Stability of an ice sheet on an elastic bed

The stability of stationary flow of a two-dimensional ice sheet is studied when the ice obeys a power flow law (Glen's flow law). The mass accumulation rate at the top is assumed to depend on elevation and span and the bed supporting the ice sheet consists of an elastic layer lying on a rigid surface. The normal perturbation of the free surface of the ice sheet is a singular eigenvalue problem. The singularity of the perturbation at the front of the ice sheet is considered using matched asymptotic expansions, and the eigenvalue problem is seen to reduce to that with fixed ice front. Numerical solution of the perturbation eigenvalue problem shows that the dependence of accumulation rate on elevation permits the existence of unstable solutions when the equilibrium line is higher than the bed at the ice divide. Alternatively, when the equilibrium line is lower than the bed, there are only stable solutions. Softening of the bed, expressed through a decrease of its elastic modulus, has a stabilising effect on the ice sheet.