The effect of basal topography on ice sheet dynamics

Classical shallow-ice theory assumes that bed topography under ice sheets has slopes comparable to the surface slope of the ice sheet. A modification of the classical steady-state theory which allows for significant bed topography on shorter length scales has recently been developed by Morland (Proc. R. Soc L. Ser. A., 456, 1711-1739 ), but his theory requires explicit integration of the ice-flow equations over the topography length scale, which may be below the grid size of typical numerical ice sheet models. Here we present a method for parameterising the effect of basal topography of wavelengths much greater than ice thickness but much smaller than the horizontal extent of the ice sheet on the bulk flow of the ice sheet. In particular, we are able to show through the use of a multiple-scales expansion technique that the effect of such topography is described by a simple correction factor applied to the classical expression for ice flux. This correction factor dispenses with the need to integrate explicitly over the topography length scale and could allow the effect of such topography to be included in numerical models with limited grid size. Examples are given for the practical implementation of this `correction factor method' in calculations of the steady-state shape of ice sheets.