We introduce the first version of the Stochastic Ice-sheet and Sea-level System Model (StISSM v1.0), which adds stochastic parameterizations within a state-of-the-art large-scale ice sheet model. In StISSM v1.0, stochastic parameterizations target climatic fields with internal variability, as well as glaciological processes exhibiting variability that cannot be resolved at the spatiotemporal resolution of ice sheet models: calving and subglacial hydrology. Because both climate and unresolved glaciological processes include internal variability, stochastic parameterizations allow StISSM v1.0 to account for the impacts of their high-frequency variability on ice dynamics and on the long-term evolution of modeled glaciers and ice sheets. StISSM v1.0 additionally includes statistical models to represent surface mass balance and oceanic forcing as autoregressive processes. Such models, once appropriately calibrated, allow users to sample irreducible uncertainty in climate prediction without the need for computationally expensive ensembles from climate models. When combined together, these novel features of StISSM v1.0 enable quantification of irreducible uncertainty in ice sheet model simulations and of ice sheet sensitivity to noisy forcings. We detail the implementation strategy of StISSM v1.0, evaluate its capabilities in idealized model experiments, demonstrate its applicability at the scale of a Greenland ice sheet simulation, and highlight priorities for future developments. Results from our test experiments demonstrate the complexity of ice sheet response to variability, such as asymmetric and/or non-zero mean responses to symmetric, zero-mean imposed variability. They also show differing levels of projection uncertainty for stochastic variability in different processes. These features are in line with results from stochastic experiments in climate and ocean models, as well as with the theoretical expected behavior of noise-forced non-linear systems.

Process-based numerical ice sheet models (ISMs) are the principal tool for projections of future mass balance of the Greenland and Antarctic ice sheets and their future contribution to sea-level rise. They simulate gravity-driven ice flow, given some climatic forcing and boundary conditions at the surface, basal, and lateral boundaries. In the past decade, a number of physically based ISMs aimed at simulating large-scale ice sheet dynamics and projecting future sea-level rise have been developed and/or substantially improved

Different ISM responses to climatic forcing stem from a variety of ISM differences related to selection of physical processes, approximation of ice flow equations, model resolution, initial conditions and geometry, numerical methods, and parameterization of poorly constrained processes

In addition to discrepancies between ISMs, another large uncertainty in ISM projections is attributable to future atmospheric and oceanic forcings

Furthermore, some glaciological processes exhibit variability on small spatiotemporal scales and thus cannot be resolved in current ISMs. Examples of such processes include iceberg calving, ice fracturing, and hydrology

Studies with simple idealized models have demonstrated that mountain glaciers and marine-terminating glaciers are sensitive to internal variability within the glacier and the climate systems

In this study, we describe the first large-scale stochastic ISM, the Stochastic Ice-sheet and Sea-level System Model (StISSM v1.0). StISSM v1.0 adds stochastic capabilities to the Ice-sheet and Sea-level System Model (ISSM;

The new stochastic capabilities are implemented within the core of the source code of ISSM. We refer readers to

Stochastic variability can be applied to a number of variables in ISSM, independently or with intervariable correlation. The stochastic variables implemented for v1.0 of StISSM encompass both climatic forcings and unresolved glaciological processes: surface mass balance (SMB), ocean forcing, calving, and subglacial water pressure. These variables were prioritized for the implementation of stochasticity because they are known to be subject to internal climate variability (SMB and ocean forcing), and/or they reflect the impact of unresolved small-scale processes (calving and subglacial water pressure)

The notion of “prescribed” means that the values of a variable are explicitly provided by the user, as opposed to calculated within ISSM. They can be prescribed as either single values or varying in space and/or time. Turning on stochasticity for such variables in StISSM v1.0 implies that Gaussian white noise is added to the prescribed values at a user-defined temporal frequency in the simulation. By definition, Gaussian white noise has zero mean and is uncorrelated in time. In StISSM v1.0, a generic variable

A common depth-dependent parameterization of floating ice melt rate uses a piecewise linear function in depth

The deterministic subglacial water pressure,

Thermal forcing, TF (K), quantifies the excess ocean temperature with respect to the freezing point of water at the interface between the front of grounded outlet glaciers and the ocean. TF thus enters in the parameterization of melt rates at the terminus of outlet glaciers,

Allowing all the

The stochastic time step corresponds to the temporal frequency at which new noise terms for the stochastic variables are computed. The only restriction on the choice of the stochastic time step is that it cannot be smaller than the main time step of the numerical model simulation, i.e., the time step used by ISSM. It is important to specify the stochastic time step separately from the simulation time step because the variability in a time series depends not only on the amplitude of the noise imposed but also on the temporal frequency at which the noise is imposed. As such, if the stochastic time step was simply set equal to the simulation time step, changing the simulation time step would modify the variability imposed by the forcings to the ice sheet. When the stochastic time step is larger than the numerical model time step, then the noise term is not changed on every model time step. Integrating the noise in this fashion, and requiring the provided noise parameters to be self-consistent with the stochastic time step, means that the ice sheet responds to a forcing with characteristics of variability unaltered by other numerical considerations. Thus, the noisy forcing frequency and amplitude remain independent of the numerical model time stepping scheme, and the latter does not influence the sensitivity of the ice sheet to stochastic variability. At this stage, StISSM v1.0 uses an identical stochastic time step for all variables modeled with additive Gaussian white noise (Eq.

The spatial dimensions of stochasticity account for the number of sub-domains of the computational domain that share the same noise terms. The stochastic fluctuations are uniformly applied in each separate sub-domain. For example, a domain could be separated into individual glacier catchments. The number of sub-domains is prescribed during the parameterization of the model and can be as large as the number of mesh elements in the domain.

StISSM v1.0 computes all noise terms according to a Gaussian distribution. The Gaussian noise can have different correlation features in space, in time, and between variables. While future work will focus on better constraining statistical distributions of variability in the processes of interest, many geophysical processes fluctuate with a Gaussian distribution when integrated over time

From the stochastic time step, each simulation time step is determined as being a stochastic model step or not. At a stochastic model step, new

Stochasticity has been implemented in the most general way possible, such that developing stochasticity for a new variable would only require reproducing the code from another variable, with minimal adaptations needed for variable names and for potential specificities of the new variable. Moreover, the stochastic noise generation is mostly implemented in a separate module, thus causing minimal interference to developments of any other aspect of ISSM. All the stochastic schemes are implemented in the C++ source code of ISSM and are integral parts of the core of the model, but the schemes are not called if stochasticity is not required by the user. The random number generator implemented in ISSM is the commonly used linear congruential generator, which is a recursive algorithm with the advantages of being fast and easy to implement

In an autoregressive (AR) process, the variable of interest evolves in time and depends linearly on its own previous values. AR models are a powerful tool in climatic time series analysis because they are discretized versions of differential equations. They capture characteristic timescales of geophysical processes, and they have been shown to characterize many complex climatic variables

We have implemented AR options in StISSM v1.0 for the three climate-related variables mentioned in Sect. 2.1: (i) SMB; (ii) deep-water melt rate,

All the variables computed via an AR model have their specific spatial dimensions and temporal setting. The spatial dimensions work in the same way as for the generic stochasticity (Sect. 2.2). The coefficients

The SMB AR scheme optionally allows for dynamic SMB–elevation feedback through prescribed altitudinal gradients of SMB. Such gradients, called SMB lapse rates, relate SMB values at individual mesh elements to varying elevation in space and/or time and are regularly applied for Greenland ice sheet simulations

In order to sample the component of irreducible uncertainty due to internal and climate variability, StISSM v1.0 allows for ensemble runs of the ice sheet model with stochastic parameterizations activated. Each ensemble is characterized by a selection of stochastic variables and a given configuration of the stochasticity (Sect. 2.1 and 2.2). The number of simulations, referred to as ensemble members, is chosen by the user; each member is then characterized by a unique stochastic realization. All the simulations for the different members can be run in parallel, allowing for efficient simulations. The runs of the different members are executed on different nodes, and each separate member run can further be parallelized on different processors using the usual ISSM parallelization capabilities

Parallelization procedure for ensemble runs.

choose stochastic variables, output variables, output frequency

configure stochasticity

launch simulation of

(

retrieve results of

We perform three sets of experiments to test and demonstrate the new capabilities of StISSM v1.0. The first set simulates a marine-terminating glacier with geometry taken from the benchmark configuration of the Marine Ice Sheet Model Intercomparison Project (MISMIP+, as described in

The MISMIP+ configuration is a well-known and thoroughly tested configuration. While this is the first study applying stochasticity to MISMIP+, our results can be compared to prior studies with a large range of different ISMs

Our MISMIP+ experiment follows the description and parameterizations of the MISMIP+ Ice1r design in

Initial steady-state configuration for the MISMIP+ experiments.

From this steady state, we perform five ensembles of transient simulations. The transient experiments are performed over a period of 500 years, with a time step of

The Idealized Quarter Ice Sheet (IQIS) experiment is performed on a square domain of

Initial steady-state configuration for the IQIS experiments.

The deterministic steady state of the ice sheet is constructed via a balance between constant positive SMB over the domain (0.4 m ice eq. yr

We use this steady state as an initial state for transient ensemble experiments with stochasticity. We perform four sets of ensemble experiments (Table

Configuration of the IQIS transient ensemble experiments.

The configuration of the IQIS transient experiments is detailed in Table

The ensemble experiments consist of 500 members and a simulation period of 500 years, and the spatial and temporal resolutions are kept identical to the final spin-up configuration. The stochastic time step is set to 1 year, such that the fluctuations imposed have an annual sampling frequency. The first set,

To demonstrate that StISSM v1.0 is readily applicable at the scale of ice sheet simulations, we simulate the evolution of the GrIS with stochastic SMB and ocean forcings. The configuration uses an initial state matched to observations but is spun up to reach a deterministic steady state before launching the transient experiments with stochasticity applied. The initial state uses the bed topography, the ice thickness and the ice mask from BedMachine v4

After this initialization, we perform a deterministic spin-up in order to reach a GrIS configuration in a steady state. We emphasize that the purpose of our simulations is not to predict future ice mass balance of the GrIS, and we do not argue that the real GrIS is in a steady state; our goal with this spin-up is to have a steady baseline against which to compare a transient ensemble. We separate the GrIS in 19 different basins following an existing delineation

Climatic forcing in each basin for the GrIS simulations. Basin numbers correspond to the delineation in

The spin-up itself is separated in two different phases, both of them using a weekly time step and the two-dimensional shallow-shelf approximation. In the first phase, we fix the ice sheet margin positions and implement free-flux boundary conditions at the ice margins, meaning that boundary ice fluxes adjust to incoming fluxes to keep margins fixed in space. We use a spatial resolution ranging from 25 km in the slowest-flowing areas to 2 km in the fastest-flowing areas. During this first spin-up phase, the modeled ice sheet adjusts to the SMB field until it reaches a steady state. The steady state of this first phase requires a dynamic equilibrium between ice flow and the SMB field and thus takes 30 000 years.

In the second phase of the spin-up, we allow for moving margins at 11 of the major outlet glaciers of the GrIS, where we parameterize ocean melt (Fig.

Calving rates applied at the 11 outlet glaciers where terminus migration is simulated and ocean melt parameterized. Each Roman numeral is associated with a corresponding glacier, shown in Fig.

Comparing the simulated GrIS state at the end of the spin-up to observations, the total ice mass and ice-covered area are

Steady-state ice velocities at the end of the second phase of the GrIS spin-up. Arabic numerals (black) show the individual basins. Roman numerals (cyan) show the outlet glaciers where ice front movement is simulated and ocean melt parameterized.

We use the GrIS final steady state as an initial state for our transient experiments with stochasticity turned on, which will cause deviations from the steady state. We perform a single transient ensemble of 200 members over 500 years, with a stochastic time step set to 1 year, thus representing annual fluctuations. While this initial state is very close to a steady state, we still perform a deterministic control run of 500 years to quantify the amount of deterministic model drift, which is minimal. In the stochastic transient ensemble, we apply stochastic fluctuations in the climate forcing fields SMB and TF. We represent both of these forcings as AR(1) processes (Eq.

In this section, we analyze the results of the transient MISMIP+, IQIS, and GrIS experiments in terms of total ice mass (Gt) evolution. While our analyses focus on a variable summed over the entire domain, we note that localized changes in ice thickness and/or ice extent occur and may be larger than the global patterns.

At the start of the MISMIP+ transient experiments, the total mass is 39 097 Gt, and the grounding line position is at

Statistics of the final ice mass distributions for the five MISMIP+ transient ensembles. The relative change is calculated with respect to the initial ice mass (39 097 Gt).

Evolution throughout the transient experiments of

Left: change in ice mass throughout the transient experiments for each ensemble member in the five MISMIP+ transient ensembles. Thick lines show the ensemble means. The right

The cause of mean gains in ice mass for the MISMIP+ transient ensembles relates to the initial grounding line position in a bed trough (Fig. 1) and to the form of the melt forcing imposed. Due to the depth-dependent melt parameterization (Eq.

Furthermore, the mean gain in ice mass is highest for the ensembles with higher

The initial state of the IQIS is in equilibrium; thus any deviation from the initial state is attributable to the stochastic fluctuations imposed in our

Statistics of the final ice mass distributions for the four IQIS ensembles. The relative change is calculated with respect to the initial ice mass (350 715 Gt).

Evolution throughout the transient experiments of

Left: change in ice mass throughout the transient experiments for each ensemble member in the four IQIS ensembles. In labels, variables between brackets denote variables with stochastic variability imposed (Table

In contrast, the results from

The perturbations

The results of

Applying decadal variability in ocean thermal forcing (

Results of the GrIS ensemble with correlated stochastic variability in SMB and TF forcings are shown in Fig.

Statistics of the ice mass distributions of the GrIS ensemble after 125, 250, 375, and 500 years of simulation. The relative change and deterministic drift are calculated with respect to the initial ice mass (2 743 269 Gt).

The skew in ice mass varies between positive and negative phases, while the ensemble mean is strongly decreasing and the ensemble spread strongly increasing after 500 years (Fig.

Evolution throughout the transient experiment of (red) the standard deviation and (green) the skewness in total ice mass for the GrIS ensemble.

Stochastic modeling is well established in climate modeling

Irreducible uncertainty is not quantified in current ice sheet model intercomparison projects

StISSM v1.0 allows for stochasticity in variables which exhibit internal variability. The features of spatiotemporal correlation can be prescribed, as well as intervariable correlations. Our model experiments show that the stochastic parameterizations implemented are functional and can be used at ice sheet scale. We have aimed at making StISSM v1.0 as user-friendly as possible, in such a way that any user familiar with ISSM should find the use of StISSM v1.0 straightforward. Ensemble runs and parallelization allow for adequate sampling of irreducible uncertainty in model simulations. In general, a StISSM v1.0 simulation run with stochastic parameterizations uses additional computational resources that are negligible compared to a corresponding deterministic simulation. As in methods for estimating the role of parameter uncertainty in ice sheet evolution

A practical question that arises concerns the number of members needed per ensemble. Here, we have used 500 members for the MISMIP+ and IQIS ensembles and 200 for the GrIS ensemble to limit computational expense. As the number of members increases, the statistics of the ensemble progressively converge to the statistics of the true underlying distribution. In other words, results from ensembles with increasingly more members converge to the results of an ensemble with infinitely many members. Convergence plots of the statistics of interest, such as the final mean, standard deviation, and skew in our case, show their progressive convergence and are a useful tool in evaluating the number of members needed. We show such an analysis of our results in Appendix C, demonstrating adequate convergence of the ensemble statistics with 100 to 150 members in our experiments. It must be kept in mind that the number of members needed to represent the true distribution depends not only on non-linearities in the system that cause larger variability between members, but also on the timescale of stochastic variability imposed and on the statistics of interest. For example, correctly estimating the 99th percentile of the distribution requires a larger ensemble than for estimating the mean, as the effective number of members influencing this statistic is smaller.

With this first version of a large-scale stochastic ISM, several future research priorities can be identified. First, the statistics of variability in the unresolved processes need to be evaluated in order for the stochastic parameterizations to capture their impacts on ice dynamics accurately. This could be achieved via theoretical, observational, and/or high-fidelity modeling studies

Our results show that ice sheets need a long period of time to converge to a statistical steady state in the presence of noisy forcing. This raises the question of how zero-mean variability during spin-up could affect the ensuing modeled initial state of an ice sheet. We have used realistic inter-annual and decadal variability in SMB and ocean forcing for our GrIS experiment, reasonably representative of the noisy climate under which the Greenland ice sheet evolves. After 500 years of our synthetic experiment, this resulted in a

This study has described the development, implementation, and testing of StISSM v1.0, the first stochastic large-scale ice sheet model. Variables with implemented stochastic parameterizations in this first version encompass climate forcing and glaciological processes that are unresolved at the spatiotemporal resolution of ice sheet models: SMB, ocean forcing, calving, and subglacial water pressure. Stochastic climate forcing captures the irreducible uncertainty in climate predictions and how it translates into projected ice sheet mass balance uncertainty. Using stochastic parameterizations for unresolved glaciological processes facilitates the quantification of the impacts of internal variability in such processes on ice dynamics. StISSM v1.0 also includes built-in statistical models for generation of stochastic variability in SMB and oceanic forcing, represented as autoregressive processes. The statistics of the stochastic variability and of the autoregressive climate models are prescribed by users and can thus be adjusted to particular user needs.

We have tested the stochastic capabilities in idealized, synthetic model experiments. These tests have demonstrated that the stochastic parameterizations are functional, and can be upscaled to realistic ice sheet configuration. Our results show that stochastic forcings cause responses of the ice sheet system in line with those observed in stochastic climate and ocean model experiments. For example, stochastic forcing causes not only variability in the final state, but also non-zero tendencies in the response, noise-induced drift, and long timescales needed for ice sheet state convergence. Even in the simple experiments proposed here, the features of the response are complex and cannot be quantified without running ensemble simulations. The response of a particular system is sensitive to the type of forcing, to the geometric configuration, and to the intrinsic non-linearity of ice dynamics. Our results thus raise important questions about representing fluctuating processes with constant deterministic parameterizations, about neglecting high-frequency climatic noise, and about ice sheet model initialization performed without imposing variability.

Our strategy for the development of StISSM v1.0 allows for potential future extensions of stochastic capabilities to other variables in a straightforward manner. In the future, calibration work will be needed to constrain the statistical models for climate forcing, as well as the variability in unresolved glaciological processes such as calving and hydrology. Such an effort will require combining observations, theory, and results from high-fidelity model experiments to understand the internal spatiotemporal variability in processes of interest. Our implementation allows for any spatial, temporal, and intervariable correlation features. StISSM v1.0 thus provides a robust modeling framework to quantify the impacts of forcings with internal variability on ice sheet mass balance.

In this section, we briefly provide additional details concerning the GrIS configuration at the end of the spin-up (Sect. 3.3). The fields of ice thickness and ice velocities are displayed in Fig.

Initial configuration from observational datasets:

Differences in

In Fig.

The correlation matrix relating noise terms for both SMB and TF in all basins of the GrIS stochastic transient runs is specified by Eq. (

Convergence of the

Convergence of the

Convergence of the

In this section, we analyze how the statistics of interest converge as the number of members per ensemble increases. The statistics of interest are the mean, the standard deviation, and the skew in final ice mass. The analyses of convergence for the MISMIP+, IQIS, and GrIS ensembles are shown in Figs.

The stochastic schemes evaluated here are currently implemented in the public release of ISSM. The code can be downloaded, compiled, and executed following the instructions available on the ISSM website:

VV led the model development, performed the model experiments, and led the writing of the manuscript. AAR supervised the work. HS contributed to the model development. VV, AAR, HS, and AFT conceived the study. LU and VV have worked on the formulation and calibration of the statistical models. All authors provided comments and suggested edits to the manuscript. All authors, and this study, are part of the Stochastic Ice Sheet Project, aimed at understanding ice sheet sensitivity to variability.

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

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Helene Seroussi was also funded by the NSF Navigating the New Arctic program. Computing resources were provided by the Partnership for an Advanced Computing Environment (PACE) at the Georgia Institute of Technology, Atlanta. We acknowledge HPC assistance from Fang (Cherry) Liu. We thank Mathieu Morlighem and Justin Quinn for providing helpful advice about ISSM. We thank Kevin Bulthuis for the implementation of the random number generator. We thank all developers of ISSM for their continuing work on model development. We acknowledge the two anonymous reviewers for their constructive comments that helped improve the quality of the manuscript. Vincent Verjans thanks John Christian for his interest in the study and for insightful discussions about ice sheet sensitivity to variability.

This research has been supported by the Heising-Simons Foundation (grant no. 2020-1965).

This paper was edited by Christopher Horvat and reviewed by two anonymous referees.