Classification of eastward propagating waves on the spherical Earth

Observational evidence for an equatorial non-dispersive mode propagating at the speed of gravity waves is strong, and while the structure and dispersion relation of such a mode can be accurately described by a wave theory on the equatorial β-plane, prior theories on the sphere were unable to find such a mode except for particular asymptotic limits of gravity wave phase speeds and/or certain zonal wave numbers. Here, an ad hoc solution of the linearized rotating shallow-water equations (LRSWE) on a sphere is developed, which propagates eastward with phase speed that nearly equals the speed of gravity waves at all zonal wave numbers. The physical interpretation of this mode in the context of other modes that solve the LRSWE is clarified through numerical calculations and through eigenvalue analysis of a Schrödinger eigenvalue equation that approximates the LRSWE. By comparing the meridional amplitude structure and phase speed of the ad hoc mode with those of the lowest gravity mode on a non-rotating sphere we show that at large zonal wave number the former is a rotation-modified counterpart of the latter. We also find that the dispersion relation of the ad hoc mode is identical to the n = 0 eastward propagating inertia–gravity (EIG0) wave on a rotating sphere which is also nearly non-dispersive, so this solution could be classified as both a Kelvin wave and as the EIG0 wave. This is in contrast to Cartesian coordinates where Kelvin waves are a distinct wave solution that supplements the EIG0 mode. Furthermore, the eigenvalue equation for the meridional velocity on the β-plane can be formally derived as an asymptotic limit (for small (Lamb Number)^-1/4) of the corresponding second order equation on a sphere, but this expansion is invalid when the phase speed equals that of gravity waves i.e. for Kelvin waves. Various expressions found in the literature for both Kelvin waves and inertia–gravity waves and which are valid only in certain asymptotic limits (e.g. slow and fast rotation) are compared with the expressions found here for the two wave types.