Summary

The behaviour of tidewater glaciers represents a key uncertainty in predictions of future cryosphere change and associated sea-level rise. They are responsible for ~50% of total mass loss from the Greenland ice sheet (Mouginot and others, Reference Mouginot2019; King and others, Reference King2020), and nearly 100% in Antarctica (Rignot and others, Reference Rignot2019), where surface melting exerts much less influence. However, these glaciers often show very heterogeneous and local responses to environmental forcing due to the complex interplay of particular atmospheric, oceanic and topographic conditions that determines their response (e.g. Seale and others, Reference Seale, Christoffersen, Mugford and O'Leary2011; Csatho and others, Reference Csatho2014; Catania and others, Reference Catania2018; Fried and others, Reference Fried2018).

At the same time, tidewater glaciers are particularly challenging environments in which to gather observations. They are among the world's fastest-flowing glaciers and drain catchments that span thousands of square kilometres. These glaciers also often have ice thicknesses of several hundred metres or more, and they tend to be heavily crevassed. In addition to this, many important processes (submarine melt, plumes, calving) happen at the ice–ocean interface, where the random nature of calving and the front's general instability represent a further barrier to observation or monitoring. As a result, studies have tended to focus on the more accessible, slower and less hazardous land-terminating ice margins, resulting in a relative lack of observational evidence for the marine-terminating glaciers that drain the ice sheet and control its mass balance.

This combination of their central role in ice-sheet dynamics and the sparse observational record means that tidewater glaciers are of particular importance as targets for modelling, in order to both predict sea-level rise from ice sheets and ice caps, and to seek to understand the controlling physical processes that govern their evolution. However, the lack of data (e.g. long-term calving observations, sediment properties, submarine melt rates, ice thickness, subglacial hydrological extent and dynamics) for model validation makes it challenging to assess model performance and to understand which processes and parameters are well-modelled and which are poorly modelled or missing entirely. The complex nature of the tidewater–glacier system, with a slew of interacting glacier and fjord processes, adds a further step in difficulty, as a model that explicitly represents all of these will necessarily be computationally very expensive.

Perhaps not surprisingly therefore, progress in developing 3-D full-Stokes-coupled models of tidewater glaciers has been relatively slow, and focused heavily on using the Elmer/Ice modelling suite (Gagliardini and others, Reference Gagliardini2013), with computing constraints only beginning to be lifted in the last decade. The first offline-coupled attempts to link ice flow, mass balance, calving, fjord circulation, submarine melt and subglacial hydrology were made by Vallot and others (Reference Vallot2018), with much work since then focusing chiefly on improving the representation of calving in glacier models (Todd and others, Reference Todd2018, Reference Todd, Christoffersen, Zwinger, Råback and Benn2019) and the notable addition of subglacial hydrology (Cook and others, Reference Cook, Christoffersen, Todd, Slater and Chauché2020). Importantly only one method to predict calving (or ‘calving law’) has been implemented in a 3-D full-Stokes setting (Todd and others, Reference Todd2018, Reference Todd, Christoffersen, Zwinger, Råback and Benn2019), a principal-stress version of the crevasse-depth law (Otero and others, Reference Otero, Navarro, Martin, Cuadrado and Corcuera2010). Although other calving laws are established in 2-D planar models or simplified 3-D ones (e.g. Choi and others, Reference Choi, Morlighem, Wood and Bondzio2018), none provide an obvious candidate for 3-D full-Stokes modelling given their requirement for tunable input parameters or lack of a physical basis (Benn and others, Reference Benn, Cowton, Todd and Luckman2017a).

Recently, the first example of a fully coupled 3-D full-Stokes model including glacier flow, meltwater plumes at the calving front, subglacial hydrology and calving has been published (Cook and others, Reference Cook, Christoffersen and Todd2021; see Fig. 1). This represents a major computational advance, with all these components of the tidewater–glacier system coupled online for the first time. Yet, the modelling still does not include all necessary processes to arrive at a full representation of a tidewater glacier, with particular issues surrounding the implementation of long-term calving and the modelled submarine melt rates. Further, little work has focused on coupling the glacier–ice-melange system, mainly because its granular nature makes it impossible to model in continuum (e.g. Burton and others, Reference Burton, Amundson, Cassotto, Kuo and Dennin2018; Amundson and others, Reference Amundson2020; Cassotto and others, Reference Cassotto, Burton, Amundson, Fahnestock and Truffer2021). In 3-D full-Stokes glacier modelling, ice-melange has always been represented by a binary backstress parameterisation (e.g. Todd and others, Reference Todd2018; Cook and others, Reference Cook, Christoffersen and Todd2021), despite increasing research showing a close reciprocal relationship between calving and ice-melange (Bevan and others, Reference Bevan, Luckman, Benn, Cowton and Todd2019; Cassotto and others, Reference Cassotto, Burton, Amundson, Fahnestock and Truffer2021; Melton and others, Reference Melton2022). This contrasts with progress on the other individual components of the system, with much modelling work having been undertaken on the fjord and plume circulation alone (e.g. Straneo and Cenedese, Reference Straneo and Cenedese2015; Cowton and others, Reference Cowton2016; Davison and others, Reference Davison, Cowton, Cottier and Sole2020), or on ice flow solely (e.g. van Dongen and others, Reference van Dongen2020; Crawford and others, Reference Crawford2021; Amundson and others, Reference Amundson, Truffer and Zwinger2022). Bringing all this smaller-scale work within a fully coupled modelling framework that includes hitherto neglected processes remains an ideal not yet achieved (Table 1).

Fig. 1. Major model components in Cook and others (Reference Cook, Christoffersen and Todd2021). Readers are directed to this paper for a full explanation of the individual components. Top panel (3-D ice-flow mesh) shows glacier surface elevation in m in grey-red, with plume melt rates in m³ a⁻¹ in grey-blue-white on the calving front (low-high in both cases). Bottom panel (2-D subglacial hydrology mesh) shows wintertime channel area in m² in black-blue-white colours, and wintertime sheet discharge in m³ a⁻¹ in purple-orange (low-high in both cases).

Table 1. Summary of how well key processes are included in current tidewater–glacier models along with the main challenge or area for improvement

Future research priorities

Given the progress in numerical modelling of tidewater glaciers over the last decade, future research priorities and directions have emerged and become clearer. In our view, the two most important are more realistic calving representations within models and greater consideration of the warm water that is brought towards the terminus by fjord circulation. Both glacier and fjord models are able to reproduce their respective environments with a reasonably good degree of accuracy, although only when they are considered individually; the difficulties lie at the point where the two systems meet and interact. This has long been recognised with regards to the necessity for high model resolution and full 3-D flow modelling around the grounding line (Pattyn and others, Reference Pattyn2012), but this is not the only important feature in this region of the tidewater–glacier system.

For calving itself, our view is that calving algorithms that simulate calving based on the stress field of the glacier (e.g. Todd and others, Reference Todd2018) are the best physically based and computationally reasonable implementation of calving currently available. However, these need to be improved further by considering crevasses as structural weaknesses consisting of damaged ice with a preferred crystal orientation fabric and lower yield strength in the direction of shear, in order to reproduce calving at observed rates. This could be done in several ways. While ice-damage functions already exist in some models (Krug and others, Reference Krug, Weiss, Gagliardini and Durand2014; Mercenier and others, Reference Mercenier, Lüthi and Vieli2019), the technical work to couple one with a full-Stokes 3-D calving algorithm has not yet been undertaken to our knowledge, nor have algorithms that track the deformational history of ice as it advects towards the ocean. It is also far from obvious how damage evolution may be incorporated into current stress-based calving laws. Unless the damage can be modelled as part of the viscous evolution of the ice and stress concentrations accounted for, a stress multiplier would need to be used that would likely end up being a tunable parameter, varying between glaciers, so would be far from ideal. Alternatively, using a Lagrangian approach, crevasse history could be explicitly tracked, something already implemented in 2-D modelling, which has been shown to lead to an increase in modelled calving compared to not including crevasse history (Berg and Bassis, Reference Berg and Bassis2022). Another possibility is to use computationally expensive models that explicitly model fracture dynamics to attempt to directly determine new calving laws or, at least, diagnostic stress states for calving (Åström and others, Reference Åström2013; Benn and Åström, Reference Benn and Åström2018). When these states then appear in continuum-flow models, calving could be assumed to have occurred, but we feel that these states will depend too heavily on the particular geometry of a given glacier to be widely applicable. We therefore consider the crevasse-history route as more promising at this point in time.

We also contend that a lack of inclusion of 3-D fjord circulation may be behind the frequent underestimation of submarine melt rates when derived from models relying on buoyant plume theory (e.g. Jenkins, Reference Jenkins2011; Slater and others, Reference Slater, Goldberg, Nienow and Cowton2016; Ezhova and others, Reference Ezhova, Cenedese and Brandt2018; Jackson and others, Reference Jackson2022). We particularly consider that lateral flow, parallel to the calving front, may be a missing element in models that would lead to more accurate melt rates were it to be added. Coupling an ice-flow model to a high-resolution fjord general circulation model would be one way of investigating this, but this would be computationally expensive. If such work were undertaken, it would certainly be best directed towards finding an inexpensive parameterisation that could be widely reproduced in existing 1-D plume models.

Overall, however, we are confident these challenges can be overcome, enabling more accurate predictions of tidewater–glacier behaviour in the forthcoming years. Only then will a fully coupled 3-D full-Stokes model of a tidewater glacier be achievable. We find it difficult to envisage an alternative modelling framework for the detailed simulation of individual tidewater glaciers discussed in this paper, as a full-Stokes approach will always be required at the grounding line at the very least. Discrete-element models (Åström and others, Reference Åström2013; Benn and others, Reference Benn2017b; Vallot and others, Reference Vallot2018) are too computationally expensive to be used for large areas or simulations of over a few weeks, and recent machine-learning emulator approaches (Jouvet and others, Reference Jouvet2021; Jouvet, Reference Jouvet2022) would have to be trained on full-Stokes simulations in the first place (and, given the heterogeneity of tidewater glaciers, might have to be retrained for each individual glacier). We therefore believe that the goal of a fully coupled 3-D full-Stokes model of a tidewater glacier remains a desirable one.