Calving of icebergs is a major mean of ice discharge for marine terminating glaciers around the world. It accounts for about half of the ice discharge of the Greenland and Antarctic ice sheets . This process causes calving front retreat, which leads to reduced basal and lateral resistive stress and results in upstream flow acceleration.

In order to assess the impact of calving on the dynamics of outlet glaciers using an ice sheet model, we need to include a dynamically evolving calving front. This requires tracking the calving front position and adjusting the boundary conditions accordingly. Addressing these issues is rather straightforward for 1-D flow-line or 2-D flow-band models , where the calving front is tracked along the flow line. However, this type of model lacks the consistent representation of lateral momentum transfer and lateral ice influx from tributaries for example, which have to be parameterized instead. This parameterization may neglect feedback effects important for simulations on decadal to centennial timescales, e.g. catchment area entrainment .

It is therefore critical to include a front tracking scheme in 2-D horizontal and 3-D models, which has been addressed by only a few ice sheet models e.g.. Various approaches to model the evolution of the shape of ice exist. Explicit methods track the position of a set of points, which represent the calving front. They require a complex technical framework to allow for geometric operations like folding and intersection of the continuum boundary, tracking singularities in curvature, and determining the position of a point in space relative to the modelled continuum.

Alternatively, the level-set method LSM; represents the continuum boundary implicitly by a contour, or “level set”, of a so-called “level-set function” (LSF). It easily handles topological changes of the modelled continuum, like splitting and merging. The LSM is based on a partial differential equation similar to the mass transport equation solved by ice sheet models. This makes the method straightforward to implement and allows for the application to continental-scale ice sheet simulations. Although the method does not necessarily conserve volume accurately, it is well established in continuum fluid mechanics . A LSM has been applied to represent the ice surface in 2-D flow-band models but not to model real ice sheets yet.

Understanding calving dynamics remains challenging because of the diversity of factors involved in calving events. Bathymetry, tides, and storm swell, as well as sea ice cover, ice mélange, and temperatures of both sea water and air are possible factors influencing calving rates. However, their effect, their respective share, and their interplay seem to vary from glacier to glacier and are not well understood . Therefore, no universal calving rate parameterization exists to date , and we rely here on user-defined calving rates. However, incorporating calving rate parameterization in the LSM should be straightforward.

Jakobshavn Isbræ is a major marine terminating glacier in West Greenland, which drains about 6.5 % of the Greenland Ice Sheet . It is characterized by two branches, which today terminate into a 30 km long ice-choked fjord (Figs.  and ). The southern branch exhibits high flow velocities, which are confined to a narrow, deep trough of about 5 km width. The trough retrogradely slopes inland to a maximum depth of about 1700 m below sea level and discharges most of the ice of the drainage basin . We refer to the fast-flowing area as “ice stream” and to the surrounding slow-moving ice as “ice sheet”. Those areas are separated by pronounced shear margins on either side of the ice stream.

Observations have shown that the fast flowing areas of Jakobshavn Isbræ exhibit a weak bed with a basal layer of temperate, soft ice . Basal sliding and shear in this layer cause most of these areas' horizontal motion. A large fraction of the ice stream's momentum is transferred to the adjacent ice sheet by lateral stress. It is thus well justified to use the 2-D shelfy-stream approximation SSA, to simulate this glacier.

Until the late 1990s, Jakobshavn Isbræ had a substantial floating ice tongue, which extended well into the fjord and was fed by both branches. The calving front position remained fairly constant from 1962 to the 1990s , and the glacier exhibited negligible seasonal variations in flow speed . In the 1990s, the glacier started a phase of acceleration, thinning, and retreat that followed the breakup of its ice tongue. Seasonal variations in calving front position and flow velocity increased sharply . Today, the glacier is one of the fastest ice streams in the world. It is still far from equilibrium and is a major contributor to global sea level rise . Observations suggest that the calving front position is a major control on the ice stream dynamics .

Various hypotheses have been proposed to explain the mechanisms behind this change. All identify the breakup of the floating ice tongue as the initial trigger of this dramatic chain of events, but different mechanisms have been proposed to explain the sustained acceleration, thinning, and retreat of the glacier. On the one hand, studies by and propose loss of buttressing and changes in basal conditions as the main cause behind the ongoing acceleration. On the other hand, argue that weakening of the lateral shear margins has significantly amplified the upstream acceleration. Several modelling studies of the glacier, which use 1-D flow-line and 2-D flow-band models, project unstable retreat of the glacier along its southern trough for up to 60 km inland within the next century . Other modelling studies argue that this type of ice stream is stable as long as it is fed by the surrounding ice sheet . However, numerical 2-D plan-view modelling efforts of Jakobshavn Isbræ so far lacked the representation of a dynamically evolving calving front. Hence, the hypotheses could not be tested in a satisfactory manner.

We present here a LSM-based framework to model the dynamic evolution of a calving front. This method is a step towards better physical representation of calving front dynamics in 2-D and 3-D ice sheet models. We describe the implementation of the method into the Ice Sheet System Model ISSM,, a parallel, state-of-the-art ice sheet model, and apply it here to Jakobshavn Isbræ in order to model its dynamic response to perturbations in calving rate.

Figure  shows calving front positions for experiments A, B1, B2, and C4. Under constant calving rate forcing, the calving front remains at a stable position after minor readjustments in the first decade of the simulation. In experiment A, the calving front undergoes gradual retreat over time due to the slowdown of the glacier caused by its ongoing thinning. When we perturb the calving rate, the calving front migrates, and higher calving rates lead to larger retreats. The retreat is highest in areas of fast flow and strongly decreases towards the ice stream margins. This yields the characteristic concave shape of a retreating calving front. The modelled calving front positions and their shape are in good agreement with observations (Fig. ). The retreat rate during continued phases of calving decreases to 0, so that the calving front reaches a new stable position 9 km upstream of its initial position (Fig. d). In experiments B and C, the calving front returns to a similar position as in the unperturbed experiment B1 within 10 to 20 years after the perturbation stops.

Figure  shows ice velocity, geometry, and strain rates for experiment C4 along two lines, which go along and across the southern trough respectively (Fig. ). During the first 20 years prior to the perturbation, the ice thickness in the floating part decreases by about 100 m (Fig. a). As the calving front retreats during the perturbation, the ice velocity increases (Fig. b), and the ice thickness adjusts accordingly (Fig. a). The ice thinning leads to a fast retreat of the grounding line in the regions of locally retrograde bed and temporarily stabilizes over local along-trough topographic maxima, referred to as “local highs”. The southern trough has many local highs, which act as pinning points and are critical for flow dynamics, in agreement with earlier results from . The acceleration of the ice stream extends tens of kilometres upstream to areas of grounded ice (Fig. b). Thinning and acceleration are strongest over the ice stream and spread out to the surrounding ice sheet in a dampened fashion. These thinning and acceleration patterns increase surface and velocity gradients, especially in the shear margins (Fig. c), where the effective strain rates gradually increase up to a factor of 4 in experiment C4 (Fig. d), which corresponds to a drop in viscosity of about 60 % (Eq. ). This substantially weakens the mechanical coupling between the ice stream and the surrounding ice sheet.

Figure  shows the intra-annual variability of ice properties at the calving front and grounding line for experiments A, B1, B2, and C4. All shown variables reflect the characteristics of the applied calving rate forcing. The constant calving rate applied in experiment A leads to a steady configuration (Fig. a and b). For an unperturbed periodic calving rate forcing (Fig. c and d), the calving front position oscillates around a constant annual mean value by ±3 km, while the grounding line position remains unchanged at kilometre 29. Ice velocities and thickness at the calving front act in phase with the calving front position, while the response of strain rates at the grounding line is delayed by about a month. The ice velocity varies over a year by ±20 %, which corresponds to about ±2 km a-1, the ice thickness by ±13 %, or ±100 m, and effective strain rates at the grounding line by ±7 %, or ±0.1 a-1.

The response to a calving rate perturbation scales with p0 and Δt. When the calving rate doubles (B2, C4), the calving front retreats initially at an average rate of 4.5 km a-1. The calving front stabilizes 9 km upstream for longer perturbations (Fig. g). The intra-annual variability of the calving front position doubles to ±6.5 km. The grounding line position is hardly affected by small calving rate perturbations, but large perturbations trigger fast retreats of several kilometres, which in turn cause drastic, but short-lived, flow accelerations (Fig. g and h). The annual average ice velocity increases by 10 %, and its intra-annual variability doubles to ±38 % (Fig. h). The mean calving front thickness decreases by 30 % towards the end of the perturbation of experiment C4 and experiences large variations up to ±75 %. This high thickness variability is due to the front retreating into areas of thick ice in summer followed by stretching and thinning during calving front advance in winter. For small perturbations, variations of effective strain rates at the grounding line quadruple to ±25 % (Fig. f). Once the calving rate perturbation stops, all variables display remarkable reversibility.

When calving is temporarily turned off (experiment B0, not shown here), the response of the glacier is reversed: the calving front advances, creating a convex ice tongue. Meanwhile, the ice stream decelerates, thickens, and the grounding line advances. After the perturbation, the glacier retreats into a state slightly thicker and faster than the one of experiment B1.

Figure a shows the evolution of the ice volume with respect to experiment B1, the control run. The glacier in experiment B1 continues to lose volume at an average rate of -22.8km3a-1 due to the ongoing geometry relaxation. In experiment A, Jakobshavn Isbræ loses an additional 0.4km3a-1, which corresponds to the gradual retreat of the calving front. Enhanced calving causes additional volume loss proportional to Δt(1-p0), the measure of the time-integrated calving rate perturbation (Fig. b). If the calving rate is doubled, the additional volume loss reaches -35.7 km3a-1 in the first year but decreases with time, as the calving front thins and retreats into areas of lower calving rates. Those numbers agree well with recent ice discharge observations . Over the first decade after the perturbation, all modelled glaciers recover 40 to 60 % of the volume deviation to the control run.

The applied calving rate determines the behaviour of the calving front and the ice stream. In our simulations, larger perturbation strengths p0 lead to faster calving front retreats. In the case of long perturbations (experiments C4 and C8), the calving front reaches a new stable position. A stable calving front position requires the calving rate to be larger than the ice velocity if the calving front advances; similarly, if the calving front retreats, the calving rate needs to be lower than the ice velocity.

Several mechanisms determine how the model responds to the calving rate forcing. First, a calving rate increase leads to a retreat of the calving front position, ice stream acceleration and dynamic thinning in the vicinity of the terminus. Second, this dynamic thinning increases surface slopes and therefore the driving stress. As the glacier locally speeds up, the ice thinning propagates upstream. The ice stream thins much faster than the surrounding ice sheet, which steepens the surface across the shear margins. Lateral inflow of ice into the ice stream hence increases until it balances the calving flux. This limits the thinning of the ice stream. Thinning of the ice stream in turn leads to grounding line retreat and reduction in basal effective pressure, both of which reduce basal drag significantly in the vicinity of the grounding line. We showed that grounding line retreat leads to short-lived but drastic increases in ice flux. This mechanism is qualitatively the same as the one described in and . Several pinning points along the retrograde trough of the southern branch, as well as the lateral stress transfer and mass influx, prevent the modelled ice stream from being prone to the marine ice sheet instability , a hypothesis which states that grounding line positions are unstable on retrograde slopes. This corroborates earlier results by , who presented examples of stable grounding line positions on retrograde beds. However, due to large uncertainties in the input data, and since some physical processes are not represented in our experiments, evaluation of this question for Jakobshavn Isbræ is beyond the scope of this study.

A third mechanism is related to the calving front lengthening during its retreat (e.g. Fig. ). The lengthening causes tributaries of the main ice stream to calve directly into the fjord, thereby increasing the calving flux Qcf (Eq. ) and thinning of the terminus vicinity.

Finally, the ice stream accelerates faster than the surrounding ice sheet, which increases strain rates at the shear margins. This reduces the ice viscosity in these areas, which mechanically decouples the ice stream from the ice sheet, allowing the ice stream to accelerate further. This positive feedback confines the initial thinning to the ice stream and is controlled by the rate at which ice enters the ice stream. This mechanism is essential for enabling ice stream acceleration tens of kilometres upstream of the grounding line, since large fractions of the ice stream's driving stress are balanced by lateral stress. This corroborates force balance arguments produced earlier by .

In experiments A and B1, we apply the same annual mean calving rate. However, due to the lack of seasonal cycle in calving rate the mechanical coupling between the ice stream and ice sheet is higher in experiment A. The ice stream velocity is therefore lower, causing gradual calving front retreat and additional ice volume loss. This illustrates that volume change estimates from models with and without seasonal cycles of calving may differ. Our results suggest that including both a dynamically evolving calving front as well as seasonal cycles are critical for accurate projections of future contributions of ice sheets to global sea level rise on decadal to centennial timescales.

Response mechanisms not covered here will likely include feedbacks with damage mechanics and thermodynamics due to the increased strain rates. During longer perturbations, ice surface lowering will probably affect the surface mass balance and the drainage basin outline.

The reversibility of the calving front configuration after the calving rate perturbation is a robust feature across all experiments. The short duration of the perturbation, the prescribed calving rates, and the geometry of the glacier are responsible for this behaviour. The volume change in all experiments never exceeds 0.1 % of the initial glacier volume in the experiments shown here. Once the perturbation stops, the surrounding ice sheet continues to replenish the ice stream, which allows for its quick recovery.

The modelled glacier response to enhanced calving is in good qualitative agreement with observations, which corroborates that calving is a major control on this glacier. The similar shape of the modelled and observed calving front suggests that calving rates are indeed proportional to its flow speed during the glacier's current retreat. However, the reversibility of the modelled calving front position is in contrast to Jakobshavn Isbræ's actual behaviour. Sustained high calving rates are therefore necessary to explain the continued retreat of the glacier, as our results suggest that the glacier would have readvanced otherwise. Accurate model input data, representation of all relevant physical processes, and incorporation of a suitable calving rate parameterization will be necessary for quantitative analysis of this dynamic ice stream.

In this study, we present the theoretical framework for coupling a LSM to ice dynamics and implement it into ISSM. The LSM proves to be a robust method for modelling the dynamic evolution of a calving front. We apply this technique to Jakobshavn Isbræ using prescribed calving rates, and we find that the glacier is highly sensitive to this forcing, which agrees well with observations.

Calving rate perturbations strongly affect the ice stream through several linked mechanisms. First, changes in calving rate cause calving front migration and alter the ice discharge. Second, the resulting thickness change at the calving front spreads out to the surrounding ice sheet. Third, thinning-induced grounding line retreat causes further ice stream acceleration and creates a positive feedback. Finally, shear margin weakening caused by the ice stream acceleration decreases lateral drag resisting ice flow. This positive feedback mechanism sustains significant acceleration of the ice stream tens of kilometres upstream of the grounding line.

The surrounding ice sheet is barely affected by short periods of enhanced calving. It stabilizes the ice stream and allows for quick reversibility of the calving front position through lateral ice influx and stress transfer once the calving rates are set back to their initial values.

Since the calving front position and dynamic lateral effects are critical to simulate and understand the behaviour of marine terminating glaciers, the inclusion of moving boundaries in 2-D plan-view and 3-D models is key for realistic sea level rise projections on centennial timescales. This method is a step towards better physical representation of calving front dynamics in ice sheet models.