The flow of ice cannot be described by the equations of fluid dynamics alone but needs to be complemented by a material-dependent constitutive equation which relates the internal forces (stress) to the deformation rate (strain rate). Numerous laboratory experiments and field measurements show that the ice deformation rate responds to stress in a nonlinear way. Under the assumptions of isotropy, incompressibility, and uni-axial stress, this observation is reflected in Glen’s flow law, which gives the constitutive equation for ice, 1ϵ˙=Aτn, where ϵ˙ is the strain rate, τ the dominant shear stress, n the flow exponent, and A the softness of ice .

Both the flow exponent and the softness are important parameters which determine the flow of ice. Usually, the exponent n is assumed to be constant through space and time. At present, there is no comprehensive understanding of all the physical processes determining the softness A. It may depend on water content, impurities, grain size, anisotropy, and temperature of the ice, among other things. Within the scope of ice-sheet modeling, A is typically expressed as a function of temperature alone: 2A=A0exp⁡-QRT′, where A0 is a constant factor, Q is an activation energy, R is the universal gas constant, and T′ is the temperature relative to the pressure melting point .

Due to pre-melt processes, the softness responds more strongly to warming at temperatures close to the pressure melting point, which is often described by a piecewise adaption of the activation energy Q , with a larger value of Q at temperatures T′>-10 ∘C. When using these piecewise defined values for Q for warm and for cold ice in the functional form of the flow law, the respective factors A0 ensure that the function is continuous at T′=-10 ∘C. A0 is therefore dependent on the values of the flow exponent n and both values of Q for cold and for warm ice.

The scalar form of Glen's flow law (Eq. ) is only valid for uni-axial stress, acting in only one direction. For a complete picture the stress is described as a tensor of order 2. The generalized flow law reads 3ϵ˙jk=A(T′)τen-1τjk, where ϵ˙jk are the components of the strain rate tensor and τjk are the components of the stress deviator, and τe is the effective stress, which is closely related to the second invariant of the deviatoric stress tensor: 4τe2=12τxx2+τyy2+τzz2+τxy2+τxz2+τyz2. Each component of the strain rate tensor depends on all the components of the deviatoric stress tensor through the effective stress τe.

Glen's flow law (Eq. ) and the softness parametrization (Eq. ) are at the center of most numerical ice-sheet and glacier models, independent of the other approximations they might use .

The simulations in this study were performed with the Parallel Ice Sheet Model (PISM) release v1.1. PISM uses shallow approximations for the discretized physical equations: the shallow-ice approximation (SIA) and the shallow-shelf approximation (SSA) are solved in parallel within the entire simulation domain. The shallow-ice approximation is typically dominant in regions with high bottom friction, such that the vertical shear stress dominates over horizontal shear stress and longitudinal stress. The shallow-shelf approximation is typically dominant for ice shelves, with zero traction at the base of the ice, and for the fast-flow regime in ice streams . PISM assumes a non-sliding SIA flow and uses the results of the SSA approximations for fast-flowing and sliding ice. In PISM, the flow law enters both the SIA and the SSA part of the velocities, as detailed in . It is possible to choose different flow exponents n for the SSA and the SIA, but the softness is the same for both approximations.

The simulations performed here use mostly the SIA mode: the geometry of a two-dimensional ice sheet sitting on a flat bed and the SIA mode serve to study the effects of changes in flow parameters on internal deformation and to separate those effects from others, such as changes in sliding. Including the shallow-shelf approximation reproduces and even enhances the effect of changes in the activation energies Q (see Sect. ).

The flow exponent n and the activation energies for warm and for cold ice, Qw and Qc, determine the deformation of the ice as a response to stress or temperature. A recent review (; see also literature in the Introduction above) reveals a broad range of potential flow parameters n, Qw, and Qc. In line with these findings, in this study the activation energy Qc is varied between 42 and 85 kJmol-1 (a typical reference value is Qc=60 kJmol-1). The activation energy for warm ice Qw is varied between 120 and 200 kJmol-1 (a reference value is Qw=139 kJmol-1). For the flow exponent n, values as low as 1 have been reported, but since many experiments and observations confirm a nonlinear flow of ice, n is varied between 2 and 4, with a reference value of n=3. The reference values above correspond to the default values in many ice-sheet models .

The flow factor A0 in the flow law must be adapted to fulfill the following conditions: first, the continuity of the piecewise defined softness A(T′) must be ensured for all combinations of Qw, Qc, and n. Secondly, a reference deformation rate ϵ˙ at the reference magnitude of the driving stress τ0 and a reference temperature T′0 should be maintained regardless of the parameters. This is because the coefficient and the power are non-trivially linked when a power law is fitted to experimental data. These conditions give 5A0,old⋅exp⁡-QoldRT′0⋅τ0nold=A0,new⋅exp⁡-QnewRT′0⋅τ0nnew, 6A0,new=A0,old⋅exp⁡-Qold-QnewRT′0⋅τ0nold-nnew. If the reference temperature is T′0<-10 ∘C, the values for cold ice A0,c and Qc are used in the equation above, or else A0,w and Qw are used. The corresponding A0,new for cold and warm ice is calculated from the continuity condition at T′=-10 ∘C. For T′0<-10 ∘C, for example, it follows that 7A0,c,new=A0,c,old⋅exp⁡-Qc,old-Qc,newRT′0⋅τ0nold-nnew, 8A0,w,new=A0,c,new⋅exp⁡-(Qc,new-Qw,new)R⋅263.15K. Here we choose τ0=80 kPa as a typical stress magnitude in a glacier and T′0=-20 ∘C. Choosing another τ0 on the same order of magnitude has only little effect on the differences in dynamic ice loss. Choosing another T′0 on the other hand influences how the softness changes with the activation energy Q; see Fig. S1 in the Supplement. With T′0 closer to the melting temperature, the difference in softness at the pressure melting point decreases and thus the ice loss is less sensitive to changes in the activation energy Q.

The study is performed in a flow-line setup, similar to , where the computational domain has an extent of 1000 km in the x direction and 3 km in the y direction (with a periodic boundary condition). The spatial horizontal resolution is 1 km. The ice rests on a flat bed of length L=900 km with a fixed calving front at the edge of the bed, such that no ice shelves can form (Fig. ). In contrast to , the temperature and the enthalpy of the ice sheet are allowed to evolve freely.

The model is initialized with a spatially constant ice thickness and is run into equilibrium for different combinations of flow parameters Qc,Qw, and n. The ice surface temperature is altitude dependent, Ts=-6∘Ckm-1⋅z-2∘C, where z is the surface elevation in kilometers. The accumulation rate is constant in space and time for each simulation. A constant geothermal heat flux of 42 mWm-2 is prescribed. In the warming experiments, for each ensemble member an instantaneous temperature increase of ΔT∈[1,2,3,4,5,6] ∘C is applied to the ice surface for a duration of 15 000 years (until a new equilibrium is reached), while the climatic mass balance remains unchanged. That means the temperature increase can lead to an acceleration of ice flow but is prohibited from inducing additional melt. This idealized forcing allows us to disentangle the effect of warming on the ice flow from climatic drivers of ice loss.

The thickness profile of the equilibrium state is similar to the Vialov profile see e.g.. However, in contrast to the isothermal Vialov profile, here the temperature of the ice is allowed to evolve freely, leading to a non-uniform softness of the ice . The extent in the x direction is given by the geometry of the setup, a flat bed with a calving boundary condition at the margin, and the height and shape of the ice sheet depend on the flow parameters n, Qw, and Qc and the accumulation rate a.

In order to gain a deeper understanding of the influences of Qc and Qw on the equilibrium shape of ice sheets, we here compare the simulated results to analytical considerations based on the Vialov profile.

At a fixed accumulation rate of a=0.5 myr-1, each flow parameter combination leads to an equilibrium state with a thickness profile similar to the Vialov profile but differences in maximal thickness and volume (Fig. a). Overall, high activation energies increase ice-flow velocities and reduce the ice-sheet volume. The activation energy for warm ice, Qw, affects the volume and the velocities more strongly than the activation energy for cold ice, Qc. A high Qw leads to softer ice close to the pressure melting point (Fig. S1) and at the base of the ice sheet, which leads to higher velocities and a lower equilibrium volume of the ice sheet, while a low Qw leads to stiffer ice close to the pressure melting point and at the base of the ice sheet, and in consequence the velocities decrease and the volume increases (Fig. b and c). For a fixed Qw, the volume appears to decrease linearly with increasing Qc and the velocity appears to increase linearly with increasing Qc.

The maximal thickness of an isothermal ice sheet can be estimated with the Vialov profile: 9hm=2n/(2n+2)L1/2(n+2)a2A(T′)(ρg)n, with the Glen exponent n, the ice-sheet extent 2L, the pressure-adjusted temperature T′, the gravity g, and the ice density ρ . The Vialov thickness of a temperate ice sheet (isothermal at the pressure melting point), where the softness is evaluated at the pressure melting point depending on the activation energies Qc and Qw (see Eq. ), gives a lower bound to the thickness, given the same geometry and flow parameters. The simulated maximal thickness is larger than the lower bound for all parameter combinations (Fig. a, lower bound indicated by a grey line), and the ratio between the maximal thickness hm from the PISM simulation to the lower bound from the Vialov profile depends on both Qw and Qc. The ratio increases with higher Qw and decreases with higher Qc (Fig. b). The ice-sheet thickness of the polythermal ice sheet, as simulated with PISM, matches well the Vialov thickness calculated with Eq. () if an effective temperature Teff′<0 ∘C is assumed. The effective temperature Teff′ that matches simulations best varies for different Qw. For Qw=120 kJmol-1, an effective temperature of Teff′=-5 ∘C matches well the equilibrium thickness of the polythermal ice sheets. For Qw=200 kJmol-1, an effective temperature of Teff′=-3.3 ∘C matches well the equilibrium thickness of the polythermal ice sheets. These differences can be partly explained by the altitude-dependent surface temperature: the maximal thickness of the ice sheets varies by approximately 800 m, which leads to a difference in ice surface temperature of approximately 4.8 ∘C between the thickest and the thinnest ice and thus influences the temperature within the ice sheet.

The relative difference of average velocities dv=(v‾-v‾0)/v‾0 spans from dv=-7 % (with a corresponding relative difference in ice-sheet volume of dvol=+10 %) for the lowest combination of activation energies to dv=+18 % with a difference in volume of dvol=-15 % for the highest combination of values for Qc and Qw (Fig. b).

In order to keep the initial ice volume largely fixed (with variations of less than 1 %) in the warming experiments, we adapt the accumulation rate for each parameter combination of Qc and Qw.

Since simulations with high activation energies Qw have a smaller equilibrium volume at the same accumulation rate than simulations with standard activation energies, the accumulation rate a is increased to maintain an equilibrium volume close to the reference value. Simulations with low activation energies Qc have a higher volume at the same accumulation rate, so the accumulation rate a is decreased. In the case of an isothermal ice sheet the maximal thickness and the volume can be computed analytically as shown above in Eq. (). In our model simulations, however, the temperature distribution within the ice can evolve freely; thus the softness is not uniform and an analytical solution cannot be found.

In order to find the right adaptation for the accumulation rates, we start from the ice profile from the isothermal approximation as a first guess and run the model into equilibrium. If the relative difference between the new equilibrium volume and the standard equilibrium volume exceeds 1 %, we further change the accumulation rate and repeat the equilibrium simulation, always starting from the same initial state. The final equilibrium states found via this iterative approach differ by a maximum of 0.8 % in ice volume (Fig. S2), and the difference in maximal thickness is less than 100 m (Fig. a and b).

For the combination of high activation energies Qw and Qc, the relative differences dx=(x-x0)/x0 of both adapted accumulation rates a and mean surface velocities v increase by more than 300 % (Fig. c and d), and for the combination of low activation energies Qc and Qw both adapted accumulation rates a and surface velocities v are approximately 50 % lower compared to the case with standard parameters. Both the accumulation rate and the velocities change in the same way since they balance each other in equilibrium. A change in accumulation rates controls the vertical velocity profile and thus influences the thermodynamics in the ice, which leads to differences in the temperatures of the ice sheet (pressure-adjusted temperature distributions shown in Fig. S4a). The change in temperature is most prominent at the top of the ice sheet, where higher accumulation rates (associated with high activation energies) lead to lower temperatures and vice versa. Thus the temperature change introduced from increased accumulation counteracts the effect of increased softness. In order to estimate how changed temperature on the one hand and changed flow parameters on the other hand impact the resulting ice softness, either one was kept fixed. The effect of the temperature changes on the ice softness is negligible, compared to parameter changes (see Fig. S4b, c, and d).

The maximal thickness of the polythermal simulated ice sheet is approximately 13–16 % larger than the lower bound estimated with a temperate ice sheet (Fig. a and b) with the same flow parameters and accumulation rates. Similar to the case with fixed accumulation rates, the simulated thickness matches the Vialov thickness well if an effective temperature T′eff<0 ∘C is assumed. The effective temperature that matches simulations best varies for different Qw, from -5 ∘C for Qw=120 kJmol-1 to -3.6 ∘C for Qw=200 kJmol-1. This difference cannot be sufficiently explained by variations in surface temperature due to the difference in ice-sheet thickness. Rather the higher effective temperatures are linked to increased flow velocities of the ice, which in turn might lead to strain heating. In simulations with a high Qw the simulated thickness has a higher discrepancy to the estimated lower bound (assuming a temperate ice sheet) than simulations with a low Qw. In contrast to the case with fixed accumulation rate (Fig. ) the ratio between the estimated and the simulated thickness depends only very little on Qc.

Disentangling the purely flow-driven ice losses from the influences of melting, different initial temperature profiles, and variations in sliding requires several conditions:

The initial volume is fixed, which here is attained through adjustment of the accumulation rate for the different flow parameter combinations as explained in Sect. .

The effect of the temperature increase is limited to warming at the ice surface, which can propagate into the interior of the ice sheet through diffusion and advection. Warming makes the ice softer and thus accelerates the flow and ice discharge. Since temperature diffusion in an ice sheet is a very slow process, we apply the temperature anomaly for a total duration of 15 000 years. The total mass balance is evaluated and compared to the standard parameter simulation after 100, 1000, and 10 000 years of warming. A new equilibrium state is reached after 10 000 years for all parameter combinations (see longer time series in Fig. S3).

In the experiments, the ice sheet loses mass for all warming levels and all parameter combinations. However, the amount and rate of the ice loss are dependent on the flow parameters. Figure  shows the ice-sheet response to a warming of 2 ∘C. For a fixed flow exponent of n=3 the fastest ice loss is observed for the flow parameter combination of Qc=85 kJmol-1 and Qw=200 kJmol-1, and the slowest ice loss for Qc=42 kJmol-1 and Qw=120 kJmol-1. Simulations with Qw=200 kJmol-1 reach a new, temperature-adapted equilibrium after only 2000 years, while simulations with lower Qw continue to lose mass.

The sensitivity to variations in flow parameters is measured via the relative differences for flow-driven ice loss dm=(Δm-Δm0)/Δm0, where the reference Δm0 is always given by the simulation with standard parameters under the same temperature increase (Fig. ). While the long-term response to warming, after 10 000 years, is not very sensitive to the particular choice of flow parameters, the rate of flow-driven ice loss is. The largest relative differences in ice loss is found in the first century after the temperature increase (Fig. a), indicating that high Qw speeds up the flow-driven ice loss. Under 2 ∘C of warming, ice loss after 100 years is enhanced more than 2-fold (i.e., increased by up to 118 %) in simulations with Qw=200 kJmol-1, while low Qw reduces the relative ice loss by up to 37 %.

The effect of the flow parameters on flow-driven ice loss upon warming is robust for different temperature increases. Ice losses and the spread in flow-driven ice loss both increase for higher warming levels (see Fig. ). For a warming of ΔT=1 ∘C the idealized ice sheet loses 0.09 % after 100 years and 0.35 % of ice after 1000 years for standard parameters. For a warming of ΔT=6 ∘C the ice sheet loses 0.46 % after 100 years and 1.89 % of ice after 1000 years for standard parameters (solid red line). For comparison, the Greenland Ice Sheet lost approximately 0.18 % of its mass in the period between 1972 and 2018 , which includes all processes: increase in flow, melting, and sliding.

The effect of flow parameter changes on the purely flow-driven ice loss after 100 years is on the same order of magnitude as the effect of surface warming by several degrees. In particular the uncertainty ranges of ice loss for warming of 2 ∘C and warming of 6 ∘C overlap (Fig. b) when solely considering the ice loss driven by changes in flow and excluding surface mass balance changes.

Variations in the flow exponent n do not change the qualitative effect of variations in activation energies Q on the ice loss. After 100 years for a temperature anomaly of ΔT=2 ∘C a higher n seems to mitigate the effect of the activation energy on differences in ice loss, while a lower n seems to enhance this effect (Fig. ). However, the effect of variations in activation energy on the average surface velocity is almost independent of the choice of the flow exponent n (Fig. ).

The influence of the activation energies Qc and Qw on ice flow is similar even with different flow exponents n. This is robust for different warming scenarios from +1 to +6 ∘C. A higher flow exponent n, which leads to a more pronounced nonlinearity in ice flow, does not enhance but reduces variations in dynamic ice loss. Compared to the nonlinear stress dependency τn in the flow law, the temperature-dependent softness A(T′)=A0⋅exp⁡(-Q/RT′) becomes less important with increasing flow exponent n.

The overall effect of uncertainties in the activation energies Q remains robust, even if an additional driver of ice loss is taken into account. In a simulation where in addition to warming of 2 ∘C we also reduce the accumulation rate by 50 %, the ice losses remain dependent on the flow parameters Qc and Qw (Fig. ; lines indicate results without a change in accumulation rate, analogous to Fig. , and squares indicate results with an additional 50 % decrease in accumulation rate). After 100 years of forcing, the relative spread of ice loss is slightly larger if accumulation changes are included. In particular, for Qw=200 kJmol-1 the relative increase of mass loss mounts from 118 to 190 %. On longer timescales, the spread in ice loss is reduced (after 10 000 years of forcing, when the ice sheet has reached a new equilibrium, the relative spread is below ±10 %).

When sliding is taken into account via the shallow-shelf approximation for sliding ice (see ) the uncertainty in flow parameters leads to relative changes in ice loss from -30 to +470 % after 100 years, which is a considerably larger spread than without sliding. The relative differences decrease with time but remain ranger than without sliding. After 1000 years the ensemble member with low activation energies have lost 40 % less ice than the standard parametrization, and high activation energies almost double the ice loss (+90 %). After 10 000 years, when the ice sheets have reached a new equilibrium, the relative differences still range from -16 to +40 % (see Fig. ).