Yanai Waves and the Classification of Westward Propagating Waves on the Rotating Spherical Earth

The elegant and seminal linear wave theory developed in Matsuno (1966) on the equatorial β-plane is the foundation of our understanding of equatorial wave theory in the ocean and atmosphere. In Matsuno’s theory Planetary (Rossby) waves and Inertia-Gravity (Poincaré) waves are derived as three distinct wave types from solutions of an eigenvalue (Schrödinger) equation for the meridional velocity, that are obtained from Linear Rotating Shallow Water Equations (LRSWE, hereafter) by eliminating the zonal velocity and height. The waves’ phase speeds are the three roots of a cubic in which one of the coefficients is the discrete eigenvalue (i.e. energy level) of the Schrödinger equation and two of these roots are negative i.e. they describe westward propagating waves. Accordingly, the mode-numbers of the three wave types are determined by the index of the corresponding discrete eigenvalues. The modes with n=0 are a special case in that one of the two westward propagating roots is associated with an unphysical zonal velocity i.e. one of the roots of the cubic equation is not a physically acceptable solution of the LRSWE. The other n=0 westward propagating wave is a mixed-mode, AKA Yanai wave, that changes from an Inertia-Gravity mode at long zonal wavenumbers to a Rossby mode at short zonal wavenumbers. Despite its wide-spread implications on the dynamics in the ocean, atmosphere and climate on earth, Matsuno’s equatorial wave theory has never been extended to spherical coordinates. With the recent successful formulation of eigenvalue equations for Planetary waves and Inertia-Gravity waves on the entire sphere that are analogous to the single planar equation derived by Matsuno on a plane we were able to derive explicit expressions for Planetary (Rossby) waves and Inertia-Gravity (Poincaré) waves on a sphere including the meridional amplitude structure and dispersion relations. In particular we were able to confirm the existence of Yanai wave on a sphere where the dynamical reason for the existence of the Yanai wave on a sphere is not the singularity of the zonal velocity. Instead, the explicit expressions for one of the n=0 westward propagating mode ceases to be valid at a zonal wavenumber where its phase speed approaches the gravity wave phase speed. In my talk I will present the various eigenvalue equations on a sphere and reconcile the ambiguity encountered when two of them overlap. In addition, I will also demonstrate how the disappearance of the superfluous westward propagating mode gives rise to the formation of the mixed-mode.