Iceberg calving from marine-terminating glaciers is an important component of the Greenland Ice Sheet mass balance. Calving accounted for a third of total mass loss between 2009 and 2012 (Enderlin et al., 2014). Moreover, calving is an important control on the flow dynamics of tidewater glaciers, reducing the backstress in the glacier snout region and leading to flow acceleration and dynamic thinning (Thomas, 2004). The acceleration of Jakobshavn Isbræ (JI, Sermeq Kujalleq in Greenlandic) by a factor of 4 since 1995, for example, is linked with its continued calving retreat following the disintegration of its ice shelf (Joughin et al., 2012).

Current models of iceberg calving fail to capture the full range of observed processes, and as such the parameterisations applied within ice sheet models are limited. Van der Veen (1996) proposed the empirical height-above-flotation criterion, whereby the glacier calves to a point where its terminus is some fixed height above the flotation thickness. Although this method successfully reproduced advance and retreat behaviour for Columbia Glacier (Vieli et al., 2001) and Helheim Glacier (Nick et al., 2009), a major shortcoming was the inability for ice to thin below the flotation thickness and form an ice tongue. A more physically based approach (Benn et al., 2007a, b) assumed that crevasses penetrating to the waterline penetrate through the full glacier thickness. This simple theory has been used in many recent modelling studies (e.g. Otero et al., 2010; Nick et al., 2013; Cook et al., 2014). However, since crevasse depths are calculated based on the equilibrium between longitudinal stretching and ice overburden pressure (Nye, 1957), calving in these models arises only as a result of ice flow dynamics. The effect of localised processes such as meltwater undercutting (Luckman et al., 2015), ice-cliff collapse (Bassis and Walker, 2012) or of super-buoyancy upon near-terminus stresses is not captured in these models.

Buoyant forces have been proposed as a driver of large calving events observed at major Greenland Ice Sheet marine-terminating glaciers. Full-depth, bottom-out calving events observed at Helheim Glacier resulted from buoyant flexure of the glacier snout and the propagation of basal crevasses (e.g. Murray et al., 2015). Wagner et al. (2016) also showed that applying a buoyant force to an elastic beam model of a glacier resulted in large basal tensile stresses, which were further amplified by the emergence of a submarine protrusion of the calving face due to sea surface melting. Buoyant forces may be at play in driving as yet unexplained calving styles.

A multiple-iceberg calving event was observed at JI in August 2009 (Walter et al., 2012) that is not fully explained by existing calving models, but which we propose is tied to buoyant force changes over the course of the multi-stage calving event. In this observation, the collapse of a subaerial portion of the ice cliff was followed minutes later by a much larger, full-depth, bottom-out calving event across the same section of the front. We consider a mechanism to explain this event, whereby a substantial portion of the snout becomes buoyant immediately following a small subaerial calving event, which we term “notch-triggered rotation”. In this mechanism, visualised in Fig. 1, the sudden increase in buoyant load causes the snout to lift and rotate. The resultant basal tensile stresses initiate basal crevassing, which rapidly propagates through the full glacier thickness. This mechanism is similar to the “footloose” mechanism investigated by Wagner et al. (2014) and earlier proposed by Scambos et al. (2005) for the breakup of tabular icebergs. However, in this study we consider the very short timescales arising from abrupt changes in the geometry and analyse the viscous stresses originating at the ice–bed interface rather than the elastic stresses resulting purely from bending.

Using a diagnostic numerical glacier model, we investigate whether notch-triggered rotation is a plausible calving mechanism. With sophisticated prognostic models, calving criteria can be tested by application to real glacier geometries (e.g. Nick et al., 2010; Krug et al., 2014) and the calving rate response to various environmental forcings can be quantified (e.g. Cook et al., 2014; Todd and Christoffersen, 2014). Simpler diagnostic models provide insight into iceberg calving mechanisms by resolving the internal stresses under instantaneously imposed geometries (e.g. Hanson and Hooke, 2000, 2003; O'Leary and Christofferson, 2013). Here we use a diagnostic model that is able to quantify changes in the stress field induced by geometrical perturbations of the ice front.

In this study we use a two-dimensional diagnostic flow-line model of an idealised glacier snout to determine whether the magnitude of stresses generated by the notch-triggered rotation mechanism is sufficient to result in calving. The mechanisms of crevasse propagation through the ice thickness are not examined. We apply the buoyant forcing in the model by cutting a notch into the subaerial ice cliff to a length ln from the waterline to the surface (Fig. 2). We use a triangular mesh with a mesh resolution at the ice–bed interface of Δx=4 m at the calving front increasing linearly to Δx=8 m at the rear of the domain. The ice flow solution is calculated using the open source, full-Stokes, finite element Elmer/Ice modelling software (Gagliardini et al., 2013).

Experiments were run for a glacier with water depth dw=900 m, terminus thickness ht=980 m, surface slope α=3∘ and C=2.371×106 Pa m-1/3 s1/3, with notches cut to varying lengths. For these experiments, full hydrostatic pressure (Eq. 7) was applied along the basal boundary. Figure 4 shows EPS mapped for the ln=100 m case, with the ln=0 m case mapped in Fig. 3 for comparison. Basal stresses are plotted for ln=0 m, ln=80 m and ln=100 m (Fig. 5).

Notch cutting caused basal ice to become ungrounded between approximately 190 and 640 m upstream of the ice front for ln=100 m (Fig. 4). Prominent stress concentrations associated with ungrounding and regrounding also appeared at the basal boundary, which were not present in the unperturbed case. The tensile stress peak centred approximately 640 m upstream of the ice front resulted from separation of basal ice from the bedrock, as the buoyant snout tended to lift. The abrupt change in basal shear stress across the grounding zone, where ice that has separated from the bedrock accelerates, gave rise to this stress peak. Further notch-cutting caused this stress peak to shift upstream and increase in magnitude. The substantial growth in concentrated stress at this location to approximately 1.4 MPa would likely result in crevasse opening.

The region of compressive stress centred approximately 190 m upstream of the ice front arose from ice regrounding on the bedrock downstream of the grounding line, due to the backstress applied on the ice front by hydrostatic pressure. An imbalance between the hydrostatic and cryostatic pressure normal to the terminus tends to warp the snout downwards (see Fig. S1 in the Supplement; Reeh, 1968), with the same effect seen at the start of prognostic model runs by Benn et al. (2017). Experiments in which the hydrostatic pressure from the pro-glacial water body was removed or the bedrock lowered downstream of the grounding line, did not include this compressive stress peak while still featuring the tensile stress peak, supporting our assertion that the compressive stress concentration resulted from basal ice regrounding.

Corresponding longitudinal velocity maps for the frontal region are shown in Fig. S2 (unperturbed) and Fig. S3 (ln=100 m). There is an acceleration of ∼2000 m a-1 following the notch cutting, resulting from the reduced basal friction in the ungrounded region.

A critical notch length lcrit was required before the glacier snout became buoyant and the tensile stress peak appeared. A sensitivity study was carried out to explore the relationship of this critical notch length to the bed stickiness and the glacier surface slope (Fig. 6) and to determine how this relationship affects the maximum basal stress (Fig. 7). Setting the notch length ln=lcrit resulted in a noisy maximum basal stress signal so we instead set ln=lcrit+25 m, which allows a coherent pattern to emerge.

Ungrounding occurred even without a notch on glaciers with very slippery beds for all surface slopes and at all values of the friction coefficient for a 2∘ surface slope. For steeper surface slopes, the critical notch length increased with bed stickiness before levelling off. For a given value of the friction coefficient, the critical notch length also increased with surface slope. Similarly, the maximum basal stress increased with both friction coefficient and surface slope. For very slippery beds the maximum stress was below the upper boundary of the tensile strength envelope, but significantly it was above the critical stress for crevasse initiation through the realistic range of friction coefficients C=106 to 107 Pa m-1/3 s1/3.

These experiments reveal a complex picture of the conditions that favour calving. An explanation for the relationship between the critical notch length and bed stickiness does not readily present itself, and this effect may warrant further investigation. The relationships of surface slope with both the critical notch length and the maximum basal stress are more easily explained. The terminus of a steeper sloped glacier is more strongly grounded, requiring the removal of more ice to render it buoyant than would be required for a more gently sloping glacier. The longer submarine foot and larger buoyant forces that result then favour larger basal stresses (e.g. Wagner et al., 2016).

Our experiments show that perturbations to the ice front geometry can induce large stress concentrations in basal ice. The magnitude and location of these stress concentrations shows a strong dependency on the basal shear stress. For a glacier snout already close to flotation, only a relatively small perturbation was required to induce large stresses. This is in line with the observed relationship between calving rate and water depth (Brown et al., 1982).

The large internal stress concentrations reported here are attributed to the requirement to balance the abrupt decrease in basal shear stress across the grounding line and are not associated with bending stresses. This distinction is based on the following observations. Firstly, the region of additional high stress is sharply focused at the glacier bed and is not balanced by a region of compressive stress at the surface as would be expected for a viscous bending moment (Mosbeux et al., 2019). Secondly, the stress concentration is compressive where the ice regrounds further downstream, whereas a bending stress at this location would also be tensile. Finally, we have shown that the form and magnitude of stress is highly dependent upon the choice of sliding law and application of basal water pressure, which would be largely irrelevant for bending stresses.

The choice of diagnostic model for a calving study was criticised by Cook et al. (2014) after their prognostic model showed much greater sensitivity to atmospheric as opposed to oceanic forcing than diagnostic models (O'Leary and Christoffersen, 2013), suggesting that this was due to the inability of a diagnostic model to respond to stress perturbations through ice deformation. However, over the short timescales of interest in this study, deformation of ice is negligible. In our experiments, measured vertical velocities for the ungrounded regions of basal ice were of the order ∼10 m a-1, equating to ∼0.1 mm of lifting over 5 min, which would have negligible effect on the stress field. Therefore, our choice of a diagnostic model is an appropriate one for this study.

As in other diagnostic studies, we did not apply a calving criterion, instead using the location of basal stress peaks as an indication of where crevasses may form. For this to result in calving on the timescale proposed requires the assumption of full-thickness crevassing on timescales much faster than those observed by, e.g. Murray et al. (2015). Given a sufficiently large buoyancy force, this assumption can be held as true, as once a crack has initiated, the tensile stress which opened that crack refocuses at the crack tip causing it to continue to propagate. As the crevasse increases in height, hydrostatic pressure acting to open the crevasse decreases at a faster rate than the ice overburden pressure acting to close it; therefore, larger basal stresses are required for full-depth crevassing than for crevasse initiation. However, once a crevasse has started to propagate and the downstream portion of the snout has begun to lift and rotate, elastic stresses further contribute to the crevasse growth in a feedback process. Benn et al. (2017) reported that glacier geometries that did not result in calving in Elmer/Ice via crevasse depth calving laws still produced large full-depth calving event when exported into HiDEM, a model representing glacier ice as a lattice of particles connected by breakable elastic beams. Further investigation of the rate and modes of crevasse propagation could integrate linear elastic fracture mechanics into a glacier model featuring basal crevasses (van der Veen, 1998) or use a model such as HiDEM in conjunction with Elmer/Ice (Benn et al., 2017).

Our study builds on that of O'Leary and Christoffersen (2013), which also explored the effect of geometrical perturbations at the ice front on the likelihood of calving. Their study found that undercutting led to larger calving events and a higher overall calving rate, which appears to be at odds with the results presented here: undercutting would reduce the buoyant load and potentially stabilise the terminus. Our results differ because the sharp transition in basal shear stress is not possible at the stress-free surface boundary applied here. Furthermore, the geometry of their model was set up to explore surface crevassing while ours was designed to explore basal crevassing. In reality a mixture of these effects may be working together to promote or prohibit calving.

Figure 1 suggests the subaerial calving event may result from undercutting by a waterline notch. Although this process is observed at some glaciers (e.g. Kirkbride and Warren, 1997; Röhl, 2006), it is questionable whether it could be a major factor in the Ilulissat Icefjord (where the original observation was made), in which the loosely bonded melange in summer may act to damp any wave-cutting action (Amundson et al., 2010). An alternative potential mechanism for triggering the subaerial calving event is provided by spontaneous collapse of the ice cliff. The maximum stable cliff height for damaged glacier ice was calculated by Bassis and Walker (2012) as 110 m, while Hanson and Hooke (2003) suggested a maximum stable height of ∼70 m based on diagnostic model experiments. The ice cliffs of JI approach 100 m but rarely exceed this height, suggesting that the inherent instability of ice cliffs may be the limiting factor and could induce calving through notch-triggered rotation.

Buoyancy in a glacier snout can also be induced by thinning due to high surface melt rates. However, the almost immediate increase in buoyant load resulting from the subaerial calving event proposed here occurs on timescales much faster than can be accommodated by ice creep, leading to a higher probability of calving (e.g. Boyce et al., 2007). The specific location of the basal stress peak varied with many parameters, including the notch length but tended to be within one ice thickness of the terminus, consistent with observations (e.g. Walter et al., 2012; Murray et al., 2015). The location of the peak stress always occurred much further back from the terminus than the cliff at the rear of the notch, leading to an amplification of the original subaerial calving event. The value of this amplifying factor cannot be accurately quantified within the limitations of a diagnostic model; however, it may present a method of linking environmental forcings to the calving rate.

There are a number of possible refinements to our model. We ignored lateral drag, which combines with basal shear stress to support the driving stress. Although lateral drag may be negligible along the flow line of wide ice streams, JI was able to form a floating tongue so it must be assumed that lateral drag is significant (e.g. Thomas, 2004). Its omission may have caused the model to overstate the dependence of basal stresses on the basal sliding law. Our model also omits the effect of temperature. The viscosity of ice and transmission of stresses are dependent on thermal gradients. JI has large vertical temperature gradients (Lüthi et al., 2002) and temperate basal ice, which are thought to play a role in its fast sliding. Warmer basal temperatures may act to damp the intensity of basal stress concentrations.

The reader should note that our model geometry is idealised. In reality, glacier beds are highly non-uniform, with variations in geometry, water and overburden pressure across a range of spatial and temporal scales. Bedrock bumps therefore play an important role in controlling the stress transmission upstream. It is plausible that these variations could result in basal stress concentrations of a similar magnitude to the mechanism discussed here.

The notch-triggered rotation mechanism was shown to be irrelevant when using the Schoof law with full water pressure, since a glacier in these conditions would tend to be vulnerable to buoyant calving anyway. This raises the question of whether the Schoof law with full water pressure provides an accurate representation of basal sliding for JI. We expect low effective pressure in the frontal region; however, given that the glacier snout is mostly grounded in the summer (e.g. Amundson et al., 2010), perfect hydraulic connectivity cannot be assumed along the ice–bed interface. Complete suppression of water pressure at the ice–bed interface resulted in a basal stress pattern very similar to the Weertman case, although it was strongly compressive everywhere due to the exclusion of water pressure in the calculation of EPS and is therefore very unlikely to result in crevassing. With Cc=q=1 and m=1/n, as in this study, it can be easily shown that large N (∼10 MPa in the absence of water pressure) leads to small χ and Eq. (9) reduces to a Weertman power law: 15τb=-As-mubm-1⋅ub.

On the other hand, limiting the basal water pressure without suppressing it completely (experiments P0 and P100) resulted in a transition case displaying similar behaviour to both the Weertman and Schoof laws; the unperturbed stress profile was similar to that produced by the Schoof law, but the perturbation resulted in a significantly larger stress peak like those produced by the Weertman law. This raises the possibility that a lightly grounded glacier snout, already in a state of basal tension, could be subjected to high enough stress by a minor subaerial calving event, like that observed at JI (Walter et al., 2012), to cause full depth crevassing and buoyant calving.

Our results show that the notch-triggered rotation mechanism does produce calving for an idealised marine-terminating glacier. Although notch-triggered rotation did not significantly affect stresses when applying the Schoof law under full hydrostatic pressure, removing the assumption of perfect hydraulic connectivity at the ice–bed interface greatly enhanced the likelihood of calving through this mechanism. Significantly, a realistic length scale for calving events, on the order of hundreds of metres and generally less than one ice thickness, naturally results from the model physics. Fast-flowing glaciers near flotation and with shallow surface slopes may be especially vulnerable to buoyant calving due to basal crevassing. The notch-triggered rotation mechanism proposed here to explain the observed calving event (Walter et al., 2012) does not replace other models of calving. Instead, it bolsters our understanding of calving by providing insight into multi-stage calving events occurring particularly on large, fast-flowing tidewater glaciers.