Linear waves in Mid-latitudes on the rotating spherical Earth

The linear waves of the Shallow Water Equations in a zonal channel in Mid-latitudes on the rotating spherical Earth are investigated analytically and numerically by solving several relevant eigenvalue problems. For baroclinic deformation radii in the ocean the phase speed of long Rossby waves in a sufficiently wide channel on the sphere can be five times that of their harmonic β-plane counterparts. The difference between the two phase speeds increases with the channel width and decreases with: 1) the latitude of the equatorward wall; 2) the radius of deformation; 3) the mode number. For Poincare (Inertia-gravity) waves the phase speed on the sphere is slightly lower than that of harmonic waves on the β-plane. The meridionally-dependent amplitude of the meridional velocity is identical for both waves and is trapped near the equatorward wall i.e., its amplitude is maximal within a few deformation radii from this wall. The phase speeds of the Kelvin and anti-Kelvin waves on a sphere are determined by the latitudes of the equatorward and poleward walls, respectively, where they attain their maximal height amplitude. Accordingly, the phase speed of the anti-Kelvin wave is larger than that of the westward propagating Poincare waves in a certain wavenumber range while the phase speed of eastward propagating Poincare waves does not approach that of the Kelvin wave even at large wavenumbers. Analytical expressions for the phase speed of trapped Poincare and Rossby waves are obtained for small deformation radii in wide channels by approximating the meridional velocity's eigenfunction by an Airy function that decays with distance from the equatorward wall. The exact latitude of the poleward wall does not affect the solution provided it is several deformation radii away from the equatorward boundary, the exact width increases with the radius of deformation