[1] The ice sheet-ice shelf transition zone plays an important role in controlling marine ice sheet dynamics, as it determines the rate at which ice flows out of the grounded part of the ice sheet. Together with accumulation, this outflow is the main control on the mass balance of the grounded sheet. In this paper, we verify the results of a boundary layer theory for ice flux in the transition zone against numerical solutions that are able to resolve the transition zone. Very close agreement is obtained, and grid refinement in the transition zone is identified as a critical component in obtaining reliable numerical results. The boundary layer theory confirms that ice flux through the grounding line in a two-dimensional sheet-shelf system increases sharply with ice thickness at the grounding line. This result is then applied to the large-scale dynamics of a marine ice sheet. Our principal results are that (1) marine ice sheets do not exhibit neutral equilibrium but have well-defined, discrete equilibrium profiles; (2) steady grounding lines cannot be stable on reverse bed slopes; and (3) marine ice sheets with overdeepened beds can undergo hysteresis under variations in sea level, accumulation rate, basal slipperiness, and ice viscosity. This hysteretic behavior can in principle explain the retreat of the West Antarctic ice sheet following the Last Glacial Maximum and may play a role in the dynamics of Heinrich events.

Increased ice velocities in Greenland are contributing significantly to eustatic sea level rise. Faster ice flow has been associated with ice-ocean interactions in water-terminating outlet glaciers and with increased surface meltwater supply to the ice-sheet bed inland. Observed correlations between surface melt and ice acceleration have raised the possibility of a positive feedback in which surface melting and accelerated dynamic thinning reinforce one another, suggesting that overall warming could lead to accelerated mass loss. Here I show that it is not simply mean surface melt but an increase in water input variability that drives faster ice flow. Glacier sliding responds to melt indirectly through changes in basal water pressure, with observations showing that water under glaciers drains through channels at low pressure or through interconnected cavities at high pressure. Using a model that captures the dynamic switching between channel and cavity drainage modes, I show that channelization and glacier deceleration rather than acceleration occur above a critical rate of water flow. Higher rates of steady water supply can therefore suppress rather than enhance dynamic thinning, indicating that the melt/dynamic thinning feedback is not universally operational. Short-term increases in water input are, however, accommodated by the drainage system through temporary spikes in water pressure. It is these spikes that lead to ice acceleration, which is therefore driven by strong diurnal melt cycles and an increase in rain and surface lake drainage events rather than an increase in mean melt supply.

Basal sliding is one of the most important components in the dynamics of fast–flowing glaciers, but remains poorly understood on a theoretical level. In this paper, the problem of glacier sliding with cavitation over hard beds is addressed in detail. First, a bound on drag generated by the bed is derived for arbitrary bed geometries. This bound shows that the commonly used sliding law, τ b = Cu m b N n , cannot apply to beds with bounded slopes. In order to resolve the issue of a realistic sliding law, we consider the classical Nye–Kamb sliding problem, extended to cover the case of cavitation but neglecting regelation. Based on an analogy with contact problems in elasticity, we develop a method which allows solutions to be constructed for any finite number of cavities per bed period. The method is then used to find sliding laws for irregular hard beds, and to test previously developed theories for calculating the drag generated by beds on which obstacles of many different sizes are present. It is found that the maximum drag attained is controlled by those bed obstacles which have the steepest slopes.

[1] We present a two-dimensional Glacier Drainage System model (GlaDS) that couples distributed and channelized subglacial water flow. Distributed flow occurs through linked cavities that are represented as a continuous water sheet of variable thickness. Channelized flow occurs through Röthlisberger channels that can form on any of the edges of a prescribed, unstructured network of potential channels. Water storage is accounted for in an englacial aquifer and in moulins, which also act as point sources of water to the subglacial system. Solutions are presented for a synthetic topography designed to mimic an ice sheet margin. For low discharge, all the flow is accommodated in the sheet, whereas for sufficiently high discharge, the model exhibits a channelization instability which leads to the formation of a self-organized channel system. The random orientation of the network edges allows the channel system geometry to be relatively unbiased, in contrast to previous structured grid-based models. Under steady conditions, the model supports the classical view of the subglacial drainage system, with low pressure regions forming around the channels. Under diurnally varying input, water flows in and out of the channels, and a rather complex spatiotemporal pattern of water pressures is predicted. We explore the effects of parameter variations on the channel system topology and mean effective pressure. The model is then applied to a mountain glacier and forced with meltwater calculated by a temperature index model. The results are broadly consistent with our current understanding of the glacier drainage system and demonstrate the applicability of the model to real settings.

Marine ice sheets are continental ice masses resting on bedrock below sea level. Their dynamics are similar to those of land-based ice sheets except that they must couple with the surrounding floating ice shelves at the grounding line, where the ice reaches a critical flotation thickness. In order to predict the evolution of the grounding line as a free boundary, two boundary conditions are required for the diffusion equation describing the evolution of the grounded-ice thickness. By analogy with Stefan problems, one of these conditions imposes a prescribed ice thickness at the grounding line and arises from the fact that the ice becomes afloat. The other condition must be determined by coupling the ice sheet to the surrounding ice shelves. Here we employ matched asymptotic expansions to study the transition from ice-sheet to ice-shelf flow for the case of rapidly sliding ice sheets. Our principal results are that the ice flux at the grounding line in a two-dimensional ice sheet is an increasing function of the depth of the sea floor there, and that ice thicknesses at the grounding line must be small compared with ice thicknesses inland. These results indicate that marine ice sheets have a discrete set of steady surface profiles (if they have any at all) and that the stability of these steady profiles depends on the slope of the sea floor at the grounding line.

Predictions of marine ice-sheet behaviour require models that are able to robustly simulate grounding line migration. We present results of an intercomparison exercise for marine ice-sheet models. Verification is effected by comparison with approximate analytical solutions for flux across the grounding line using simplified geometrical configurations (no lateral variations, no effects of lateral buttressing). Unique steady state grounding line positions exist for ice sheets on a downward sloping bed, while hysteresis occurs across an overdeepened bed, and stable steady state grounding line positions only occur on the downward-sloping sections. Models based on the shallow ice approximation, which does not resolve extensional stresses, do not reproduce the approximate analytical results unless appropriate parameterizations for ice flux are imposed at the grounding line. For extensional-stress resolving "shelfy stream" models, differences between model results were mainly due to the choice of spatial discretization. Moving grid methods were found to be the most accurate at capturing grounding line evolution, since they track the grounding line explicitly. Adaptive mesh refinement can further improve accuracy, including fixed grid models that generally perform poorly at coarse resolution. Fixed grid models, with nested grid representations of the grounding line, are able to generate accurate steady state positions, but can be inaccurate over transients. Only one full-Stokes model was included in the intercomparison, and consequently the accuracy of shelfy stream models as approximations of full-Stokes models remains to be determined in detail, especially during transients

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[1] Understanding the dynamics of marine ice sheets is integral to studying the evolution of the Antarctic ice sheet in both the short and long terms. An important component of the dynamics, grounding line migration, has proved difficult to represent in numerical models, and most successful attempts have made use of techniques that are only readily applicable to flow line models. However, to capture the stress transmission involved in another important component, the buttressing of a marine ice sheet by its ice shelf, the transverse direction must also be resolved. We introduce a model that solves the time-dependent shelfy stream equations and makes use of mesh adaption techniques to overcome the difficulties typically associated with the numerics of grounding line migration. We compare the model output with a recent benchmark for flow line models and show that our model yields an accurate solution while using far less resources than would be required without mesh adaption. We also show that the mesh adapting techniques extend to two horizontal dimensions. Experiments are carried out to determine how both ice shelf buttressing and ice rises affect the marine instability predicted for an ice sheet on a foredeepened bed. We find that buttressing is not always sufficient to stabilize such a sheet but collapse of the grounded portion is still greatly delayed. We also find that the effect of an ice rise is similar to that of narrowing the ice shelf.