The construction of approximate Schrödinger eigenvalue equations for planetary (Rossby) waves and for inertia–gravity (Poincaré) waves on an ocean-covered rotating sphere yields highly accurate estimates of the phase speeds and meridional variation of these waves. The results are applicable to fast rotating spheres such as Earth where the speed of barotropic gravity waves is smaller than twice the tangential speed on the equator of the rotating sphere. The implication of these new results is that the phase speed of Rossby waves in a barotropic ocean that covers an Earth-like planet is independent of the speed of gravity waves for sufficiently large zonal wavenumber and (meridional) mode number. For Poincaré waves our results demonstrate that the dispersion relation is linear, (so the waves are non-dispersive and the phase speed is independent of the wavenumber), except when the zonal wavenumber and the (meridional) mode number are both near 1.

A B S T R A C T 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.

No abstract

The equivalent depth of a rotating fluid layer determines the phase speed of all free (i.e. unforced) linear waves that propagate in it. The equivalent depth can be estimated from the eigenvalues of a second‐order differential equation for the vertical dynamics where the mean temperature profile is a coefficient. The eigenfunctions dictate the vertical structure of the modes. This work combines analytic and numerical solutions of the vertical structure equation for various mean temperature profiles and for various combinations of top and bottom boundary conditions relevant to two atmospheric configurations: troposphere and combined troposphere‐stratosphere. Our formulation provides a clear definition of the barotropic mode and the countable baroclinic modes (including the special n = 0 mode). The barotropic mode exists only when the lower boundary condition is the vanishing of the vertical velocity (i.e. when a solid boundary bounds the layer from below) and the equivalent depth of this mode is about 10 km. The n = 0 baroclinic mode does not exist in a layer whose thickness exceeds a threshold value of about 20 km and therefore this mode does not exist in the combined troposphere‐stratosphere layer. The upper boundary condition affects the eigenvalues much more so than the details of the temperature profile, as the details of the temperature profile only affect the equivalent depth in the troposphere. Increasing the height of the upper boundary has little effect on the barotropic mode, but strongly influences the phase speed and vertical structure of the baroclinic modes; a general circulation model with a lid at or below the stratopause where, for example, planetary Rossby waves are present will therefore be incapable of correctly simulating the interaction of these waves with the mean flow. The values of the equivalent depth of baroclinic modes are approximately 10% of the actual layer's depth in all realistic cases.

No abstract