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Variational formulation and finite element type solution for free oscillations of a rotating laterally heterogeneous earth Moon, Wooil

Abstract

A variational type, finite element method is proposed for the study of normal modes of a rotating, laterally heterogeneous Earth. This method is very powerful and, in theory, overcomes all the limitations of the perturbation method. However, the present size of computer memory puts severe limitations on the use of this method. The Lagrangian energy integral is derived for a general configuration of a perfectly elastic continuum in both shperical and oblate ellipsoidal coordinate systems. The assumed solutions for the displacements and the perturbation of the gravitational potential are formed by tensor products of the cubic Hermite basis functions of three coordinate variables. The undetermined coefficients of these assumed solutions are solved by the minimization of the Lagrangian energy integral by a Rayleigh-Ritz technique. The results of the numerical solution of this approach show that (1) this algorithm is very efficient and promising in one-dimensionalized normal mode problems, (2) that the degenerate frequency of [sub n]W[sub ℓ][sup m] splits into (2ℓ+1) components, even for the non-rotating, homogeneous spherical Earth model, due to the interpolation scheme of the azimuthal basis functions and (3) because the numerical spectral splitting is very large, the effects of self-gravitation and rotation cannot be examined clearly in this study. However, lateral heterogeneities which break the symmetry of the physical shape of the Earth greatly affect the normal mode spectra.

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