= 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 $H$ gives
$$
-\psi_y(Q_b)_x+\psi_x(Q_b)_y=F-r\zeta.
$$
Define the <topographic potential-vorticity pseudovelocity>
$$
\boxed{\widetilde{\mathbf u}=\widehat{\mathbf z}\times\nabla_h(f/H)=\left(-\partial_y(f/H),\partial_x(f/H)\right).}
$$
Its 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> yields
$$
\boxed{\widetilde{\mathbf u}\cdot\nabla_h\psi=\frac rH\nabla_h^2\psi-F-r\frac{\nabla_hH\cdot\nabla_h\psi}{H^2}.}
$$
The <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,
$$
\boxed{\psi=\Phi(f/H),}
$$
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.
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