Full and reduced model solutions of unsteady axi-symmetric ice sheet flow over a flat bed

Abstract

A finite element algorithm is presented for the solution of the unsteady free-surface problem governed by the full, axi-symmetric, slow, linearly-viscous fluid flow equations for an isothermal ice sheet on a flat bed. A class of linearly viscous Reduced Model, Shallow Ice Approximation, exact solutions is also constructed, for which the relative error is of order of the very small surface slope squared. The finite element algorithm is shown to reproduce the exact solutions to a high degree of accuracy for a series of examples involving expanding and contracting ice sheet spans, and increasing and decreasing thicknesses, which is an important test of its validity. The stability of various equilibrium ice sheet configurations is shown to depend on the manner in which the surface accumulation distribution is prescribed. If it is prescribed purely as a function of surface elevation the equilibrium is unstable, but if it is prescribed purely as a function of radial position the equilibrium is stable. Different weightings of the prescription, in terms of elevation and horizontal location, are investigated to determine how the stability of the steady-state solutions depends on the weighting parameter \(\beta\). It is found that for a range of values of \(\beta\) between two limits points on the solutions branches, no steady-state solutions exit. In contrast, direct solution of the Reduced Model equations yields corresponding solution branches meeting at a unique bifurcation point, with steady-state solutions existing for all values of \(\beta\).