Free Oscillations of the Laterally Inhomogeneous Earth: Quasi-degenerate Multiplet Coupling

Luh, Peter C.

doi:10.1111/j.1365-246x.1973.tb06526.x

Cited by 46 publications

(29 citation statements)

References 8 publications

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“…Madariaga (1971) and Luh (1973) have studied the effects of arbitrary but small lateral variations in a SNREI Earth's material properties on its normal mode spectrum. In the course of these studies, both authors have developed selection rules that describe which surface spherical harmonic coefficients in the expansion of these lateral variations can perturb a given mode.…”

Section: (411)mentioning

confidence: 99%

“…By fairly simple arguments and the application of the inversion properties of poloidal and toroidal fields we show, in Appendix D, how to arrive at these selection rules. What this approach does not give, and Madariaga (1971) and Luh (1973) do, is an estimate of how much each mode is perturbed.…”

Section: (411)mentioning

confidence: 99%

“…The perturbation theoretic treatment of this case is known as quasi-degenerate perturbation theory. Dahlen (1969) and Luh (1973) have studied this problem. When u is comparable to or smaller than R, dme will not be nearly diagonal and we must seek other ways of generating solutions to (5.48)-(5.49).…”

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confidence: 99%

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The Scalar Equations of Infinitesimal Elastic-Gravitational Motion for a Rotating, Slightly Elliptical Earth

Smith

1974

Geophysical Journal International

175

167

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We derive the infinite set of coupled ordinary differential equations over radius that govern the infinitesimal free elastic-gravitational oscillations of a rotating, slightly elliptical Earth with an isotropic perfectly elastic constitutive relation and a hydrostatic prestress field. We show how the symmetries of such a body restrict the most general form of the displacement eigenfunctions. We discuss situations in which finite sets of coupled equations may yield good approximate eigenfunctions and describe the sense in which the solution to such a set approximates the solution of the infinite set. Finally we describe in some detail the equations governing a particular class of finite expansions and show how these may be put in a form convenient for numerical solution. The method used to convert tensor equations to scalar equations is an extension of the generalized surface spherical harmonic expansion of Phinney & Burridge and should be useful in other non spherically-symmetric applications.

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Section: (411)mentioning

confidence: 99%

Section: (411)mentioning

confidence: 99%

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The Scalar Equations of Infinitesimal Elastic-Gravitational Motion for a Rotating, Slightly Elliptical Earth

Smith

1974

Geophysical Journal International

175

167

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“…are zeroth-order mass density, angular-velocity vector of Earth's steady rotation, displacement vector, and a linear operator that embodies the potential energy of the Earth in deformation. With the usual definition and notation of inner-product space (Dahlen 1972;Luh 1973a), consider the following two inner-product equations (after equation (2.1) is Fourier-transformed with respect to time so that a/at is replaced by iw):…”

Section: Theorymentioning

confidence: 99%

“…Backus & Gilbert (1961), MacDonald & Ness (1961, and Pekeris, Alterman & Jarosch (1961) introduced Earth's rotation into the problem; Dahlen (1968) studied the effect of ellipticity of figure on a rotating Earth; Dahlen (1969) then proceeded further by considering the problem of quasi-degeneracy of these two perturbations ;Zharkov & Lyubimov (1970a, b) and Madariaga (1971Madariaga ( , 1972 treated the first-order ordinary degenerate perturbation of lateral inhomogeneities on a spherically symmetric, non-rotating Earth; and Luh (1973a) gave a progress report (call it Report I hereafter) on quasi-degeneracy of a general lateral perturbation. This present study represents an extension of Report I by incorporating Earth's rotation into the problem and by presenting numerical results of first-order perturbation theory.…”

Section: Introductionmentioning

confidence: 99%

Normal Modes of a Rotating, Self-gravitating Inhomogeneous Earth

Luh

1974

Geophysical Journal International

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This present study focuses on three topics: (i) formulation of the firstorder perturbation problem of the Earth's normal modes; (ii) formulation of a variational method based on the solutions of first-order quasidegenerate 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.( O 2 5 -Y ) ( U Y v) = 0. Note that w2 I and Y here differ from the ones in equation (2.3). There po and C were functions of three-dimensional space co-ordinates; here they are functions of

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Low frequency seismology and three-dimensional structure — observational aspects

Masters

Ritzwoller

1988

Mathematical Geophysics

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Free Oscillations of the Laterally Inhomogeneous Earth: Quasi-degenerate Multiplet Coupling

Cited by 46 publications

References 8 publications

The Scalar Equations of Infinitesimal Elastic-Gravitational Motion for a Rotating, Slightly Elliptical Earth

The Scalar Equations of Infinitesimal Elastic-Gravitational Motion for a Rotating, Slightly Elliptical Earth

Normal Modes of a Rotating, Self-gravitating Inhomogeneous Earth

Low frequency seismology and three-dimensional structure — observational aspects

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