The impact of recent and future calving events and ice-shelf thinning on the Larsen C ice shelf

. The Antarctic Peninsula has seen rapid and widespread changes in the extent of its ice shelves in recent decades, including the collapse of the Larsen A and B ice shelves in 1995 and 2002, respectively. In 2017 the Larsen C ice shelf Ice Shelf (cid:58) (LCIS) lost around 10% of its area by calving one of the largest icebergs ever recorded (A68). This has raised questions about the structural integrity of the shelf and the impact of any changes in its extent on the ﬂow of its tributary glaciers. In this work, we used an ice ﬂow model to study the instantaneous impact of changes in the thickness and extent of the LCIS on ice dynamics, and in particular on changes in the grounding line ﬂux (GLF). We initialised the model to a pre-A68 calving state, and ﬁrst replicated the calving of the A68 iceberg. We found that there was a limited (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) instantaneous impact on upstream ﬂow – with speeds increasing by less than 10% across almost all of the shelf – and a 0.5 (cid:58)(cid:58)(cid:58)(cid:58) 0.28% increase in GLF. This result is supported by observations of ice velocity made before and after the calving event. We then perturbed the ice-shelf geometry through (cid:58) a series (cid:58)(cid:58) of (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) instantaneous, idealised calving and thinning experiments of increasing magnitude. We found that signiﬁcant changes 10 to the geometry of the ice shelf, through both calving and thinning, resulted in limited (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) instantaneous (cid:58) changes in GLF. For example, to produce a doubling of GLF from calving, the new calving front needed to be moved to 5 km from the grounding line, removing almost the entire ice shelf. For thinning, over 200 m of the ice-shelf thickness had to be removed across the whole shelf to produce a doubling of GLF. the instantaneous increase in GLF ( 607 %) after removing the entire ice shelf allowed us to quantify the total amount of buttressing provided by the LCIS. From this, we identiﬁed that the region of 15 the ice shelf in the ﬁrst 5 km downstream of the grounding line provided over 80% of the buttressing capacity of the shelf. This is due to the large resistive stresses generated in the narrow, local embayments downstream of the largest tributary glaciers.

:::: mass :: in :::::::: response :: to ::: the :::::::::::: perturbations. : Our first objective was to model the response of the LCIS and its tributaries to the calving of the A68 iceberg and validate these results with observations. We then studied the ::::::::::: instantaneous : GLF response to a series of idealised ice-shelf calving events. From quantifying the maximum GLF response, we determined the total amount of buttressing provided by the ice shelf, and examined what proportion of this total is generated by different regions of the shelf.
We simulate the loss of contact of the ice shelf from the Bawden and Gipps ice rises ::::::: (outlined :::: alnd ::::::: labelled :: in :::: Fig. :: 1), again 70 examining the impact on GLF. And finally, we systematically perturb the thickness of the ice shelf by increasing amounts, again aiming to understand how much the ice-shelf geometry needs to change before a significant response in GLF is produced.

Ice flow model
We used the Úa ice flow model (Gudmundsson et al., 2012), which solves the vertically integrated, shallow-shelf approximation 75 (e.g. MacAyeal, 1989) using the finite element method on an unstructured mesh. Úa has been used in both idealised (e.g. Gudmundsson et al., 2012; and realistic (e.g. De Rydt et al., 2015;Minchew et al., 2017;Hill et al., 2018;Reese et al., 2018;Gudmundsson et al., 2019) settings to examine the response of grounded ice to perturbations in the ice shelf. It has also been tested in recent model intercomparison projects (Pattyn et al., 2013;Cornford et al., 2020).
The equation solved in Úa for the vertically integrated balance of stresses is 80 ∇ xy · (hR) − t bh = ρ i gh∇ xy s + 1 2 where is the resistive stress tensor, τ ij are the components of the deviatoric stress tensor, ∇ xy = (∂ x , ∂ y ) T , t bh is the horizontal component of the basal traction, h is the ice thickness, s is the ice surface elevation, ρ i is the vertically integrated ice density In this work, we solved the equations for stress balance in a diagnostic, or time-independent mode, together with Glen's flow law, the constitutive equation linking the stress field in the ice to deformatioṅ ij = Aτ (n−1) τ ij (3) where˙ ij are the components of the strain rate tensor, τ is the second invariant of the deviatoric stress tensor, given by 90 τ = τ ij τ ij /2 (4) and the rate factor, A -which depends on ice properties including temperature, crystal fabric and damage -was optimised using inverse methods (see Sect. 2.3). We chose :: set the creep exponent n = 3 :: as :: is ::::::: standard :: in ::::::: ice-flow ::::::::: modelling.
For the optimisation procedure used to initialise the model (see Sect. 2.3) we used the MEaSUREs InSAR-based Antarctic Ice Velocity v2 data set :::::::::::::::: (Rignot et al., 2017) Mouginot et al., 2012). For model validation (see Sect. 3.1) we used ice velocity measurements generated from Sentinel-1 SAR data which were provided by ENVEO. This data set consisted of monthly maps of tide-corrected ice velocities over the LCIS and its tributaries from October 2014 -September 2019.

Model initialisation
To generate ::: the initial conditions the rate factor, A : , : and basal slipperiness parameter, C, were optimised by minimising the misfit between observed and modelled ice velocity through inverse methods widely used in glaciology (e.g. MacAyeal, 1993).

Calving experiments
From the initial conditions derived from the initialisation procedure, the geometry of the ice shelf was perturbed whilst holding all other parameters constant, generating a new stress field. This yielded an instantaneous change in the modelled ice velocity, which we compared to the initial velocity field and from which we calculated changes in GLF.

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The first experiment undertaken was to replicate the calving of the A68 iceberg, the extent of which was derived from Landsat 8 images. To perform the experiment, the mesh elements within the region that calved were removed, thereby relocating the model boundary to the new calving front. Through this procedure, the remaining elements of the mesh were left unchanged and any interpolation errors avoided.
In addition to the A68 calving event, a series of idealised calving experiments were conducted. The calving front was moved 165 progressively nearer to the GL by removing mesh elements using a 'distance from the main grounding line' metric (mapped in Fig. 4c).
It is important to note that in these calving experiments (and indeed in all experiments in this study) no perturbation was applied to the nodal values of any element which crossed the main GL. This meant that there was no change in driving stresses across the GL, and that the GL location remained fixed in all experiments. This ensured that any change in GLF was due solely 170 to changes in the buttressing provided by the ice shelf.

Ungrounding and thinning experiments
To simulate the ungrounding of the LCIS from the Bawden and Gipps ice rises, the bed topography was lowered so that the ice shelf became afloat without changing the ice thickness. This experiment was carried out for the ungrounding from the two ice rises individually, and then for a 'combined' ungrounding, in which both contacts were removed simultaneously.

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Finally, we explored the GLF response to perturbations in ice thickness. The ice thickness at nodes belonging to elements that were fully afloat (again, to ensure no change in driving stress across the GL) was progressively reduced until the whole ice shelf had a thickness of only 1 m. The 1 m thin layer of ice across the shelf was maintained for computational reasons, and the results are insensitive to a further reduction in the minimum ice thickness. Locally this thinning was done both proportionally, i.e. the ice thickness was reduced by a given fraction of the total thickness at each node, and uniformly where thickness was 180 reduced across the whole ice shelf by the same fixed amount. The subsequent changes in ice velocity and GLF were calculated for each step in the series of experiments.

A68 calving
In response to the removal of the A68 iceberg from the model domain, there was an ::::::::::: instantaneous : increase in ice velocity 190 immediately upstream of the new calving front of up to ∼ 100 ma −1 (Fig. 3a). The spatial extent of this velocity response was limited, and across almost all of the ice shelf the change in velocity was smaller than 10% -even becoming negative in the region of the shelf to the north of the Gipps ice rise ::: Ice :::: Rise. The changes in velocity did not extend throughout the whole ice shelf, and as such there was almost no modelled increase in GLF (0.5 ::: 0.28%) due to this calving event.

Idealised calving experiments
The impact of moving the calving front progressively closer to the GL on the GLF can be seen in Fig. 4a and b. It shows that a 205 retreat of the calving front from its present day position back into the embayment produced a limited instantaneous impact on the GLF. The calving front had to be retreated to 40 km from the GL to induce a 10% increase in GLF. For a doubling of GLF, the calving front had to be positioned 5 km from the grounding line, removing almost all of the ice shelf in the process.
The maximum GLF increase (607%) -from the complete removal of the ice shelf -can be thought of as representing the total buttressing provided by the LCIS : in ::: its :::::: current ::::::::::: configuration : to its grounded tributary glaciers. By comparing the GLF 210 increase ::::::::::: instantaneous ::::::: increase :: in :::: GLF for each idealised calving experiment to the maximum GLF increase from complete iceshelf removal, we are able to calculate the proportion of the total buttressing that remains after each perturbation experiment.
Therefore, Fig. 4a and b show what proportion of the total buttressing is provided by each section of the ice shelf removed in the series of calving perturbations. From this, we see that over 95% of the total buttressing is provided by ice in the first 25 km downstream of the GL, and that over 80% is generated in the first 5 km of ice immediately downstream of the GL. and (d) show the modelled ice speed before and after the calving event along two flowlines in the shelf. The mean, observed speed before and after July 2017, together with shading representing the 2σ variability, is also plotted.

Ungrounding experiments
In the modelled response to the ungrounding of the LCIS from the Bawden ice rise there was a significant local ::::::::::: instantaneous increase in velocity upstream of the ice rise ( Fig. 5a) of ∼ 200 ma −1 (and an even greater increase for the ice that was previously grounded). This represents a ∼ 50% increase in ice velocity in this region. However, as with the A68 calving, this instantaneous velocity response is spatially limited and there is a just a 1% increase in GLF from this perturbation.
By calculating the ice-shelf mass removed in each experiment we were able to compare the two approaches (Fig. 6c). We see from the 'uniform' experiment curve, that there is a large response in GLF to a small change in mass removed towards the 245 end of the series of perturbations, when the thickest ice is being significantly perturbed. The maximum GLF increase of 502% is identical in both the 'uniform' 1,500 m thinning and the 'proportional' 100% thinning experiments, as expected. The initial linear regimes in both sets of thinning perturbations are discussed in Appendix B.
Our first objective was to model the ::::::::::: instantaneous : response of the LCIS to the calving of the A68 iceberg and compare the results to observations. The limited change in ice-shelf velocities, and lack of change in the GLF, suggests that this part of the ice shelf provided almost no buttressing. This finding is in agreement with the work of Fürst et al. (2016), who classified this region as 'passive ice', and with the map of 'buttressing flux response number' that Reese et al. (2018) produced. Borstad et al. (2017 hypothesised four potential calving events based on the trajectory in which the rift that eventually formed the A68 255 iceberg was growing, and modelled the response to these events. Their 'Scenario 2' is most similar to the calving event that eventually occurred, and our modelled results are in close agreement with theirs in both spatial pattern and amplitude (see Fig.   2d in Borstad et al., 2017).
The model results shows a decrease in ice velocity following the calving event in the region of the shelf just to the north of the Gipps ice rise ::: Ice :::: Rise. This is likely to be an artefact of the method used to perform the experiment. The rift that 260 eventually formed the A68 iceberg had been present in the ice shelf for over a decade, and grew significantly during 2014 and 2016 (Jansen et al., 2015;Borstad et al., 2017). Therefore, the dynamic response to the detaching of the nascent A68 iceberg will have already taken place in this region, and this response is included in the ice velocity data used to initialise our model.
When comparing our result with observations, we found that the modelled response was smaller than the internal variability in the monthly ice velocity data available. Consequently, we were unable to validate the details in spatial pattern and amplitude of our modelled response. However, the observations do show that the mean ice velocity across the shelf before and after the calving event remained unchanged, or at least smaller than the measurement errors. This demonstrates that the calving of the 270 A68 had little or no dynamic impact on the system, supporting ours and previous work that predicted such a response, or rather lack thereof.
This previous work has ::: The :::::::: previous :::: work ::: of :::::::::::::::: Reese et al. (2018) ::: and :::::::::::::::: Zhang et al. (2020) measured the response in GLF to small perturbations in ice-shelf geometry, but by removing the entire ice shelf and calculating the instantaneous response in GLF we are able to quantify the total amount of buttressing that the LCIS provides. This allowed us to examine what proportion 295 of the total buttressing capacity is provided by different regions in the ice shelf. We find that over 95% of the buttressing is generated by the ice in the first 25 km downstream of the GL, and that over 80% comes from the first 5 km of ice directly downstream of the GL. The :::::: primary : reason for this is that the LCIS geometry is characterised by a number of small, narrow embayments where the main tributary glaciers flow into the ice shelf. It is in these regions that the largest resistive stresses are generated, which dominate the buttressing capability :::::: capacity : of the shelf as a whole.

Ice-shelf thinning and ungrounding
We also set out to examine the GLF response to ice-shelf thickness perturbations. Fig. 6a shows that the response in GLF to thinning is approximately linear as a function of the amplitude of the thickness perturbation, as long as the amplitude is less than about 100 m (this is further explored in Appendix B). For amplitudes larger than about 100 m, the response becomes progressively more non-linear, something that is also very evident when the GLF response is plotted as a function of the ice-response from ice-shelf thinning to 475%, whilst reducing it further to 0.001 m, only increases the maximum GLF response to 505% from the 502% modelled with a 1 m minimum ice thickness.
To put the levels of ice-shelf thinning presented in Fig. 6 into context, the maximum basal melt rates observed under the LCIS are on the order of 2 ma −1 (Adusumilli et al., 2018), but typical values across most of the shelf are lower than this. Assuming 320 the maximum basal melt rate was applied across the whole ice shelf, and that the impact of the thinning is considered in an instantaneous sense, our results suggest around 15 years of thinning is required to produce an increase in GLF of 10%, and over 100 years of thinning to produce a doubling of GLF.
The dynamic ::::::::::: instantaneous response to removing the basal contacts at the Bawden and Gipps ice rises has previously been modelled with different methods to ours. Borstad et al. (2013) only modelled the floating ice shelf, and therefore simulated a 325 loss of contact at the ice rises by manually adjusting their inferred ice-viscosity parameter (the equivalent of our rate factor, A). Fürst et al. (2016) modelled the grounded ice as well as the shelf, and chose to set their basal friction coefficient to zero at the ice rises -removing the basal traction -rather than adjusting the ice or bed geometry. Despite these differences in approach, we found that our results are very similar in both spatial pattern and amplitude to these previous studies. In the experiment in which both ice rises were removed the ::::::::::: instantaneous change in GLF was 2.2%. This suggests that, whilst these two ice rises 330 may exert a significant control on the flow of the shelf upstream of the pinning points, they do not exert a strong mechanical control on the ice flux at the GL, and only contribute a small amount to the total buttressing capability :::::: capacity : of the shelf : , :::: given :::: that :::: their ::::::: removal :::: only ::::::: affected ::: the ::::::: stresses : at ::: the :::: GL :::::: enough :: to :::: raise ::: the :::: GLF ::: by ::::: 2.2% ::::::::::::: instantaneously.
The experiments conducted here are highly idealised in nature. We only considered 'uniform' or 'proportional' thinning perturbations to the shelf, and calving front locations were determined by a 'distance to the grounding line' metric. This 335 resulted in some unlikely calving front positions. More realistic calving experiments, using a physically based calving law or metric, could be used to model the response to more plausible ice-shelf configurations.

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In this study we have examined the instantaneous response of the LCIS and its tributaries to both observed and idealised perturbations to the ice-shelf geometry. We found that the calving of the A68 iceberg in July 2017 produced a limited change (mostly < 10%) in ice velocities in the shelf and had almost no ::::::::::: instantaneous : impact (a 0.5 ::: 0.28% increase) on the GLF. This finding is supported by observations which show no evidence of a change in velocity due to the calving event, and this furthermore confirms earlier work that suggested that the region that calved was largely 'passive ice'.

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Through further, idealised calving experiments we found that a significant retreat of the calving front to 25 km downstream of the GL (removing over 50% of the ice-shelf mass) only produced a 13% ::::::::::: instantaneous : increase in GLF. Further retreat of the calving front to 5 km from the GL was needed to produce a doubling of GLF. By calculating the total buttressing provided by the LCIS -through modelling the ::::::::::: instantaneous increase in GLF due to a complete collapse (607%) -we deduced that over 95% of the buttressing capability ::::::: capacity of the LCIS is provided by ice within 25 km of the GL, in the narrow embayments 355 downstream of the main tributary glaciers. We further found that over 80% of the buttressing is generated in first 5 km of ice downstream of the GL.
For the 'uniform' perturbations, initially the deviation from an exact linear response is below the straight line plotted through 385 the origin and the 0.1 m thinning point. This suggests that increasing thickness perturbations produce a relatively smaller increase in GLF when uniform thinning is applied. However, Fig B2 shows that when the shelf is thinned in proportion to the total ice-shelf thickness at each node, this behaviour is suppressed, and the relative response in GLF steadily increases as the proportion of the ice-shelf thickness removed increases.
The way in which the GLF response to thinning deviates below the initial linear regime is an interesting phenomena that 390 has yet to be explained, but has been observed in previous studies on other ice shelves (e.g. Fig. S2 in Reese at al., 2018). It suggests that when a uniform perturbation to ice-shelf thickness is applied across a shelf, over a certain range of perturbation size (here ∼ 1−50 m), the relative increase in GLF is progressively reduced. From the lack of evidence of this behaviour in the proportional thinning experiments, we can see that is related to the distribution of the thickness perturbation across the shelf, and the cause of this is still unknown. and 5 uniform thinning) with the 5 different meshes are shown in Fig. C1. From this we can see that the response in GLF is consistent across the 5 different mesh resolutions for each experiment, and therefore any mesh dependence of our results is 405 negligible.