Inertial modes in near-spherical geometries

Abstract

We propose a numerical method to compute the inertial modes of a container with nearspherical geometry based on the fully spectral discretization of the angular and radial directions using spherical harmonics and Gegenbauer polynomial expansion, respectively. This allows to solve simultaneously the Poincaré equation and the no penetration condition as an algebraic polynomial eigenvalue problem. The inertial modes of an exact oblate spheroid are recovered to machine precision using an appropriate set of spheroidal coordinates. We show how other boundaries that deviate slightly from a sphere can be accommodated for with the technique of equivalent spherical boundary andwe demonstrate the convergence properties of this approach for the triaxial ellipsoid.