Melt response to PIIS calving

As well as an ice dynamic response, there may be changes to melt rates on PIIS following ice front retreat. This is because the topographic blocking by the combination of a seabed ridge beneath PIIS and the ice shelf itself reduces the amount of relatively warm Circumpolar Deep Water able to reach the cavity inshore of the ridge, thereby restricting the amount of melting that can take place (Dutrieux and others, Reference Dutrieux2014; De Rydt and others, Reference De Rydt, Holland, Dutrieux and Jenkins2014). Ice front retreat might relax this topographic barrier and thus result in altered melt rates on PIIS.

To investigate this possibility, Bradley and others (Reference Bradley, Bett, Dutrieux, De Rydt and Holland2022a) performed numerical experiments in which they explicitly resolved the ocean cavity circulation and ice shelf melting using the MITgcm (Marshall and others, Reference Marshall, Hill, Perelman and Adcroft1997) in a geometry accurately resembling PIG. A full description of the model setup, experiments, and results can be found in Bradley and others (Reference Bradley, Bett, Dutrieux, De Rydt and Holland2022a). Six experiments were performed in total, each featuring a different ice front position (Fig. 2a), while the grounding line position and ice thickness in areas of shelf not removed were fixed. Comparing melt rates between experiments with different ice front positions offers insight into the melt response to calving: Bradley and others, Reference Bradley, Bett, Dutrieux, De Rydt and Holland2022a found that, while the maps of melt rate display complex patterns of change upon ice front retreat (Fig. 2b), the mean melt rate close to the PIIS grounding line increases monotonically with retreat (Fig. 2c). This means that, assuming that nothing else about the geometry changes, ice front retreat always enhances melting. This enhancement results from both an increase in the amount, and temperature, of relatively warm water crossing the seabed ridge, as well as changes in the cavity circulation following ice front retreat (Fig. 1) (Bradley and others, Reference Bradley, Bett, Dutrieux, De Rydt and Holland2022a).

Figure 2.

(a) Ice front positions used in experiments designed to assess the melt response of the PIIS to calving. Each experiment corresponds to a different ice front position as labelled: ice fronts labelled 2009 and 2020 indicate the ice front position in those years, while ice fronts labelled F1–F4 correspond to hypothetical future ice front positions. The solid black line indicates the 2009 grounding line from Joughin and others (Reference Joughin, Smith and Holland2010). The dashed line roughly indicates the centreline of the cavity, along which the calved length – the difference between the ice front in the respective experiments and the 2009 ice front – is measured. Mean melt rate values shown in (c) are calculated over the shaded pink region. The background image is a Sentinel 2 mosaic from November 2020. (b) Simulated melt rate in the 2009 Pine Island geometry (first panel) and cumulative (i.e. measured to the first panel) melt rate anomalies (other panels). (c) Percentage enhancement in melt rate as a function of calved length measured relative to the 2009 geometry. Values correspond to those shown as text labels in (b).

Ice dynamic response to PIIS calving

In addition to changes in basal melting, calving causes the ice sheet to adjust mechanically to the loss of a section of its restraining ice shelf. We refer to this as the ice-dynamic response. To facilitate a comparison between the melt and ice-dynamic responses to calving, we consider mass conservation:

(1)$${\partial h\over \partial t} = - \dot{m} - \nabla.( h{\bf u}) .$$

Here h is the ice thickness, u the depth-averaged ice velocity, $\dot {m}$ the basal melt rate (positive indicates ice removal). Surface accumulation is small compared to melting on PIIS (e.g. Nakayama and others, Reference Nakayama, Hirata, Goldberg and Greene2022) and is therefore ignored.

Instantaneous adjustments to the rate of change ice thickness, ∂h/∂t, consist of two components: changes in the melt rate (first term on the right hand side of (1)) and changes in the flux divergence (second term). Calving induces changes in both of these: changes in melting occur because of a dynamical adjustment in the ocean circulation, whereas changes in flux divergence occur because of a dynamical adjustment in the ice flow. Here, we compare these contributions by running a series of ice sheet model experiments and comparing the modelled flux divergence response to calving with the melt response described above. We note, however, that this is an inherently coupled system – a coupled ice-ocean model must be used to assess the transient response – and comment on this in the ‘Outlook’ section below.

To facilitate the comparison between melting and ice-dynamic contributions to changes in ∂h/∂t, we used the Úa ice sheet model (Gudmundsson and others, Reference Gudmundsson, Paolo, Adusumilli and Fricker2019; Gudmundsson, Reference Gudmundsson2022), with the setup as described by De Rydt and others (Reference De Rydt, Reese, Paolo and Gudmundsson2021), to determine changes in ice velocity and flux divergence in response to changes in ice front position, according to those shown in Fig. 2a. Úa solves the vertically integrated formulation of the momentum equations on an unstructured mesh using the finite element method. Basal slipperiness and ice viscosity parameters were obtained using a commonly applied optimisation procedure, as described in detail by De Rydt and others (Reference De Rydt, Reese, Paolo and Gudmundsson2021).

Figure 3e shows modelled ice velocity anomalies relative to the modelled 2009 ice velocity, which is shown in Fig. 3a. Upon retreating the ice front from its 2009 position to its 2020 position, the mean ice velocity increases by approximately 400 m/yr (Fig. 3i), which is consistent with observations (Joughin and others, Reference Joughin, Shapero, Smith, Dutrieux and Barham2021). Further ice front retreat of PIIS is expected to induce significant further acceleration, with a velocity response that is approximately linear in the loss of ice shelf area (Fig. 3i): the model predicts an approximately 115 m/yr ice speed-up per 5 km length of ice shelf removed. (For context, the current retreat rate of the PIIS front is 5 km/yr (Joughin and others, Reference Joughin, Shapero, Smith, Dutrieux and Barham2021) and the mean (predominantly melt-driven) speed-up of PIIS between 1970 and 2010 was 40  m/yr².) Note that this result is in contrast to a similar analysis applied to the Larsen C ice shelf (Mitcham and others, Reference Mitcham, Gudmundsson and Bamber2022), which indicated that progressive loss of ice shelf area results in a highly non-linear response of the grounding line flux, with the largest acceleration linked to loss of ice within 10 km of the grounding line. This emphasises the importance of the entire central portion of PIIS for buttressing of the PIG.

Figure 3.

Comparison of the instantaneous ice dynamic and melt responses to PIIS ice front retreat. (a) Modelled PIG ice velocity and (e) velocity anomalies following ice front retreat (ice front retreat from left to right). (b)–(d) Negative basal melt rate $-\dot {m}$, negative flux divergence $-FD = -\nabla. ( h {\bf u})$, and thinning rate $-\dot {m} -\nabla. ( h {\bf u})$ (i.e. the sum of (b) and (c)), alongside (f–h) responses following ice front retreat. Note the different colour bars in (f) and (g–h). (i) Mean velocity perturbation measured over the inner cavity (pink box in Fig. 2a), relative to the experiment with the 2009 ice front. (j) As in (i) but for the melt, flux divergence and total (sum of the melt and flux divergence) contributions. Note that the melt rates shown in (b) and (f) are as in Fig. 2b, but Fig. 2b uses a slightly different grounding line position (the grounding line shown here is from 2016 (De Rydt and others, Reference De Rydt, Reese, Paolo and Gudmundsson2021), while Fig. 2 shows a 2009 grounding line (Joughin and others, Reference Joughin, Smith and Holland2010)).

Figures 3b–d show, respectively, the negative melt rate, negative flux divergence, and their sum – the effective thinning rate – in the 2009 ice front experiment, alongside anomalies of these quantities in the calving scenarios (f–h, respectively). The large ice velocity response is also borne out in the flux divergence response, which is an order of magnitude larger than the corresponding melt response in many places (noting the different limits on the colour bars in Figs. 3f and g–h). Equivalently, the patterns of thinning rate anomalies (Fig. 3h) are highly similar to the patterns of flux divergence anomalies (Fig. 3g). Although the patterns of flux divergence anomalies are highly variable, featuring regions of large positive and negative anomalies, the mean flux divergence response in the inner cavity region (the pink box in Fig. 2a) is positive and increasing with ice front retreat (Fig. 3j), indicating that flux divergence changes following ice front retreat always promote a more positive dh/dt. This is consistent with increased ice advection into the shelf concomitant with increased ice velocity. However, this positive net flux divergence contribution to the rate response is outweighed by the negative net melting contribution (Fig. 3j): our simulations suggest that the instantaneous response to PIIS calving is always further thinning. This highlights the crucial role that changes in melting following ice front retreat might play: without a change in melting following ice front retreat, the instantaneous response would promote ice shelf thickening (red line in Fig. 3j is positive); however, as a result of the changes in melting, we expect further ice shelf thinning following ice front retreat (purple line in Fig. 3j is negative).

Outlook

Although the analysis included in this paper does not provide quantitative predictions of the transient evolution of PIIS following calving, the instantaneous analysis is highly informative. Most pertinently, it demonstrates the importance of calving on changes in PIIS buttressing and hence flow across the grounding line. We have shown that all areas of the PIIS are important for buttressing PIG, in contrast to many other regions of Antarctica in which only ice shelf areas close to grounding lines provide strong buttressing (Fürst and others, Reference Fürst2016). As well as this, the instantaneous analysis demonstrates a large immediate PIIS velocity response to calving (on the same order of magnitude as changes over the past 10 years Mouginot and others, Reference Mouginot, Rignot and Scheuchl2014), which would be expected to lead to significant changes on longer (decadal) timescales, as well as explicitly demonstrating that ice shelf melt rates may depend on ice front position, which no present day parametrization of melting accounts for (Bradley and others, Reference Bradley, Rosie Williams, Jenkins and Arthern2022b). Finally, it demonstrates that the melt response to calving could enhance the impact of calving on the ice dynamics. We also note that satellite data (Joughin and others, Reference Joughin, Shapero, Smith, Dutrieux and Barham2021) suggests that the significant ice acceleration over the period 2017–2020 was synchronous with prolonged ice front retreat over the period, following a seven year period with little acceleration; this suggests that the immediate response to calving is comparable to, or may even dominate over, the background decadal trend in speed-up. However, a longer observational record is required to decompose responses on different timescales following such calving.

Due to the geometric feedbacks between melting, ice velocity, and calving shown above, investigating the post-instantaneous response of PIIS to ice front retreat in detail requires the use of a coupled ice-ocean model with a damage-calving scheme included (a ‘coupled ice-ocean-calving’ model). Coupled ice-ocean models have only recently begun to emerge (e.g. De Rydt and Gudmundsson, Reference De Rydt and Gudmundsson2016; Seroussi and others, Reference Seroussi2017; Favier and others, Reference Favier2019; Smith and others, Reference Smith2021), with most ice sheet projections still relying on parametrizations of melting (e.g. Bradley and others, Reference Bradley, Rosie Williams, Jenkins and Arthern2022b), which are unable to capture the important feedbacks between calving and melting. The inclusion of calving schemes within ice sheet models is a nascent field, and, to the authors’ knowledge, there are no extant coupled ice-ocean-calving models. Since such models are not yet available, the instantaneous approach taken here remains the best option to assess the importance of calving for changes in ice-shelf buttressing and hence flow across the grounding line.

The potential imminence of PIIS's decline, and understanding the implications of such, should provide urgent motivation to the modelling community to develop coupled models with moving ice fronts. There are, however, significant computational challenges to overcome before such models are ready (Asay-Davis and others, Reference Asay-Davis, Jourdain and Nakayama2017). There is no uniform ‘grand-challenge’ here, rather individual models face specific difficulties. Initially a delicate treatment of boundary conditions (e.g. Albrecht and others, Reference Albrecht, Martin, Haseloff, Winkelmann and Levermann2011) was adopted to deal with moving ice fronts, while more recently, a level set method has been adopted fairly widely (Bondzio and others, Reference Bondzio2016). Moving boundaries are problematic for ocean models since new grid cells are opened, possibly instantaneously. It remains unclear how to robustly implement calving in ocean models (Asay-Davis and others, Reference Asay-Davis, Jourdain and Nakayama2017); progress has, however, been made on similar problems relating to grounding lines (another moving boundary in ice-ocean models) either by including a porous fluid layer beneath the ice (Goldberg and others, Reference Goldberg2018), or by interpolating quantities into new grid cells in a physically consistent way (De Rydt and Gudmundsson, Reference De Rydt and Gudmundsson2016). Besides the ongoing development in the numerical implementation of moving ice fronts, the community must also improve and validate calving parametrizations, which describe where calving should occur based on other model diagnostics. Calving laws, including that which gives rise to the marine ice cliff instability (DeConto and Pollard, Reference DeConto and Pollard2016), add significant uncertainty into future SLR projections (Edwards and others, Reference Edwards2019) but remain contested and largely unvalidated.

Despite our lack of transient simulations, we can speculate on the longer-term implications of the modelled PIIS response to ice front retreat. Firstly, we have shown that the average instantaneous response is further ice shelf thinning; since enhanced ice shelf thinning promotes further calving (Liu and others, Reference Liu2015), there is the potential for a retreat-melting feedback loop in which ice front retreat enhances melting, which in turn promotes enhanced calving and thus ice front retreat, potentially encouraging collapse of the PIIS. Ice shelf collapse might additionally be expedited by a retreat-damage feedback loop: the simulated ice acceleration that accompanies ice front retreat might enhance ice shelf damage (e.g. Sun and others, Reference Sun, Cornford, Moore, Gladstone and Zhao2017) and thus precondition the shelf to calve further, leading to ice front retreat (e.g. Lhermitte and others, Reference Lhermitte2020). Finally, ice acceleration would be expected to be accompanied by thinning, which has the potential to alter the cavity geometry and influence the melt rate (Nakayama and others, Reference Nakayama, Hirata, Goldberg and Greene2022). In particular, thinning that further increases the gap between the seabed ridge and ice shelf might increase the flux of relatively warm water across the seabed ridge and thus increase melt rates close to the PIIS grounding line (De Rydt and others, Reference De Rydt, Holland, Dutrieux and Jenkins2014; Bradley and others, Reference Bradley, Bett, Dutrieux, De Rydt and Holland2022a).

The recent acceleration and retreat of PIG is alarming and the possibility of the collapse of its restraining ice shelf now appears more likely than ever before. We have shown that future ice shelf front retreat is expected to lead to significant acceleration of the adjoining grounded ice, which might additionally promote further calving via a damage-acceleration-calving feedback loop. The acceleration of the grounded ice may be exacerbated by an increase in ice shelf melting in response to ice front retreat, with this melt response promoting further thinning and calving. An extreme acceleration of PIG, as suggested by our simulations, would undoubtedly have significant consequences for future SLR contributions from the entire WAIS, which operates as a connected system of glaciers together holding approximately 5.3 m of SLR equivalent of ice (Morlighem and others, Reference Morlighem2020). Given the possibility of significant near-future acceleration of PIG, a research priority must be to better understand the response of the entire WAIS to abrupt acceleration of its constituent glaciers. More generally, such acceleration and possible collapse represents an extreme scenario with far-reaching consequences; the implications of such high consequence events warrants a significant research effort, particularly as their likelihood is expected to increase in a warming world.