Toroidal modes of a simple laterally heterogeneous sphere

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Abstract
Two methods which have been proposed for calculating the normal modes of laterally heterogeneous models of the Earth are first-order degenerate perturbation theory and the Rayleigh-Ritz variational procedure. The purpose of this paper is to compare these two techniques in a calculation of some low-order toroidal modes of a simple laterally heterogeneous body—a sphere consisting of two hemispheres with different elastic properties. Both methods express the eigenfunctions of the test case as linear combinations of degenerate eigenfunctions of the corresponding laterally averaged model. Our results show, however, that solutions generated by the more accurate variational technique can contain significant contributions from modes from several degenerate multiplets, whereas each solution given by first-order degenerate perturbation theory contains the effects of modes from only one degenerate multiplet. Furthermore, the variational procedure yields solutions which reflect the presence of lateral heterogeneity of odd angular order, but first-order degenerate perturbation theory does not. These differences are potentially important for free oscillation studies of realistic, laterally heterogeneous models of the Earth: our findings suggest that the Rayleigh-Ritz variational procedure, or a comparably accurate method, may be required for such work.