Circular topographic anticyclone ($q=-fϵ1_{r<a}$)

For an upstream irrotational current on an f-plane in a rigid layer of height $h$, a circular region of height $h(1-ϵ)$ has the displayed relative vorticity if its fluid retains upstream potential vorticity $f/h$. To leading order in the height contrast, take $u=-{\Psi }_y$, $v={\Psi }_x$ and $\Psi =-Uy+\psi$. Solving the Poisson equation for $\psi$ gives $\psi =-C{r}^2/2$ inside and $\psi =C{a}^2[\log (a/r)-1/2]$ outside, where $C=fϵ/2$. The disturbance is clockwise for positive $fϵ$. Variable-depth transport and the transition across the rim introduce higher-order corrections to this leading columnar inversion.

For the circular topographic anticyclone in a positive uniform current $U$, the opposing disturbance speed is at most $Ca=fϵa/2$. Below the displayed threshold $u$ remains positive everywhere, so no streamline can close. Above it an interior stagnation point at $(0,-U/C)$ is a center, while the exterior point $(0,-C{a}^2/U)$ is a saddle. At equality the two meet at the rim. This is the onset criterion; once closed contours form, the potential vorticity of trapped fluid is not determined solely by its present upstream conditions.