On the computation of the inertial modes in spheroids

Summary The explicit inertial modes in spheres and oblate spheroids, owing to their clear and concise mathematical formulations, have been applied in many geophysical and astrophysical studies. In contrast, the implicit inertial modes are rarely used because of their mathematical complexity. Due to the presence of factorials and double factorials inherited from the associated Legendre polynomials, the computation of explicit inertial modes becomes intractable at high orders. Based on the implicit inertial modes, this research, for the first time, develops a new algorithm that enables fast computation of the inertial modes in spheres and spheroids of arbitrary eccentricity even at high orders. In addition, it offers an efficient approach to computing the geostrophic polynomials, which are a set of special inertial modes with zero frequency in spheres and spheroids. In this new algorithm the inertial modes and the half-frequencies are expressed as functions of the associated Legendre polynomials and their first derivatives with respect to the modified oblate spheroidal coordinates. Several numerical experiments demonstrate the efficiency of this new algorithm. It is also verified that both the non-penetrable boundary condition and the incompressible condition are satisfied by the numerical results produced by this algorithm.