Hydrostatic equilibrium, gravity waves, and inertial waves in an inhomogeneous rotating sphere.

Abstract

Small perturbations to the hydrostatic state of a rotating, self-gravitating mass composed of two fluids of different density are investigated for small values of the angular velocity. The hydrostatic state of the system is calculated through use of the method of domain perturbations. It is shown that the oblateness of the inner and external boundaries is different. The domain perturbation procedure is also employed in the study of small perturbations to the hydrostatic state. Scaling methods are used to show that the equations governing internal and external gravity waves are separable and can be treated using standard methods. It is found that the frequencies are even functions of the angular velocity and, for small angular velocities, are independent of rotational effects up to and including terms of the order of the angular velocity. The equations governing low frequency inertial oscillations are analyzed using an asymptotic WKB approach. A solution for the case of motion in a homogeneous sphere is obtained and is found to be in close agreement with results in the literature obtained through use of a different method. The same method is used for the case of motion in a spherical annulus, and it is found that inclusion of an inner core increases the frequency of the inertial waves.