Abstract

The reduced model for a large ice sheet flow is uniformly valid when the bed topography is flat or has slopes relative to the horizontal no greater than order ϵ, where ϵ2 is a very small dimensionless viscosity based on the geometry and flow parameters. Real beds will have much greater slopes, of order unity in places. Large-scale numerical simulations of ice sheet evolution are currently based on the reduced model, and their validity when significant basal topography is present, the common situation, is not certain. Numerical solution of the full slow viscous flow equations on the unknown sheet domain, where the surface is stress-free and subject to a kinematic accumulation/melt condition, is much more difficult. No established algorithms exist at present. Here attention is focused on the most simple configuration of steady plane flow of an incompressible linearly viscous fluid, assuming isothermal conditions so that the temperature dependent rate factor is constant. It allows the application of complex potential representations for a solution of the full equations superposed on a flat bed reduced solution in order to satisfy non-slip conditions on a humped bed. A particular class of conformal mappings of a humped bed contour onto a flat bed result in boundary conditions which can be solved exactly by Cauchy integral methods to yield explicit integral representations for the potentials and related physical variables. Evaluation is necessarily numerical, but accurate. This method is applied to various humped contours with amplitude increasing from zero to determine when the reduced solution, ignoring the bed topography, starts to lose accuracy, and how far from the humped section this error persists. It is demonstrated that bed humps do influence the velocity field. © 1998 John Wiley & Sons, Ltd.

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