The effect of arbitrarily small rigidity on the free oscillations of the Earth

Abstract

The system of propagator equations for an elastic solid becomes singular as the shear modulus becomes vanishingly small. In computational applications there is severe loss of precision as the limit of zero shear modulus is approached. The use of perturbation theory to address the effect of very small shear modulus, using the fluid state as a basis, is unsatisfactory because certain phenomena, e.g., Rayleigh waves, cannot be represented. Two approximate methods are presented to account for the singular perturbation. Since most of the Earth is nearly neutrally stratified, in which case the motion is nearly irrotational, one can impose the irrotational constraint and obtain a modified and reduced system of propagator equations. This system does not have the singular perturbation. In the second method the transition zone between a fluid and a solid is represented as an infinitesimally thin, Massive, Elastic Interface (MEI). The boundary conditions across the MEI are dispersive and algebraic. The limit of zero shear modulus is non-singular.