Solution of the whole earth geodynamic equation in a slightly elliptical coordinate system

Abstract

The normal mode problem of the Earth is extremely complex and difficult mathematically. One numerical approach is a straight application of the finite element solution of the geodynamic partial differential equations with proper boundary conditions. However, it has been learned that this is very inefficient numerically and also in a scientific context. In this paper it is shown that if we use the generalized spherical harmonics as normal mode eigenfunctions, the problems arising from the rotation and ellipticity of the Earth can be solved rather easily without imposing too severe limitations on the terrestrial spectral range of current interest. The reduced equation of motion, with coupling coefficients, is solved with piecewise continuous trial functions and the resulting quadratic generalized eigenvalue problem is solved by an inverse iteration approximation.