We considered an incompressible, isothermal, and gravity-driven ice flow. The ice exhibits non-linear viscosity, and the flow is ruled by the Stokes equations, which reads div(σ)+ρig=0div(u)=0, where σ represents the Cauchy stress tensor, g the gravity force vector, ρi the density of ice, and u the velocity vector. The Cauchy stress tensor can be expressed as a function of the deviatoric stress tensor S and the cryostatic pressure p with σ = S - pI and p = - tr(σ)/3. Ice rheology is represented by a non-linear Norton–Hoff-type flow law called Glen's flow law, which can be expressed as S=2ηϵ˙. This equation links the deviatoric stress tensor S to the strain rate tensor ϵ˙. The effective viscosity η is written as η=12(EA)-1/nIϵ˙2(1-n)/n, where Iϵ˙22 represents the square of the second invariant of the strain rate tensor, A is the fluidity parameter, and E is an enhancement factor. A complete description of the model can be found in .

The ice-flow model described above was coupled with a calving model based on damage and fracture mechanics. Damage mechanics were used to describe the slow degradation of the mechanical properties of ice under the stress field, averaged at a mesoscale. Linear elastic fracture mechanics (LEFM) were used to describe the brittle initiation and propagation of crevasses.

Our damage model is inspired from the work of and relies on damage mechanics . The level of isotropic damage in the ice is quantified by a scalar variable D called the damage variable, which equals 0 for undamaged ice and tends to 1 for fully damaged ice. In order to avoid singularity when D = 1, an upper bound is set such that D cannot exceed 0.7, accounting for the fact that D = 0.6 usually refers to fully damaged ice see. Damage is advected with the ice flow and it follows ∂D∂t+u∇D=B⋅χifB⋅χ>00otherwise, where B is a numerical parameter called damage enhancement factor. Damage increase depends on the stress state: χσI,σth,D=max0,σI(1-D)-σth. Here σI is the maximum principal stress and σth represents a stress threshold for damage initiation. χ is called the damage criterion and quantifies the damage source term. Then, damage alters the deviatoric part of the Cauchy stress tensor S by introducing an effective deviatoric stress: S̃=S(1-D). This new effective stress reduces the effective viscosity of the ice through the expression of the enhancement factor that enters Eq. (): E=1(1-D)n. The depth of crevasse fields can be represented using a damage contour and a stress history of the ice can be recorded.

This damage model was coupled with an LEFM model inspired from, which was used to represent the rapid propagation of crevasses that characterizes calving events. In LEFM, the initiation of crevasse propagation depends on the stress intensity factor KI. To trigger propagation, the stress intensity factor, which depends on the size of the initial flaw and on the stress field, must be higher than the ice toughness KIc. An initial crevasse depth provided by the previously computed damage field was used to compute the stress intensity factor. Once propagation is initiated, the stress intensity factor is computed at sea level. If the stress intensity factor is higher than an arrest criterion KIa, the crevasse continues to propagate until it reaches the bottom of the glacier and triggers calving. Crevasse propagation may be facilitated by surface meltwater entering the crevasse. However, our model currently does not incorporate this process especially because of the lack of field observation that would be required to constrain it seefor details.

Among the numerical parameters required to run the model, three have to be calibrated, and are discussed below: the damage critical value Dc, the stress threshold σth, and the damage enhancement factor B. The damage contour given by D = Dc corresponds to the depth of pre-existing flaws (from which LEFM is applied), σth is the load that has to be applied to trigger ice damaging, and B quantifies the rate at which damage increases. The criterion for calving was initially proposed by for the calculation of penetration depth of surface crevasses. It was then expanded by in order to incorporate the growth of basal crevasses. The main difference with our model is that the crevasse propagation, in the formulation of , does not rely on linear elastic fracture mechanics.

The model summarized here is described in detail in (along with all sensitivity tests) and implemented in the finite element open-source model Elmer/Ice see.

We wanted to generalize our conclusions to a wide range of two-dimensional synthetic flow line glacier geometries of time-varying length (L) and thickness (H). To this end, we built 60 geometries that depend on five parameters: the inlet ice flux (Finlet), water depth (Hw), and damage parameters (σth, B, Dc). These parameters were sampled using a Latin hypercube sampling (LHS) method, which ensures that each probability distribution in the model is evenly sampled, using a given number of simulations and a given number of parameters to sample. The glaciers were built up from ice slabs initially grounded on a linear analytical prograde slope (1 %). The choice of prograde slopes is motivated by the fact that dynamical instabilities arising from retrograde sloped glaciers make intercomparison of a large set of geometries difficult. The meshes comprise ∼ 7000 quadrilateral elements. Horizontally, the refinement is higher at the front (10 to 15 m) and in the vicinity of the upper surface (2 to 3 m) to account for the processes that occur at the calving front as well as the production of damage and advection at the upper surface. The setup is illustrated in Fig. .

Boundary conditions are the same as those given in . In addition, some specific conditions are given below.

At the bed, the glacier can be either grounded or floating. The grounding line position is obtained through the resolution of a contact problem . The basal friction is linearly decreasing along the flow from 1.5 × 10-2 MPa m-1/3 a1/3 at the inlet boundary to 1.0 × 10-4 MPa m-1/3 a1/3 at x = 10 km. Feedbacks on basal friction arising from changes in glacier geometry are not studied here.

For the purpose of comparison, we chose to apply submarine melting and ice mélange forcing on glaciers in a quasi-steady-state (QSS) mode, i.e. their front has to stabilize within a given range lower than the length of one calving event. Among the 60 simulations generated by the LHS sampling, 20 had this feature and are listed in Table . Other geometries either advanced too far without calving or collapsed because of prolific calving. The sets of damage parameters with which a QSS was reached generally differed slightly from those calibrated in . B ranged from 1.5 to 3 MPa-1 and σth from 0.01 to 0.11 MPa compared with 0.5 to 2 MPa-1 and 0.01 to 0.2 MPa respectively in. The explanation for these differences is straightforward: the geometries studied here flow on a linear bedrock with no bumps or roughness. Consequently, except near the front, no high-velocity gradients appeared in the upper surface. Damage is consequently more difficult to initiate than in cases of rough bedrock and consequently has to be promoted. In addition, since thinner glaciers are subject to less internal stress, they require parameters that promote damaging, unlike thicker glaciers (see Table ).

For the sake of clarity, out of the 20 representative geometries, unless otherwise specified, in Sects.  and , we only use one to illustrate the model's response. This terminus-floating geometry (hereafter referred to as Geo 9) is shown in bold in Table . However, the conclusions obtained in this study are robust against all the geometries considered, as discussed in Sects.  and .

The 20 setups summarized in Table  were run for 7 years to reach the QSS discussed above (spin-up). After this spin-up, for each setup we imposed eight perturbations in melting or ice mélange as well as a control run (CR), in which the glacier continues its QSS evolution (i.e. without any melting or ice mélange forcing). These forcings were maintained for 5 years, after which they were removed, and we let the geometries evolve freely for 5 more years (relaxation period). In total, we performed 180 simulations of 17 years each.

The perturbations described in Sect.  and  are listed in Table . Perturbations in submarine frontal melting are named U1 to U4 (for “undercutting”). Ice mélange perturbations are named S1 to S4 (for “Sikussak”, the Greenlandic word for ice mélange).

The simulation starts on 1 January (time = 0), when the prescribed melt rate is set to its minimum value. The distribution of maximum melt rate for U1 to U4 and for the control run is given in Fig. a. The control run was not subject to any melt rate and its front position never moved by more than a few tens of metres (see Fig. b). Figure  shows the shape of the glacier under perturbation U3 in the middle of the summer season (day 173). Simulations U1 to U4 produced slight oscillations, resulting in a slight advance of the terminus compared to the control run, but it never moved more than 400 m downstream. These advances may seem counterintuitive as most research suggests that submarine melting causes the front to retreat. However, our model revealed that when an advance takes place, it is not triggered by the same mechanism in all the setups. It is related to (i) a decrease in the frequency of calving events (this process is described in Sect.  below) and/or (ii) a torque effect caused by the retreat of the lowest point of the front (due to melting) and the advance of the highest point. In the case presented here, the advance of the front is due to a decrease in the frequency of calving events. Its geometry is illustrated in Fig. .

As soon as the forcing was removed, all the fronts reached their QSS position within a few months, except in simulation U4 (whose specific behaviour is discussed in Sect. ). The ice velocity at the front varied within a range of 200 m yr-1, which is very similar to the natural variation in the control run after a calving event (see Fig. 3c).

Considering the contribution of melting and calving to ice loss, the volume of calved ice always appears to be larger than the melted volume. During summer, melting accounted for up to ∼ 23 % of the total mass loss (Fig. ). Comparing winter and summer suggests that an increase in the intensity of undercutting does not significantly alter the total loss: more ice is melted but less ice is calved, meaning the cumulated volume does not vary significantly over the seasons.

To account for the different geometries, we summarized them as a function of their QSS mean thickness and velocity at the terminus for a given realistic forcing in melting (U2) (Fig. ). The conclusions drawn above were qualitatively confirmed for all the other geometries: mean ice loss during the melting season was comparable to the ice loss during the rest of the year, whatever the size and velocity of the glacier. In summer, the calved volume was reduced by the increasing melt rate, but the cumulative loss remained unchanged.

To measure the impact of ice mélange, we ran simulations S1 to S4 (Table ), as well as the control run. Figure a shows the ice mélange intensity. Figure b shows changes in the position of the front as a function of time in the four corresponding experiments and in the control run (dashed black line). Two types of behaviour were observed during the first 5 years. In winter, the ice mélange strengthened, and calving frequency decreased or stopped. As a consequence, the front advanced. In summer, the decay of the ice mélange back force was immediately followed by a rapid sequence of large-scale iceberg calving events. In all the perturbation simulations, the glacier front was always located further downstream than in the control run. Each winter, the gaps between the positions of the S1–S4 fronts and that of the control run increased to reach a value of 500 m in S1 and 3 km in S4. Moreover, in perturbations S2, S3, and S4 after each year, the front did not retreat back to its QSS position, suggesting a consequence for interannual mass loss. These behaviours are consistent with observations, confirming the hypothesis that a strong mélange reduces calving discharge .

Figure c shows the ice velocity at the front using the same colour scale. The position of the terminus was inversely correlated with the velocity of the ice. The advance of the front in winter led to a decrease in ice velocity due to the increasing buttressing effect of the glacier sliding against the fjord walls and, to a lesser extent, to increasing back pressure with increasing ice mélange strength (e.g. S3 and S4). As can be seen for the three highest back forces (S2, S3, S4), when the mélange collapsed, the ice flow at the front accelerated to a faster speed than the maximum in the control run. The increase in speed can be explained by the following chronology: first, the release of the mélange back force accelerated the ice flow; second, after the first calving event was triggered, the resulting geometry was a high vertical ice cliff. Velocity vectors were no longer parallel to sea level and a torque appeared, leading to a force imbalance that further increased the ice velocity at the front. The red inset in Fig.  shows the first year of the forcing and underlines this phenomenon: at stage 1, mélange strength was maximum and its decay accelerated ice flow. At stage 2, the major front retreat in simulation S4 occurred, further accelerating the flow.

Such rapid acceleration of the flow was observed during calving events at the front of the Jakobshavn glacier in May 2007 by . Near the front, GPS stations recorded an increase in ice velocity from 11 315 m a-1 to 12 775 m a-1 in 20 days, during which the glacier underwent three calving events that were attributed to an adjustment in the stress field. The freshly calved glacier was shorter, the buttressing effect was reduced, and glacier flow accelerated . monitored the magnitude of the speedup at 550 m a-1 during the days following the break up of the ice mélange at the terminus of Store Glacier. Our results are in agreement with these variations in measured velocity.

Figure  shows the mass loss for all geometries for perturbation S2. Whatever the glacier geometry, the loss in summer was greater than in winter, as the winter ice mélange layer reduced calving activity. However, the same back force does not have the same effect on small and large glaciers. In the case of smaller glaciers, it completely prevents calving; in the case of larger geometries, it only decreases the iceberg discharge. This explains why in the winter inset in Fig. , the thickest glaciers show the smallest contrast between winter and summer. This feature is also visible in Fig. b: large ice mélange intensities prevent calving (light blue, yellow, and red curves), while smaller intensities simply reduce calving frequency (dark blue). This consideration reinforces the need for better knowledge of the properties of ice mélange.

According to , to prevent the rotation of a calved iceberg away from the terminus, the required back force is between ∼ 1.0 × 107 N m-1 and 10.0 × 107 N m-1, depending on the glacier flotation and the inclination angle of the iceberg.

Concerning perturbations caused by the ice mélange, our model suggests that calving ceases as soon as the applied back force reaches a given value. We investigated model sensitivity to the applied back force by evaluating the value of this threshold. To this end, we isolated the pairs of parameters (σbmax, h) for which the winter season was characterized by the absence of calving events. For each of the five winter seasons, we then calculated the back pressure applied when calving ceased using Eq. (). Multiplying this back-stress by the thickness of the ice mélange gives a back force per metre of lateral width. The corresponding distribution for the 45 values is given in Fig. , of which the mean value is around 1.1 × 107 N m-1. Thus, the value of the back force that prevents the iceberg rotation as calculated by and the one which prevents fracture propagation are on the same order of magnitude.

Our coupled ice flow and calving model enabled us to distinguish between the different processes that culminate in iceberg calving that could be affected by the ice mélange. In the first stage, development of the crevasse field is determined by the damage criterion χ, which quantifies the damage increase in the ice. Figure a shows changes in the position of the terminus over a period of 150 days, i.e. a full winter season. Three key events are highlighted by diamond symbols. The red diamond corresponds to a situation in which the glacier is about to calve, the yellow diamond illustrates a case where the glacier is subject to ice mélange, and the blue diamond corresponds to a situation in which the glacier has just calved. Figure b shows the value of χ along the upper surface where the tensile stress is the highest. During the ice mélange season (yellow curve), damage production is slightly lower than that in the pre- and post-calving situations but remains positive. This means that ice mélange reduces the production of damage but that the effect is too weak to completely halt damage to the ice.

Following damage to the ice, three criteria have to be fulfilled to trigger calving: the condition on damage contour D = Dc, the initiation of fracture propagation at a depth given by the Dc contour, and propagation to sea level. These criteria are shown in Fig. c, where the stress intensity factor is represented on the vertical axis. The horizontal extent of the solid coloured lines shows the length of the damage envelope (the front is located on the right side of the figure): a longer one illustrates a more extended crevasse field. As expected, the solid blue curve is almost nonexistent: as calving has just occurred, the crevasse field is not deep enough to apply LEFM model (D < Dc) or the stress intensity factor is lower than the propagation threshold (KI < KIc). In contrast, the solid red and yellow lines show that the surface is sufficiently damaged to reach the criterion D = Dc over a larger surface area. Wherever the stress intensity factor becomes higher than the ice toughness (KI > KIc), crevasse propagation begins. This condition was satisfied in the case of the red and yellow lines near to the upper surface, so propagation can begin. The criterion KI > KIa must then be validated at sea level. The stress intensity at sea level is represented by the coloured dashed curves. The red curve satisfies this criterion at some points, meaning that the calving event can begin. However, when the ice mélange layer is present (yellow dashed line), this criterion is not fulfilled, and as a consequence the crevasse cannot propagate down to sea level.

Regarding model sensitivity to the thickness of the ice mélange, we observed that a thinner layer associated with a stronger back force reduced the calving rate more than a thicker layer associated with a weaker back force, with the same total back force (data not shown). This is because the thinnest layer of ice mélange is concentrated at sea level, which significantly reduces the stress intensity factor at this depth, thereby preventing crevasse propagation.

These mechanisms could explain the behaviour observed in Fig. b. For simulations S2, S3, and S4, the figure shows that the decay of the mélange layer was followed by a “cascade” of calving events. However, the glacier did not immediately retreat to its QSS position. This rate of retreat depends on the degree of damage at the surface, which depends on the driving force of the ice mélange. This suggests that a stronger or longer winter season could alter the position of the front over a period of more than 1 year, when one considers the “stress history” of the glacier, and does not only rely on a one-off record of the stress balance.

The results presented in this section are in direct contrast with those of . These authors observed no remarkable changes in the position of the front unless they applied a back force of 50.0 × 107 N m-1, although they simulated a difference of 25 m in the longitudinal extent of the crevasse field near the front for smaller values of ice mélange (5.0 × 107 N m-1). Conversely, the results of clearly agree with ours, as they simulated a comparable advance of the front (∼ 1.5 km) with a back force close to ours in simulation S2. Finally, for an applied force of the same order of magnitude, our model shows that the ice mélange acts sooner than suggested by by preventing the propagation of the fracture down to the glacier base.

Figure a shows the maximum difference in the surface along-flow component of the deviatoric stress tensor Sxx between the U2 and the CR simulations, in the vicinity of the front (< 600 m) during the middle of the first summer period. Undercutting grounded glaciers slightly increased the tensile stress at the upper surface (red dots). It increased the frequency of calving events but reduced the duration of each event (see red dots and crosses in Fig. b). Conversely, in the case of floating glaciers, the surface adjustment of the tongue decreased the tensile stress compared with the control run (Fig. a, blue dots). Consequently, the frequency of calving events decreased slightly, but the distance the front retreated at each event increased slightly (blue dots and crosses in Fig. b).

Concerning the behaviour of the grounded geometry of Helheim Glacier, stated that the melting of the front has relatively little effect on the position of the front, unless the prescribed melt rates are extremely high (up to 20 m day-1). On Store Glacier, modelled a slight increase in frequency with undercutting, as well as a decrease in the amplitude of the retreat of the calving front, and they attributed this interannual stability to the glacier's topographic setting. As the geometry of was grounded for most of the melt season, their modelling results are in agreement with ours. Here, it is worth to be mentioned that we only managed to observe a front retreat with especially high melt rates (≥ 12 m day-1). Incorporating melting from underneath the floating ice tongue would probably make the glacier front to retreat.

In our simulations, another difference appeared between grounded and floating glaciers. In Sect. , we showed that for most perturbations, after the relaxation period, glacier fronts usually reached their QSS position. However, this was not the case for the glacier undergoing perturbation U4 illustrated in Fig. b. Indeed, its front rapidly advanced further downstream than the others and appeared to stabilize at an extent of 6.2 km, compared with 6.0 km for the other fronts. When extended to other geometries, the same result also was obtained in some of the ice mélange experiments. However, it only concerned glaciers with a floating tongue and only occurred under the strongest forcings.

Concerning these processes, we propose an explanation for this phenomenon (see Fig. ). The melting perturbation applied on the glacier front affects the shape of the floating tongue (Fig. b). It reduces its area along with the subsequent buttressing effect. As a consequence, the whole glacier accelerates and thins, and the grounding line retreats (Fig. c). The ice flux at the grounding line is therefore modified, and a new equilibrium is established that relies on interactions between the ice flow, damage production, and the calving law. Considering the ice mélange, the concept is similar but the process is reversed (Fig. d). As the ice mélange prevents the floating tongue from calving, the area of the tongue increases. Consequently, the glacier slows down and thickens, and the grounding line advances (Fig. e). Again, a new equilibrium may be established with an associated quasi-steady state front dynamics.