 Uniform-current obstruction to constant shallow-water QG potential vorticity

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.