Steady plane isothermal linearly viscous flow of ice sheets on beds with moderate-slope topography

The reduced model for the flow of a large ice sheet is uniformly valid when the bed topography is flat or has slopes relative to the horizontal of order no greater than epsilon, where epsilon(2) is a very small dimensionless viscosity based on the geometry and how parameters. The reduced model is given by the leading-order balances of an asymptotic expansion in epsilon. Real ice-sheet beds will have much greater slopes, of order unity in places, but commonly of moderate magnitude, say delta = 0.2, corresponding to 11 degrees, over large regions. The length s over which a moderate slope extends may be as small as the sheet thickness, or considerably greater, subject to the restriction that the amplitude a = delta s of the local topograph does not exceed the sheet thickness. Then, in addition to epsilon, there are two further independent parameters from the trio delta, s and a. An asymptotic expansion is constructed for steady plane linearly viscous isothermal flow over bed topography, such that epsilon << delta << 1; and the leading-order terms, an enhanced reduced model, are determined explicitly. First-order correction terms are also determined explicitly when s is much greater than the sheet thickness. Examples are computed in the latter case for a variety of bed forms involving isolated moderate-slope zones, and for a wavy-bed form of moderate slope. Comparisons between the leading-order solutions and the standard reduced-model flat-bed solutions are made, and the effects of the first correction terms are shown. It is found that moderate bed slopes with linearly viscous isothermal flow do not induce a significant correction, so that the enhanced reduced model provides a good approximation.