The influence of rotation on the free oscillations of the Earth

Abstract

We pursue an abstract investigation of the theory of the infinitesimal free elasticgravitational oscillations of a fairly general rotating Earth model. By considering in some detail the transition to the non-rotating case, we are able to delineate certain of the intrinsic effects of rotation on the normal mode eigensolutions, and to show how profoundly rotation alters the fundamental mathematical and physical properties of these eigensolutions. In particular, we show that the displacement eigenfunctions of a rotating Earth model are not mutually orthogonal, and that the corresponding normal modes of oscillation cannot in general be represented by pure standing waves. We consider the excitation of the normal modes of oscillation of a rotating Earth model by a transient imposed body force distribution, and we show that the complex dynamical amplitude of each normal mode may, in many geophysical applications, be determined separately, in spite of the lack of orthogonality among the displacement eigenfunctions. The calculation of the associated static response after the decay of the normal modes of oscillation is, on the other hand, complicated considerably by the absence of orthogonality. We specifically examine the influence of rotation on the zero-frequency rigid body translational and rotational modes of any non-rotating Earth model, and show how to account for the corresponding rigid body modes of any rotating Earth model in excitation calculations.