We use the ice sheet model Úa , which implements the vertically-integrated Shallow Shelf Approximation . The model solves simultaneously for the transient changes in ice thickness and ice velocities using a fully implicit time integration. Our domain covers the ASE region using inland boundaries based on the MEaSUREs Antarctic Boundaries , with smoothing applied. We use geometry from BedMachine Antarctica v3 , from which we derive the initial calving front. A shallow section of the bed downstream of the PIG grounding line is lowered to avoid an unrealistic initial advance and regrounding. Details can be found in Appendix . Our bed geometry is shown in Fig. a. A Dirichlet condition is used to set all velocities along the inland boundary to zero, since the domain boundary generally follows the edges of drainage basins. The initial mesh is created using Mesh2D , with a resolution of 1 km at the grounding line, becoming coarser further upstream up to 10 km. An adaptive meshing scheme is applied such that the mesh is refined to 1 km around the grounding line and calving front as they move.

The densities are given the values of 917 kg m⁻³ for ice and 1027 kg m⁻³ for ocean water, consistent with the BedMachine dataset. The flow follows Glen's law with exponent n=3 and basal sliding follows a Weertman power law : 1τ=C-1/m‖vb‖1/mvb‖vb‖ where τ is the basal drag, m=3 is a sliding exponent and vb is the basal velocity. C is a sliding parameter, which is inverted for along with the rate factor A from Glen's law. We follow the inversion process detailed for Úa in .

The calving rate in Úa is defined as the difference between the retreat rate and the ice velocity normal to the calving front. The implementation of calving via a level-set method is presented in Appendix .

Following the ISMIP6-2300 experimental protocol , we begin our simulations in 2015 and run to 2300. Surface accumulation and basal melting, which we hereafter refer to together as climate forcing, are derived from ocean-atmosphere coupled simulations as specified in the protocol, using the local quadratic melting parameterisation with median MeanAnt calibration as set out in . The equation for local quadratic melting, directly from the source, is 2m(x,y)=γ0×(ρswcpwρiLf)2×{max⁡TF(x,y,zdraft)+δTsector,0}2, where ρsw is sea water density, ρi is ice density, cpw=3974Jkg-1K-1 is the specific heat of sea water and Lf=3.34×105Jkg-1 is the fusion latent heat of ice. TF(x,y,zdraft) is the thermal forcing provided by the ocean-atmosphere models, and relies on the ice draft in the ice model. The coefficient γ0 and temperature correction δTsector are used for calibration. Full details of the MeanAnt calibration which determines values for γ0 and δTsector are found in , and summarised in Fig. 3 within that paper. We do not reproduce the method here.

The seven “Tier 1” experiments comprise the CMIP5 RCP8.5 scenario outputs from CCSM4 and HadGEM , the CMIP5 RCP2.6 scenario output from NorESM with repeated forcing after 2100, the CMIP6 SSP5-8.5 scenario outputs from CESM2 and UKESM , an additional UKESM output with repeated forcing after 2100 and a Control which applies constant thermal forcing throughout the simulation. The repeated forcing is taken from the 2080–2100 period, sampled randomly between 2100 and 2300 to avoid repeating the same forcing pattern.

Experiments are started with the present-day geometry of BedMachine. We prescribe constant, uniform retreat rates along the entire calving front of the ASE, ranging from 0 to 1 km a⁻¹, in steps of 0.1 km a⁻¹. The retreat rate of 0 km a⁻¹ is specifically included as a control, in which the calving front does not move. Each retreat rate value is used in an experiment forced by the Control forcing, with further experiments for 0, 0.5 and 1 km a⁻¹ run with each of the forcing scenarios. We refer to experiments by their climate forcing and retreat rate in the format “Forcing_RR#”, so for example the simulation using Control forcing with a retreat rate of 0.5 km a⁻¹ would be Control_RR0.5.

We only allow calving on fully floating elements, since calving of grounded termini around Antarctica is minimal and it would be unrealistic to apply the same retreat rate universally. This means that grounded ice is not removed, but any ice which comes afloat due to changing dynamics and geometry during the simulation is then subject to the prescribed retreat rate.

Additional experiments are run to identify whether behaviours can be attributed to particular parts of the calving front. This involves splitting the calving front into three sections as displayed in Fig. ; PIG, Thwaites and Crosson/Dotson. Experiments are then run using the Control forcing in which each of these sections is allowed to retreat individually, while the rest of the calving front remains in place.

An overview of all the simulations run is given in Table .

We calculate the sensitivity of negative changes in water-equivalent volume above flotation (VAF) – expressed as a contribution to mean sea level rise (SLR) – to prescribed retreat rates (RR), by quantifying an SLR-RR sensitivity, Υ, defined as 3Υ=-δVAFδRR×Ao, where δVAF and δRR are the differences between the VAF and RR values in two different experiments at the same model time, and Ao=3.614×108 km² is the global surface area of the ocean . The SI units of Υ are seconds, but to aid physical interpretation we express Υ in millimeters of sea level rise per metre of frontal retreat per year, i.e. as mm (ma-1)-1.

Figure b–h shows the difference in ice speed between Control_RR0.5 and Control_RR0 at various points in time during the simulation. These are presented alongside the bed geometry (Fig. a) to help in interpreting aspects of the ice evolution, particularly with respect to where pinning points are located. This example demonstrates common features of the response of the ASE to calving front retreat across our ensemble. Generally the introduction of a calving front retreat rate leads to greater speeds and more loss of ice, but in some regions the ice becomes slower or thicker compared to Control_RR0. In the remainder of this section we summarise the responses of the three main regions within the domain.

Despite the central flow from the main trunk of PIG being faster when the retreat rate is higher, there is almost no grounding line movement here until all the floating ice downstream is removed (Fig. g). However, “PIGlet” speeds up in response to forced calving front retreat, and by 2100 the grounding line in Control_RR0.5 has already retreated in a way that does not occur in Control_RR0 during our simulation timeframe (Fig. e). This grounding line retreat causes “PIGlet” to merge with Eastern Thwaites, driving further retreat.

Thwaites ice shelf shows two types of response, with the eastern section being thicker and slower in Control_RR0.5, while the western section flows faster. As Eastern Thwaites merges with “PIGlet” and undergoes grounding line retreat (Fig. e), the entire catchment ends up flowing faster. Western Thwaites does not undergo a significantly different grounding line migration compared to Control_RR0 until later in the simulation, after 2100 (Fig. f).

The Crosson and Dotson ice shelves display very little reaction to a forced retreat rate initially, only starting to speed up significantly between 2100 and 2150 when the ice shelf loses contact with Bear Island (Fig. f). Even then, the outlet glaciers do not show a large increase in speed compared to Control_RR0 until further towards the end of the simulation – after 2200 – as contact is lost with pinning points further upstream and the ice shelf is almost entirely removed (Fig. h).

We use -ΔVAF to refer to the loss of VAF compared to the initial state in a single experiment. Introducing a prescribed retreat rate leads to -ΔVAF increasing, compared to RR0. We consider the -ΔVAF values of different retreat rates in comparison to that of RR0 (using the same climate forcing), in order to determine the additional contribution of the prescribed retreat rate. When doing so, we refer to this as -ΔVAFadd. We give values as an equivalent contribution to mean sea level rise in millimetres (mm SLR), for ease of interpretation.

The magnitude of -ΔVAFadd depends on the retreat rate in a monotonic relationship, shown in Fig. a–b for chosen years during the simulation, using the Control climate forcing. The relationship shows transitions between different gradients in -ΔVAFadd, becoming steeper as the retreat rate increases in a piecewise-linear relationship. With a 1 km a⁻¹ retreat rate, -ΔVAFadd is 21.4 mm SLR by 2100, and 239 mm SLR by 2300.

Figure c–d shows the value of our SLR-RR sensitivity, which is proportional to the gradient of the curves in Fig. a–b, calculated between each 0.1 km a⁻¹ step in retreat rate. A somewhat piecewise relationship can be seen here, with visible transitions between different states of sensitivity. Such a transition occurs around 0.6 km a⁻¹ in 2060, which then occurs around 0.5 km a⁻¹ by 2080 and closer to 0.4 km a⁻¹ in 2090. Such transitions in this relationship are not always clear, but can be seen occurring at lower retreat rates over time. A second transition can be seen at around 0.9 km a⁻¹ in 2100, and then although not obvious in the curve for 2150, appears again around 0.4–0.5 km a⁻¹ in 2200.

Figure a displays -ΔVAF for each of the RR0, RR0.5 and RR1 experiments using the different climate forcings. For most of the simulation time, -ΔVAF has a similar range over the different climate forcings regardless of the retreat rate. At 2100, the range is 8.08 mm SLR for RR0, 8.65 mm SLR for RR0.5 and 9.76 mm SLR for RR1. At 2200, the ranges are 37.77, 44.25 and 46.97 mm SLR respectively. By 2300, the range of RR1 has a higher value at 92.27 mm SLR, compared to 69.96 and 68.18 mm SLR for RR0 and RR0.5, respectively. We compare these ranges – the sea level rise response to climate forcings – to the differences between responses to retreat rates as shown in Fig. b. We see that changing from the lowest to the highest climate forcing produces a difference in response roughly equivalent to changing from Control_RR0 to Control_RR0.5 (7.99 mm SLR at 2100, 34.75 mm SLR at 2200, 88.75 mm SLR at 2300), or from Control_RR0.7 to Control_RR1 (8.46 mm SLR at 2100, 47.94 mm SLR at 2200, 93.02 mm SLR at 2300). This shows that the full range of the response to changing our choice of climate forcing is less than the range of the response to changing our choice of calving front retreat rate.

Figure b shows -ΔVAFadd for all Control simulations along with the RR0.5 and RR1 experiments for all climate forcings. -ΔVAFadd follows a similar trajectory for each climate forcing case when using RR0.5, and the same is true for the RR1 simulations (as shown by the shaded areas covering the full range of these sets of simulation outputs). At 2100, for RR0.5, all climate forcings have -ΔVAFadd in the range 7.16 mm SLR ±11.5 %, and for RR1 the range is 19.41 ±10.5 %. At 2200, these ranges are 28.54 mm SLR ±22.3 % and 104.37 mm SLR ±9.4 %, and at 2300 they are 71.69 mm SLR ±23.8 % and 208.19 mm SLR ±14.9 %. So the uncertainty in the magnitude of -ΔVAFadd is never higher than 24 %.

Our results show that for a given prescribed non-zero retreat rate, and for the range of climate forcings in Tier 1 of ISMIP6-2300, -ΔVAFadd does not scale with the magnitude of -ΔVAF using RR0 (i.e. in the simulations with no calving front retreat). By this we mean that regardless of the magnitude of VAF responses to climate forcing, any particular increase in the retreat rate always causes similar additional VAF loss, as demonstrated by the shaded areas in Fig. . This demonstrates that VAF loss in models due to calving front retreat is not heavily dependent on the climate forcing. Therefore, the results displayed in Figs.  and are not entirely unique to the chosen climate forcing, which allows us to make clear statements about the impact of frontal retreat rate on the future mass loss of the region.

We suggest that this finding could potentially apply more generally to Antarctic ice shelves. Since it holds over the ASE domain, which contains a variety of ice shelf configurations from the unconstrained ice tongue of Thwaites to the highly buttressed ice shelves of Crosson and Dotson, it is possible that the choice of climate forcing, at least within the range used for ISMIP6-2300, is not strongly related to the modelled response to calving front retreat for any ice geometry. This could be the subject for a dedicated future study, and care should of course be taken in making generalisations. An obvious caveat is that in our case there is no feedback between the ice and the ocean or atmosphere. In the real world, or a coupled simulation, the rate of calving is particularly likely to impact ocean circulation and change the thermal forcing under ice shelves. Thus this finding is only relevant to stand-alone ice sheet models such as the one we use.

The climate forcings themselves produce very different results in the RR0 experiments, with some including periods of increasing VAF (Fig. a). In these cases (CCSM, CESM and UKESM) the ice is thickening upstream of the grounding line, and the melt rate distribution on the ice shelf is not concentrated as close to the grounding line as it is in the cases which do not display this behaviour. It is notable that this increase in VAF does not happen before 2100 in any case, nor at all for the UKESM case with repeated forcing after 2100 (which of course does not undergo any large changes after 2100 as some of the other forcings do). This could demonstrate some limitations in the use of these extrapolated climate forcing products beyond a certain time, as grounding lines evolve and the geometry moves further away from the state used in the ocean models. This is something that could be investigated in future, but for our purposes it serves to demonstrate that even with this wide range of behaviour, -ΔVAFadd is remarkably similar between cases.

The relationship between retreat rates and -ΔVAFadd exhibits a somewhat piecewise behaviour (Fig. ). The discrete steps can be attributed to the system passing certain buttressing thresholds, such as loss of contact with pinning points, which change the ice dynamics significantly enough to cause an increase in -ΔVAF across the domain. In the absence of any such buttressing thresholds, the response to an increase in prescribed retreat rates remains quite linear, seen in Fig.  as constant values of Υ as the retreat rate varies. This linearity is particularly obvious in earlier years of simulation, and becomes less obvious as time progresses and more complexities are introduced by the changing geometry, adding more noise to the signal in our SLR-RR sensitivity. Nevertheless, relatively flat sections can be seen, for example, from 0.1 to 0.4 km a⁻¹ and again above 0.5 km a⁻¹ in 2200, in which cases the value of Υ remains at around 75 and 150 respectively, with a transition between the two states.

There are at least two distinct thresholds which we can confidently identify in the ASE system. First, in the grounded area between Thwaites and Pine Island ice shelves, there is a peak in the bedrock geometry which is above sea level, as can be seen in Fig. a. As the grounding line retreats, this becomes an important pinning point which buttresses “PIGlet”. Eventually, the ice loses contact with what is by that point a small island, resulting in the Pine Island and Thwaites calving fronts merging into one as Thwaites appears to be driven into retreat by “PIGlet”. The loss of contact with this pinning point, and the resultant speed-up of Eastern Thwaites, can be seen in Fig. d–e. The timing of this lines up with Fig. c, in which a transition in the sensitivity is seen around RR0.5 in 2080. Later in the simulation, the grounding line retreat at the main trunk of PIG in Fig. f–g is also instigated from the west, and may not happen without the collapse of “PIGlet”. This threshold is never reached without calving front retreat in Control_RR0, but is reached with even the smallest prescribed retreat rate in Control_RR0.1.

A second major threshold is in the Dotson and Crosson ice shelves. They take a longer time to show major changes, but when the ice shelves lose contact with Bear Island, after about 60 km of calving front retreat, the ice flows faster and extra grounding line retreat is seen. The speed and grounding line retreat then decrease again until contact is lost with further bedrock peaks upstream of the initial grounding line. In Fig. , the loss of contact with Bear Island and increase in speed occurs between panels (e) and (f), the following reduction in speed in panel (g) and further loss of buttressing leading to greater grounding line retreat in panel (h). The timing of this threshold appears to line up with Fig. c–d, in which a threshold appears in 2100 around RR0.9 and by 2200 is around RR0.4. However, the curve for 2150 does not show a clear signature to strengthen this connection. At such an advanced stage of the simulation, this could be due to several competing signals as different thresholds of varying sizes are reached. For example, several small pinning points and embayments are exposed as the grounding lines retreat, particularly around the Dotson and Crosson ice shelves, which could each be seen as a small threshold affecting buttressing.

The grounding line of the main trunk of PIG does not show any difference in position until the calving front approaches very close to it, at which point the glacier speeds up significantly. This is another threshold in the system, and occurs just after contact is lost with the Bear Island, so could be a major competing signal masking the signature in 2150 discussed above. The behaviour of Pine Island is consistent with the findings of , and , that much of PIG's buttressing is provided by the ice closest to the grounding line.

To clarify the differing effects of individual regions of the calving front, we ran simulations with RR0.5 and RR1 in which parts of the calving front, indicated in Fig. , were retreated individually, while keeping the rest of the calving front fixed. The results, displayed in Fig. , show that retreat of the PIG calving front is responsible for the largest long-term changes, while retreat of the Dotson and Crosson calving fronts causes the least difference.

The effect of the entire calving front retreating is initially almost identical to the sum of the three individual experiments, but diverges as the simulations continue beyond about 2100. This roughly coincides with the point at which the ice shelves of Thwaites and Pine Island have been entirely calved away. Changes in geometry near the present-day zero-velocity boundaries as the ice sheet evolves could be the cause of this. The present-day boundaries are used to determine where the calving front is allowed to retreat in the individual regional experiments, but these boundaries would likely not be in the exactly the same location in future, particularly around the “PIGlet” ice flow.

Our results are in agreement with other recent modelling studies e.g. that the Thwaites ice shelf does not currently contribute much to the glacier dynamics. In our case, the additional VAF loss when retreating the ice front is not significant in the early stages as the ice shelf is calved away, at only 0.19 mm SLR in 2050 for RR0.5 before rising towards 2100. The changing position of the grounding line appears to be the more important factor, and causes ice loss after the first few decades. looks specifically at the buttressing provided by Thwaites Ice Shelf, finding that fluxes across the grounding line can either increase or decrease locally when the floating ice is removed, suggesting a negligible net impact of buttressing. used a buttressing flux response approach following , finding a mixture of positive and negative responses along the Thwaites grounding line.

With the comparisons detailed in Sect. , we have demonstrated that the variability in our experimental -ΔVAF outputs using retreat rates between 0 and 1 km a⁻¹ is greater than that of using the range of climate forcing from the ISMIP6-2300 Tier 1 experiments. This means that accurately representing calving front movement in predictive simulations can be a problem of comparable importance to accurately representing melt rates.

There is currently no consensus on a suitable calving law to use for predictive modelling, despite several options being available as outlined in the introduction. Calving can alternatively be applied as a prescribed retreat rate, as we have done here to systematically investigate the process. However, future retreat rates are unknown so predictions using this method will be unreliable. Retreat rates assumed from historical rates can be highly variable depending on the timescales used. For example, historical calving front positions of Pine Island presented in show an average retreat rate over their entire record from 1973 to 2020 of 0.6 km a⁻¹. But the calving front was in about the same position in 2015 as in 1973, so 5 years earlier the average retreat rate would have been almost zero. On the other hand, the most recent period of frontal retreat from 2013–2020 gives 6 km a⁻¹. This observational example presents a far greater degree of uncertainty than the comparatively small range of retreat rates in our experiments.

Due to the high uncertainty in future retreat rates and the importance of this process in model evolution, we propose that explicitly testing calving mechanisms, either using existing proposed laws or prescribed retreat rates, should be a feature of future model intercomparisons. By not including such tests, a large amount of potential variability between models could be missed. More ice flow models are now able to include calving, as demonstrated by the ongoing CalvingMIP project , so this is now a more practical option than it may have been in the past.