The mixed Rossby–gravity wave on the spherical Earth

This work revisits the theory of the mixed Rossby–gravity (MRG) wave on a sphere. Three analytic methods are employed in this study: (a) derivation of a simple ad hoc solution corresponding to the MRG wave that reproduces the solutions of Longuet‐Higgins and Matsuno in the limits of zero and infinite Lamb's parameter, respectively, while remaining accurate for moderate values of Lamb's parameter, (b) demonstration that westward‐propagating waves with phase speed equalling the negative of the gravity‐wave speed exist, unlike the equatorial β‐plane, where the zonal velocity associated with such waves is infinite, and (c) approximation of the governing second‐order system by Schrödinger eigenvalue equations, which show that the MRG wave corresponds to the branch of the ground‐state solutions that connects Rossby waves with zonally symmetric waves. The analytic conclusions are confirmed by comparing them with numerical solutions of the associated second‐order equation for zonally propagating waves of the shallow‐water equations. We find that the asymptotic solutions obtained by Longuet‐Higgins in the limit of infinite Lamb's parameter are not suitable for describing the MRG wave even when Lamb's parameter equals 104. On the other hand, the dispersion relation obtained by Matsuno for the MRG wave on the equatorial β‐plane is accurate for values of Lamb's parameter as small as 16, even though the equatorial β‐plane formally provides an asymptotic limit of the equations on the sphere only in the limit of infinite Lamb's parameter. This article provides explicit expressions for the dispersion relation and meridional structure of the mixed Rossby–gravity (MRG) wave on a sphere and the accuracy of these analytic expressions is confirmed numerically. Unlike the equatorial β‐plane, on a sphere the westward‐propagating gravity wave is not excluded from the MRG spectrum. The approximation of the governing equations by a Schrödinger equation confirms that the MRG wave connects the n = 0 Rossby mode at large zonal wavenumber with the n = 0 zonally symmetric mode.