Recent advances in linear wave theory on the spherical earth

Abstract

In Cartesian coordinates the formulation of a Schrödinger eigenvalue equation for zonally propagating waves of the Linear Rotating Shallow Water Equations (LRSWE) has yielded the most accurate and concise expressions for the waves’ meridional amplitude structure and dispersion relation. In contrast, in spherical coordinates the complicated form of the LRSWE has not enabled a formulation of a Schrödinger equation for these waves which is the reason why no explicit expressions were developed for these waves on the spherical earth. In recent years approximate Schrödinger equations were developed for waves of the LRSWE in spherical coordinates which yielded highly accurate analytic expressions (as judged by comparing them with numerical solutions of the original equations) for the countable number of modes of both Inertia-Gravity (AKA Poincaré) waves and Planetary (AKA Rossby) waves. The formulation of approximate Schrödinger equations that are valid in different asymptotic limits has not only provided explicit expressions for these types of waves but it has also demonstrated that no equatorial Kelvin waves exist on a sphere and that the Mixed Rossby-Gravity (MRG) mode does not have a singularity at the wavenumber where its speed equals the phase speed of gravity waves, i.e. the spectrum of the MRG mode on a sphere is continuous throughout.