Linear Waves in Midlatitudes on the Rotating Spherical Earth

Abstract

Abstract The linear waves of the shallow water equations in a zonal channel in midlatitudes 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 5 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, and 3) the mode number. For Poincaré (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—that is, 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 Poincaré waves in a certain wavenumber range, whereas the phase speed of eastward-propagating Poincaré waves does not approach that of the Kelvin wave even at large wavenumbers. Analytical expressions for the phase speed of trapped Poincaré 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 and the exact channel width increases with the radius of deformation. For a sufficiently small radius of deformation, such as that observed in the ocean, the solution is trapped, even for very narrow channels, and the phase speed is only slightly larger than that of harmonic waves.