Antarctic ice sheet GLIMMER model test and its simplifiedmodel on 2-dimensional ice flow

Xueyuan Tang a,*, Zhanhai Zhang a, Bo Sun a, Yuansheng Li a, Na Li a,Bangbing Wang a,b, Xiangpei Zhang a,b

a Polar Research Institute of China, Shanghai 200136, Chinab College of Geo-exploration Science and Technology, Jilin University, Changchun 130026, China

Received 4 June 2007; received in revised form 20 July 2007; accepted 20 July 2007

Abstract

The 3-dimensional finite difference thermodynamic coupled model on Antarctic ice sheet, GLIMMER model, is described. An ide-alized ice sheet numerical test was conducted under the EISMINT-1 benchmark, and the characteristic curves of ice sheets under steadystate were obtained. Based on this, this model was simplified from a 3-dimensional one to 2-dimensional one. Improvement of the dif-ference method and coordinate system was proposed. Evolution of the 2-dimensional ice flow was simulated under coupled temperaturefield conditions. The results showed that the characteristic curves deriving from the conservation of the mass, momentum and energyagree with the results of ice sheet profile simulated with GLIMMER model and with the theoretical results. The application prospectof the simplified 2-dimensional ice flow model to simulate the relation of age-depth-accumulation in Dome A region was discussed.� 2007 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in

China Press. All rights reserved.

Keywords: Antarctic ice sheet; GLIMMER model; Numerical simulation; 2-Dimensional ice flow

1. Introduction

The Antarctic ice sheet contains 90% of global ice vol-ume throughout the world. It is sufficient to raise sea levelover 65 m. Therefore, even if there is a slight fractionalchange of its volume, significant effect will be brought tothe global environment. It is estimated that annual solidprecipitation exchanged between Antarctic ice sheet andocean is equivalent to 6 mm of change of sea level [1]. Thisinput is approximately equivalent to the thawed volume ofice shelf floating on the sea. In fact, the current behavior ofAntarctic ice sheet may be the inevitable result of the pastclimate history. In particular, these changes are closely con-nected with the last interglacial cycle. It is not hard to seethat the future trend of Antarctic ice sheet is deeply associ-

ated with the current climate change when being affected bythe past climate. Therefore, to evaluate the future changeof Antarctic ice sheet practically and to distinguish thelong-term evolution background of ice sheet from thehuman activities causing the climate change seem extremelyimportant. To disclose various physical processes of icesheet with numerical simulation model has already becomean important method of studying the evolution history andfuture change of ice sheet. The advantage of numerical sim-ulation method is that it can overcome the restrictions ofobservation in respect of time and space and generate var-ious thermodynamic information of the whole ice sheet,showing the physical processes of interior and bottom ofice sheet that cannot be obtained directly by observationand disclosing the historical evolution state of ice sheet.In fact, numerical simulation models may be the only fea-sible tools to explain and forecast future change of ice sheet[2–4].

1002-0071/$ - see front matter � 2007 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited

and Science in China Press. All rights reserved.

doi:10.1016/j.pnsc.2007.07.014

* Corresponding author. Tel.: +86 21 58713308; fax: +86 21 58711663.E-mail address: tangxueyuan@pric.gov.cn (X. Tang).

Available online at www.sciencedirect.com

Progress in Natural Science 18 (2008) 173–180

In the past two decades, many mathematical modelssimulating ice sheet behavior were developed, for instance,Nye–Dansgaard–Johnsen Model [5–7], the 1-dimensionalflowline model, was used to estimate the evolution of icesheet in age; the vertically integrated plane model with cou-pled isostasy and simple ice shelf structure developed byOerlemans [8,9] and Budd and Smith [10]. The large-scalemodeling of the Antarctic ice sheet was first brought upby Budd in 1971, this model is based on the assumptionthat ice sheet is in steady state and ice flow is always alonglow terrain [11]. In order to learn the feedback mechanismof Antarctic ice sheet on climate, considering the 3-dimen-sional ice sheet thermodynamic model coupling ice temper-ature field is an efficient path, the models with significanteffect in this respect include the ice sheet model developedby Huybrechts and Oerlemans [3]. This model has becomecomparatively mature now after developed by others [12–14]. With this model, some specific simulation results havebeen obtained. It is found through simulation that the mostimportant factor controlling the rise of sea level is westernAntarctic ice sheets. Once the Western Antarctic ice sheetswere all thawed, sea level would rise by 7 m, while meltingof the whole Antarctic ice sheets may make the sea levelrise by 73 m [4,12,15–19]. Under the framework of EIS-MINT, Huyberchts model was revised twice in 1999 and2002. Now it has already been an integral part of the Ant-arctic ice sheet 3-dimensional thermodynamic standardcoupled model GLIMMER [15,20,21].

GLIMMER model is an increasingly perfect standardmodel of Antarctic ice sheet. It originally results from TonyPayne’s work in the research of GENIE project [22–25].Our study makes an idealized test through GLIMMER,the 3-dimensional finite difference ice sheet model, to learnthe ice flow thermodynamic characteristics of ice sheets andsimplifies this complicated nonlinear coupled model into a2-dimensional ice flow model. We aimed at finding a math-ematical model that can simulate various thermodynamiccharacteristic curves of East Antarctic Dome A regionand testing the feasibility of this model in simulating‘‘depth-age-accumulation’’ of Dome A.

2. GLIMMER model and EISMINT-1 test

GLIMMER ice sheet model actually is a set of librariesused to simulate ice sheet evolution. It can be run indepen-dently or as a sub-module of Earth System Model (ESM).The purpose of developing this model is to generate a stan-dard ice sheet model available for other earth system mod-els. By 2006, this model had already passed the formaltesting conducted by Hagdorn’s team under EISMINT-1and EISMINT-2 benchmark [26].

2.1. Principle of GLIMMER model

The GLIMMER model adopts the shallow ice approxi-mation. According to this approximation, the gradient ofthe bedrock and ice surface is considered to be very small

and the normal stress can be ignored. The approximationof horizontal shear stress (sxz,syz) is: sxz ¼ �qgðs� zÞ os

ox ;

syz ¼ �qgðs� zÞ osoy, the relation between ice viscosity and

viscous force is determined with the law of nonlinear ice

flow law: _eiz ¼ 12

ouioz þ

ouzoi

� �¼ AðT Þjsjn�1siz; i ¼ x; y, where _eiz

is strain rate, s is the second-order invariant of shear stresstensor, and n is the constant of law of ice flow. The coeffi-cient A(T) of law of ice flow meets the Arrhenius relation:

AðT Þ ¼ fA0 exp �QRT

� �, where f is the reinforced factor of

Arrhenius relation, an adjustment parameter reflectingthe anisotropy of impurities in ice fabrics and ice particlesin ice flow; A0 is a constant independent of temperature; R

is the universal gas constant; Q is the activation energy ofdeformation. The equation system of continuity reflectingthe conservation of mass and energy of ice sheet can beobtained therefrom.

With Jenssen’s transformation [6,27], the vertical frame-work is converted to r coordinate. Under (x,y,r) coordi-nate system, the ice thickness evolution equationdepicting mass conservation of ice sheet is obtained:

ohot¼ �divðD � gradðsÞÞ þM

where D ¼ hR 1

0cðrÞdr is the diffusion coefficient, in which

cðrÞ ¼ �2ðqgÞnhnþ1jgradðsÞjn�1 R r1

Arndr, h is the ice thick-ness, s = s(x,y, t) is the surface elevation, M is the surfacemass balance, and div is the divergence operator.

The thermodynamic equation of ice temperature can bederived from conservation of energy:

oTot¼ k

qCph2

o2Tor2� < u; gradðT Þ > þ 1

hoTorðw� wgridÞ

þ rgCp

<ouor; gradðsÞ >

where T is the absolute temperature, u is the horizontalvelocity field, w is the vertical velocity, wgrid is the verticalvelocity at the grid point, q is the ice density, Cp is the heatcapacity of ice, Æ æ the Euclidean inner product, and grad isthe gradient operator on (x,y) plane. Therefore, at ice sheetbottom r = 1 and at ice sheet surface r = 0. Obviously, sin-gularity will show up at the edge of ice sheet (h = 0) in thethermodynamic equation.

GLIMMER model adopts (x,y,r) coordinate systemand solves the equation system for ice sheets with finite dif-ference methods. In practical computation, the evolutionequation of ice thickness adopts Crank–Nicolson semi-implicit difference scheme, the diffusion coefficient D

adopts nonlinear Picard iteration; the gradients in theequation of ice temperature distribution are approximatedwith its central difference after expansion through Taylorseries. The model has two sets of staggered grids in hori-zontal direction. Temperature, thickness and vertical veloc-ity are calculated on integer grids. Horizontal ice velocityand gradients are calculated on checkered half-integergrids. The number of grids in horizontal and vertical

174 X. Tang et al. / Progress in Natural Science 18 (2008) 173–180

directions can be set according to actual demand, and theselection of vertical grid spacing under r coordinate systemis irregular (see [27]).

2.2. Test of GLIMMER model under EISMINT-1

benchmark

EISMINT-1 is a simple test suite benchmark designatedby European Ice Sheet Modeling Initiative (EISMINT)that all ice sheet models have to pass. One of its keyassumptions is that the law of ice flow is independent oftemperature field, namely in Arrhenius relation, the factorA = constant A0. This benchmark includes two boundaryconditions: fixed boundary and moving boundary. In orderto simplify the GLIMMER ice sheet model, EISMINT-1fixed boundary condition was adopted to simulate 2-dimensional ice flow state directly.

Surface temperature forcing: Tsur = 239 + 8 · 10�8d3,where d = max{jx � xsummitj,jy � ysummitj} is the maxi-

mum horizontal distance from ice sheet surface to itscenter.

Annual accumulation rate of surface mass: M = 0.3m/yr.

Periodic time-dependent forcing: DT ¼ 10 sin 2ptT ;

DM ¼ 0:2 sin 2ptT , where T = 40000 years, that is a Milan-

kovitch period. The selection of this period is to surveythe evolution of idealized ice sheet in Medio-Pleistoceneof Quaternary.

Boundary condition: ice temperature at upper boundarywas set as surface temperature; ice at lower boundary washeated through geothermal flux and sliding friction,namely: oT

or

��r¼1¼ � Gh

k �Hsbuð1Þ

k , where sb = �qgh Æ grad(s) isthe bottom shear stress, u(1) is the bottom ice velocity.At the bottom, if the melting point of ice T0 is reached(namely T P T0), the ice temperature shall be a constantT* = T0; the excess heat is converted into melting rateS ¼ k

qLoT �

oz � oToz

� �, where L is the latent heat at melt ratio.

According to the Ref. [15], we selected an idealized icesheet setting the initial condition at zero ice thickness,and the integral domain at flat-bottom square ice bed atzero elevation with a size of 1500 km · 1500 km, where(xsummit,ysummit) = (750 km, 750 km), square region is31 · 31, step length is 50 km as the horizontal grids. Thereare 11 vertical layers; the integrating time is 200,000 yearsand the step 10 years; the boundary condition is Dirichlet,namely at boundary (i, j) the ice thickness H(x,y, t) = 0,where i = 1,31 or j = 1,31.

2.2.1. Test results and analysis

Under the above EISMINT-1 condition, when the icesheet evolved to a steady state, the distribution of ice thick-ness showed a symmetrical distribution of axis and center,and the ice thickness does not decrease from boundary tocenter (Fig. 1(a)). Based on symmetry, the distribution ofice thickness can be reflected via a profile passing throughthe center of ice sheet; seen from the profile of ice sheettaken, under steady state, the thickness at uppermost of

ice sheet is stable around 3394 m, and gradient of ice thick-ness steps down from boundary to center. According tocentral symmetry and considering the spatial distributionthat depends on ice sheet state variable, only a part fromice sheet profile boundary to center was needed. Therefore,five reference points on the profile connecting grid (1,16) togrid (16,16) were selected: grid (4, 16), grid (7,16), grid(10,16), grid (13,16) and grid (16,16) (Fig. 1(b)).

By the test results, the behavior on the profile of icesheet ice flow conforms to our basic knowledge on ice sheetevolution: within a comparable range, the temperatureapproximately decreases from boundary to center at sameheight of ice sheet (with ice bottom as zero). Only when thedepth reaches about 1000 m, the variation of temperatureat all places is small, and for any fixed points, the ice tem-perature steps down first and then up with increase ofdepth (Fig. 2(a)). In vertical direction, the ice flow velocitysteps up with increase of height, therefore the vertical flowvelocity on ice sheet surface is the highest (as grid (4, 16) inFig. 2(b)). From the test we assumed that the ice sheet bot-tom velocity was zero, and from the calculation results, weknew that the average vertical ice flow rate and the gradientat the same height were lowered gradually on the profile

Fig. 1. Ice thickness distribution of EISMINT-1 fixed boundary test(50 km per grid). (a) Distribution of ice thickness in steady state ice sheet;(b) distribution of ice thickness of ice flow profile parallel to x-axisthrough ice sheet center.

X. Tang et al. / Progress in Natural Science 18 (2008) 173–180 175

from boundary to center. In addition, at uppermost of icesheet, the vertical velocity was very small (less than 0.3m/yr). Therefore, based on Dansgaard–Johnsen ice flowmodel [6,28], simulating the time scale at ice sheet domewas greatly simplified and reliable (Fig. 2(b)).

The test results also showed that the ice sheet began tobe stable after the model integrated 40000 years, which isa time forcing period of EISMINT-1; meanwhile the far-ther to the ice sheet center, the lower the surface tempera-ture was, and for a fixed place, the surface temperature hada process of falling first and then rising (Fig. 3(a)). Thechange of horizontal velocity with height was similar tothat of vertical velocity, namely increasing with theincrease of height, until to its highest at maximum height(namely surface of ice sheet). A same height, from bound-ary to center of ice sheet, the horizontal velocity steppeddown; while at the center (namely at grid (16,16)), the hor-izontal velocity from ice sheet bottom to the surface wasclose to zero (Fig. 3(b)). These findings allowed us to sim-ulate the relation of age and depth at Dome A region withDansgaard–Johnsen ice flow model. In fact, by Dansg-aard–Johnsen model, the horizontal flow velocity at thesimulated domain had to approach zero and the verticalice velocity was also required to be lower. Here our test

showed that any theoretical dome meets all these condi-tions, while Dome A is a typical dome [29].

3. Simplification of GLIMMER model to the 2-dimensionalice flow model

In the simulation of the ‘‘age-depth-accumulation’’ rela-tion in Dome A region, the input parameters includingmaximum depth and estimate of velocity field at the domewere used, while the current survey on this region could notprovide sufficient information. Therefore, it was necessaryto give its approximation with numerical simulationmethod. In order to get estimate of surface velocity, wegave some relevant assumptions for EISMINT-1 bench-mark under GLIMMER model and simplified it into anice flow model simulating 2-dimensional ice flow state.After coupling temperature field, the thickness, velocityand temperature distributions of a dome profile were sim-ulated under some idealized conditions. Though the simpli-fied equation system became simpler than the GLIMMERmodel, the model still contained all reasonable structuresthat an ice flow thermodynamic model of ice sheet should

Fig. 2. Vertical distribution of temperature and ice velocity of EISMINT-1 fixed boundary test. (a) Vertical distribution of temperature under steadystate; (b) vertical velocity distribution of ice flow. Fig. 3. Vertical distribution of surface temperature and horizontal

velocity under EISMINT-1 fixed boundary condition. (a) Variation oftemperature with time on ice sheet surface; (b) variation of horizontalvelocity of ice sheet profile with height (here U ¼

ffiffiffiffiffiffiffiffiffiffiffihu; ui

pÞ.

176 X. Tang et al. / Progress in Natural Science 18 (2008) 173–180

satisfy [30,31]. Therefore, to optimize the parameterizedscheme and improve the model should be able to simulatethe ‘‘age-depth- accumulation’’ relation of Dome A.

3.1. GLIMMER model is simplified to a 2-dimensional ice

flow model

According to some physical assumptions from Van derVeen [2], Paterson [6] and EISMINT-1 [15], a 2-dimen-sional ice flow thermodynamic coupled model was devel-oped with physical constants listed in Table 1. It wasassumed that: flow is only in horizontal direction (x-direc-tion) and vertical direction (z-direction); ice flow mainlycomes from horizontal shear stress and a relation connect-ing stress and strain rate, namely nonlinear Glen flow law;change of ice temperature comes from heat conduction invertical direction, heat advection in vertical and horizontaldirections and heat dissipation of ice flow; ice is consideredas incompressible; parameters and coefficients in Glen flowlaw are given via Arrhenius relation, i.e., taking it as afunction on temperature; ice thickness is set at zero attwo ends of a defined interval, which is equivalent to theissue of fixed boundary; bottom is assumed to be flat, non-deformative and zero-elevated; at the bottom, the verticalmelt is ignored, and therefore the vertical velocity of bot-tom ice body can be assumed to be zero; the accumulationrate of surface snow is assumed to be zero (ice equivalent);sliding is ignored at ice bottom; the surface temperature ofice is in linear relation with the cube of center distance of itsposition; and the bottom temperature boundary conditionis determined by geothermal flux. By these assumptions,GLIMMER model is simplified into a 2-dimensionalmodel coupling temperature field, and its boundary condi-tions and standardized data set of ice sheet and climate arein accordance with EISMINT-1; the difference is that tem-perature field is coupled in extra and some algorithms dif-ferent from GLIMMER model are adopted.

Let H(x, t), T(x,z, t), u(x,z, t) and w(x,z, t) be distribu-tion functions of depth, temperature, horizontal velocityand vertical velocity of ice sheet profile, respectively.

x 2 [ � L,L] is the defined interval of ice sheet profile inhorizontal direction. z > 0, t P 0, they are vertical coordi-nate and time coordinate, respectively, let uðx; z; tÞ ¼2f ðqgÞnAðT Þ oH

ox

�� ��n�1ðHðx; tÞ � zÞn,

Dðx; tÞ ¼R Hðx;tÞ

0 uðx; n; tÞðhðx; tÞ � nÞdn.

Then the following ice sheet equation system can beobtained by the assumptions:

Ice thickness evolution equation :oHot

¼ M þ o

oxD

oHox� ubH

� �; ð1Þ

Horizontal ice velocity : u

¼ � oHoxðx; tÞ

Z z

0

uðx; n; tÞdnþ ubðx; tÞ; ð2Þ

Vertical velocity : wðx; z; tÞ

¼ o

oxoHotðx; tÞ

Z z

0

uðx; z; tÞðz� nÞdn

� �

� zoub

oxðx; tÞ þ wbðx; tÞ; ð3Þ

Temperature evolution equation :oTotþ u

oToxþ w

oToz

¼ kqCp

o2T

oz2þ gðh� zÞ

Cp

oHox

ouoz

����

����; ð4Þ

Incompressible :ouoxþ ow

oz¼ 0; ð5Þ

Arrhenius relation : A ¼ fA0 exp�QRTþ 3cðT r � T Þj

� �; ð6Þ

Zero elevation at bottom : Hð�L; tÞ¼HðL; tÞ¼ 0;L¼ 750 km;

ð7ÞVertical velocity of bottom : wb ¼ wðx; 0; tÞ ¼ 0; ð8ÞSurface accumulation : M ¼ 0:3 m=yr; ð9ÞHorizontal velocity of bottom : ub ¼ uðx; 0; tÞ ¼ 0; ð10ÞSurface temperature of ice sheet : T ðx; h; tÞ

¼ 239þ ð8� 10�8Þjxj3: ð11Þ

Boundary condition of bottom temperature: If T ðx; 0; tÞ 6eT ðx; 0; tÞ, then oT

oz ðx; 0; tÞ ¼ � Gk . In other cases, T ðx; 0; tÞ ¼

eT ðx; 0; tÞ;

where eT ðx; z; tÞ ¼ T r � bðh� zÞ representing melting

point of ice at depth of z: ð12Þ

3.2. Difference scheme and algorithm of the 2-dimensional ice

flow model

In the equation system (1)–(12), the critical ones are theice thickness evolution Eq. (1) and the ice temperature evo-lution Eq. (4). For Eq. (1), numerical calculation refers tothe Crank–Nicolson finite difference mode [27] of GLIM-MER model, with iteration via semi-implicit difference

Table 1Physical constants in the model

Physical value Physical quantity

q = 910 kg/m3 Ice densityg = 9.81 m/s2 Gravity accelerationR = 8.321 J/molK Gas constantA0 = 2.948 · 10�9/Pa3s Glen law constantTr = 273.39 K Tristate critical value of waterCp = 2009 J/kgK2 Water heat capacityQ = 7.88 · 104 J/mol Creep activation energyb = 8.7 · 10�4 K/m Variable rate of melting point with depthG = 0.042 J/m2s Geothermal fluxj = 1.17 Constant in Arrhenius relationk = 2.10 J/mKs Thermal conductivity of icec = 0.16612 Kj Arrhenius constant

Data come from Refs. [6] and [15].

X. Tang et al. / Progress in Natural Science 18 (2008) 173–180 177

and the diffusion coefficient approximated by central differ-ence. For Eq. (4), in GLIMMER model, singularityappears at the boundary of ice sheet and this problemcan be avoided by directly adopting rectangular coordinatesystem (x,y,z). For discretization in vertical direction, thegrids are not necessary to be with same step size proposedby Paterson [6], and Payne and Dongelmans’s [32] isadopted. The boundary conditions are directly discretizedwith Taylor Series.

3.3. Test results and analysis

The initial conditions of ice flow were determined via theequation proposed by Vialov and Weertman [6]. It is

hðxÞ ¼ 23=8 5M2ðqgÞ3A0

� �1=8

L43 � x

43

� �38

(Fig. 4(a), real line). The

ice sheet profile after evolution of 25000 years is shownas broken line in Fig. 4(a). At the dome, the thickness ishigher than theoretical value, reaching about 3600 m, butas for the whole profile, the gradient of ice thickness isincreasing. In addition, because the theoretical curve ofice sheet is under steady state, the overall deviationbetween the theoretical curve and the simulated one isnot big (Fig. 4(a)).

Three positions (A: x = 0, B: x = 0.5L, C: x = 0.9L)were selected to analyze the vertical temperature of icesheet profile at different places (Fig. 4(a)). The simulationshowed that the vertical distribution of temperaturearound the central zone of ice sheet stepped up withincrease of depth while in other positions the temperatureincreased first and then decreased with increase of depth(Fig. 5). This is the same as the simulation results ofGLIMMER model under EISMINT-1 (Fig. 2(a)). The spa-tial distribution of temperature on the whole ice sheet pro-file (Fig. 4(b)) shows that around the dome, thecharacteristic of surface temperature distribution is rela-tively simple. At the ice sheet boundary, the surface tem-perature distribution is complicated. The internaltemperature distribution of ice sheet has a trend of increas-ing with increase of depth. The simulation results alsoshowed that, around ice dome (x = 0) the horizontal veloc-ity of ice flow was close to 0, and the horizontal and verti-cal velocity (Fig. 6) agreed with the simulation results ofGLIMMER model (Figs. 2,3). This has meaningful signif-icance for verification of the horizontal shear at different

Fig. 4. Ice sheet profile – thickness and temperature distribution. (a)Theoretical thickness distribution of profile and comparison with simu-lation results; (b) temperature distribution of ice sheet profile understeady-state.

Fig. 5. Vertical distribution of temperature of ice sheet profile (to threereference lines in Fig. 4(a): x = 0, x = 0.5L, x = 0.9L).

Fig. 6. Vertical distribution of horizontal ice flow velocity at dome(reference line in Fig. 4(a): x = 0).

178 X. Tang et al. / Progress in Natural Science 18 (2008) 173–180

depths on ice sheet velocity profile. The results of simula-tion showed that after GLIMMER model was simplifiedinto the 2-dimensional ice flow model, practical surfacevelocity and vertical velocity distribution of ice sheet profilecould be simulated so long as model parameters andboundary condition settings were optimized further andfactual observation data were combined.

4. Discussion and conclusion

The present study makes an idealized ice sheet simula-tion test with a 3-dimensional coupled model for Antarcticice sheet – GLIMMER model under EISMINT-1 bench-mark, gets various evolution curves expressing conserva-tion of mass, momentum and energy of ice sheet, andindicates that under fixed boundary condition, the physicalcharacteristic curves when ice sheet is in a steady state arein accordance with our current knowledge on Antarctic icesheet on the whole. In order to consider further simulationof the relation of ‘‘depth-age-accumulation’’ at Dome Aregion, we simplify the GLIMMER model into a 2-dimen-sional ice flow model with coupled temperature field andtest it under some brief assumptions. The change of eleva-tion, temperature field and velocity field of ice flow profilein evolution steady state of ice sheet are obtained quantita-tively. The results are similar to the calculation results ofGLIMMER model.

Meanwhile, when solving the equations of the model,our study also provides improvement on the original 3-dimensional model difference scheme, taking into consider-ation the stability of simulation calculation. Such a methodallows the vertical axis to discretize more freely and cancompare the effect of step size on simulation results at dif-ferent time. This helps to analyze the change details of theevolution process of ice sheet better.

Of course, the case treated here is still an idealized state.It is not sufficient to draw the evolution details of a practi-cal region (such as Dome A) yet, because the model lacks aparameter optimization process for a specific region. Inaddition, the choice of initial and boundary conditions isalso an important issue. Nevertheless, the simplified modelcan already simulate directly ice thickness, temperaturefield, velocity field and various gradients. The results showthat the 2-dimensional ice flow model can accurately simu-late the ice deformation around ice dome and the fluiddeformation characteristics dependent on temperature. Byusing the model, we try to simulate the relation of‘‘depth-age-accumulation’’ at Dome A region so as tomake some quantitative analysis on mass balance of Ant-arctic ice sheet and its effect on sea level change.

Acknowledgements

This work was jointly supported by National NaturalScience Foundation of China (Grant Nos. 40476005 and40233032), the Ministry of Science and Technology,China (Grant No. 2005DIB3J114), Polar Science Youth

innovational Foundation, PRIC (Grant No. JDQ200602),China National Bureau of Oceanography Youth ScienceFoundation (Grant No. 2007219).

References

[1] Warrick RA, Provost CL, Meier MF, et al. Changes in sealevel. Cambridge and New York: Cambridge University Press;1996 , [p. 359–405].

[2] Van der Veen CJ, Whillans IM. Force budget: (I) theory andnumerical methods. J Glaciol 1989;35:53–60.

[3] Huybrechts P, Oerlemans J. Evolution of the East Antarctic ice sheet:a numerical study of thermo-mechanical response patterns withchanging climate. Ann Glaciol 1988;11:52–9.

[4] Orerlemans J, Jonker PJ. The heat budget of the Antarctic ice sheet.Zeitscher Gletsch Glazialgeol 1985;21:291–9.

[5] Nye JF. Correction factor for accumulation measured by thethickness of the annual layers in an ice sheet. J Glaciol 1963;4:785–8.

[6] Paterson WSB. The physics of glaciers. 3rd ed. Oxford: PergamonPress/Elsevier Ltd; 1994, 15,243:275–9.

[7] Dansggard WSB, Johnsen SJ. A flow model and a time scale for theice core from Camp Century, Greenland. J Glaciol1969;8(53):215–23.

[8] Oerlemans J. A model of the Antarctic ice sheet. Nature1982;297(5967):550–3.

[9] Oerlemans J. Response of the Antarctic ice sheet to a climatewarming: a model study. J Climatol 1982;2:1–11.

[10] Budd WF, Smith IN. Large-scale numerical modelling of theAntarctic ice sheet. Ann Glaciol 1982;3:42–9.

[11] Budd WF, Jenssen D, Radok U. Derived physical characteristics ofthe Antarctic ice sheet. ANARE Interim Report, Series A(IV).University of Melbourne, Melbourne: Meteorology DepartmentPublication; 1971, p. 221–5.

[12] Huybrechts P, De Wolde J. The dynamics response of the Greenlandand Antarctic ice sheets to multiple-century climatic warming. JClimate 1999;12(8):2169–88.

[13] Ritz C, Rommelaere V, Dumas C. Modeling the evolution of theAntarctic ice sheet over the last 420000 years: implications foraltitude changes in the Vostok region. J Geophys Res2001;106(D23):943–64.

[14] Budd WF, Coutts B, Warner RC. Modelling the Antarctic andnorthern-hemisphere ice-sheet changes with global climate throughthe glacial cycle. Ann Glaciol 1998;27:153–60.

[15] Huybrechts P, Payne AT. The EISMINT benchmarks for testing ice-sheet models. Ann Glaciol 1996;23:1–12.

[16] Van Der Veen CJ. Fundamentals of glacier dynamics. Rotter-dam: A.A. Balkema; 1999, [163: 215–33, 261–99].

[17] Huybrechts P, LeMeur E. Predicted present-day evolution patterns ofice thickness and bedrock elevation over Greenland and Antarctica.Polar Res 1999;18(2):299–308.

[18] Huybrechts P. A 3-D model for the Antarctic ice sheet: a sensitivitystudy on the glacial-interglacial contrast. Climate Dyn 1990;5:79–92.

[19] Huybrechts P. Formation and disintegration of the Antarctic icesheet. Ann Glaciol 1994;20:336–40.

[20] MacAyeal DR, Rommelaere V, Huybrechts P, et al. An ice-shelfmodel test based on the Ross ice shelf. Ann Glaciol 1996;23:46–51.

[21] Payne AJ, Huybrechts P, Abe-Ouchi A, et al. Results from theEISMINT model intercomparison: the effects of thermomechanicalcoupling. J Glaciol 2000;46(153):227–38.

[22] Payne AJ, Baldwin DJ. Thermo-mechanical modelling of theScandinavian ice sheet: implications for ice-stream formation. AnnGlaciol 1999;28:83–9.

[23] Payne AJ. Dynamics of the Siple Coast ice streams, West Antarctica:results from a thermomechanical ice sheet model. Geophys Res Lett1998;25:3173–6.

X. Tang et al. / Progress in Natural Science 18 (2008) 173–180 179

[24] Payne AJ. A thermomechanical model of ice flow in West Antarctica.Climate Dyn 1999;15:115–25.

[25] Payne AJ, Sugden DE, Clapperton CM. Modeling the growth anddecay of the Antarctic Peninsula ice sheet. Q Res 1989;31:119–34.

[26] Hagdorn M. Basically glimmer-test runs GLIMMER using EIS-MINT 1&2 Climate. <http://forge.nesc.ac.uk/pipermail/glimmer-dis-cuss/2004-November/000060.html> [accessed 09.08.06].

[27] Hagdorn M, Rutt I, Payne AJ. Glimmer 0.5.6 Documentation.<http://forge.nesc.ac.uk/projects/glimmer> [accessed 19.12.05].

[28] Siegret M, Hindmarsh R, Hamilton G. Evidence for a large surfaceablation zone in central East Antarctica during the last ice age. Q Res2003;59:114–21.

[29] Qin DH, Ren JW, Kang JC, et al. Primary results ofglaciological studies along an 1100 km transect from Zhongshanstation to Dome A, East Antarctic ice sheet. Ann Glaciol 2000;31:198–204.

[30] Greve R. A continuum-mechanical formulation for shallowpolythermal ice sheets. Philos Trans R Soc 1997;355:921–74.

[31] Hutter K. Thermomechanically coupled ice-sheet response: cold,polythermal, temperature. J Glaciol 1993;39(131):65–86.

[32] Payne AJ, Dongelmans PD. Self-organization in the thermome-chanical flow of ice sheets. J Geophys Res 1997;102(B6):12219–33.

180 X. Tang et al. / Progress in Natural Science 18 (2008) 173–180