Influence of Bed Topography on Steady Plane Ice Sheet Flow

Abstract

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 no greater than the magnitude e of the surface slope, where ε ² defines a very small dimensionless viscosity based on the geometry and flow parameters. The reduced model is given by the leading order balances of an asymptotic expansion in e. Real ice sheet beds will have much greater slopes, of order unity in places, but commonly of moderate magnitude δ over large regions, such that ε ≪ δ ≪ 1. 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 = δs of the local topography does not exceed the sheet thickness. Then, in addition to ε, there are two further independent parameters from the trio δ, s and a. An asymptotic expansion is derived for steady plane non-linearly viscous flow with a prescribed temperature field over moderate slope bed topography. The leading order balances define an enhanced reduced model, which is constructed explicitly, and the first order correction terms have magnitude δ/s when s is dimensionless with unit the sheet thickness. The first and second order correction problems are derived. It is shown that their structure is distinct for topography scales s = 1 and s ≫ 1. When s = 1, the differential equations of the correction problems retain an elliptic structure, but when s ≫ 1, explicit depth integration of the momentum balances is possible, the crucial simplifying property of the leading order reduced model. The leading order, enhanced reduced model, is solved in the simpler case of linearly viscous isothermal flow for a particular bed form for a range of the parameters δ, s and a, to illustrate the influence of the enhancement, and also show the distributions with height of the deviatoric stresses and horizontal and vertical velocity in the region of the moderate bed slope.