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
studies.
Above: ill-behaved secular determinant (thin line) and
corrected secular determinant
plotted as a function of s (i.e. in the Laplace domain),
for a one-layer Earth model
at spherical harmonic degree l=5. The ill-behaved determinant
coincides with
the secular determinant encountered in numerical calculations. The
corrected one is the result of
our analytical approach. A root of the corrected secular determinant
that coincides with the negative inverse Maxwell time of the
model is dubbed a ``Maxwell mode''; these modes turn out not to carry
any energy and therefore should not be considered true modes of the
Earth. From Boschi, Tromp and O'Connell, "On Maxwell singularities in
post-glacial rebound" (GJI, 1999).
Harvard seismology: 3-D Earth Structure
Lapo Boschi,
Department of Earth and Planetary Science, Harvard University
Last
modified: Mon Mar 6 18:19:38 EST 2000