Open AccessThe pattern of eigenfrequencies of radial overtones which is predicted for a specified Earth-model
Author(s) -
E. R. Lapwood,
Ryôsuke Satô
Publication year - 2010
Publication title -
annals of geophysics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.394
H-Index - 60
eISSN - 2037-416X
pISSN - 1593-5213
DOI - 10.4401/ag-4832
Subject(s) - classification of discontinuities , overtone , discontinuity (linguistics) , amplitude , spherical harmonics , physics , legendre function , radius , mathematical analysis , asymptote , wavenumber , legendre polynomials , normal mode , mathematics , geometry , quantum mechanics , computer security , computer science , vibration , spectral line
In 1974 Anderssen and Cleary examined the distribution of eigenfrequencies
of radial overtones in torsional oscillations of Earth-models.
They pointed out that according to Sturm-Liouville theory this distribution
should approach asymptotically, for large overtone number m,
the value nnz/y, where y is the time taken by a shear-wave to travel
along a radius from the core-mantle interface to the surface, provided
elastic parameters vary continuously along the radius. They found that,
for all the models which they considered, the distributions of eigenfrequencies
deviated from the asymptote by amounts which depended on
the existence and size of internal discontinuities. Lapwood (1975) showed
that such deviations were to be expected from Sturm-Liouville theory,
and McNabb, Anderssen and Lapwood (1976) extended Sturm-Liouville
theory to apply to differential equations with discontinuous coefficients.
Anderssen (1977) used their results to show how to predict the pattern
of deviations —called by McNabb et al. the solotone effect— for a
given discontinuity in an Earth-model.
Recently Sato and Lapwood (1977), in a series of papers which will
be referred to here simply as I, II, III, have explored the solotone effect
for layered spherical shells, using approximations derived from an exacttheory which holds for uniform layering. They have shown how the
form of the pattern of eigenfrequencies, which is the graph of
S — YMUJI/N — m against m, where ,„CJI is the frequency of the m"'
overtone in the I"' (Legendre) mode of torsional oscillation, is determined
as to periodicity (or quasi-periodicity) by the thicknesses and velocities
of the layers, and as to amplitude by the amounts of the discontinuities
(III). The pattern of eigenfrequencies proves to be extremely sensitive
to small changes in layer-thicknesses in a model.
In this paper we examine a proposed Earth-model with six surfaces
of discontinuity between core boundary and surface, and predict its
pattern of eigenfrequencies. When seismological observations become
precise enough, and can be subjected to numerical analysis refined
enough, to identify the radial overtones for large m, this pattern of
eigenfrequencies will prove to be a severe test for any proposed model,
including he one which we discuss below
of radial overtones in torsional oscillations of Earth-models.
They pointed out that according to Sturm-Liouville theory this distribution
should approach asymptotically, for large overtone number m,
the value nnz/y, where y is the time taken by a shear-wave to travel
along a radius from the core-mantle interface to the surface, provided
elastic parameters vary continuously along the radius. They found that,
for all the models which they considered, the distributions of eigenfrequencies
deviated from the asymptote by amounts which depended on
the existence and size of internal discontinuities. Lapwood (1975) showed
that such deviations were to be expected from Sturm-Liouville theory,
and McNabb, Anderssen and Lapwood (1976) extended Sturm-Liouville
theory to apply to differential equations with discontinuous coefficients.
Anderssen (1977) used their results to show how to predict the pattern
of deviations —called by McNabb et al. the solotone effect— for a
given discontinuity in an Earth-model.
Recently Sato and Lapwood (1977), in a series of papers which will
be referred to here simply as I, II, III, have explored the solotone effect
for layered spherical shells, using approximations derived from an exacttheory which holds for uniform layering. They have shown how the
form of the pattern of eigenfrequencies, which is the graph of
S — YMUJI/N — m against m, where ,„CJI is the frequency of the m"'
overtone in the I"' (Legendre) mode of torsional oscillation, is determined
as to periodicity (or quasi-periodicity) by the thicknesses and velocities
of the layers, and as to amplitude by the amounts of the discontinuities
(III). The pattern of eigenfrequencies proves to be extremely sensitive
to small changes in layer-thicknesses in a model.
In this paper we examine a proposed Earth-model with six surfaces
of discontinuity between core boundary and surface, and predict its
pattern of eigenfrequencies. When seismological observations become
precise enough, and can be subjected to numerical analysis refined
enough, to identify the radial overtones for large m, this pattern of
eigenfrequencies will prove to be a severe test for any proposed model,
including he one which we discuss below