On oscillations of the Earth's fluid core

Abstract

Summary. In the conventional spheroidal and toroidal representation, the differential equations governing long-period free oscillations of the Earth’s fluid core form an infinitely coupled system. Because of the mathematical complexity, solutions of this system are customarily attempted by numerical integration. The approach, however, necessitates a severe truncation of the coupling chain and this, in turn, renders it difficult to interpret the results. An alternative approach is sought in this work. By invoking the ‘solenoidal flow’ approximation which neglects the dilatation, and the ‘subseismic’ approximation which neglects the flow pressure respectively, we have succeeded in developing analytic solutions for the inertial and gravitational oscillations. This results in a significant simplification of the eigenvalue problem because the infinitely coupled system of differential equations is now reduced to an algebraic one. More importantly, the analytic solutions reveal immediately the roles played by the rotation and density stratification in core dynamics. We find that the inertial oscillations, which arise from the Earth’s rotation, are essentially independent of the core’s density stratification. Thus the fluid core can be approximated by a homogeneous, incompressible and non-gravitating inviscid Newtonian fluid with only minor modifications. The gravitational oscillations, on the other hand, are governed by both the rotation and density stratification. In a non-rotating earth, gravitational oscillations are not possible for neutrally or unstably stratified cores. Because of rotation, however, gravitational oscillations become possible for any density stratification in the core, and the eigenfrequencies are divided into alternating allowed and forbidden zones. We show that in each allowed zone, there exist an infinite number of gravitational oscillations with their eigenfrequencies approaching asymptotically the eigenfrequency of a corresponding inertial oscillation. Both the solenoidal flow and subseismic approximations ignore the fluid flow arising from the variation in gravitational potential. To correct for this neglect, as well as taking into account the deformation in the solid part of the Earth, an asymptotic formulation is developed. It is shown that the eigenfrequencies can be determined without explicitly solving for the ‘correction’