Stable Shallow Water Vortices over Localized Topography

It is shown that a sufficient condition for stability by P. Ripa, based on the monotonicity of the flow potential vorticity (PV), can be used to prove linear stability of isolated shallow water vortices over localized topographic features. Stable axisymmetric vortices over axisymmetric topography that satisfy Ripa's condition are explicitly constructed by using a simple two-step, fully analytic approach. First, for a given velocity profile, the topography is found that yields a steady-state, constant-PV solution of the shallow water equations. Then, this topography is slightly modified to obtain new steady solutions, with monotonic PV, that satisfy Ripa's stability criterion. Application of this procedure shows that modest depressions (elevations) can stabilize cyclones (anticyclones) with small Rossby and large Burger numbers and velocity profiles similar to those observed in mesoscale oceanic vortices. The stabilizing topographic features have radial sizes comparable with that of the vortex (about twice the radius of maximumspeed) and maximum vertical size, normalized to the unperturbed fluid depth, from 2 to 3.3 times the Rossby number for the profiles considered. The upper limit corresponds to a Gaussian profile, whereas the lower limit is approached by a velocity profile that is linear inside the vortex core and a cubic polynomial outside. Finally, it is argued that a similar stabilization mechanism holds for two-dimensional (2D) flows, and a method for the construction of stable 2D shallow water vortices over 2D topography is outlined that is analogous to that used for the axisymmetric problem. In the 2D case, however, it is generally not possible to obtain stable equilibria analytically.