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ABSTRACT

Despite the accurate formulation of Laplace's Tidal Equations (LTE) nearly 250 years ago, analytic solutions of these equations on a spherical planet that yield explicit expressions for the dispersion relations of wave solutions have been found only for slowly rotating planets, so these solutions are of no relevance to Earth. Analytic solutions of the LTE in a symmetric equatorial channel on a rotating sphere were recently obtained by approximating the LTE by a Schrödinger equation whose energy levels yield the dispersion relations of zonally propagating waves and whose eigenfunctions determine the meridional structure of the amplitude of these waves. A similar approximation of the LTE on a sphere (with no channel walls) by a Schrödinger equation yields accurate analytic solutions for zonally propagating waves in the parameter range relevant to a baroclinic ocean, where the ratio between the radius of deformation and Earth's radius is small. For sufficiently low (meridional) modes the amplitudes of the solutions vanish at some extra-tropical latitudes but this is not assumed abinitio. These newly found solutions do not restrict the value of the zonal wavenumber to be smaller than the meridional wavenumber as is the case in previous theories on a slowly rotating sphere.