On Maxwell singularities in postglacial rebound

Abstract

We investigate the problem of finding the numerous relaxation times associated with the postglacial rebound of a layered Maxwell earth model. In general, these relaxation times are the roots of a secular polynomial. When a numerical approach is followed, this polynomial can be very ill behaved, with a number of singularities that coincide with the Maxwell times associated with the model rheology. This problem becomes dramatically evident when the rheological profile of the model is continuous or includes a large number of uniform layers (these two cases are basically the same when the solution is computed numerically). In order to understand the physical meaning of such Maxwell singularities, we perform a comparison between the numerical approach and the existing analytical solution to the problem of the postglacial relaxation of an incompressible, self-gravitating, N-layer, spherical Maxwell earth. We show that the analytical method does not suffer from the Maxwell singularity problem, and give a theoretical explanation of the ill behaviour of the secular polynomial computed in numerical studies.