to the insulation ice-core interpretation, Polar Meteorol. Glaciol., 16, 42–52, 2002b. effect and might be responsible for high basal temperatures, Payne, A., et al., Results from the EISMINT model intercomparsion: The effects of thermomechanical coupling, J. Glaciol., 46, 227–238, 2000. hence ensuring the presence of a subglacial lake. Press, W., S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., Cambridge Univ. Press, New York, 1992. [40] Acknowledgments. This paper forms a contribution to the Raymond, C., Deformation in the vicinity of ice divides, J. Glaciol., 29, Belgian Research Programme on the Antarctic (Federal Office for Scien- 357–373, 1983. tific, Technical and Cultural Affairs), contract EV/03/08A, and a contribu- Ritz, C., A. Fabre, and A. Letre´guilly, Sensitivity of a Greenland ice sheet tion to the ‘‘European Project for Ice Coring in Antarctica (EPICA)’’, a joint model to ice flow and ablation parameters: Consequences for the evolu- European Science Foundation/European Commission (EC) scientific tion through the last glacial cycle, Clim. Dyn., 13, 11–24, 1997. programme, funded by the EC under the Environment and Climate Saito, F., Development of a three dimensional ice sheet model for numerical Programme and by national contributions from Belgium, Denmark, France, studies of Antarctic and Greenland ice sheet, Ph.D. thesis, Univ. of Germany, Italy, Netherlands, Norway, Sweden, Switzerland, and the United Tokyo, Tokyo, 2002. Kingdom. Saito, F., A. Abe-Ouchi, and H. Blatter, Effects of the first order stress gradients to an ice sheet evaluated by a three-dimensional thermomechanical coupled model, Ann. Glaciol., 37, in press, 2003. References Schøtt Hvidberg, C., Steady-state thermomechanical modelling of ice flow Albrecht, O., Dynamics of glaciers and ice sheets: A numerical model near the centre of large ice sheets with the finite- element method, Ann. study, Ph.D. thesis, Swiss Fed. Inst. of Technol., Zurich, 2000. Glaciol., 23, 116–123, 1996. PATTYN: A NEW 3-D HIGHER-ORDER ICE SHEET MODEL EPM 4 - 15

Siegert, M., and J. Ridley, An analysis of the ice-sheet surface and subsur- Van der Veen, C., A numerical scheme for calculating stresses and strain face topography above the Vostok Station subglacial lake, central East rates in glaciers, Math. Geol., 21, 363–377, 1989. Antarctica, J. Geophys. Res., 103, 10,195–10,207, 1998. Van der Veen, C., and I. Whillans, Force budget: I. theory and numerical Siegert, M., R. Kwok, C. Mayer, and B. Hubbard, Water exchange between methods, J. Glaciol., 35, 53–60, 1989. the subglacial Lake Vostok and the overlying ice sheet, Nature, 403, Whillans, I., and C. Van der Veen, The role of lateral drag in the dynamics 643–646, 2000. of Ice Stream B, Antarctica, J. Glaciol., 43, 231–237, 1997. Van der Veen, C., Longitudinal stresses and basal sliding: A comparative study, in Dynamics of the West Antarctic Ice Sheet, edited by C. Van der ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ Veen and J. Oerlemans, pp. 223–248, Kluwer Acad., Norwell, Mass., F. Pattyn, Department of Geography, Vrije Universiteit Brussel, Pleinlaan 2, 1987. B-1050 Brussel, Belgium. ([email protected])