= Uniform-current obstruction to constant shallow-water QG potential vorticity
{title2=$\overline Q=f+Uy/L_R^2$}
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.
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