Analytical visco-elastic relaxation models

In the past, relaxation processes employing PREM or 1066B-stratified earth models have been solved numerically, either directly in the time domain (initial value models) or in the Laplace-transformed domain (normal mode models). Solving N-layer stratified models analytically has only been performed for small values of N. We present a brief outline of the analytical method of general radially stratified N-layer, self-gravitating, Maxwell rheological models. The analytical models allow for the relaxation times of the myriad of relaxation modes to be determined with very high accuracies, as the secular determinant from which they derive is an analytical function. We show by explicitly comparing results on a 5 and 30-layer incompressible Maxwell model with a convex mantle viscosity profile, that the differences in the results between the two models are small when the 5-layer model has the volume-averaged structural and rheological properties of the 30-layer model. The fact that in both models the fluid values of the Love numbers reach the same expected limits leaves no room for hypothetical so-called non-modal contributions. The application to compressible linear rheologies of this generalized analytical normal mode method is more involved, but straightforward.