In this study we use a finite-element model Elmer/Ice to solve the full-Stokes equations for a viscous fluid. A rheology following Glen's law is used, with the temperature held constant through the whole of the ice sheet at -15 ∘C. We use a two-dimensional flow line geometry with the bedrock shape the same as that used in a recent model intercomparison project , featuring a section of retrograde slope, described by

Bx=729-2184.8x750km2+1031.72x750km4-151.72x750km6, where x is distance from the inland boundary and B is bedrock elevation relative to sea level.

The grounding line position is solved in the model through a contact problem, taking into account the geometry of the lower surface with respect to the bedrock, the effective pressure at the base of the grounded ice and the buoyancy of the ice in contact with the ocean.

We use a basal sliding relation that is based on the theory of sliding with cavitation and has been implemented in Elmer/Ice . The basal friction is related to the sliding velocity by τbN=Cmax⁡χ1+αχq1/n, where τb is the basal shear stress; ub is the basal ice sliding velocity, with χ=ubCmax⁡nNnAS and α=q-1q-1qq. AS is the sliding parameter in the absence of cavitation; n=3 is the Glen's law exponent; Cmax⁡=0.1 is the maximal value of τb/N, bounded by the maximum slope of the bedrock; and q=1 is the exponent controlling the post-peak decrease. The effective pressure N is calculated based on the assumption of full connectivity between the subglacial hydrologic system and the ocean. This sliding relation is henceforth referred to as the “cavitation sliding relation”.

The other sliding relation used in this study is a non-linear, Weertman type friction law of the form τb=CWubm, where m=1/n, τb and ub the basal shear stress and ice velocity respectively, and CW is a constant friction coefficient. This sliding relation is henceforth referred to as the “Weertman sliding relation”.

A buttressing-like force due to friction along glacier side walls is included by adding a body force into the force balance using a parameterisation relating lateral resistance to the rheological parameters of the ice and an ice shelf embayment width described by . The body force is given by

f=-K|u|mlr-1u, where mlr=1/n is the lateral resistance exponent, with n as the usual Glen's law parameter. The resistance parameter K is given by K=(n+1)1/nWn+1n(2A)1/n, with A being the fluidity parameter of the ice. W is a parameter corresponding to a channel half-width which we use to modify the lateral drag during the experiment from being initially high (i.e. low W) and then decreased by changing to a high value of W as a means of forcing the glacier to retreat.

The experiments presented here used a horizontally uniform mesh resolution with a 500 m element size for simulations using the cavitation sliding relation and 250 m element size for simulations using the Weertman sliding relation. Tests were carried out at coarser resolutions to test for resolution dependence (see Appendix  for details and Appendix  for a table of model parameter values). The mesh is extruded in the vertical direction to 20 equally spaced layers in all simulations.

The retreat experiments examine the behaviour of an ice sheet retreating across a section of retrograde slope when using the cavitation sliding relation, which incorporates a dependency of basal shear stress on effective pressure at the base of the ice. We begin with a 2-D ice sheet grown from a uniform initial thickness of 100 m. During the initial 5000 years of spin-up the parameterised buttressing is set to zero. The buttressing is then linearly increased from zero to an effective channel half-width of 100 km over the next 5000 years, resulting in a high buttressing force. The model is then continued until the grounding line position rests on the seaward side of the bedrock overdeepening (at approximately 1400 km from the ice divide). During this initial spin-up period the top surface accumulation rate is set to 1myr-1, to reduce the total run time for the spin-up. After 10 000 years the accumulation rate is then decreased to 0.3myr-1 and held at this value while the ice sheet stabilises, and it remains at this value throughout the retreat experiments. We determine that the spin-up has finished and the ice sheet has reached a steady state when there has been no change in the grounding line position and the mesh velocity, determining the change in the top and bottom free surfaces, remains less than 0.001myr-1 over 10 000 years, resulting in a total spin-up time of 25 000 years.

We run a series of experiments where we trigger retreat of the glacier through a reduction in the buttressing force by linearly increasing the channel half-width parameter over 10 years to reduce the buttressing force, using values of W equal to 250, 300, 350, 375, 400 and 450 km. An infinitely wide channel corresponds to the case of no lateral drag, and the values of W used in the experiments should be considered as simply providing a range of values for buttressing. Simulations are then run for 2500 years, with 0.1-year time steps with no further forcing change applied after the initial buttressing adjustment.

We carry out a similar retreat experiment using the Weertman sliding relation (Eq. ). In this experiment we again spin up the ice sheet, initially for 5000 years with no buttressing and accumulation of 1myr-1, increasing the buttressing with W=100 km linearly over 5000 years. The accumulation rate is then reduced to 0.3myr-1, and the model is left to evolve for a further 10 000 years until the top and bottom free surfaces show minimal change, resulting in a total spin-up time of 20 000 years.

Recently, showed that far from the grounding line (i.e. for large values of height above flotation) the Weertman and cavitation sliding relations give an approximately equivalent relationship between basal velocity ub and basal sheer stress τb. For this study the Weertman friction coefficient used corresponds to a similar value of τb given by the cavitation sliding relation in the high effective-pressure limit (inland), while also resulting in the initial position of the grounding line being within a few kilometres for both sets of experiments.

To trigger a retreat of the grounding line, we again reduce the buttressing by increasing W from 100 km to values equal to 350, 400 and 500 km linearly over a period of 10 years and then continuing to let the simulation run without further forcing. A time step of 0.5 years was used throughout the experiment.

The position of the grounding line is tracked over time during the channel half-width increase and through the continuation of the model run (Fig. a). In most simulations using the cavitation sliding relation, the grounding line has retreated across the retrograde slope within the first 1000 years, and by 2500 years grounding lines and surface slopes are stabilising. The simulations with higher buttressing (i.e. a smaller forcing perturbation relative to the pre-retreat state) take longer to retreat and stabilise.

The simulations with W=350 and 375 km feature a temporary regrounding of the ice shelf during retreat on the retrograde bedrock. The temporary regrounding occurs approximately 200 km downstream of the original (henceforth “upstream”) grounding line. The upstream grounding line continues to retreat during regrounding of the shelf.

This regrounding is likely caused by the downstream advection of thicker interior ice, which is mobilised by the buttressing reduction. In general, dynamic thinning of the ice shelf due to reduced buttressing competes with thickening due to downstream advection of thicker interior ice to give either a net thinning or thickening in the shelf. For simulations with parameterised channel half-width W≥400 km the peak ice discharge comes early (approximately 300 years; see also Fig. b), and dynamic thinning is sufficient to prevent regrounding. For simulations with W≤300 km a sharp peak in discharge is not seen (Fig. b), and downstream advection of thicker ice is slow, too slow to overcome dynamic thinning in the shelf. Thus the simulations with W=350 and 375 km represent a key region of input space in which regrounding may occur. We refer to this as “dynamic regrounding” to distinguish it from regrounding due to bedrock uplift (described in Sect. ).

The position of the flux gate in Fig. b (1200 km from the ice divide) is chosen as it is located where the regrounding occurs. The flux reaches a maximum as the grounding line approaches the inland end of the retrograde-bedrock region and decreases as the grounding line migrates up the prograde slope. Similarly, we see a reduction in the sliding speed of the ice across the grounding line as shown in Fig. c. For the cases where dynamic regrounding occurs we see a temporary reduction in the flux and sliding speed, but this reduction is not sufficient to stabilise retreat.

Figure  presents a more detailed analysis of retreat and dynamic regrounding for W=350 km. It can be seen that the slope of the lower surface of the ice shelf is similar to the retrograde bedrock slope, and this corresponds to a very shallow water column under the shelf. The implication is that only a very small amount of thickening is needed to cause regrounding. The shallow water column is common to all retreat simulations with the cavitation sliding relation (not shown).

Figure c shows basal shear stress, the point of inflection and the grounding line. The point of inflection is with respect to the upper surface height of the ice sheet and indicates the switch from a convex ice sheet surface (inland, which includes most of the grounded ice) to concave. It is identified here through calculation of the maximum gradient of the upper surface. Figure c shows that the point of inflection corresponds to the maximum in basal shear stress. Downstream of this line basal shear stress drops rapidly to zero, due to the dependence on effective pressure at the bed, N, in the cavitation sliding relation (Eq. ). The grounding line is persistently a few tens of kilometres downstream of the point of inflection. The implications of the point of inflection and basal shear stress pattern for transition zones and ice sheet profiles with different sliding relations will be discussed further in Sect. . Basal shear stress is also very low under the regrounding region, again due to the dependence on N.

In experiments where the Weertman sliding law is used we see no temporary regrounding of the ice sheet. The ice shelf develops a thinner profile than with the cavitation sliding relation and a markedly different shape, as shown in Fig. a. A strongly concave shape is evident immediately downstream of the grounding line, in contrast to the more linear ice shelf profile when using the cavitation sliding relation. As a result of this difference, water column thickness is much larger during retreat than for the cavitation sliding relation.

In Fig. b we see high-frequency changes in velocity, with each jump in velocity likely to correspond to grounding line movement between neighbouring mesh elements. Ungrounding of an element significantly reduces the basal friction, allowing speedup. This does not happen with the cavitation sliding law (Fig. c) because the dependence on effective pressure means that the basal shear stress is in any case close to zero for elements that are newly ungrounded.