• Abe-Ouchi, A., 1993: Ice sheet response to climatic changes: A modelling approach. Zürcher Geogr. Schr.,54, 134 pp.

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  • ——, and K. Hutter, 1991: Polythermal conditions in arctic glaciers. J. Glaciol.,37, 261–269.

  • Braithwaite, R. J., and O. B. Olesen, 1989: Calculation of glacier ablation from air temperature, West Greenland. Glacier Fluctuations and Climatic Change, J. Oerlemans, Ed., Kluwer, 219–233.

  • Budd, W. F., and I. N. Smith, 1982: Large-scale numerical modelling of the Antarctic ice sheet. Ann. Glaciol.,3, 42–49.

  • Calov, R., 1994: Das thermomechanische Verhalten des grönländischen Eisschildes unter der Wirkung verschiedener Klimaszenarien—Antworten eines theoretisch-numerischen Modells. Ph.D. dissertation, Institut für Mechanik, Technische Hochschule Darmstadt, 171 pp. [Available from Institut für Mechanik, Technische Hochschule Dannstadt, Hochschulstr. 1, D-64289 Dannstadt, Germany.].

  • ——, and K. Hutter, 1996: The thermomechanical response of the Greenland ice sheet to various climate scenarios. Climate Dyn.,12, 243–260.

  • Fabré, A., A. Letréguilly, C. Ritz, and A. Mangeney, 1995: Greenland under changing climates: Sensitivity experiments with a new three-dimensional ice sheet model. Ann. Glaciol.,21, 1–7.

  • Fastook, J. L., and J. E. Chapman, 1989: A map-plane finite-element model: Three modeling experiments. J. Glaciol.,35, 48–52.

  • Fowler, A. C., and D. A. Larson, 1978: On the flow of polythermal glaciers, Part I, Model and preliminary analysis. Proc. Roy. Soc. London, Ser. A,363, 217–242.

  • Funk, M., K. Echelmeyer, and A. Iken, 1994: Mechanisms of fast flow in Jacobshavns Isbræ, West Greenland. Part II: Modeling of englacial temperatures. J. Glaciol.,40, 569–585.

  • Greve, R., 1995: Thermomechanisches Verhalten polythermer Eisschilde—Theorie, Analytik, Numerik. Ph.D. dissertation, Institut für Mechanik, Technische Hochschule Dannstadt, 226 pp. [Available from Shaker Verlag, Postfach 1290, D-52013 Aachen, Germany.].

  • ——, 1997: A continuum-mechanical formulation for shallow polythermal ice sheets. Philos. Trans. Roy. Soc. London, Ser. A, in press.

  • ——, and K. Hutter, 1995: Polythermal three-dimensional modelling of the Greenland Ice Sheet with varied geothermal heat flux. Ann. Glaciol.,21, 8–12.

  • Herterich, K., 1988: A three-dimensional model of the Antarctic Ice Sheet. Ann. Glaciol.,11, 32–35.

  • Hutter, K., 1982: A mathematical model of polythermal glaciers and ice sheets. J. Geophys. Astrophys. Fluid Dyn.,21, 201–224.

  • ——, 1983: Theoretical Glaciology; Material Science of Ice and the Mechanics of Glaciers and Ice Sheets. D. Reidel, 510 pp.

  • ——, 1993: Thermo-mechanically coupled ice-sheet response. Cold, polythermal, temperate. J. Glaciol.,39, 65–86.

  • ——, H. Blatter, and M. Funk, 1988: A model computation of moisture content in polythermal glaciers. J. Geophys. Res.,93 (B10), 12 205–12 214.

  • Huybrechts, P., 1992: The Antarctic Ice Sheet and environmental change: A three-dimensional modelling study. Ber. Polarforschung,99, 241 pp.

  • ——, 1994: The present evolution of the Greenland ice sheet: An assessment by modelling. Global Planet. Change,9, 39–51.

  • ——, A. Letréguilly, and N. Reeh, 1991: The Greenland Ice Sheet and greenhouse warming. Palaeogeogr., Palaeoclimatol., Palaeoecol.,89, 399–412.

  • Jenssen, D., 1977: A three-dimensional polar ice-sheet model. J. Glaciol.,18, 373–389.

  • Johnsen, S. J., and Coauthors, 1992: Irregular glacial interstadials recorded in a new Greenland ice core. Nature,359, 311–313.

  • Lee, W. H. K., 1970: On the global variations of terrestrial heat flow. Phys. Earth Planet. Inter.,2, 332–341.

  • Letréguilly, A., P. Huybrechts, and N. Reeh, 1991a: Steady-state characteristics of the Greenland Ice Sheet under different climates. J. Glaciol.,37, 149–157.

  • ——, N. Reeh, and P. Huybrechts, 1991b: The Greenland Ice Sheet through the last glacial–interglacial cycle. Palaeogeogr., Palaeoclimatol., Palaeoecol.,90, 385–394.

  • Lliboutry, L., and P. Duval, 1985: Various isotropic and anisotropic ices found in glaciers and polar ice caps and their corresponding rheologies. Ann. Geophys.,3, 207–224.

  • Mahaffy, M. W., 1976: A three-dimensional numerical model of ice sheets: Test on the Barnes ice cap, Northwest Territories. J. Geophys. Res.,81 (6), 1059–1066.

  • Morland, L. W., 1984: Thermo-mechanical balances of ice sheet flows. J. Geophys. Astrophys. Fluid Dyn.,29, 237–266.

  • Müller, I., 1985: Thermodynamics. Pitman Advanced Publishing Program, 521 pp.

  • Oerlemans, J., 1982: Response of the Antarctic Ice Sheet to a climatic warming: A model study. J. Climatol.,2, 1–11.

  • Ohmura, A., 1987: New temperature distribution maps for Greenland. Z. Gletscherkd. Glazialgeol.,23, 1–45.

  • ——, and N. Reeh, 1991: New precipitation and accumulation maps for Greenland. J. Glaciol.,37, 140–148.

  • Paterson, W. S. B., 1991: Why ice-age ice is sometimes “soft.” Cold Reg. Sci. Technol.,20, 75–98.

  • ——, 1994: The Physics of Glaciers. 3d ed. Pergamon Press, 480 pp.

  • Reeh, N., 1991: Parameterization of melt rate and surface temperature on the Greenland Ice Sheet. Polarforschung,59, 113–128.

  • Schönwiese, C.-D., 1992: Klima im Wandel. Tatsachen, Irrtümer, Risiken. Deutsche Verlags-Anstalt, 223 pp.

  • Svendsen, B., and K. Hutter, 1996: A continuum approach for modelling induced anisotropy in glaciers and ice sheets. Ann. Glaciol.,23, 262–269.

  • ——, and ——, 1997: A continuum model for induced anisotropy in polycrystals. Quart. J. Mech. Appl. Math., in press.

  • View in gallery

    Measured surface topography h of the present Greenland Ice Sheet (Letréguilly et al. 1991; in km MSL). The spacing between the isolines is 200 m. The dashed heavy line indicates the ice margin.

  • View in gallery

    Final state of run 1: topography of the ice surface (in km MSL). The spacing between the isolines is 200 m. The dashed heavy line indicates the ice margin.

  • View in gallery

    Final state of run 1: horizontal mass flux. Double arrow length corresponds to tenfold mass flux. The dashed heavy line indicates the ice margin, the isolines represent the surface topography in 500 m intervals.

  • View in gallery

    Final state of run 1: homologous temperature at the ice base (in °C). The spacing between the isolines is 3°C. Open diamond symbols indicate positions where the basal ice is at the pressure melting point, yet with no temperate layer above; full diamonds (full circles, see → in the plot) indicate positions where there is a basal layer of temperate ice with a melting (freezing) CTS. The dashed heavy line indicates the ice margin.

  • View in gallery

    Final state of run 1: transect at y = −2280 km. Top: ice velocity; middle: homologous ice temperature (in °C); bottom: thickness of the basal layer of temperate ice [open circles: cold ice base; open diamonds: temperate ice base with no temperate ice layer above; full diamonds: basal layer of temperate ice with a melting CTS; full circles (see → in the plot): basal layer of temperate ice with a freezing CTS].

  • View in gallery

    Run 1: time evolution of ΔTma, hmax, Vtot, Hmax, Vtemp, Ht,max, Ai,b, and At,b.

  • View in gallery

    Same as Fig. 2, but for the final state of run 10.

  • View in gallery

    Same as Fig. 4, but for the final state of run 10.

  • View in gallery

    Same as Fig. 6, but for run 11.

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    Same as Fig. 6, but for run 12.

  • View in gallery

    Same as Fig. 6, but for run 13.

  • View in gallery

    Same as Fig. 6, but for run 13. A 4-kyr time window around t =150 kyr is zoomed out.

  • View in gallery

    Air temperature scenario for the last 250 kyr, deduced from the GRIP data. Top: unfiltered data. Bottom: filtered data (2-kyr averages). The assumed time limits for the Eemian interglacial (−132 kyr; −114.5 kyr) and the Holocene interglacial (−11 kyr; 0 kyr) are indicated by the dashed lines.

  • View in gallery

    Same as Fig. 6, but for runs 14 (dashed) and 15 (solid).

  • View in gallery

    Same as Fig. 2, but for run 15. (a) t = −127 kyr (EIVM), (b) t = −18 kyr (LGM), and (c) t = 0 kyr (today).

  • View in gallery

    Same as Fig. 4, but for run 15. (a) t = −127 kyr (EIVM), (b) t = −18 kyr (LGM), and (c) t = 0 kyr (today).

  • View in gallery

    Same as Fig. 5, but for run 15. (a) t = −127 kyr (EIVM), (b) t = −18 kyr (LGM), and (c) t = 0 kyr (today).

  • View in gallery

    Same as Fig. 6, but for runs 16 (dotted), 17 (dashed), and 18 (solid).

  • View in gallery

    Same as Fig. 2, but for run 18, t = 5 kyr.

  • View in gallery

    Same as Fig. 4, but for run 18, t = 5 kyr.

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Application of a Polythermal Three-Dimensional Ice Sheet Model to the Greenland Ice Sheet: Response to Steady-State and Transient Climate Scenarios

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  • 1 Department of Mechanics, Technische Hochschule Darmstadt, Darmstadt, Germany
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Abstract

Steady-state and transient climate-change computations are performed with the author’s three-dimensional polythermal ice sheet model Simulation Code for Polythermal Ice Sheets for the Greenland Ice Sheet. The distinctive feature of this model is the detailed consideration of the basal temperate ice layer, in which the water content and its impact on the ice viscosity are computed; its transition surface to the cold ice region is accounted for by continuum-mechanical jump conditions on this interface. The simulations presented include steady states subject to a range of physical parameters and two different climates (present and glacial conditions), as well as three types of transient scenarios, namely (i) sinusoidal Milanković-period forcing, (ii) paleoclimatic forcing from the Greenland Ice Core Project core reconstruction, and (iii) future greenhouse warming forcing.

Corresponding author address: Dr. Ralf Greve, Department of Mechanics, Technische Hochschule Darmstadt, D-64289 Darmstadt, Germany.

Email: greve@mechanik.th-darmstadt.de

Abstract

Steady-state and transient climate-change computations are performed with the author’s three-dimensional polythermal ice sheet model Simulation Code for Polythermal Ice Sheets for the Greenland Ice Sheet. The distinctive feature of this model is the detailed consideration of the basal temperate ice layer, in which the water content and its impact on the ice viscosity are computed; its transition surface to the cold ice region is accounted for by continuum-mechanical jump conditions on this interface. The simulations presented include steady states subject to a range of physical parameters and two different climates (present and glacial conditions), as well as three types of transient scenarios, namely (i) sinusoidal Milanković-period forcing, (ii) paleoclimatic forcing from the Greenland Ice Core Project core reconstruction, and (iii) future greenhouse warming forcing.

Corresponding author address: Dr. Ralf Greve, Department of Mechanics, Technische Hochschule Darmstadt, D-64289 Darmstadt, Germany.

Email: greve@mechanik.th-darmstadt.de

1. Introduction

The earth’s cryosphere consists of several components. Ice sheets are extended ice masses with a base resting on solid land; they have formed by accumulated snowfall in the course of the millennia. Ice stored in the present ice sheets (essentially Antarctica and Greenland) represents by far the biggest part of today’s ice volume on the earth; this paper is concerned with a new modeling approach to the Greenland Ice Sheet. Alpine glaciers develop in the same way; however, they cover alpine regions of much less extent, typically valleys, and therefore their contribution to the cryosphere is small. Ice shelves are floating ice masses that are fed from the seaward mass flux of an ice sheet; they exist typically in large bays of an ice-covered continental shield and are therefore insignificant for the Greenland Ice Sheet. Sea ice is superficially frozen sea water, and soil ice is frozen water in the ground as it occurs in permafrost regions.

Ice appearing in glaciers and ice sheets exists in two fundamentally different states (when additional tracers of salt and sediment are neglected). Cold ice is characterized by temperatures below the pressure melting point and can be described as a density-preserving, viscous and heat-conducting one-component fluid; the large ice sheets on the earth consist mainly of this type of ice. However, the temperature of temperate ice is exactly at the pressure melting point, so that this ice may have incorporated small quantities of water. As opposed to cold ice, it must therefore be regarded at least as a two-component fluid. In ice sheets, regions of temperate ice may exist in thin, near-basal layers, with significant consequences on the flow behavior. Glaciers and ice sheets that are made up of cold as well as temperate regions are referred to as polythermal (e.g., Paterson 1994).

In the past 20 yr several models for the numerical simulation of ice sheets were developed and, with the high speed performance of modern computers, became economically feasible. An early, vertically integrated model due to Mahaffy (1976) was applied to the Barnes Ice Cap in the Canadian Arctic. The first genuinely three-dimensional model, due to Jenssen (1977), was used to model the Greenland Ice Sheet, albeit with a very low spatial resolution because of the limited computer capacities at that time. Numerous models of increasing sophistication followed, with applications to the Greenland and the Antarctic Ice Sheets, the glacial Laurentide Ice Sheet, the hypothetic glacial Tibetian Ice Sheet, and so on (Budd and Smith 1982; Oerlemans 1982; Herterich 1988; Fastook and Chapman 1989; Huybrechts 1992; Abe-Ouchi 1993; Calov 1994; Fabré et al. 1995; Calov and Hutter 1996). Especially remarkable are the simulations of the Antarctic Ice Sheet carried out by Huybrechts (e.g., Huybrechts 1992), where the coupled ice sheet–ice shelf–lithosphere problem is modeled with high spatial resolution.

All these models have in common that they either neglect the possible presence of temperate ice regions or account for them by a posteriori resetting simulated temperatures that exceed the pressure melting point to the pressure melting point (this is referred to as “cold-ice method” in the following). However, this is an oversimplified approach, because it does not account for the fact that with the cold and temperate ice regions two different phases are present, separated by a phase transition surface (the cold-temperate transition surface or CTS) for which jump conditions of mass, momentum, and energy must be fulfilled (Hutter 1983; Müller 1985). Moreover, the cold-ice method does not allow for calculation of the water content in temperate ice that has a pronounced influence on the ice viscosity (Lliboutry and Duval 1985).

In this study, the polythermal ice sheet model SICOPOLIS (Simulation Code for Polythermal Ice Sheets), designed by Greve (1995), is applied to the Greenland Ice Sheet, and a variety of different steady-state and transient climate change simulations will be discussed. This model is based on the continuum–mechanical theory of polythermal ice (Fowler and Larson 1978; Hutter 1982, 1993; Greve 1995, 1997). Its uniqueness is the determination of the CTS position by accounting for and satisfying the several jump conditions at the CTS. It entails accomodation of unsteady temperature gradients and water contents at the CTS, as well as the calculation of the water content in the temperate ice region and its influence on the ice viscosity. Polythermal models were already applied to parallel-sided slab problems and two-dimensional stationary glaciers (Hutter et al. 1988; Blatter 1991; Blatter and Hutter 1991; Funk et al. 1994); here, the polythermal calculation is carried out for the full time-dependent problem in three dimensions. In detail, SICOPOLIS computes the time evolution of the extent and thickness of the ice sheet as a whole as well as of the basal temperate ice layer; the velocity and temperature fields in the cold ice regions; the water content in the temperate ice regions and the age of the ice (the time since its accumulation on the ice surface). The model is driven by external forcing given as (i) surface temperature, (ii) surface accumulation–ablation function, and (iii) geothermal heat flux from below.

2. Polythermal ice sheet model SICOPOLIS

SICOPOLIS is a code that solves the polythermal ice sheet equations as described by Greve (1995, 1997). The complete set of field equations, boundary, and transition conditions will not be repeated here, but the main prognostic equations are listed below. The model equations are subjected to the shallow ice approximation (Hutter 1983; Morland 1984); that is, they are scaled with respect to the aspect ratio ϵ (ratio of typical thickness to typical length) and only lowest-order terms are kept.

Ice thickness:
i1520-0442-10-5-901-e1
where x, y are the horizontal Cartesian coordinates; z is the vertical Cartesian coordinate [elevation above mean sea level (MSL)]; t the time; h the z-coordinate of the ice surface; b the z-coordinate of the ice base or lithosphere surface, respectively; H is the ice thickness; qx, qy are the components of the horizontal mass flux; as is the accumulation–ablation function at the ice surface; Pwb is the basal melting rate; and ρ the density of ice.
Lithosphere response to ice load:
i1520-0442-10-5-901-e2
where τV is the time lag for the lithosphere response; b0 the position b for the relaxed lithosphere surface without ice load; and ρa the density of the asthenosphere.
Temperature (cold ice regions):
i1520-0442-10-5-901-e3
where T is the temperature; vx, vy, and vz are the components of the ice velocity; c is the specific heat of ice; κ the heat conductivity of ice; E the creep enhancement factor; A(T′) the creep rate factor for cold ice, dependent on the homologous temperature T′ = TTmelt; and f(σ) the creep function for cold ice, dependent on the effective shear stress σ.
Water content (temperate ice regions):
i1520-0442-10-5-901-e4
where ω is the water content, that is, the mass fraction of water in the mixture; L the latent heat of ice; At(ω) the creep rate factor for temperate ice; ft(σ) the creep function for temperate ice; D(ω) the water drainage function; and CCC the Clausius–Clapeyron correction, a small term not given explicitly (see Greve 1995, 1997).
Age of the ice (cold and temperate ice regions):
i1520-0442-10-5-901-e5
where A is the age of the ice and DA is the numerical diffusivity that is needed for reasons of numerical stability.
Temperature (lithosphere):
i1520-0442-10-5-901-e6
where κr is the heat conductivity of the lithosphere; ρr the density of the lithosphere; and cr the specific heat of the lithosphere.

The numerical solution is based on a finite-difference integration technique, for which the σ-transformation, mapping of vertical columns onto the [0, 1] interval, is carried out. In so doing, the three model domains, cold ice, temperate ice, and lithosphere, are treated separately in order to simplify the implementation of the boundary and transition conditions at the interfaces. For the integration of the ice thickness equation (1), a modified ADI (alternating direction implicit) scheme is used, in which one of the two horizontal directions is implicitly discretized in an alternating fashion. Equations (3)–(6) are solved by applying implicit discretization of the vertical derivatives and explicit discretization of the horizontal derivatives; the horizontal advection terms are stabilized by upwind differencing. The choice of these procedures has the advantage that they are more stable than purely explicit schemes, and only tridiagonal systems of linear equations must be solved. Because of the different ranges of stability, two different time steps are used: a smaller value Δt for the solution of Eqs. (1) and (2), and a larger value Δ̃ for the solution of Eqs. (3)–(6).

If for any ice column the presence of an internal CTS, that is, a basal temperate ice layer of nonzero thickness, is predicted by the simulation, the position of the CTS is determined by an iterative trial-and-error procedure. In this case the temperature problem for the upper cold-ice region is subjected to three boundary conditions, a Dirichlet condition at the free ice surface and a Dirichlet as well as a Neumann condition at the CTS. Since one boundary condition at the CTS suffices to solve the parabolic temperature Eq. (3) for cold ice, the remaining condition can be used to determine the unknown CTS position. Therefore, the procedure is to calculate repeatedly the temperature field for the affected column by applying only the Neumann condition at the CTS (derived from the energy jump relation), and shifting the CTS in the vertical until the Dirichlet condition (temperature continuity, i.e., the CTS temperature must equal the pressure melting point) is equally fulfilled. This is discussed in detail by Greve (1995).

3. Modeling the Greenland Ice Sheet

a. Present state of the Greenland Ice Sheet

For all simulations presented below, the position of the relaxed lithosphere surface without ice load, b0, needs to be known in advance. This quantity can be calculated approximately under the assumption that the present lithosphere surface btoday is close to the equilibrium with the present ice load Htoday = htodaybtoday,
i1520-0442-10-5-901-e7
Here, btoday and htoday are based on data of Letréguilly et al. (1991a) that were interpolated on a 40-km grid by Calov (1994). The present surface topography htoday is shown in Fig. 1.

The present climate conditions are imposed on the ice sheet as a mean annual air temperature immediately above the ice Tma and as an accumulation–ablation function as (difference of snowfall rate S and surface melting rate M). As for Ttodayma, we use the parameterization given by Reeh (1991) that is based on measurements of Ohmura (1987). Ohmura and Reeh (1991) also published data on the snowfall rate Stoday; Calov’s (1994) interpolation of those data on a 40-km grid is applied for our study. The melting rate M is parameterized with a degree-day model (Braithwaite and Olesen 1989; Reeh 1991) as implemented by Calov (1994; see also Calov and Hutter 1996).

The remaining boundary condition at the lower boundary of the lithosphere layer is the prescription of the geothermal heat flux; we use the standard value for precambrian shields, Qgeoth = 42 m Wm−2 (Lee 1970).

b. Standard boundary conditions for varying climate scenarios

In order to model the Greenland Ice Sheet under climate conditions different from those at the present, specifications as to how this input varies must also be given. For the mean annual air temperature Tma, we assume that the present spatial distribution is preserved; thus, the modification is expressed by a purely time-dependent term. This yields an expression for Tma as a function of geographic latitude ϕ, elevation above sea level z, and time t:
i1520-0442-10-5-901-e8
For the snowfall rate S, we assume a spatially uniform reduction of S by 50% when ΔTma = −10°C, corresponding to the climate minimum; furthermore, we postulate a linear connection between S and ΔTma, yielding
i1520-0442-10-5-901-e9
Here, S is expressed as a function of geographic latitude ϕ, geographic longitude λ, and time t. The connection between the coordinates on the earth surface ϕ, λ and the model plane x, y is provided by a stereographic projection.

All other parameters are assumed to have the same functional form as for the present state. With these specifications any climate scenario is defined by prescribing the purely time-dependent function ΔTma(t).

c. Specification of physical quantities

Here, the physical quantities used for the simulations are given. These are the following in detail.

Creep law for cold and temperate ice: Glen’s flow law (Paterson 1994),
fσftσσn−1n
Rate factor for cold ice: Arrhenius law (Paterson 1994),
ATA0eQ/R(T0+T′)
with T0 = 273.15 K, the activation energies
i1520-0442-10-5-901-e12
and the universal gas constant R. The two temperature regimes are connected by choosing
AT−25−1−3
this determines the coefficients A0 for T′ < −10°C and T′ ≥ −10°C, respectively.
Rate factor for temperate ice: According to Lliboutry and Duval (1985),
AtωATω
is a suitable form for this.
No-slip condition for the cold ice base: This requires
vsl0
for the basal sliding velocity vsl.
Sliding law for the temperate ice base: This is chosen as a Weertman-type power law:
i1520-0442-10-5-901-e16
where Csl is the sliding coefficient, t the basal shear traction within the basal plane, g the gravity acceleration, and ρgH the overburden pressure. According to Calov (1994) and Calov and Hutter (1996), Eq. (16) implies
vslCslHh2h.
These authors obtained Csl = 6 × 104 a−1 as a result of flow optimization for present climate conditions, a value we adopt here.
Water drainage function: In Greve (1995) the introduction of this quantity is discussed in detail. A reasonably realistic representation was found to be
i1520-0442-10-5-901-e18
This is a simple, one-parameter form for D(ω); ωmax plays the role of a threshold value for the water content in temperate ice, any water surplus is assumed to be instantaneously drained into the ground. Further quantities are listed in Table 1.

4. Discussion of the steady-state simulations

The simulations discussed below were conducted with a horizontal resolution of Δx = Δy = 40 km. With the stereographic projection, Greenland lies within the domain x (−720 km, 920 km), y (−3400 km, −600 km), resulting in 42 × 71 grid points. The vertical resolution is 51 grid points in the cold ice region and 11 grid points in the temperate ice region as well as in the lithosphere. The time steps are Δt = 10 a and Δ̃ = 100 a. Simultaneous calculation of the lithosphere temperature is switched off, and the geothermal heat flow is imposed at the ice base in order to shorten the time required for reaching steady state, which is not influenced by the thermal state of the lithosphere.

Apart from the parameter studies presented in this section, Greve and Hutter (1995) already discussed the impact of a variation of Qgeoth by ±30%, covering approximately the range of uncertainty of that quantity (Lee 1970).

a. Steady-state control run

As a first application of SICOPOLIS to the Greenland Ice Sheet, run 1 is discussed now. This is a steady-state run under present climate conditions [ΔTma(t) ≡ 0°C]; the simulation covers 100 ka, an integration time regarded as sufficient to reach stationary conditions. As initial conditions for the topography the present values btoday and htoday are used, the initial temperature is uniformly set equal to −10°C, and for the initial age of the ice the value 15 kyr is used.

As for the creep enhancement factor E [see Eqs. (3) and (4)], it is taken into account that glacial ice is less viscous than interglacial ice. The exact cause for this difference is unclear, but it is argued that dust content and/or induced anisotropies contribute to this phenomenon (cf. Paterson 1991, 1994; Svendsen and Hutter 1996, 1997). Hence, the enhancement factor E is coupled to the age A of the ice. With the transition between Wisconsin ice age and Holocene interglacial set at 11 kyr BP, we choose
i1520-0442-10-5-901-e19
Possible variations of the fluidity of Eemian or even older ice are not considered here because this ice can only exist in very thin layers near the ice base, where reliable calculation of the age is not ensured, as it is necessary to introduce a small amount of numerical diffusion in solving the purely advective age equation (5) (Huybrechts 1994).

The results of run 1 are depicted in Figs. 2–6. When comparing the present surface topography obtained from data (Fig. 1) with that modeled (Fig. 2), in general good agreement both for the surface topography itself and for the ice margin is observed; the basic features are thus reproduced by the simulation. However, the north dome is 124 m too high (3371 instead of 3247 m); the south dome is 51 m too high (2960 instead of 2909 m); and their positions are slightly shifted when compared with those of the measurements. Furthermore, the simulated ice margin tends to be more advanced toward the coast; especially close to the north and east coast run 1 produces small ice tongues into actually ice-free land. Therefore, with 1.725 × 106 km2 the simulated ice-covered area Ai,b is slightly larger than in reality, for which it is 1.682 × 106 km2.

Figure 3 shows the mass flux (vertically integrated horizontal velocity). It follows the direction of steepest surface slope, in agreement with the findings of the shallow ice approximation. Moreover, the flux away from the north dome follows pronounced drainage areas, which are separated by regions with distinctly reduced ice flow. By contrast, the drainage of the south dome appears to be more regular. This difference is caused by the distribution of the basal temperate ice regions (Fig. 4), on which the ice is allowed to slide. They surround the south dome as an almost continuous band; north of the Arctic circle, however, basically four temperate patches exist that form the drainage regions. This behavior demonstrates impressively the importance of temperate ice for the dynamic behavior of ice sheets.

It is further noticed that of all basal grid points overlain by a temperate ice layer of nonzero thickness only one corresponds to a CTS with freezing conditions (Fig. 4, marked by an arrow). This is because, typically, the thickness of the basal temperate ice layer gradually increases from the interior ice sheet region downstream (toward the margin), and then decreases sharply in the immediate vicinity of the margin. This can clearly be seen for West Greenland in Fig. 5, which shows a west–east transect for the ice sheet at y = −2280 km. Because of the rapid decrease of the ice thickness, freezing conditions at the CTS that are expected close to the margin cannot be resolved in general. Incidentally, the relatively thick predicted temperate ice layer that is paired with large ice velocities in West Greenland in this transect coincides with the presence of a very fast ice stream in this area (“Jacobshavns Isbræ”). Funk et al. (1994), by applying a polythermal 2D streamline model, equally obtain a temperate ice layer for this ice stream.

Figure 6 depicts the temporal evolution for the temperature forcing ΔTma, the maximum ice elevation MSL hmax (taken at the north dome), the total ice volume Vtot, the maximum ice thickness Hmax, the temperate ice volume Vtemp, the maximum thickness of the temperate ice layer Ht,max, the ice covered basal area Ai,b, as well as the basal area covered by temperate ice At,b. A striking feature is the early occurrence of strong peaks for those quantities that are connected with temperate ice, Vtemp, Ht,max, and At,b; they coincide with troughs of the topography quantities hmax, Vtot, and Hmax. This behavior is apparently an effect of the arbitrarily chosen initial conditions (present topography and isothermal temperature distribution) and thus does not have any counterpart in the actual history of the Greenland Ice Sheet.

In order to check the sensitivity of the modeled steady state to such initial conditions, run 1 was recomputed under the assumption of an ice-free initial state with relaxed lithosphere (b = b0). The difference of the final states after 100 ka of model time is very small for all above investigated quantities (Greve 1995). This result is also interesting insofar as it demonstrates that the Greenland Ice Sheet is not a mere relict of the last (Wisconsin) ice age, but builds up equally under present climate conditions (as specified by us).

b. Cold-ice-method simulation

In order to investigate to what extent the polythermal physics influences the simulated ice sheet, the polythermal mode in SICOPOLIS was switched off in run 2. This simulation was therefore conducted with the conventional cold-ice method: In the entire ice sheet body the temperature equation (3) for cold ice is solved, and calculated temperatures above the melting point are artificially reset to the pressure melting point afterward. This method neither accounts for the energy jump relation connecting temperature gradient, water content, and ice flux at the CTS (Stefan condition), nor for the water content in temperate ice and its influence on the ice fluidity. Apart from this, run 2 corresponds to the control run 1.

Table 2 shows for run 2 the final values of ΔTma, hmax, Vtot, Hmax, Vtemp, Ht,max, Ai,b, and At,b compared with those of run 1. The temperate ice volume Vtemp and the maximum thickness of the temperate ice layer Ht,max are much larger than for the control run, whereas the basal area covered by temperate ice At,b is almost unchanged. The reason is that, because of the implicit solution technique in the vertical in the cold-ice-method simulation, near-basal temperatures above melting shift the temperature profile in the entire affected ice column toward too large values. Since adjustment to pressure melting is possible only after solving the system of linear equations for the affected column, too high temperatures below pressure melting can no longer be corrected. Since this only affects the vertical direction, there is no direct influence on the value of At,b.

Moreover, the time evolution of Ht,max appears to be discontinuous in run 2 (Greve 1995). This is so because the cold-ice method does not allow to separately map the cold and temperate regions onto the [0, 1] interval in the vertical, so that the CTS must be placed on discrete grid points of the numerical cold ice region.

Despite these considerable differences, the impact of the polythermal mode on the ice sheet as a whole is moderate, because the absolute amount of temperate ice is very small. Nevertheless, a slight trend can be realized: for run 2 hmax, Vtot, and Hmax are smaller by 1%, and Ai,b is 0.2% larger than in run 1. Obviously, on the whole in run 2 the ice is slightly less viscous. This results from the larger englacial temperatures in run 2 and the consequentially enhanced deformability, which apparently overcompensates the reducing effect of the water content on the ice viscosity that is accounted for in the control run.

c. Simulation with constant enhancement factor

For run 3 the age-coupled enhancement factor as given in Eq. (19) was replaced by a spatially and temporally constant enhancement factor, E = 3, which corresponds to glacial, more readily deformable ice. The results are listed in Table 2.

The differences between run 3 and run 1 are very small, even though for run 1 the interglacial value E = 1 is assigned to the larger part of the ice volume. The reason is that the ice flow is basically determined by the mechanical properties of the near-basal ice, because that ice is subjected to the largest shear stresses that influence the shear rates overproportionally due to the nonlinear flow law. Since near-basal ice is old (glacial) ice, E = 3 is assigned to this region for both runs 1 and 3. Larger differences in the order of a few percent are merely discernible for the temperate ice volume Vtemp and the maximum thickness of the temperate ice layer Ht,max, with larger values for run 3. This is so because for run 3 the ice sheet as a whole is slightly warmer (due to the larger enhancement factor of interglacial ice and the implied enhanced strain heating), to which the very thin temperate ice layer reacts most sensitively. This correlates also with the reduced total ice volume Vtot by about 1%.

d. Variation of the threshold value for the water content

Runs 4 and 5 were carried out with varied threshold value ωmax in the drainage law (18); for run 4 ωmax = 0.5% was used; for run 5 ωmax = 2%. Table 2 shows the results together with those of the control run (ωmax = 1%).

As above, the ice sheet as a whole remains essentially unchanged, with a slight trend toward a smaller ice sheet when ωmax increases, the differences being in the order of per mille. The influence on the temperate ice layer (Vtemp, Ht,max), however, is pronounced: the larger ωmax, the less temperate ice remains. The reason is probably that, due to the strong increase of the rate factor At(ω) with water content [Eq. (14)], larger values of ωmax entail larger ice velocities within and in the vicinity of temperate ice regions, leading to intensified downward advection of cold ice; this reduces the thickness of the temperate ice layer. These slightly larger ice velocities also explain the trend toward a volume reduction of the ice, although the decrease of the temperate ice volume acts in the opposite way.

e. Variation of the enhancement factor for glacial ice

Runs 6 and 7 were conducted with varied enhancement factor for glacial ice [age A ≥ 11 kyr; Eq. (19)]; for run 6 E = 2 was assigned, for run 7 E = 4. The value E = 1 for interglacial ice with A < 11 kyr was kept in all cases. In Table 2 the results are listed together with those of the control run (E = 3 for glacial ice).

This is the first of all parameter variations that lead to a distinct impact on the entire ice sheet. Since by increasing the enhancement factor E the ice becomes more easily deformable, Vtot, hmax, and Hmax decrease, with the change of the ice volume Vtot covering 7% of the standard value. Simultaneously, the temperate ice layer grows, owing to the intensified strain heating. However, the basal area covered by temperate ice At,b decreases slightly, possibly because of the counteracting effect of stronger downward advection of cold ice due to larger ice velocities.

f. Variation of the sliding coefficient

In runs 8 and 9 the effect of a variation of the sliding coefficient Csl in the sliding law (17) was studied. For run 8, Csl = 5 × 104 yr−1 was employed and for run 9 Csl = 7 × 104 yr−1, while the standard value of run 1 was Csl = 6 × 104 yr−1. The results are again given in Table 2.

As for the case with varied enhancement factor (runs 6 and 7), a reduction of ice volume occurs with increasing sliding coefficient; however, the deviation of 1% from the standard value is surprisingly small. Apparently, the volume-reducing influence of the intensified basal sliding is nearly compensated by ice hardening due to enhanced downward advection of cold near-surface ice and consequentially lower englacial temperatures. In this process, the noticeable retreat of the basal area covered by temperate ice At,b certainly plays a role as well, because basal sliding is connected with the presence of a temperate ice base. The influence of enhanced advection is further responsible for the decrease of the temperate ice layer (Vtemp, Ht,max).

g. Simulation with glacial climate conditions

As a final steady-state simulation, run 10 is presented; it was conducted under glacial climate conditions. These are parameterized by ΔTma = −10°C and therefore, according to Eq. (9), S = Stoday/2. The enhancement factor was set equal to its glacial value E = 3 uniformly in space and time. The results of this simulation are depicted in Figs. 7 and 8, and Table 2 summarizes the final state.

Comparison of Fig. 7 with the corresponding Fig. 2 of the control run shows that the surface topography in the inner ice sheet regions is basically unchanged; this is further demonstrated by the similar values of hmax and Hmax (Table 2). The reasons are the counteracting effects of lower englacial temperatures and reduced snowfall rate for the glacial ice sheet, which obviously cancel out each other to a large extent. However, the lower surface temperatures make surface melting virtually unimportant, so that the ice margin reaches everywhere on the coast line, as opposed to the present ice sheet.

The retreat of temperate regions (Fig. 8), caused by the lower temperatures, has the effect that the ice velocities are generally smaller in run 10 than in run 1. However, the location of the main drainage areas is essentially unchanged, but the resulting mass fluxes are considerably smaller (Greve 1995). As stated above, the impact of this on the surface topography is almost compensated by the reduced snowfall rate.

It is striking that run 10 is the only one of the steady-state simulations that produces a noticeable change of the ice-covered basal area Ai,b when compared with the control run. Apparently, for the position of the ice margin the specification of the accumulation–ablation function as is decisive. This behavior is plausible: the decision whether a certain land point of the model domain is ice-covered or not depends to the first order only upon the balance of snowfall and melting. Effects of ice dynamics are not important as long as the ice has not reached a certain thickness because for very thin ice the ice flow is virtually negligible due to the nonlinear flow law. This furnishes the possibility to inversely determine paleoclimatic accumulation–ablation balances from geologically reconstructed ice margins of the past, basically independent of unprecisely known parameters like the geothermal heat flux Qgeoth, the enhancement factor E or the sliding coefficient Csl.

5. Discussion of the transient simulations

The setup for these is essentially the same as for the steady-state runs (see the beginning of section 4). Exceptions are (a) the time steps for the runs with Milanković-period forcing and Greenland Ice Core Project (GRIP) data forcing (sections 5a and 5b) are Δt = 5 a, Δ̃ = 100 a, and those for the greenhouse-warming runs (section 5c) are Δt = 5 a, Δ̃ = 50 a; (b) the evolution of the lithosphere temperature according to (6) is taken into account.

a. Transient simulations with Milanković forcing

Runs 11, 12, and 13 were conducted with a sinusoidal forcing of the air temperature, oscillating between the interglacial value ΔTma = 0°C and the glacial value ΔTma = −10°C. In doing so, the periods tMil = 20 (run 11), 40 (run 12), and 100 kyr (run 13), corresponding approximately to the three Milanković cycles (Schönwiese 1992), were used. Expressed in a formula, this yields
i1520-0442-10-5-901-e20
For the enhancement factor, the constant, glacial value E = 3 was chosen, because it does not seem very meaningful to introduce a coupling to the age of the ice when applying an artificial forcing like (20). As initial condition the previous steady-state run 3 under present climate conditions, carried out with constant E = 3, was used. The iterations cover 200 kyr model time, so that at least two entire periods are reproduced.

The time evolutions of the simulated ice sheet for runs 11, 12, and 13 are depicted in Figs. 9, 10, and 11, respectively. In the response functions of all plotted quantities, surface temperature forcing ΔTma, maximum elevation MSL hmax, total ice volume Vtot, maximum ice thickness Hmax, volume of temperate ice Vtemp, maximum thickness of basal temperate ice layer Ht,max, basal ice-covered area Ai,b, and basal area covered by temperate ice At,b, the forcing period is reproduced; however, especially for Vtot, Vtemp, and Ht,max, distinct overtones are found. In the time evolution of Ht,max of run 11 the first overtone (with a period of half the forcing period) even predominates the forcing period.

For all three model runs the time series for the total ice volume Vtot shows a characteristic, saddle-shape course: starting from a maximum of the forcing temperature ΔTma, Vtot first increases with falling ΔTma, soon reaches an intermediate maximum, then decreases slightly, and begins to increase again, however, not before the ΔTma minimum is reached. The main Vtot maximum of a cycle is assumed approximately in the middle of the rising ΔTma flank, followed by a rapid decrease until the next ΔTma maximum is reached. This behavior is connected with that of the ice-covered basal area Ai,b: The ice-volume increase with the onset of falling ΔTma is not due to the ice sheet becoming thicker (Hmax is even decreasing during this phase), but due to an increase of the ice-covered basal area. This continues until the entire available land area is glaciated, marked in the corresponding Ai,b plots by reaching the cut-off maxima. This point coincides with the intermediate maximum of Vtot; after that the spread of the ice sheet is stopped and Vtot decreases slightly. Only after having passed the ΔTma minimum, Vtot starts rising again due to the connected increase of the snowfall rate, first with still entirely glaciated land area. The main Vtot maximum of a cycle is reached when, due to the simultaneously intensified surface-melting rate, the ice-covered basal area starts to decrease again, which is followed by a rapid volume reduction that leads into the main Vtot minimum of a cycle, taken approximately at the same time as the ΔTma maximum.

The time series of the quantities describing temperate ice, Vtemp, Ht,max, and At,b, appear for all three simulations somewhat noisy, and the noise amplitude is increasing with increasing forcing period (from run 11 to run 13). To scrutinize this phenomenon, a 4-kyr time window around the air temperature minimum at t = 150 kyr is repeated in Fig. 12 with enlarged timescale; it shows run 13, for which the noise is most pronounced. Apparently, we are concerned with a superposition of numerical noise from time step to time step (Δ̃ = 100 yr) with a real eigenmode of the system, whose period is slightly greater than 1 ka. This periodicity can also be found in the behavior of Vtot and Ai,b. On the other hand, examination of a 4-kyr time window at the end of the steady-state control run under present climate conditions (run 1) only reveals the irregular numerical noise; a superposed periodicity cannot be found in this case.

b. Transient simulations with GRIP forcing

In this section, with runs 14 and 15, two paleoclimatic simulations are presented, the model time reaching from 250 000 yr BP (t = −250 kyr) to today (t = 0), and thus the two last glacial–interglacial cycles are covered (in chronological order: Holstein interglacial, Illinois glacial, Eemian interglacial, Wisconsin glacial, Holocene interglacial). The air-temperature forcing for these simulations is deduced from δ18O data of the GRIP ice core (Summit, Greenland) that contains the climate information reaching farthest back into the past (S. Johnsen 1994, personal communication; Johnsen et al. 1992). For reasons of numerical stability, the original data were subjected to a 2-kyr averaging because otherwise the iterations became unstable at abrupt positive peaks of the temperature forcing. This problem seems to be due to the polythermal calculation; when applying the cold-ice method these instabilities do not appear. However, this filtering should influence the result only minimally because high-frequency temperature oscillations at the ice surface merely have a small penetration depth into the ice. Figure 13 shows the original and the filtered air temperature reconstruction from the GRIP data, together with the assumed time limits of the glacials and interglacials.

Runs 14 and 15 differ from each other by the choice of the enhancement factor. For run 14 the constant glacial value E = 3 is applied, whereas for run 15 a coupling between enhancement factor and age of the ice is taken into account. The decision whether a certain ice particle represents glacial or interglacial ice depends here on its time of accumulation tacc, that is, the moment when it had settled on the ice surface. This accumulation time is calculated from the age A and the current time t via tacc = tA. Ice that is older than Eemian ice is assumed to be glacial ice; the Eemian interglacial itself is assumed to lie between −132 kyr ≤ t < −114.5 kyr, based on intersections of the GRIP temperature curve with ΔTma = 0°C (Fig. 13). As done for the steady-state simulations, the transition from the Wisconsin glacial to the Holocene interglacial is set at 11 000 yr BP, that is, t = −11 kyr. This yields the following assignment for the enhancement factors:
i1520-0442-10-5-901-e21

In order to obtain a suitable initial state, in both cases a 100-kyr, steady-state run was carried out before. It uses the constant enhancement factor E = 3; for the temporally constant air-temperature forcing the initial value of the GRIP data ΔTma(−250 kyr) was applied to ensure a smooth transition into the transient scenario.

Figure 14 shows the temporal evolution of the respective ice sheet for runs 14 and 15. During the Wisconsin glacial remarkable differences appear; run 15 produces in this period an ice sheet of distinctly larger thickness and volume. The reason for this is that in run 15 the smaller value E = 1 is assigned to the ice accumulated during the Eemian. As long as this stiffer ice is close to the surface, its influence on the flow behavior of the ice sheet is small; however, as it is advectively transported downward, it increasingly reduces the ice flow due to its larger viscosity. Consequently, the mass loss due to calving is smaller than for run 14, resulting in a larger ice volume during the Wisconsin glacial. During this period also, differences in the time series of Vtemp and Ht,max are discernible. Between approximately −115 and −92 kyr both values are substantially larger for run 14 than for run 15. Later, between approximately −92 and −80 kyr, a distinct peak in Ht,max appears in run 15 that has no correspondence in Vtemp and is therefore a mere local phenomenon.

Toward the end of the Wisconsin glacial, the near-basal Eemian ice in the ice sheet is largely driven out and replaced by Wisconsin ice, with the effect that the results of the two simulations converge again. For the period from the Last Glacial Maximum (LGM; t = −18 kyr) until today virtually no differences appear; in particular for the simulated present state of the ice sheet, it is insignificant which enhancement scenario is applied.

The evolution of the basal ice-covered area Ai,b is identical for both simulations. This supports the assertion stated in the discussion of run 10 that the ice-covered area is mainly determined by the accumulation–ablation function as and that the internal dynamics is merely of minor importance for the behavior of Ai,b.

The most striking feature in Fig. 14 is the strong decrease of the total ice volume during the Eemian at t = −127 kyr to Vtot = 2.681 × 106 km3; it corresponds to 82.9% of the simulated present value, equally appearing for both simulations. This abrupt loss in total volume is accompanied by similar losses of Vtemp, Ht,max, Ai,b and At,b; however, it has no correspondence in the time series of hmax and Hmax. During this Eemian ice volume minimum (EIVM), both quantities take even larger values than at the LGM, which is characterized by a maximum of the ice-covered basal area and a large ice volume. Apparently, the interior of the ice sheet close to the north dome reacts primarily to changes of the snowfall rate, with the effect that higher air temperatures, and thus larger snowfall rates, entail larger values for hmax and Hmax (whereas the trend for the ice volume is the opposite). This becomes further evident by the similarity of the form of the time signals for these two quantities with the driving signal for ΔTma.

Figures 15–17 show for run 15 the ice surface topography (Fig. 15), the basal temperature together with regions at pressure melting (Fig. 16), and a west–east transect through the ice sheet (Fig. 17) as time slices for the simulated EIVM (t = −127 kyr), the simulated and verified (e.g., by marine sediment cores) LGM (t = −18 kyr), and the present state (t = 0 kyr). Furthermore, Table 2 lists values of the stated variables for the present state of run 15, the steady-state control run 1 under present climate conditions, and, if available, data.

For the EIVM, in the southwest of the ice sheet a conspicuous narrowing of the ice sheet is predicted (Fig. 15a), that is also paralleled in the northeast where the ice margin remains far behind the present one. When looking at the temperate regions (Fig. 16a), freezing conditions prevail at the CTS for comparatively many grid points, quite contrary to the two other time slices (Figs. 16b,c), for which only melting conditions are found. Some of these grid points are not situated close to the ice margin, where, owing to the flow geometry, freezing conditions are expected, but farther inland. This is because the volume of temperate ice strongly decreased immediately before the EIVM (Fig. 14), so that the CTS moved downward, which favors freezing conditions also in the inner ice regions. Figure 17a reveals distinct temperature inversions in the upper half of the ice sheet as a consequence of the stored englacial temperature information from the Illinois glacial. However, in the uppermost layer the isotherms appear to be bent open owing to the reduction by some degrees of the air temperature in the millennia immediately prior to the EIVM.

The LGM is characterized by a simulated ice sheet that covers the entire available land area of Greenland (Fig. 15b). This result holds with the proviso, however, that the sea level was not adjusted. It is known that the sea level at LGM was approximately 130 m lower than today. It likely causes a connection between the northwestern part of the Greenland Ice Sheet and the adjacent Laurentide (North American) Ice Sheet (Schönwiese 1992). Because of the Wisconsin ice age that had prevailed for a long time before the LGM, the temperature information penetrated to the ice base, causing the lowest basal ice temperatures of all the three time slices (Fig. 16b); consequently, the extent of the temperate regions is relatively small. The temperature inversion in Fig. 17b is primarily due to the always effective transport of cold surface ice from central Greenland downward and outward by advection; however, a certain contribution also results from the onset of a rise in air temperature.

As for the simulated present state it is seen that, since the LGM, the ice retreated partly from the coast (Fig. 15c); the temperature at the base increased in general and so did the extent of temperate ice regions (Fig. 16c). In Fig. 17c temperature inversions are less pronounced and restricted to the regions close to the margin, where they are caused by the advection effects described above. The glacial climate information can still be found but at great depth only, because about 10 000 yr of relatively constant interglacial climate conditions have already passed.

Table 2 shows that run 15, when compared with the steady-state control run 1, provides an ice sheet of larger volume and extent and lower temperatures (the latter manifest in a smaller extent of temperate regions). This is an immediate consequence of the impact of the Wisconsin ice age on the present state and demonstrates that steady-state simulations are inadequate for realistic simulations of present large ice sheets (Greenland, Antarctica). It is further remarkable that the agreement between run 15 and the data is worse than between run 1 and the data, in apparent contradiction to this statement. The obvious reason is the uncertainty in the parameterization of the surface melting rate M. It was fitted by Calov (1994) to optimally match steady-state simulations, making the lesser agreement for the transient run 15 understandable.

Similar calculations with cold-ice models were already described by other authors. Letréguilly et al. (1991b) carried out a paleoclimatic simulation covering 150 000 yr of climate history, driven by a surface-temperature reconstruction from surface-ice samples of the ablation zone in central–west Greenland. Their results correspond qualitatively to ours; however, the ice volume retreat during the Eemian interglacial turns out to be slightly larger, and another, even more distinct retreat is found at t = −100 kyr. This is because in their study (i) the snowfall in warmer climates is not allowed to exceed the present value, and (ii) the temperature reconstruction entails a very warm interstadial approximately at t = −100 kyr that does not have an equivalent counterpart in the GRIP reconstruction. Fabré et al. (1995) used the GRIP reconstruction to conduct a simulation over 250 000 yr, as is done here, and find an ice-sheet response that is less influenced by the changing climate than in this study, although the trend is very similar. The main reason for this discrepancy is the exponential coupling of air temperature and snowfall forcing used by these authors, counteracting the impact of changing temperature on the ice sheet more than the linear coupling Eq. (9) used here.

c. Transient simulations on the impact of an anthropogenically enhanced greenhouse warming

Due to the accumulation of several trace gases in the atmosphere caused by mankind (water vapor, carbon dioxide, methane, and CFC) and the subsequently enhanced greenhouse warming, the temperature of the earth’s atmosphere will probably rise in the order of some degrees in the near future. This entails the risk of partial melting of the present large ice sheets on timescales of 102–104 yr, accompanied by a sea level rise to which Antarctica would contribute with approximately 65 m and Greenland with 7 m in case of an entire disintegration (Schönwiese 1992). In order to shed light on this problem, runs 16–19 were designed; they assume a sudden air temperature increase of ΔTma = 2°C (run 16), 4°C (run 17) and 6°C (runs 18 and 19), respectively, and simulate the resulting response of the Greenland Ice Sheet from today (t = 0) until 5000 yr into the future (t = 5 kyr). In order to examine the impact of the polythermal calculation on the results, the polythermal mode was switched off in run 19, so that this run is subjected to the conventional cold-ice method (see section 4b).

As initial condition for runs 16–18 the final state of the steady-state control run 1 under present climate conditions is applied; run 19 is started with the final state of the steady-state run 2 (same as run 1 but using the cold-ice method). The coupling between the enhancement factor and the age of the ice corresponds to Eq. (21), but with no ice older than Wisconsin ice occuring, so that only the transition from Wisconsin to Holocene ice at tacc = −11 kyr is relevant. In particular, relation (21) means that E = 1 is assigned to the ice accumulated during the model time.

Figure 18 shows the temporal evolution of the simulated ice sheet for the three different scenarios (runs 16–18). Evidently, for run 16 the impact of the rise of the air temperature is comparatively small during the model time; in particular, the ice volume Vtot merely decreases by 4.9%, corresponding to a sea level rise of 34 cm. For runs 17 and 18, however, a strong decrease is evident: the volume of the molten ice after 5 kyr is 37.6% (run 17) and as large as 87.2% (run 18), with equivalent sea level rises of 2.6 m and 6.1 m, respectively. The ice volume decrease takes place approximately linearly in the first millennium for all three cases, the resulting sea level changes during this period being 1.47 cm century−1 (run 16), 6.44 cm century−1 (run 17), and 15.7 cm century−1 (run 18), respectively.1

The tendency of the behavior of the temperate ice volume Vtemp is the same in all cases: an initial increase is followed by a decrease below the initial value. The initial increase is due to the elevated surface temperature, which makes the ice sheet as a whole slightly warmer so that the CTS moves upward. This trend does not continue because the ice volume decreases, with the consequence that, first, the internal heat production due to strain heating is reduced, and second, the lithosphere rises due to the relief, both causing the near-basal ice temperatures to decrease. Connected with this, one can further see a somewhat noisy course of the quantities relevant for the temperate ice: Vtemp, Ht,max, and At,b. This is due to the fact that the fast changes here are numerically problematic for the temperate ice, which reacts very sensitively in general.

In Figs. 19 and 20, for run 18 the topography and the temperature at the ice base are depicted for the remainder of the ice sheet at the end of the simulation (i.e., after 5000 yr of model time). The main ice mass is situated in a band between 72°N and 76°N; the dome is shifted toward the east coast. It is striking that in the vicinity of the new dome the elevation of the ice surface hardly decreased. This is because the altitude of the ice base is very high in this region, leading to the low basal temperatures in Fig. 20; this has a stabilizing effect on the ice sheet due to the connected adhesion at the base.

Table 2 displays the simulated ice sheet parameters after 5000 model yr for runs 18 (ΔTma = 6°C; polythermal method) and 19 (ΔTma = 6°C; cold-ice method). Like the comparison between the steady-state runs 1 (polythermal method) and 2 (cold-ice method), we find that the resulting volume of temperate ice Vtemp and maximum thickness of the temperate ice layer Ht,max are much larger for the cold-ice-method run than for the polythermal run. Here the impact on the ice sheet as a whole is much more pronounced than it was for the steady-state runs. For the cold-ice-method run 19, the total ice volume Vtot is 7% smaller, the basal ice-covered area Ai,b 3.5% smaller, and the basal area covered by temperate ice At,b 5.3% smaller than for the polythermal run 18. Apparently, to model adequately the retreat process of an ice sheet in response to high air temperatures, application of the polythermal method is important. A reason for this may be that, as discussed above, immediately after the onset of the air-temperature increase the volume of temperate ice grows strongly, so that the amount of temperate ice in the ice sheet is considerably larger than for present climate conditions. Therefore, its influence on the dynamics of the entire ice sheet is greater, and it becomes more important to properly deal with the presence of temperate ice, as attempted with the polythermal method.

In these predictions, a major uncertainty is the future development of the snowfall rate, that is here linearly coupled to the air-temperature change according to Eq. (9). In the literature (Huybrechts et al. 1991; Letréguilly et al. 1991a; Fabré et al. 1995), two different approaches were used for analogous simulations, namely, an unchanged snowfall and a snowfall rate increasing exponentially with ΔTma; Eq. (9) is in between these two cases. Fabré et al. (1995) demonstrated that a steady-state simulation with ΔTma = 5°C yields an ice sheet that almost vanishes for constant, present snowfall rate and an ice sheet that is insignificantly different from today’s when forced by an exponentially increasing snowfall. This shows that reliable prognoses are not possible with simple parameterizations of snowfall rates. Improvements can only be provided from detailed precipitation modeling within GCM simulations that can derive the snowfall rate in principle from the dynamic behavior of the hydrologic cycle.

6. Conclusions

In the present paper, the application of a polythermal three-dimensional ice sheet model to the Greenland Ice Sheet was discussed. A variety of simulations was presented, subject to different model parameters and climate scenarios. Special emphasis was put on the behavior of the basal layer of temperate ice. It was demonstrated that the model can be applied to a variety of problems, steady-state as well as time-dependent, and that it provides meaningful and interpretable results.

However, it is not possible at the moment to check the model results with regard to the basal temperate ice layer quantitatively against field data; this is due to a lack of observations and measurements on that subject on large ice sheets. On the one hand, this is a severe shortcoming of the polythermal modeling approach presented here; on the other hand, our computations should have made clear that, physically, the polythermal mode cannot be dispensed with in time-dependent scenarios. This should serve as an impetus to the applied glaciologist.

Acknowledgments

The author wishes to thank Prof. Kolumban Hutter for his instructions when conducting this work and for carefully reading and correcting the manuscript in terms of clarity and writing; Dipl.-Phys. Georg Bauer, Dr. Stefan Diebels, and Dipl.-Phys. Magnus Weis for their time-consuming efforts to keep the institute’s workstations operating; Dr. Reinhard Calov for his aid in developing the numerical solution technique for the model equations; Prof. Inez Fung for helpful editorial comments; Dr. Sigfus Johnsen for providing the δ18O data of the GRIP core; Dr. Anne Letréguilly for providing the surface-topography and bedrock-topography data of Greenland; and two anonymous reviewers for their comments on an earlier draft of this paper. This research is part of the author’s Ph.D. thesis. It was funded by a grant of the Studienstiftung des Deutschen Volkes.

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Fig. 1.
Fig. 1.

Measured surface topography h of the present Greenland Ice Sheet (Letréguilly et al. 1991; in km MSL). The spacing between the isolines is 200 m. The dashed heavy line indicates the ice margin.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 2.
Fig. 2.

Final state of run 1: topography of the ice surface (in km MSL). The spacing between the isolines is 200 m. The dashed heavy line indicates the ice margin.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 3.
Fig. 3.

Final state of run 1: horizontal mass flux. Double arrow length corresponds to tenfold mass flux. The dashed heavy line indicates the ice margin, the isolines represent the surface topography in 500 m intervals.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 4.
Fig. 4.

Final state of run 1: homologous temperature at the ice base (in °C). The spacing between the isolines is 3°C. Open diamond symbols indicate positions where the basal ice is at the pressure melting point, yet with no temperate layer above; full diamonds (full circles, see → in the plot) indicate positions where there is a basal layer of temperate ice with a melting (freezing) CTS. The dashed heavy line indicates the ice margin.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 5.
Fig. 5.

Final state of run 1: transect at y = −2280 km. Top: ice velocity; middle: homologous ice temperature (in °C); bottom: thickness of the basal layer of temperate ice [open circles: cold ice base; open diamonds: temperate ice base with no temperate ice layer above; full diamonds: basal layer of temperate ice with a melting CTS; full circles (see → in the plot): basal layer of temperate ice with a freezing CTS].

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 6.
Fig. 6.

Run 1: time evolution of ΔTma, hmax, Vtot, Hmax, Vtemp, Ht,max, Ai,b, and At,b.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 7.
Fig. 7.

Same as Fig. 2, but for the final state of run 10.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 8.
Fig. 8.

Same as Fig. 4, but for the final state of run 10.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 10.
Fig. 10.

Same as Fig. 6, but for run 12.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 11.
Fig. 11.

Same as Fig. 6, but for run 13.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 12.
Fig. 12.

Same as Fig. 6, but for run 13. A 4-kyr time window around t =150 kyr is zoomed out.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 13.
Fig. 13.

Air temperature scenario for the last 250 kyr, deduced from the GRIP data. Top: unfiltered data. Bottom: filtered data (2-kyr averages). The assumed time limits for the Eemian interglacial (−132 kyr; −114.5 kyr) and the Holocene interglacial (−11 kyr; 0 kyr) are indicated by the dashed lines.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 14.
Fig. 14.

Same as Fig. 6, but for runs 14 (dashed) and 15 (solid).

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 15.
Fig. 15.

Same as Fig. 2, but for run 15. (a) t = −127 kyr (EIVM), (b) t = −18 kyr (LGM), and (c) t = 0 kyr (today).

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 16.
Fig. 16.

Same as Fig. 4, but for run 15. (a) t = −127 kyr (EIVM), (b) t = −18 kyr (LGM), and (c) t = 0 kyr (today).

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 17.
Fig. 17.

Same as Fig. 5, but for run 15. (a) t = −127 kyr (EIVM), (b) t = −18 kyr (LGM), and (c) t = 0 kyr (today).

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 18.
Fig. 18.

Same as Fig. 6, but for runs 16 (dotted), 17 (dashed), and 18 (solid).

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 19.
Fig. 19.

Same as Fig. 2, but for run 18, t = 5 kyr.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Fig. 20.
Fig. 20.

Same as Fig. 4, but for run 18, t = 5 kyr.

Citation: Journal of Climate 10, 5; 10.1175/1520-0442(1997)010<0901:AOAPTD>2.0.CO;2

Table 1.

Compilation of standard physical quantities used for the simulations in this study. For references see Greve (1995).

Table 1.
Table 2.

Data for the present Greenland Ice Sheet; results for runs 1–6, 7–10, 15, 18, and 19.

Table 2.
1

These statements are based on the above-mentioned sea level equivalent of 7 m for the entire ice volume and the assumption of a linear relationship between ice volume decrease and sea level rise.

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