Simple Methods to Extract Analytical Solutions for Rossby Waves on Continuously Stratified Zonal Background Flows

Observed westward phase speeds of Rossby waves in a subtropical region dominated by stratified eastward jets in the South Indian Ocean are about five times the traditional linear theory for propagation of Rossby waves in a stratified ocean. To date no simple analytical generalization of the standard dispersion relation with the inclusion of the effect of baroclinic eastward currents has been obtained, even for flat surface and bottom boundary conditions. One reason may be due to a kind of entanglement between the vertical profile of the jet and the phase speed eigenvalue, which makes this a nonlinear albeit separable eigenvalue problem even for linear theory. A quasi-exact entangled solution, separable by a Modified Bessel Transform, makes possible simple generalizations of the standard dispersion relation, away from critical layers. An interesting application of the method to Rossby waves over the South Indian Coutercurrent at 26\textdegree S using a parabolic approximation to the zonal velocity around its minimum at 500 m is offered here. The phase speed enhancement $\gamma=c/c_{0}$, where $ c $ is the eigenvalue of the vertical velocity and $ c_{0} $ the phase speed given by the standard theory for the first baroclinic mode depends on a single parameter $s=(\pi^{2}/12)U_{zz}(\beta N^{2}/f^{2})^{-1} $, where $ U_{zz} $ is the curvature of the vertical profile of the mean zonal velocity. For $ (s/\gamma)\geq{1} $, the enhancement is simply given by $\gamma= s^{2/3} $, while for $ (s/\gamma)\ll{1} $, $\gamma= 1+s $. Examples near the critical layer and wavelengths of the order of the mean local Rossby radius (baroclinic instability) is also discussed in this context. The effects of adding sharp mid-oceanic ridges to the boundary conditions are also discussed.