Meltwater drainage through ice sheets has recently been a key focus of glaciological research due to its influence on the dynamics of ice sheets in a warming climate. However, the processes, topologies and products of ice sheet hydrology are some of the least understood components of both past and modem ice sheets. This is to some extent a result of a disconnect between the fields of theoretical, contemporary observational and palaeo-glaciology that each approach ice sheet hydrology from a different perspective and with different research objectives. With an increasing realisation of the potential of using the past to inform on the future of contemporary ice sheets, bridging the gaps in the understanding of ice sheet hydrology has become paramount. Here, we review the current state of knowledge about ice sheet hydrology from the perspectives of theoretical, observational and palaeo-glaciology. We then explore and discuss some of the key questions in understanding and interpretation between these research fields, including: 1) disagreement between the palaeo-record, glaciological theory and contemporary observations in the operational extent of channelised subglacial drainage and the topology of drainage systems; 2) uncertainty over the magnitude and frequency of drainage events associated with geomorphic activity; and 3) contrasts in scale between the three fields of research, both in a spatial and temporal context The main concluding points are that modem observations, modelling experiments and inferences from the palaeo-record indicate that drainage topologies may comprise a multiplicity of forms in an amalgam of drainage modes occurring in different contexts and at different scales. Drainage under high pressure appears to dominate at ice sheet scale and might in some cases be considered efficient; the sustainability of a particular drainage mode is governed primarily by the stability of discharge. To gain better understanding of meltwater drainage under thick ice, determining what drainage topologies are reached under high pressure conditions is of primary importance. Our review attests that the interconnectivity between research sub-disciplines in progressing the field is essential, both in interpreting the palaeo-record and in developing physical understanding of glacial hydrological processes and systems.

We investigate the accuracy and robustness of one of the most common methods used in glaciology for the discretization of the p-Stokes equations: equal order finite elements with Galerkin Least-Squares (GLS) stabilization. Furthermore we compare the results to other stabilized methods. We find that the vertical velocity component is more sensitive to the choice of GLS stabilization parameter than horizontal velocity. Additionally, the accuracy of the vertical velocity component is especially important since errors in this component can cause ice surface instabilities and propagate into future ice volume predictions. If the element cell size is set to the minimum edge length and the stabilization parameter is allowed to vary non-linearly with viscosity, the GLS stabilization parameter found in literature is a good choice on simple domains. However, near ice margins the standard parameter choice may result in significant oscillations in the vertical component of the surface velocity. For these cases, other stabilization techniques, such as the interior penalty method, result in better accuracy and are less sensitive to the choice of the stabilization parameter. During this work we also discovered that the manufactured solutions often used to evaluate errors in glaciology are not reliable due to high artificial surface forces at singularities. We perform our numerical experiments in both FEniCS and Elmer/Ice.

The basal sliding of glaciers and ice sheets can constitute a large part of the total observed ice velocity, in particular in dynamically active areas. It is therefore important to accurately represent this process in numerical models. The condition that the sliding velocity should be tangential to the bed is realized by imposing an impenetrability condition at the base. We study the, in glaciological literature used, numerical implementations of the impenetrability condition for non-linear Stokes flow with Navier's slip on the boundary. Using the finite element method, we enforce impenetrability by: a local rotation of the coordinate system (strong method), a Lagrange multiplier method enforcing zero average flow across each facet (weak method) and an approximative method that uses the pressure variable as a Lagrange multiplier for both incompressibility and impenetrability. An analysis of the latter shows that it relaxes the incompressibility constraint, but enforces impenetrability approximately if the pressure is close to the normal component of the stress at the bed. Comparing the methods numerically using a method of manufactured solutions unexpectedly leads to similar convergence results. However, we find that, for more realistic cases, in areas of high sliding or varying topography the velocity field simulated by the approximative method differs from that of the other methods by approx. 1% (two dimensional flow) and  > 5% when compared to the strong method (three-dimensional flow). In this study the strong method, which is the most commonly used in numerical ice sheet models, emerges as the preferred method due to its stable properties (compared to the weak method in three dimensions) and ability to well enforce the impenetrability condition.