Progress in Numerical Modeling of Antarctic Ice-Sheet Dynamics
Numerical modeling of the Antarctic ice sheet has gone through a paradigm shift over the last decade. While initially models focussed on long-time diffusive response to surface mass balance changes, processes occurring at the marine boundary of the ice sheet are progressively incorporated in newly developed state-of-the-art ice-sheet models. These models now exhibit fast, short-term volume changes, in line with current observations of mass loss. Coupling with ocean models is currently on its way and applied to key areas of the Antarctic ice sheet. New model intercomparisons have been launched, focusing on ice/ocean interaction (MISMIP+, MISOMIP) or ice-sheet model initialization and multi-ensemble projections (ISMIP6). Nevertheless, the inclusion of new processes pertaining to ice-shelf calving, evolution of basal friction, and other processes, also increase uncertainties in the contribution of the Antarctic ice sheet to future sea-level rise.
KeywordsIce-sheet modeling Antarctica Marine ice Sheet instability
Unlike atmospheric modeling, Antarctic continental-scale ice-sheet modeling only fully emerged at the beginning of the 1990s (e.g., [68, 70]). Initially, such models were employed at rather coarse resolution (∼50 km) to investigate ice-sheet changes during glacial-interglacial cycles [69, 105] or on longer time scales . At that time, ice sheets were believed to be a slow component of the climate system with a highly diffusive response to surface mass balance change. However, while the possibility of rapid continental change was advocated several decades before [81, 126], ice-sheet models were unable to deal with rapid changes, which was clearly identified as a limitation for Antarctic ice-sheet modeling .
Due to the increase of satellite data observations witnessing rapid mass loss, ice-sheet modeling needed a paradigm shift, which came through an improved insight into grounding-line (limit between the grounded ice sheet and the floating ice shelf) physics and the way to represent this in ice-sheet models [67, 95, 111, 123]. Since then, Antarctic ice-sheet modeling has taken a great leap forward based on an improved understanding of key processes and the development of assimilation methods leading to the ability to reproduce observed ice-sheet changes and by making refined future projections.
This paper reports on the recent progress made since the Fifth IPCC Assessment Report (AR5, ), focusing on advances in understanding key glacial processes, the way these processes are included in ice-sheet models and how this has affected century-scale projections of future Antarctic ice-mass change.
Advances in Process Understanding and Modeling
Damage and Calving
Iceberg calving (IC) occurs when ice chunks break off from the edge of a glacier, mostly from floating ice shelves in Antarctica, where IC is directly responsible for almost half of the ice mass loss [33, 103]. IC also modulates buttressing induced by ice shelves [36, 49] and therefore indirectly contributes to upstream grounded ice speed up and subsequent sea level rise (SLR) contribution . The large amount of ice lost through IC is common for Antarctica , but its representation and quantification in models is hampered by the difficult access to field sites, a high variability in time and space, and its inherent discontinuous nature, as opposed to the continuum approach mostly used in models. Until recently, calving rates were essentially based on empirical relationships (an excellent review is given in ), making them unable to be extended to other regions.
Physically-based discrete approaches were recently developed, in which the ice body is made of discrete particles linked to each other by bonds that can break off when undergoing a too high stress. While initially developed along flow lines [8, 11], Åström et al.  coupled IC with a continuum three-dimensional ice-flow model. However, the high computational cost of the discrete approach limits its use to ideal cases or small areas.
Physically based continuum approaches are mostly based on the work of Benn et al. , which states that ice calves when a crevasse extents vertically from the surface down to below sea level. A crevasse penetration depth is calculated based on the pioneering work of Nye , and depends on the equilibrium between an opening term being longitudinal stretching, and a closing term being cryostatic pressure. Studies applying this criterion (and derivations of it) to Greenland [26, 86, 87] and Antarctica [32, 100, 101] could qualitatively reproduce coherent variations of the calving front. However, the approach is based on an instantaneous stress balance field, thus not considering accumulated weaknesses in the ice advected with the ice flow, nor the stress concentration at their tip on their vertical propagation.
This issue was addressed by recent studies that applied continuum damage mechanics (CDM) for simulating crevasses. The CDM theory represents initial ice micro-defects and their vertical development as crevasses, which in turn weakens the ice through damage and decreases ice viscosity. This approach was used to describe the crevasse field of non changing Antarctic glaciers [2, 21, 22, 23, 35, 102]. Applied to evolving glaciers, this approach was improved by Krug et al. , who added the concept of Linear Elastic Fracture Mechanics in which the crevasse propagation depends on the fracture toughness at the crevasse tip, and is advected with the ice flow. This work, along a 1D-flowline of Helheim glacier in Greenland, was extended in two horizontal dimensions by Sun et al.  for the damage part (initiation and advection). Using the concept of plastic necking, Bassis and Ma  adopted a complementary approach suggesting that initial narrow cracks at the bottom of cold ice shelves can be enlarged under extensional stress. The morphology of the resulting wide crevasse can then be modulated by either ice accretion or sub-shelf melting. This model was applied to Antarctic ice shelves (Ross, Filchner-Ronne, Amery and Larsen C).
The Antarctic ice sheet is surrounded by topographic highs emerging from the Antarctic continental shelf . They pin the ice shelf from below, hence contributing to ice-shelf buttressing. These pinning points can eventually lose contact with the ice-shelf bottom as a result of current ice-shelf thinning , which makes them crucial to be considered in ice-sheet models in order to provide correct predictions of Antarctic contribution to SLR.
About half of those pinning points (ice rumples, rises, or promontories) are smaller than 100 km 2 , making them non-resolvable in continental-scale ice-sheet models, which have to parametrize them by a local increase in ice-shelf basal friction  within floating ice-shelf grid cells. However, high-resolution ice-sheet models need to be employed to physically represent pinning points accurately. In essence, pinning points have two effects: if the ice shelf becomes pinned (due to ice-shelf thickening for instance), then the grounding line has a tendency to advance and stabilize the system [41, 55]. However, the presence of a pinning point does not necessarily protects ice shelves from unstable retreat, but may lead to a slow down of grounding-line retreat rates . The latter study also found that ice rises are formed during a deglaciation phase, i.e., when the grounding line retreats and the ice shelf locally remains stuck on a bedrock high. Pinning points should also be considered in ice-sheet model initialization with inverse methods [15, 43]; their absence otherwise alters the initial buttressing state since pinning points locally slow down ice velocity .
Relative sea level change also influences the stability of marine ice sheets , as ice-sheet mass loss causes a falling of sea level at the grounding line due to bedrock rebound and sea-surface water migrating away . Such a sea-level fall reduces the ice flux at the grounding line by decreasing its ice thickness, hence stabilizing the ice sheet. Gravitationally self-consistent sea-level models incorporating Maxwell viscoelastic deformation of the solid Earth have recently been coupled to ice-sheet models to investigate their effect on millennial time scales [19, 62] as its importance for the stability of the West-Antarctic ice sheet on these time scales has been demonstrated . However, given the thin elastic lithosphere and low-viscosity upper mantle in West Antarctica, this effect may even be important on centennial time scales .
Tides significantly affect the dynamics of Antarctic ice streams, increasing or decreasing their speed when they are high or low, respectively, which is induced by a change in stresses at the grounding line. Their effect on ice flow can extend hundreds of kilometers upstream from the grounding line . Few studies have investigated the tidal effect on ice-sheet dynamics so far ([107, 108] for ideal cases and  for an Antarctic ice shelf). All of them applied a nonlinear viscoelastic law for ice rheology, as opposed to the traditional viscoplastic Glen’s flow law, which is less relevant than the former when the change in stress—due to tidal movement—occurs over a very short (hourly) time scale.
Sub-shelf melting is responsible for more than half of the ice mass loss at the margins of the Antarctic ice sheet [33, 103]. As with IC, sub-shelf melting decreases the buttressing capacity of ice shelves via loss of pinning points. In particular, this has been presumably been the trigger of the observed acceleration of large Antarctic outlet glaciers in the Amundsen Sea sector during the last decade [42, 73].
Sub-shelf melting can either be parametrized or computed through the coupling with an ocean model. Parametrizations relate sub-shelf melting to ocean temperature and ice-shelf depth , either in a linear or a quadratic fashion (e.g., [32, 128]), which leads to higher melting close to the grounding line. Other parametrizations relate sub-shelf melting to ice-shelf depth and distance to the grounding line [28, 43, 64, 85], to the ice-shelf and cavity depths , or more recently by using melt rates from a plume model that are extended spatially using physically motivated scalings depending upon local slope and ice draft .
More accurate representations of sub-shelf melting can be achieved by the coupling to an ocean model, which should be an improvement compared to simple parametrizations, since it accounts for the transfer of heat, freshwater and momentum between the two bodies. The principle of coupling is simple: sub-shelf melting is computed by the ocean model, which is used by the ice-sheet model to modify the cavity beneath the ice shelf; the new cavity, in turn, constrains the ocean model, and so on. Although this seems straightforward, the problem remains highly complex because of high computational costs and different grid resolutions that hamper exchanges between the two models. So far, this coupling has been done between a plume ocean model and a 1D flow line  or a 2D plan-view ice shelf [113, 125], hence without considering the grounded ice sheet (the grounding line is thus fixed). Gladstone et al.  coupled a 1D time-dependent flow-line model to a box model for ice-shelf cavity circulation and made centennial predictions for Pine Island Glacier. 2D plan-view coupling with ice-sheet models has been realized for ideal cases [30, 57], and more recently for Pine Island Glacier . The time steps employed by both models differ by few orders of magnitude, making the coupling asynchronous.
Despite the lack of direct observations of sub-shelf melting limiting the development of parametrizations as well as ocean model validation, highly detailed mapping of sub-shelf melt has become gradually available [16, 39, 82, 103]. Nevertheless, ice/ocean interaction seems to drive the current loss of Antarctic ice, which motivates future developments into coupled models in line with increased computing power .
Ice-Sheet Model Intercomparisons
The concept of MISI (Fig. 1b), first proposed 40 years ago [121, 126], provided a basis to hypothesize a possible collapse of West Antarctica as a consequence of anthropogenic global warming . However, the MISI theory was challenged by Hindmarsh , considering that ice shelves were too weak (and therefore mechanically uncoupled from the ice sheet) to affect the force balance of the grounded ice sheet. This led to the hypothesis of neutral equilibria that were neither stable or unstable . The resulting non-uniqueness of numerically-derived ice-sheet profiles, and the failure to recognize the importance of model resolution, confounded attempts to reproduce consistent dynamical behavior among different models . MISI theory was also disputed by observations of an apparently balanced ice sheet, but that was before several ice shelves collapsed in the Antarctic Peninsula, leading to increased glacier discharge , and glaciers started to retreat in the Amundsen Sea Embayment (ASE) [104, 117, 127]. Schoof  gave mathematical formulation of the MISI theory put forward by Weertman , dispelling the idea that ice sheets and ice shelves are mechanically uncoupled. This theory showed that the ice-sheet margin should be treated as a moving boundary problem that requires two constraints, i.e., a hydrostatic floatation constraint and a flux constraint that satisfies the momentum balance for ice-sheet and ice-shelf flow (mechanical coupling).
This had a profound impact on ice-sheet model development, pushing models to conform to known (analytical) solutions , which led to international model intercomparisons, such as MISMIP  and MISMIP3d . These numerical experiments demonstrated that in order to resolve grounding-line migration in marine ice-sheet models, a sufficient high spatial resolution needs to be applied, since membrane stresses need to be resolved across the grounding line to guarantee mechanical coupling. The latter condition needs at least a model based on the so-called Shallow-Shelf approximation (SSA) that basically only includes membrane stresses, thereby neglecting vertical shearing. However, MISMIP3d demonstrated that such models wrongly estimate ice-sheet mass changes compared to models that include vertical shearing and membrane stresses (so-called higher-order and full-Stokes models). Several approximations (vertically integrated) have recently been developed that include those components in one way or the other to optimize computational efficiency [24, 27, 54, 99, 112].
Alternatively, some models  have implemented the boundary-layer theory as a parametrization in a large-scale model, with the main advantage that high spatial resolutions can be avoided. However, subsequent intercomparisons, such as MISMIP3d, showed that while the general behavior complies with theory, transient behavior may deviate .
A recent intercomparison  tackles challenging geometries of ice shelves and grounding lines (MISMIP+), where grounding lines may reach stable positions on retrograde slopes  due to ice-shelf buttressing. Asay-Davis et al.  take it even further and propose an experiment of a coupled ice sheet-ocean system, where the ocean model provides sub-shelf melt distribution, making the ice shelf thin and the grounding line retreat, and the new cavity shape is then returned to the ocean model. This challenging setup is the first intercomparison of real ice-sheet/ocean coupling and will become a benchmark for future modeling attempts.
Finally, understanding the past to give insights into the future has motivated the PLISMIP-ANT intercomparison project , focussing on the Antarctic ice-sheet behavior during the late-Pliocene warm period.
Based on the MISMIP series, several further improvements in both numerical treatment and understanding of grounding-line migration have emerged. Most studies, and the MISMIP series is not an exception, use the Weertman friction law (WFL) to represent basal velocity as a power-law function of basal shear stress . Both terms are linked by a basal friction coefficient, which is low for high sliding areas such as ice streams and high across the interior ice sheet. Due to the lack of measurements, this coefficient is currently determined by the use of inverse methods (see “Model Initialization”).
The dependence on spatial resolution, as mentioned in “Ice-Sheet Model Intercomparisons” decreases with reduced change in basal shear stress across the grounding line . The inherent abrupt change in basal friction occurring across the grounding line with the WFL – zero friction below the ice shelf – thus requires high spatial resolution (e.g., < 1 km for Pine Island Glacier; ) for an accurate representation of grounding-line migration (Fig. 1c). Therefore, a series of ice-sheet models have implemented a spatial grid refinement, mainly for the purpose of accurate data assimilation [28, 50, 83], but also for further transient simulations where the adaptive mesh approach enables the finest grid to follow the grounding-line migration [27, 28]. However, MISMIP led to the development of subgrid interpolation schemes enabling grounding-line migration without necessarily decreasing the grid size drastically at the grounding line [29, 45, 114].
To circumvent the somewhat non-physical discontinuous jump in friction between grounded and floating ice obtained through the WFL, Tsai et al.  derived a grounding line boundary layer solution for a Coulomb friction law (CFL), in line with the solution of Schoof  for WFL. According to CFL, basal shear stress is proportional to the effective pressure (difference between ice overburden and water pressure) and a friction coefficient, the latter related to the rheological properties of the till. Near the grounding line, till is assumed cohesionless and hydrostatically connected to the ocean. The CFL therefore ensures a smooth transition of basal shear stress across the grounding line, reducing it to zero because of the vanishing effective pressure (Fig. 1c). This physically represents the hydrological connection between the subglacial system and the ocean in the transition zone . The CFL condition makes marine ice sheets also more sensitive to climate perturbations because of the zero effective pressure condition at the grounding line (contrary to the WFL condition) and a greater dependence on ice thickness [77, 93, 122].
A key aspect of projecting future Antarctic mass loss with dynamical ice-sheet models is related to the initial state of the model. Since ice-sheet models were initially applied for palaeo-climatic studies on long time scales (see “Introduction”), initialization was generally obtained from a long spinup time leading to a steady-state ice sheet (both in terms of geometry and thermodynamics). However, for predictions on shorter time scales (decades to centuries), a stable spinup generally leads to an ice-sheet geometry far different from the one currently observed , which is one of the reasons why such ice-sheet models may respond differently than observations suggest. Moreover, using a steady-state for initializing the ice sheet prevents models from properly accounting for the dynamical mass losses observed over the last decade, as the present-day ice sheet is not in steady state .
Diagnosing whether grounding lines are stable or unstable demands more than characterizing bedrock slopes (inland sloping beds may potentially lead to MISI). Ice flux across the grounding line also depends on ice rheology, basal sliding conditions and ice-shelf buttressing (e.g., ). Such diagnosis requires model initialization and parameter estimation. New developments in data assimilation methods led to improved initializations in which the initial ice-sheet geometry and velocity field are kept as close as possible to observations by optimizing other unknown fields, such as basal friction coefficient (Advances in Process Understanding and Modeling) and ice stiffness (accounting for crevasse weakening and ice anisotropy; [4, 5, 28, 78, 83, 84]). Motivated by the increasing ice-sheet imbalance of the ASE glaciers over the last 20 years , and supported by the recent boom in satellite data availability, data-assimilation methods are progressively used to evaluate unknown fields using time-evolving states accounting for the transient nature of observations and the model dynamics [51, 56, 58, 59].
The increase in computational efficiency enabling high spatial resolution modelling, high-resolution datasets of bedrock topography and surface velocity, longer time series on ice-sheet changes, and the improved initialization of ice-sheet models are now allowing ice-sheet modelling to move away from the slow-diffusive response over millennium time scales toward robust predictions on decadal time scales, hindcasting and potentially reanalysis. For the first time, Antarctic ice-sheet models have become part of the Coupled Model Intercomparison Project (CMIP6) under ISMIP6, where the focus on ice-sheet model initialization is currently underway .
A few years ago, the first international initiative to produce multi-model average projections of the contribution of Greenland and Antarctica to SLR, SeaRISE, was launched [18, 88]. Based on the premise that the combined results are more robust than the evaluation of a single model, it estimated that the contribution of the Antarctic ice sheet to future SLR during the next century will likely be within -20 to + 185 mm. However, Antarctic SeaRISE projections should be regarded with caution, as a lack of physics in a number of participating models most likely biased the projections . Furthermore, irreversible ice loss induced by MISI could not be ruled out, and if initiated, would substantially increase Antarctic contribution to SLR. Limited process understanding and lack of evidence at the time of publication of IPCC AR5, reduced the confidence in the possibility of MISI occurrence, so that SLR was estimated to be limited to less than one meter during the twenty-first century .
Since 2013, ice-sheet modelling studies have suggested that Pine Island Glacier  and Thwaites Glacier , which are currently the Antarctic glaciers experiencing the largest mass loss , may already be engaged in a MISI. A sustained retreat in the ASE is to be expected even without additional forcing from climate change [6, 44, 115]. Furthermore, Wilkes and Aurora basins in East Antarctica exhibit a similar topographic configuration with a retrograde bed slope upstream the current position of the grounding line, but it would require substantial atmospheric forcing (temperature anomaly > 5 °C) to trigger MISI in the near future [1, 61, 80].
0.11 ± 0.11
0.49 ± 0.20
1.05 ± 0.30
Pliocene: 10-20 m
0.02 ± 0.13
0.26 ± 0.28
0.64 ± 0.49
Pliocene: 5-15 m
0.25 ± 0.23
5.69 ± 1.00
15.65 ± 2.00
Pliocene: 10-20 m
0.19 ± 0.42
3.97 ± 1.97
13.11 ± 3.04
Pliocene: 5-15 m
Conclusions and Outlook
Over the last decade, Antarctic ice-sheet models have been greatly improved, especially regarding grounding-line migration and model initialization, and through a better understanding of processes such as iceberg calving, sub-shelf melting, ocean coupling and solid-Earth effects. From a theoretical viewpoint, we now have the ability to verify marine ice-sheet models, albeit that certain effects, such as buttressing, cannot be quantified/verified accurately. A danger with intercomparisons such as MISMIP, however, is that models, solely by their participation, automatically get approved while still oversimplifying the underlying mechanisms.
The availability of continent-covering satellite datasets over longer time spans has opened up the opportunity for model validation and initialization at unprecedented scale. Once these time series become sufficiently long enough, hindcasting with ice-sheet models may be envisaged.
Crucial processes of ice-shelf breakup (hydro-fracturing) and calving front/cliff stability still need to be further explored. While such mechanisms aid at explaining past changes in the Antarctic ice sheet, they do show a higher sensitivity to forcing, and hence lead to a significant larger mass loss [32, 100].
While the early models developed during the 1990s where thoroughly tested via the EISMINT intercomparisons for thermomechanical ice-sheet models [71, 98], several new-generation models lack the thermodynamic part as they strongly focus on abrupt changes at the marine boundary. However, since recent studies (e.g., ) point to significant high mass loss even on centennial time scales, thermomechanical aspects should not be neglected.
Finally, we deliberately did not treat advances in basal hydrology and basal characteristics (apart from initialization of basal friction fields) of the Antarctic ice sheet, simply because very few advances have been made. This may be deplorable, because on longer time scales or due to rapid ice changes, basal conditions may well change significantly and our knowledge on basal conditions remains far from complete.
Compliance with Ethical Standards
Conflict of Interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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