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

• Main Authors: E. R. LAPWOOD, R. SATO
• Format: Article
• Language: English
• Published: Istituto Nazionale di Geofisica e Vulcanologia (INGV) 1977-06-01
• Series: Annals of Geophysics
• Online Access: http://www.annalsofgeophysics.eu/index.php/annals/article/view/4832

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