On inertial waves and oscillations in a rapidly rotating spheroid

The problem of fluid motions in the form of inertial waves or inertial oscillations in an incompressible viscous fluid contained in a rotating spheroidal cavity was first formulated and studied by Poincare (1885) and Bryan (1889). Upon realizing the limitation of Bryan's implicit Solution using complicated modified spheroidal coordinates, Kudlick (1966) proposed a procedure that may be used to compute an explicit solution in spheroidal coordinates. However, the procedure requires an analytical expression for the N real and distinct roots of a polynomial of degree N, where N is a key parameter in the problem. When 0 less than or equal to N less than or equal to 2, an explicit solution can be derived by using Kudlick's procedure. When 3 less than or equal to N less than or equal to 4, the procedure cannot be practically used because the analytical expression for the N distinct roots becomes too complicated. When N > 4, Kudlick's procedure cannot be used because of the non-existence of an analytical expression for the N distinct roots. For the inertial wave problem, Kudlick thus restricted his analysis to several modes for 1 less than or equal to N less than or equal to 2 with the azimuthal wavenumbers 1 less than or equal to m less than or equal to 2. We have found the first explicit general analytical solution of this classical problem valid for 0 less than or equal to N < infinity and O less than or equal to m < infinity. The explicit general solution in spheroidal polar coordinates represents a possibly complete set of the inertial modes in an oblate spheroid of arbitrary eccentricity. The problem is solved by a perturbation analysis. In the first approximation, the effect of viscosity on inertial waves or oscillations is neglected and the corresponding inviscid solution, the pressure and the three velocity components in explicit spheroidal coordinates, is obtained. In the next approximation, the effect of viscous dissipation on the inviscid solution is examined. We have derived the first explicit general solution for the viscous spheroidal boundary layer valid for all inertial modes. The boundary-layer flux provides the solvability condition that is required to solve the higher-order interior problem, leading to an explicit general expression for the viscous correction of all inertial modes in a rapidly rotating, general spheroidal cavity. On the basis of the general explicit solution, some unusual and intriguing properties of the spheroidal inertial waves or oscillation are discovered. In particular, we are able to show that [GRAPHICS] where u is the velocity of any three-dimensional inviscid inertial waves or oscillations in an oblate spheroid of arbitrary eccentricity and integral(v) denotes three-dimensional integration over the volume of the spheroidal cavity.