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.