Modeling ice sheet instabilities is a numerical challenge of potentially high real-world relevance. Yet, differentiating between the impacts of model physics, numerical implementation choices, and numerical errors is not straightforward. Here, we use an idealized North American geometry and climate representation (similarly to the HEINO (Heinrich Event INtercOmparison) experiments –

The use of ice sheet models has grown by at least 1 order of magnitude over the last 2 decades. The relevance of such modeling studies to the actual physical system can be unclear without careful consideration and testing of numerical aspects and implementations. This is especially true when modeling the highly non-linear ice sheet surge instability, which has significant implications not only for the ice sheet itself but also for the climate. In fact, it is often difficult to assess whether model results are physically significant (effects of physical system processes), a consequence of model-specific numerical choices, or a combination of both. Whether ice sheet instabilities observed in numerical simulations are the result of physical instabilities of the underlying continuum models or of spurious effects of the discretization and numerical implementation of said models has long been debated (e.g.,

Binge–purge ice stream cycling was first introduced in the glaciological literature by

As a result of the physics involved and the behaviors expected, modeling of ice stream surge cycling is challenging. The challenges entail, among others, rapid surge onset, high ice velocities, and non-linear (thermo-viscous, hydraulic, and thermo-frictional) feedbacks. In addition to the physical complexity, further challenges arise in the numerical modeling of ice stream surge cycling, whether in terms of model choices (e.g., choice of mechanical model, thermal modeling of the substrate, accounting for sub-glacial hydrology) and/or in terms of their numerical implementation (e.g., grid size, convergence under grid refinement).

Our focus here is on the challenges arising from numerical modeling, both those related to the physical system being modeled and those related to the numerical implementation. The effects of different approximations of the Stokes equations have been previously addressed

The discretization and related numerical implementation choices (e.g., grid resolution and grid orientation) have been shown to affect numerical results (e.g.,

An additional level of complexity in the modeling of ice sheet surge cycling arises from the fact that small perturbations of the initial or boundary conditions can significantly vary the surge characteristics

Herein, we disentangle the effects of numerical choices (e.g., grid size) and physical system processes (e.g., sub-temperate basal sliding) on ice sheet surges via numerical experiments.

In terms of ice flow models, we primarily use the 3D Glacial Systems Model with hybrid shallow-shelf–ice physics (GSM,

In terms of different numerical choices, the impact on model results is usually determined by calculating the model error in relation to the exact analytical solution. However, the theory behind the surge instability is not fully developed (no analytical solution exists) in the context of a spatially extended 3D system, thus precluding systematic benchmarking of numerical models.

To overcome this issue and to provide at least a minimum estimate of the numerical model error, we first determine minimum numerical error estimates (MNEEs). This is a minimal threshold to resolve whether a change in surge characteristics due to changes in the model configuration is significant (see Sect.

Equipped with these tools, we set out to tackle the research questions detailed in Sect.

In this subsection, we detail the key research questions that we address through numerical experiments. Following the above-described structure in the description of the results, the research questions are divided into three sub-categories: minimum numerical error estimates (MNEEs), sensitivity experiments, and convergence study.

What is the threshold of MNEEs in the two models (Sect.

We examine the significance of different model configurations to the surge characteristics. We are particularly interested in model configurations affecting the basal temperature and thus the surge behavior. Therefore, we first discuss the change in surge characteristics due to a bed thermal model (

Is the inclusion of a bed thermal model a controlling factor for surge activity (Sect.

Except for PISM, all models in the HEINO (Heinrich Event INtercOmparison) experiments did not include a bed thermal model

Do different approaches to determining the grid cell interface basal temperature significantly affect surge behavior, and if yes, which one should be implemented (Sect.

On a staggered grid (commonly Arakawa C grid;

How much of the ice flow should be blocked by upstream or downstream cold-based ice, or equivalently, what weight should be given to the adjacent minimum basal temperature (Sect. S8.1 in the Supplement)?

At relatively coarse horizontal grid resolutions (e.g.,

How different are the model results for different basal- temperature ramps? And what ramp should be used (Sect.

Another issue that is often ignored is the basal-sliding thermal-activation criterion. Based on the results of

An additional argument for sub-temperate sliding can be made on numerical grounds for coarse horizontal grid resolutions. It is unlikely that an entire grid cell reaches the pressure-melting point within one time step (e.g.,

Experimental work

We use basal-temperature gradients in fine-resolution runs and approximations of the sub-grid warm-based connectivity between the faces of, e.g., a

Does the abrupt transition between a soft and hard bed significantly affect surge characteristics (Sect.

An abrupt transition from hard bedrock to soft sediment (as, e.g., used in the HEINO experiments;

How does a non-flat topography affect the surge behavior (Sect.

Given the topographic lateral bounds of the Hudson Strait, we examine the effects of a non-flat topography on the surge characteristics.

What is the effect of a simplified basal hydrology on surge characteristics in the GSM (Sect.

The implementation of a fully coupled basal-hydrology model changes the basal drag and, therefore, has the potential to affect the surge characteristics. A basal-hydrology model coupled to an effective-pressure-dependent sliding law or a Coulomb-plastic bed (as in PISM) introduces a positive feedback such that larger sliding speeds increase frictional heating and thus meltwater availability, which further weakens the bed and leads to even faster sliding. Different basal-hydrology process representations have been proposed in the literature (e.g., a 0D (

How significant are the details of the basal-hydrology model to surge characteristics in PISM (Sect. 8.2)?

PISM surge characteristics are compared for local and mass-conserving horizontal-transport hydrology models.

What are the differences (if any) in surge characteristics between local basal hydrology and a basal-temperature ramp as the primary smoothing mechanism at the warm–cold-based transition zone (Sect. S8.3)?

While both sub-glacial hydrology and a basal-temperature ramp provide a means for a smooth increase in sliding velocities, these processes operate in slightly different temperature regimes. The basal-temperature ramp enables sub-temperate sliding, and the maximum velocities occur once the pressure-melting point is reached. In contrast, a local basal-hydrology model increases sliding velocities once the basal temperature reaches the pressure-melting point (basal melting), and basal-ice velocities further ramp up with decreasing effective pressure (ice overburden pressure minus basal water pressure). Note that sub-glacial hydrology is not an alternative for a basal-temperature ramp. The ramp is still needed to prevent refreezing even when a description of sub-glacial hydrology is included

Do model results converge (decreasing differences when increasing horizontal grid resolution – Sect.

Incorporating the findings of the above experiments, we study numerical convergence with respect to horizontal grid resolution for surge cycling. By convergence, we mean decreasing differences between simulations when increasing the resolution.

The 3D thermo-mechanically coupled Glacial Systems Model (GSM) has developed over many years (e.g.,

The GSM is run with an idealized down-scaled North American geometry (Fig.

The surface mass balance forcing is then determined by

The GSM is initialized from ice-free conditions. The coarsest horizontal grid resolution is

Modified ISMIP–HEINO geometry

While

To partially offset the limitations of the zeroth-order approximations, the GSM uses hybrid SIA–SSA ice dynamics

We configure the GSM with a

Temperature ramps for different values of

Each GSM experiment is run with an ensemble based on five input parameter vectors. The current idealized setup encompasses a maximum of eight input parameters (Table

Model parameters are listed with respect to their purpose or category. Ice sheet model – ISM. Hydrology parameters used when running the GSM with local basal hydrology. Additional (non-regular) input parameters that are usually set to a fixed value. The default values of the 3.125 km horizontal grid resolution reference setup are shown as bold values (in brackets) for the additional parameters.

The reference setup (Table

In contrast to the GSM, the Parallel Ice Sheet Model (PISM) is not specifically developed for glacial-cycle ensemble modeling. Therefore, the two models use distinct sets of numerical optimizations for computational speed. To minimize the model dependency of our analysis, experiments are also carried out with v2.0.2 of PISM.

Similarly to the GSM, PISM is a 3D thermodynamically coupled ice sheet model, and the SSA is used as a sliding law once the sliding velocity exceeds

For stability reasons, the PISM adaptive time-stepping ratio (used in the explicit scheme for the mass balance equation) was reduced to

The default sliding law in PISM is a purely plastic (Coulomb) model, where

Comparison between the GSM and PISM reference setup.

The PISM configuration encompasses six model input parameters (Table

Parameters used to generate the PISM input fields.

The

A PISM ensemble parameter restriction arose as experiments carried out with PISM only show oscillatory behavior for small yield stresses

The resulting very slippery beds enabled occasional maximum sliding velocities of up to

As for the GSM, we carry out one-factor-at-a-time sensitivity experiments branching off the PISM reference setup (Table

The quantities being analyzed are the number of surges, the surge duration, the ice volume change during a surge, and the period between surges (Fig.

Pseudo-Hudson Strait ice volume of a GSM model run with visual illustration of the surge characteristics used to compare different model setups. The horizontal grid resolution is

In addition to the surge characteristics, the root mean square error (RMSE) and mean bias are calculated as a percentage deviation from the reference (pseudo-Hudson Strait) ice volume time series for all setups (each parameter individually) and are then averaged over the five parameter vectors (Eqs. S3 and S4). The full run time is considered (no spin-up interval).

We compare different model setups by calculating the percentage difference between the reference setup and all other setups for every parameter vector individually and then average this difference over all parameter vectors. Crashed runs are not considered, and runs with less than two surges require special treatment (see Sect. S5 for further details on the analysis).

In the GSM, the whole pseudo-Hudson Strait (Fig.

For PISM, a large fraction of the pseudo-Hudson Strait area is only ice covered when a surge occurs (e.g., Fig.

We compute the new minimum numerical error estimates (MNEEs) threshold by examining the model response to changes in the model configuration that are not part of the physical system. The MNEEs are defined as the percentage differences in surge characteristics when applying a stricter (than default) numerical convergence in the GSM and changing the number of processor cores used in PISM. The differences between PISM runs with different numbers of processor cores can be caused by, for example, a different order of floating-point arithmetic operations and the processor-number-dependent preconditioner used in PISM

While the MNEEs are useful for our purpose, we wish to emphasize that they can not replace proper model verification and validation and are missing uncertainties due to, e.g., different approximations of the Stokes equations and other physical processes not included in the models. Nonetheless, they provide a minimum estimate of the numerical model error, which is still a significant improvement over ignoring this issue entirely.

Before analyzing ensemble characteristics, it is crucial to understand how surges initiate, propagate, and terminate. Surges in the GSM originate at the pseudo-Hudson Strait mouth (

Basal-ice velocity for parameter vector

The surge propagates nearly symmetrically until the pseudo-Hudson Bay area is reached (

Since the GSM setup and climate forcing are symmetric about the horizontal axis in the middle of the pseudo-Hudson Strait (

Basal-ice velocity for parameter vector

Surges in PISM originate at the ice sheet margin in the soft-bedded pseudo-Hudson Strait (exact position varies between runs) and propagate towards the center of the pseudo-Hudson Bay (

Once the warm-based area connects with the margin (

Due to the differences in model setups, physics, and numerics (Table

Surge characteristics of the GSM (

Differences in surge characteristics (compared to the reference setup) are considered to be significant when they exceed the MNEEs given in Tables

To determine a minimum significant threshold in the GSM, we re-run a set of GSM runs with

The largest differences between simulations occur for the mean period (

Percentage differences (except first column) of surge characteristics between GSM runs with regular and stricter numerical convergence and increased maximum iterations for the ice dynamics loops at

MNEEs in PISM are determined by comparing runs with different numbers of cores. Although most parameter vectors show similar results at the beginning of the runs, minor differences can slowly accumulate and lead to significant discrepancies in surge activity by the end of the run (Fig. S18). The largest differences occur for the number of surges (

Percentage differences of surge characteristics (except first row) between the PISM reference setup and setups with different numbers of cores at

The differences in surge characteristics between different numbers of cores can be minimized (but not removed entirely) by decreasing the relative Picard tolerance in the calculation of the vertically averaged effective viscosity (

Low levels of surface temperature noise have previously been shown to cause chaotic behavior in the mean periods of oscillations

In contrast to the commonly used explicit time step coupling between the thermodynamics and ice dynamics in glaciological ice sheet models, we test the impact of approximate implicit time step coupling via an iteration between the two calculations for each time step. The implicit coupling decreases the mean duration and pseudo-Hudson Strait ice volume change (

Here, we discuss differences in surge characteristics due to changes in the model setup. An overview of the results can be found in Figs.

Percentage differences in surge characteristics compared to the GSM reference setup for model setups discussed in Sect.

Percentage differences in surge characteristics compared to the PISM reference setup for model setups discussed in Sect.

First, we examine the effects of a

Advection of cold ice near the end of a surge rapidly decreases the basal-ice temperature and, therefore, increases the temperature gradient between the basal ice and the bed. In GSM runs with the

Heat flux at the base of the ice sheet (positive from bed into ice) and basal-ice temperature for a grid cell in the center of the pseudo-Hudson Strait (grid cell center at

With only one bed thermal layer (

Another modeling choice that affects the thermal activation of basal sliding is the approach to determining the basal temperature at the grid cell interface. The most straightforward approach to determining the basal temperature with respect to the pressure-melting point at the grid cell interface (

TpmInt, on the other hand, calculates the basal temperature at the interface (

The last approach (TpmTrans) attempts to represent heat transfer from sub-glacial hydrology and ice advection by accounting for extra warming above the pressure-melting point, given by

The GSM reference setup (no hydrology) uses TpmTrans. The additional heat embodied in

The most straightforward approach, TpmCen, leads to

Here we examine the effect of different basal-temperature ramps (thermal-activation criteria for basal sliding) at

When running the GSM at

Percentage differences in surge characteristics compared to the GSM reference setup (

Except for the three widest ramps, the mean ice volume bias is less than

We compare the different temperature ramps at

At

A more physically-based approach to determining an appropriate scale-compensating temperature ramp stems from our motivation for research question

However, this upscaling analysis does not account for the connectivity between the faces of, e.g., a

Furthermore, the upscaling results depend on the bed properties (soft sediment vs. hard bedrock) and the specific scenario (surge vs. quiescent phase). Therefore, we only consider patches within the pseudo-Hudson Strait area during surges. Due to the limited storage capacity for the 10-year output fields, only the first

Warm-based fraction (basal temperature with respect to the pressure-melting point at

The upscaling results agree well with the score analysis at

The effects of a smooth sediment transition zone (instead of an abrupt transition from hard bedrock (

The abrupt transition from hard bedrock to soft sediment (pseudo-Hudson Bay and Hudson Strait) in the GSM reference setup and the corresponding difference in basal-sliding coefficient provide an additional heating source due to shearing between slow- and fast-moving ice. This additional heat appears to foster the propagation of small surges along the transition zone (e.g.,

Similarly to the GSM results, the PISM percentage differences between a smooth (reference setup) and abrupt sediment transition show no significant effect except for a

Pseudo-Hudson Strait ice volume for GSM parameter vector

Adding a

Comparing the results for two different widths of the topographic transition zone (

While imposing a non-flat topography fosters surges in both models, the increase in mean ice volume change is much larger in PISM (

Since the topography will vary from ice stream to ice stream, we stick to a flat topography for the remaining experiments.

The effects of adding a simple local basal-hydrology model to the GSM are examined here. The local basal hydrology sets the basal water thickness by calculating the difference between the basal-melt rate and a constant basal-drainage rate (rBedDrainRate in Table

The basal water thickness (

When running the GSM with the local sub-glacial hydrology model, intermediate values are used for all three parameters (the effective bed roughness scale

Adding the local basal hydrology model to the GSM increases the mean ice volume change and duration by

Since the local hydrology model effectively increases the basal-sliding coefficient, we test if this impact can be replicated simply by increasing the sliding coefficients (Table

The effect of an experiment is considered to be insignificant when the change in surge characteristics is smaller than the MNEEs (Sect.

In this section, we examine the horizontal grid resolution dependence of the GSM and PISM model results. Model results are considered to be converging when the differences in surge characteristics decrease with increasing horizontal grid resolutions.

Significant differences in surge characteristics occur when changing the horizontal grid resolution. These differences can be as large as a highly oscillatory behavior at

Percentage differences in surge characteristics compared to the GSM reference setup (

We compare the differences in surge characteristics for different basal-temperature ramps at each resolution (Fig.

All three basal-temperature ramps lead to similar differences in surge characteristics at

Similarly to the results presented for the GSM, the ice volume RMSE and mean bias show convergence under systematic grid refinement (Table S26). However, for the three resolutions examined here, the PISM surge characteristics show convergence for the mean duration and ice volume change but not the number of surges and mean period (Table S25). Note that four out of nine runs at

This section summarizes our modeling results in the context of the research questions outlined in Sect.

What is the threshold of MNEEs in the two models?

The MNEEs can be as large as

In contrast to the findings of

Is the inclusion of a bed thermal model a controlling factor for surge activity?

Including a

Do different approaches for determining the grid cell interface basal temperature significantly affect surge behavior, and if yes, which one should be implemented?

The choice of approach for determining the basal temperature at the grid cell interface significantly changes the surge characteristics. Without considering additional heat transfer to the grid cell interface (as an attempt to represent heat contributions from sub-glacial hydrology and sub-grid ice advection), the number of surges decreases by at least

This additional heat transfer to the grid cell interface is comparable to spreading

How different are the model results for different basal-temperature ramps? And what ramp should be used?

Similarly to

Does the abrupt transition between a soft and hard bed significantly affect surge characteristics?

Incorporating a smooth transition zone with two different widths (

How does a non-flat topography affect the surge behavior?

Imposing a non-flat topography leads to significantly longer and stronger surges (Figs.

What is the effect of a simplified basal hydrology on surge characteristics?

Activating the local basal-hydrology model (including the addition of effective-pressure dependence into the sliding law) in the GSM significantly increases the surge duration and amplitude (Fig.

How much of the ice flow should be blocked by upstream or downstream cold-based ice, or equivalently, what weight should be given to the adjacent minimum basal temperature?

Changing the weight of the adjacent minimum basal temperature for the basal-sliding temperature ramp in the GSM yields a maximum difference of

How significant are the details of the basal-hydrology model to surge characteristics in PISM?

Incorporating a mass-conserving horizontal-transport hydrology model does not significantly change the surge characteristics in PISM (Fig.

What are the differences (if any) in surge characteristics between local basal hydrology and a basal-temperature ramp as the primary smoothing mechanism at the warm-based–cold-based transition zone?

Once included, the local basal hydrology is the primary smoothing mechanism. However, since the two smoothing mechanisms operate in different temperature regimes, a basal-temperature ramp (representing sub-temperate sliding) cannot be replaced by a basal-hydrology scheme (as in, e.g.,

Do model results converge (decreasing differences when increasing horizontal grid resolution)?

In general, both models exhibit convergence under systematic horizontal grid refinement for the overall ice volume (mean bias, Tables S19, S23, and S24), but the solution is not fully converged at the finest resolutions tested. However, while all surge characteristics converge for the GSM (Table S18), PISM results do not show convergence for the number of surges and mean period (Table S25). This clearly illustrates that mean ice volume and, consequently, mean ice thickness, as presented, e.g., in

While other studies examining thermally induced ice streaming do not find a strong resolution dependence

Even though the studies are not directly comparable, the results of

Within the limitations of hybrid SIA–SSA ice dynamics, we investigate the effect of ice sheet model numerics and discretization choices on surge characteristics often neglected in ice sheet modeling studies. We show how to reduce numerical and discretization sensitivities given finite computational resources and then how to determine the significance of model results given residual computationally unavoidable numerical sensitivities for surge cycling contexts. In particular, our analyses offer guidance on minimizing the resolution dependency by implementing a resolution-dependent basal-temperature ramp for basal-sliding thermal activation and increasing confidence in model results by determining minimum numerical error estimates (MNEEs). Based on these MNEEs, our results indicate that surge characteristics are significantly affected by the inclusion of a basal-hydrology model. Not including the dampening effect of a bed thermal model on basal-temperature changes, as has been the tendency in idealized process studies, overestimates the surge amplitude. The key takeaways of this study are the physical modeling choices and numerical sensitivities that must be considered when numerically modeling ice stream surge oscillations.

The GSM source code (v01.31.2023) and run instructions are available at

The supplement related to this article is available online at:

KH and LT conceptualized the ideas behind this study. All the authors (KH, LT, and EM) were involved in designing the experimental setup of the GSM. KH designed the experimental setup for PISM and performed the modeling analysis for both models under the supervision of LT. All the authors contributed to the results, interpretation, and writing of the paper.

The contact author has declared that none of the authors has any competing interests.

The views and opinions expressed here are those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank Andy Aschwanden, Ed Bueler, and Constantine Khrulev for their support with the Parallel Ice Sheet Model (PISM). We thank Ed Bueler and Daniel F. Martin for the fruitful discussions about the bed thermal model and the numerical tolerances, respectively. We also thank Florian Ziemen and Clemens Schannwell for the insightful discussions on modeling Heinrich-event-like surges. Finally, we thank two anonymous reviewers and the handling topic editor Ludovic Räss for their constructive comments.

This research has been supported by an NSERC Discovery grant (grant no. RGPIN-2018-06658), the Canadian Foundation for Innovation, and the German Federal Ministry of Education and Research (BMBF) as a Research for Sustainability initiative (FONA) through the PalMod project. Elisa Mantelli was supported by the European Union (ERC-2022-STG, grant no. 101076793).

This paper was edited by Ludovic Räss and reviewed by two anonymous referees.