Waves on a Sphere: Equatorial Kelvin Wave Replaced by an Inertia-Gravity mode

As has been shown by Matsuno in his 1968 seminal study of wave theory on the equatorial beta-plane, Kelvin waves are derived as a special (i.e. degenerate or singular) solution of the Shallow Water Equations (SWE) by setting to zero the meridional velocity component. It so happens that the remaining two variables, the zonal velocity component and the height variations, can be determined so as to provide a solution to the three scalar SWE. In terms of the corresponding eigenvalue equation that determines the phase speed and meridional amplitude structure of the waves Kelvin waves belong to the null space of the equation i.e. to the degenerate case where the eigenfunction vanishes identically. In contrast to this planar construction setting the meridional velocity component to zero does not yield a solution of the SWE on a sphere so it is unclear whether there exist physically relevant solutions of the eigenvalue equation that belong to the null space of this equation. Numerical solutions of the SWE for zonally progating waves show that there exists a non-dispersive mode that passes through the origin of the frequency-wavenumber plane but it slope exceeds the phase speed of gravity waves and the meridional velocity component does not vanish. In my talk I will show that this mode is the n=0 Inertia-Gravity and not a Kelvin wave, which implies that Kelvin waves are a special solution of the SWE only in Cartesian Coordinates. These theoretical findings have no implications for observations of Equatorial Kelvin waves in the atmosphere or the ocean since the accuracy of these observations is to crude to determine whether the observed phase speed equals that of gravity waves or deviates slightly from it.