Normal Modes of a Rotating, Self-gravitating Inhomogeneous Earth

Abstract

This present study focuses on three topics: (i) formulation of the first- order perturbation problem of the Earth's normal modes; (ii) formulation of a variational method based on the solutions of first-order quasi- degenerate perturbation calculation; and (iii) numerical results for both ordinary degenerate and quasi-degenerate multiplets using formulation (i). The Earth considered is a rotating, laterally-inhomogeneous Earth. To zeroth order, however, it is approximated by a spherically symmetric, non-rotating, elastic and isotropic sphere. Lateral perturbations and Earth's rotation are treated as perturbations. Examples of lateral perturbations are Earth's ellipticity of figure and lateral inhomogeneities. Because of the symmetry in the Earth model, unperturbed modes form multiplets with varying degrees of degeneracy. For almost all of these multiplets, the perturbation calculation need be carried out to first order. When one multiplet and another nearly coincide in their frequencies, first-order ordinary degenerate perturbation method must be replaced by first-order quasi-degenerate perturbation method. Since the latter is a more general form of the former, the first-order perturbation problem is completely solved. The perturbations considered in this study consist of Earth's rotation, ellipticity of figure, a continent-ocean discontinuity near the Earth's surface, and an artificially contrived discontinuity about the core-mantle interface. When the effect due to rotation becomes dominant, the first-order calculation becomes insufficient, and a variational method is posed. Such a method can be formulated from quasi-degenerate solutions. The numerical results show that ellipticity corrections are important and should always be included in the computations of the Earth's normal modes, that lateral inhomogeneities in the Earth's deep interior become an important perturbation for high-overtone multiplets, and that Coriolis coupling between a spheroidal and a toroidal multiplet overshadows all other types of coupling.