 Solution

Let $Q_b=f/H$. At sufficiently small Rossby number, retain advection of this full background shallow-water potential vorticity while dropping the higher-order advection of $\zeta /H$. For uniform-depth Sverdrup balance, this also requires relative-vorticity advection to be small against planetary-vorticity advection; small Rossby number alone does not guarantee this when $\beta$ is very small. Multiplying the resulting steady potential vorticity equation by $H$ givesDefine the topographic potential-vorticity pseudovelocityIts sign is important: it is opposite to the coefficient vector on the left of the preceding equation. Substituting the ocean transport streamfunction expression for relative vorticity yieldsThe pseudovelocity follows contours of $f/H$ and is not the fluid velocity; its units are inverse length squared per time because $\psi$ is a transport streamfunction. If $r=F=0$, the ocean transport streamfunction is constant along each connected background potential vorticity contour. Locally, at regular points,with arbitrary differentiable $\Phi$. Different disconnected components of a level set may carry different functions until boundary conditions identify them. If $f/H$ is constant over an open region, the leading equation gives no restriction there; critical points similarly require regularity and global matching rather than division by a vanishing gradient.