Ice sheet models (ISMs) are increasingly being used in process studies, sensitivity studies and projections of marine ice sheet (MIS) future behaviour , and model intercomparison projects (MIPs) to investigate the ice sheet response to ocean forced basal melting of ice shelves are currently in their design phase .

Past ISM studies have shown inconsistent grounding line behaviour at typical resolutions . Inconsistencies were very large (grounding line discrepancies of ≈ 100 km) for grid resolutions of ≈ 10 km and typically still not converged for grid resolutions of ≈ 1 km. Studies in which simulations are carried out at multiple different mesh resolutions usually demonstrate convergent behaviour, but very fine resolution is often needed to approach a converged solution . Practical solutions have been suggested, such as parameterising the flux of ice across the grounding line as a function of ice thickness , parameterising the grounding line position at sub-grid resolution , or implementing adaptive mesh refinement to provide very high resolution at and near the grounding line . These solutions all have limitations, and the computational cost of running a sufficiently high-resolution ISM to robustly represent grounding line motion remains high, even with adaptive refinement.

However, model-based MIS studies (e.g. ) typically use a simple basal traction prescription, or “sliding relation” , which neglects the impact of effective pressure at the bed (or equivalently “height above buoyancy”) on basal shear stress. The inclusion of pressure dependency (reviewed by ) provides a physical motivation for smoothing out what is otherwise a large step change in basal drag across the grounding line. This was proposed over 30 years ago (e.g. ) and may affect the resolution requirements for successful grounding line modelling. Recent treatments of basal traction that also vanish smoothly at the grounding line include  and .

Furthermore, the implications of imposing basal melting (note that in the current study “basal melting” refers always to melting under the ice shelf and not the grounded part of the MIS) on resolution requirements have not been explicitly investigated.

In the current study, we assess the impact of choosing between different sliding relations, and between different approaches to parameterising basal melting, on model resolution requirements in a Stokes flow ice dynamic model.

We use the ice dynamic model Elmer/Ice . The Stokes equations for a viscous fluid with non-linear rheology are solved using the finite element method over a two-dimensional flow-line domain (one vertical and one horizontal dimension) in which lateral drag is parameterised  according to channel width, W, and a contact problem is solved to determine the evolving grounding line position .

The rheology follows Glen's law  with viscosity calculated using a temperature-dependent Arrhenius law . Temperature is held constant at -10 ∘C for all simulations in the current study.

We implement alternative sliding relations (Sect. ) and a basal melt parameterisation (Sect. ) in Elmer/Ice.

Our main criterion for assessing the results is resolution dependency. The model used here, Elmer/Ice, has been demonstrated in previous studies (e.g. ) to give convergent behaviour with resolution: its output approaches a self-consistent solution as resolution is made increasingly fine. In the current study we do not attempt to demonstrate convergence in all cases (indeed convergence is certainly not achieved in all cases), but instead consider the dependence on resolution across the three resolutions used (Table ), under the premise that weaker dependence on resolution is an indicator of being closer to the converged solution. The causes of strong resolution dependency in the current study will be discussed in Sect. .

Specifically, we consider experiments in which the grounding line position differs between simulations of different resolution by distances of approximately the same magnitude as the size of a single element not to have significant dependency on resolution. Conversely, we consider experiments with grounding line differences of several element sizes or greater to have significant dependence on resolution. For example, differences in grounding line position of the order of 100 km between simulations at different resolutions are considered to indicate significant resolution dependency, whereas differences of the order of 1 km are not. Similarly, when we say “near to the grounding line” we are also talking in terms of element size. For example “high melt near the grounding line” can be interpreted as “high melt within a very small number of elements of the grounding line”.

We focus mainly on the evolution of grounding line position. The spin-up simulations (SPIN and ADV, Table ) do not vary significantly with resolution, and so our analysis focuses on retreat and re-advance simulations. The fact that the spin-up simulations show very little dependency on resolution is not an indicator that the retreat simulations should show equally low dependency on resolution. Previous studies have shown that ice sheet models often demonstrate much higher resolution dependency in retreat than advance, or vice versa . This discrepancy between retreat and advance experiments implies multiple possible steady states for a given forcing .

However, the ice geometry in the spin-up simulations does vary significantly with choice of sliding relation (Fig. ). The steady-state ADVA_SR1 profiles have their steepest surface slope close to the grounding line due to the step change in basal drag from grounded to floating ice (Figs.  and ). The steady-state SR2 and SR4 profiles (Fig. ) have their greatest surface slope further inland where the overburden pressure becomes important. The steady-state SR3 profile is similar to SR1 towards the ice sheet interior but is thinner due to having lower drag close to the grounding line. It is very similar to SR2 near the grounding line.

Figure  also shows the shear stress component of the Cauchy stress tensor, σxz, for SR1 and SR3. Given the low gradient of the bed, σxz at the bed is an approximation to basal drag. Shear stress peaks at the grounding line for SR1 due to the step change in basal drag. The inclusion of dependence on effective pressure at the bed via z* in SR3 leads to a gradual change and a much larger transition zone.

The impact of choice of sliding relation on the way the modelled ice sheet responds to changing resolution is shown for retreat simulations in Fig. . The sliding relations featuring a step change in basal drag across the grounding line (SR1, SR4a and SR4b, shown in teal, black and blue respectively – the top three families of curves in the upper plot) do not exhibit consistent behaviour with resolution, with the RB_SR1_R0 simulation in particular showing no retreat of the grounding line after the buttressing reduction. Note that of these three simulations the magnitude of this basal drag step change is smallest for SR4b and largest for SR1. The results in Fig.  are consistent with a smaller step change in basal drag being indicative of better convergence with resolution, similar to a previous result when using a “shelfy-stream” ice sheet model . However, even SR4b still shows significant resolution dependency, indicating that much finer resolution than is considered here would be required for a reliable simulation. The sliding relations in which basal drag goes smoothly to zero as the grounding line is approached (SR2 and SR3, shown in green and red respectively) show the most consistent behaviour with resolution in experiments RB and RHW.

The purpose of the RHW experiment (Fig. , lower panel), as distinct from the RB experiment, is to test dependence on resolution in the presence of basal melting. Because SR1 and SR4a showed strong resolution dependence already in RB they have been omitted from the RHW experiments. In general the consistency across different resolutions appears weaker in the case of the melt-induced retreat simulations (RHW) than the reduced buttressing simulations (RB). The ice sheet profiles at the end of the RHW simulations are outlined in grey in Fig.  for sliding relations SR2, SR3 and SR4b.

The SR3 retreat simulations are unique in exhibiting an overshoot: after a strong initial grounding line retreat a small advance is seen. In the RHW_SR3_R2 simulation damped oscillations can be seen. The reason for this behaviour is not clear, but the lower-resolution simulations fail to exhibit this behaviour, indicating at least some resolution dependency in this experiment. The grounding line positions in the RHW_SR3 simulations also do not show a monotonic progression with increasing resolution.

Figure  shows more clearly the resolution dependency in the final grounding line positions for the RB and RHW experiments, after 40 ka. Again, R1, R4a and R4b show much stronger resolution dependence than R2 and R3.

We now look more closely at the impact of basal melting on resolution dependency. We compare retreat and re-advance simulations. These experiments involve the lower melt and greater buttressing scenarios, applied to different starting configurations (see Table ). We also investigate the impact of the water column scaling, in which zero melt is approached close to the grounding line. Sliding relation SR2 is used for these experiments as it has shown much weaker resolution dependency than SR1 and SR4 and has not shown difficult-to-interpret behaviour such as the damped oscillations in RHW_SR3_R2.

Figure  shows both retreat (RL and RLW, red lines) and advance (AL and ALW, black lines) simulations with water column scaling either on (RLW and ALW) or off (RL and AL). The advance simulations have identical inputs to the corresponding retreat simulations in all respects except for initial conditions. Note that while the majority of these simulations were run for 20 ka, the AL simulations were run for 40 ka because 20 ka was not long enough to approach a steady state.

In the presence of water column scaling the advance and retreat simulations approach the same grounding line position at all resolutions, showing no significant resolution dependency (right-hand plots of Fig. ). This is consistent with the premise that a unique solution exists, which might be expected behaviour on a linear down-sloping bed , although this has not been proven in the presence of buttressing and basal melting that depend on ice shelf geometry.

However, where there is a large step change in basal melt across the grounding line (AL and RL, left-hand plots of Fig. ), the advance and retreat grounding lines do not approach the same final position. Behaviour is strongly resolution dependent, especially in the re-advance experiment. It is unclear whether retreat and re-advance simulations would eventually converge to the same solution, and finer-resolution simulations would be required to determine this with confidence.

Dependency on resolution appears to be stronger in the case of advance experiments than retreat experiments. This is in sharp contrast to the SPIN and ADVA experiments, which are a kind of advance experiment (in that the grounding line position is advancing through the simulation toward its final position), in which no significant resolution dependency was observed. This suggests that it is specifically the melting which causes resolution dependence and that it causes greater resolution dependence in advance than in retreat.

The step changes in grounding line position during the early stages of retreat (Fig.  upper panels) are typically indicative of a single element retreat for RLW_SR2 but are typically multiple element retreat steps in RL_SR2.

As in previous studies with different ice dynamic models (e.g. ), a step change in basal drag across the grounding line causes strongly resolution-dependent behaviour in the current study using the Elmer/Ice finite element Stokes flow model. A large step change causes stronger resolution dependency than a smaller step change. A comparable resolution dependency on basal melt is shown in the current study: a step change in basal melt across the grounding line causes significant resolution-dependent behaviour, worse for larger step changes. Cases demonstrating strong resolution dependence at the resolutions presented here are of low interest to the current project, which aims to identify situations where such resolutions are viable. Much weaker resolution dependence is found in the current study in the case where both basal drag and basal melt approach zero as the grounding line is approached from landward and seaward respectively.

A change in value of sliding coefficient for a given sliding relation can also impact on resolution requirements . But since SR1 will typically give a global maximum basal shear stress at the grounding line, and SR2 will typically give a global minimum basal shear stress at the grounding line, it is expected (and this is the result of comparing  to the current study) that choice of sliding relation has much greater impact on resolution requirements than the magnitude of the sliding coefficient.

The results of the melting experiments have important implications for application of model studies to real marine ice sheet systems. We have shown that, even when the ice sliding relation permits resolution-independent simulations at the widely achievable resolutions used in the current study, this situation can be negated by the abruptness of spatial onset of ice shelf basal melting. In RHW_SR2, RLW_SR2 and ALW_SR2 experiments, where the onset of basal melting was gradual due to the scaling factor Sw (Eq. ), acceptable behaviour was observed over the sequence of resolutions we explored. However, even in low melt rate scenarios, the absence of this gradual transition gave rise to much more significant resolution dependence and a failure of retreat and readvance simulations to arrive at a unique grounding line location. Clearly more studies are required to explore the influence of abrupt spatial onset of melting. As discussed earlier (Sect. ) high melt rates are observed within tens of kilometres of the grounding lines of major Antarctic outlet glaciers, with the likelihood that such melt rates occur immediately adjacent to the grounding line in the presence of strong sub-glacial outflows. Accordingly, marine ice sheet systems with low surface slopes near the grounding line (indicating low basal drag approaching the grounding line) and with low basal melting near the grounding line (such as might be the case in the absence of strong sub-glacial outflow) would likely be more easily achievable targets for modelling studies at the resolutions explored in the present study. For model studies of less tractable systems, very high resolution would be needed near the grounding line. While sub-grid parameterisations for grounding line position or cross-grounding line ice flux have been developed (e.g. ), there is clearly a new challenge to handle the influence of onset of basal melting on the near grounding line dynamics. Furthermore, parameterisations that work well in the absence of ice shelf basal melting will need to be tested in the presence of melting and may need to be modified.

The basal melt parameterisations presented here, in particular the choice of whether or not to implement water column scaling, are intended to provide opposing end members in terms of melt distribution near the grounding line. The current study has demonstrated the impact this choice has on required model resolution but does not advocate a particular melt parameterisation. Similarly, the sliding laws SR1 and SR2 are opposing end members in terms of basal shear stress near the grounding line. The choice of sliding relation has been shown to impact on resolution requirements, but a specific sliding relation is not advocated here. The choice of both melt parameterisation and sliding relation should be governed by the physical processes, not by numerical convenience. Our aim has been to demonstrate that different physical systems can have different resolution sensitivities.

The sliding relations presented here in which dependence on effective pressure at the bed is incorporated would have a stronger physical justification if used in conjunction with a computer model for sub-glacial hydrology, to replace the assumption that the hydrologic system is everywhere in contact with the ocean with a physically justifiable effective pressure distribution. It might be expected that, for the case of efficient channelised sub-glacial drainage , a strong hydrologic connection to the ocean may exist. However, in such a case, there may be very high local variations in basal water pressure, resulting in sticky spots (relatively low basal water pressure and hence high basal shear stress) in between active channels. To simulate such a system a very high model resolution would need to be used to represent basal processes, and potentially also for the grounding line, if these sticky spots are present close to the grounding line. For the case of less efficient “distributed” drainage , a lower resolution would suffice for the hydrology system, and perhaps also for the grounding line, since there would likely be uniformly high basal water pressures (i.e. low effective pressure) near the grounding line. Studies of grounding line behaviour in a coupled hydrology–ice sheet model would be of great benefit to further understand this issue.

The results from the current study appear to be in conflict with the findings of , who found that imposing a fixed length transition zone near the grounding line (similar to that proposed by ), where the basal drag is scaled linearly to zero as the grounding line is approached (from landward), did not significantly reduce the resolution requirements. There are, however, a number of significant differences between the current study and , such as the use of a direct physical motivation to impose the drag reduction in the current study, rather than imposition of linearity. We speculate that the key factor is that the imposed linear transition zone of is typically of the same order of magnitude as the element size, meaning that the step change in basal drag across the grounding line, while moderately reduced, is not reduced by an order of magnitude or more, as in the current study for SR2 and SR3. The effect of incorporating dependence on basal effective pressure on the basal stress gradient approaching the grounding line is evident for the current study in Fig. . The transition zone is several hundred kilometres for SR3. A future study with further simulations will be needed to fill the gap in experiment design between the two studies to confirm whether this difference in transition zone size is the actual explanation for the differences in resolution dependence between the two studies.

We have demonstrated that resolution requirements for marine ice sheet simulations with an evolving grounding line are highly sensitive to the physical implementation of both basal sliding and ice shelf basal melting. In particular a large step change in either basal drag or basal melting across the grounding line can cause strong dependence of model behaviour on resolution.

Any marine ice sheet modelling studies whose outcomes involve a moving grounding line should demonstrate convergent behaviour with resolution over the region of parameter space relevant to their experimental setup, bearing in mind that basal drag and basal melt can both cause resolution dependence and that resolution dependence may differ for an advancing and a retreating grounding line.

A significant implication of the current study is that conducting transient Stokes flow simulations of whole marine ice sheets, such as century-scale simulations of the West Antarctic Ice Sheet for example, is a potentially tractable problem where evidence supports both basal drag and basal melting decreasing smoothly to zero as the grounding line is approached from respectively grounded and floating regions. Conversely, if there is a sharp onset of basal melting immediately beyond the grounding line, high resolution might be required regardless of the character of the basal sliding relation.