Generalized Charney–Stern and Fjortoft Theorems for Rapidly Rotating Vortices

A generalized Charney–Stern theorem for rapidly rotating (large Rossby number) baroclinic vortices, such as hurricanes, is derived based on the asymmetric balance (AB) approximation. In the absence of dissipative processes, a symmetrically stable baroclinic vortex is shown to be exponentially stable to nonaxisymmetric perturbations if a generalized potential vorticity gradient on theta surfaces remains single signed throughout the vortex. The generalized potential vorticity gradient involves the sum of an interior potential vorticity gradient associated with the symmetric vortex and surface contributions associated with the vertical shear of the tangential wind. The AB stability formulation is then shown to yield Fjortoft's theorem as a corollary. In the modem view of shear instabilities the theorems admit simple interpretation. The Charney–Stern theorem represents a necessary condition for the existence of counterpropagating Rossby waves associated with the radial potential vorticity gradient, while Fjortoft's theorem represents a necessary condition for these waves to phase lock and grow in strength. Potential application of these results as well as limitations of the slow-manifold approach are briefly discussed.