Vortex Rossby waves on smooth circular vortices: Exact results and application to tropical cyclone and polar vortex interiors

A complete theory of the linear initial-value problem for Rossby waves on a class of smooth circular vortices in both f-plane and polar geometries is presented. Although restricted to the interior region of barotropically stable circular vortices possessing a single extrema in tangential wind, the theory covers all azimuthal wavenumbers. The nondimensional evolution equation for potential vorticity is shown to depend on only one parameter, G, involving the azimuthal wavenumber, the basic state radial potential vorticity gradient, the interior deformation radius, and the interior Rossby number.

In Hankel transform space the problem admits a Schrodinger's equation formulation that permits a qualitative and quantitative discussion of the interaction between vortex Rossby wave disturbances and the mean vortex. Using results from the theory of Lie groups a nontrivial separation of variables can be achieved to obtain an exact solution for asymmetric balanced disturbances covering a wide range of geophysical vortex applications including tropical cyclone, polar vortex, and cyclone/anticyclone interiors. The expansion for square summable potential vorticity comprises a discrete basis of radially propagating sheared vortex Rossby wave packets with nontrivial transient behavior. The solution representation is new. At the extrema of these stable vortices and for a certain range of azimuthal wavenumbers, the Rossby wave dynamics are shown to become nonlinear for all initial conditions.

Linear and nonlinear numerical experiments will be presented in order to demonstrate the usefulness of the theory. For a monopolar TC-like vortex whose strength corresponds roughly to that of a tropical storm, linear theory is shown to remain uniformly valid in time in the vortex core for all azimuthal wavenumbers. Examples elucidate the Rossby wave spin up mechanism proposed previously. For the case of the polar vortex, however, wavenumber one disturbances in the continuous spectrum are predicted to develop a nonlinear evolution for realistic polar-night stratospheric-jet configurations and wave breaking is demonstrated to occur within the dynamics of the continuous spectrum.