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

Abstract

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.

Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth's interior

“…In analytic formulations of nutation theory [e.g., Sasao et al, 1980;Mathews et al, 1991aMathews et al, , 1991b, a tidal constituent is expressed as the real part of (r) is real and proportional to r n and the asterisk denotes complex conjugation. In approaches where nutations are treated and computed as part of the complete displacement field produced by the TGP [e.g., Smith, 1974;Wahr, 1981aWahr, , 1981bWahr, , 1981c] the convention of Cartwright and Tayler [1971] is employed. It differs from that of (C1) in that the space-time dependence is through …”

Geophysical applications of very-long-baseline interferometry

“…Much of the theory used to model the behavior of a realistic, nonrigid Earth was developed by Wahr in a series of classic papers that began in 1981. Using a normal-mode expansion technique developed by Smith (1974), Wahr worked out the response of a nonrigid, stratified, elliptical, oceanless, rotating Earth with a fluid core to effects such as nutation (1981b), Earth tides (1981a), and the pole tide (1985). Recent work has focused on the effects of barometric pressure loading on the surface of the Earth (Van Dam and Wahr, 1987) and the effects of the Earth's solid inner core on its rotation (De Vries and Wahr, 1991;Matthews et al, 1991aMatthews et al, , 1991bHerrings a/., 1991).…”

Tides and nutation of the earth: I. Models of an earth with an inelastic mantle and homogeneous, inviscid, liquid core

“…The substitution of these decompositions into the equations of motion yielded infinite systems of ordinary differential equations that were integrated by means of truncating them (i.e., replacing an infinite system with a finite one). As noted before (see, e.g., Smith, 1974), this approach entails errors that are due to the reduction of infinite systems to finite ones, and these errors can hardly be taken into account. The main difficulty stems from the fact that the boundary-value problem at hand is an ill-conditioned (in Hadamard's sense) hyperbolic problem with only one boundary condition (instead of two conditions as typical of equations of this type), which is, however, specified at a closed surface.…”