Uniform-current obstruction to constant shallow-water QG potential vorticity ($\overline{Q}=f+Uy/L_R^2$)

On a constant f-plane with finite Rossby deformation radius, a geostrophic uniform current $(U,0)$ has streamfunction $-Uy$ and a sloping free surface. The full shallow-water quasi-geostrophic potential vorticity is $Q=f+{\nabla }^2\Psi -\Psi /L_R^2+fb/H$, so a flat-bottom background has the displayed nonconstant value. Omitting its gradient is not a harmless streamfunction gauge choice. A uniform-current, constant-PV inversion must explicitly supply a compensating background, external balance, or a rigid-lid limit.

In a shallow-water potential-vorticity inversion with the background gradient explicitly compensated, uniform PV gives $({\nabla }^2-L_R^{-2}){\psi }^{\prime }=-fb/H$. The displayed Gaussian function solves it for $b=ϵ[4(1-r^2/a^2)+a^2/L_R^2]e^{-r^2/a^2}$. This is not, without that compensation, a steady solution of the standard finite-depth constant-f current problem: its residual is $U{\psi }_x^{\prime }/L_R^2$. Closed streamlines can occur while Rossby number and surface displacement remain small.

For total streamfunction $\Psi =-Uy+A{e}^{-(x^2+y^2)/a^2}$, the maximum opposing zonal disturbance speed is $\sqrt{2/e}|A|/a$. Below the displayed threshold the zonal velocity never changes sign and streamlines cannot close. Above it two stagnation points appear on the symmetry axis: one has definite Hessian and is a center, the other has indefinite Hessian and is a saddle. Closed contours surround the center. Equality is a degenerate onset, not a finite-area closed-cell regime.