Boschi, Tromp and O'Connell: On Maxwell singularities in post-glacial rebound
We investigate the problem of finding the numerous
relaxation times associated with the post-glacial 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 post-glacial 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
ill-behaved secular polynomial (blue line) and corrected secular polynomial (orange line), plotted as a function of the Laplace transform variable s (corresponding to the frequency in the case of Fourier-transformation) for a two-layer Earth model. The ill-behaved secular polynomial coincides with the secular polynomial encountered in numerical calculations. The corrected one is the result of our analytical approach. Notice that the singularity in the ill-behaved polynomial is replaced by a zero-crossing in our approach. Such zero-crossings turn out to always coincide with the Maxwell time (ratio of rigidity to viscosity) of one of the model's layers, and are therefore dubbed "Maxwell modes". Maxwell modes do not carry any energy, and should not be considered true relaxation modes of the Earth.
From Boschi, Tromp and O'Connell, "On Maxwell singularities in
post-glacial rebound" (GJI, 1999).
modified: Thu Jan 21 11:40:48 CET 2010