The pattern of eigenfrequencies of radial overtones which is predicted for a specified Earth-model

Abstract

In 1974 Anderssen and Cleary examined the distribution of eigenfrequenciesof radial overtones in torsional oscillations of Earth-models.They pointed out that according to Sturm-Liouville theory this distributionshould approach asymptotically, for large overtone number m,the value nnz/y, where y is the time taken by a shear-wave to travelalong a radius from the core-mantle interface to the surface, providedelastic parameters vary continuously along the radius. They found that,for all the models which they considered, the distributions of eigenfrequenciesdeviated from the asymptote by amounts which depended onthe existence and size of internal discontinuities. Lapwood (1975) showedthat such deviations were to be expected from Sturm-Liouville theory,and McNabb, Anderssen and Lapwood (1976) extended Sturm-Liouvilletheory to apply to differential equations with discontinuous coefficients.Anderssen (1977) used their results to show how to predict the patternof deviations —called by McNabb et al. the solotone effect— for agiven discontinuity in an Earth-model.Recently Sato and Lapwood (1977), in a series of papers which willbe referred to here simply as I, II, III, have explored the solotone effectfor layered spherical shells, using approximations derived from an exacttheory which holds for uniform layering. They have shown how theform of the pattern of eigenfrequencies, which is the graph ofS — YMUJI/N — m against m, where ,„CJI is the frequency of the m"'overtone in the I"' (Legendre) mode of torsional oscillation, is determinedas to periodicity (or quasi-periodicity) by the thicknesses and velocitiesof the layers, and as to amplitude by the amounts of the discontinuities(III). The pattern of eigenfrequencies proves to be extremely sensitiveto small changes in layer-thicknesses in a model.In this paper we examine a proposed Earth-model with six surfacesof discontinuity between core boundary and surface, and predict itspattern of eigenfrequencies. When seismological observations becomeprecise enough, and can be subjected to numerical analysis refinedenough, to identify the radial overtones for large m, this pattern ofeigenfrequencies will prove to be a severe test for any proposed model,including he one which we discuss below.