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Potential-vorticity conservation still determines the final surface throughThe unbounded problem only required decay or matching as $|y|$ and $x$ become large. The coast adds a boundary condition. In the adjusted geostrophic flow,so $\eta$ must be constant along the connected wall. Odd symmetry and the arbitrary common height datum set that constant to zero:This is a Dirichlet condition on $\eta$, even though it originated as no normal flow.

The initial disturbance does not instantaneously know this along the whole coast. A southward coastal Kelvin wave for $f>0$ carries the pressure signal and establishes the constant wall height behind its wavefront. In the final streamline sketch, the quasi-geostrophic streamfunction is proportional to ${\eta }_f$. Far from the wall the contours and current resemble the unbounded east--west front; near $x=0$ those contours bend through a right angle and run along the coast, so the incident geostrophic current turns into a southward boundary current instead of crossing the wall.

The eastern wall has no normal flow, so the quasi-geostrophic streamfunction is constant there. Choose $\psi (L,y)=0$. Integrating the Sverdrup relation givesSince geostrophic balance gives $\psi =g\eta /f_0$,If this interior solution were extended to both walls, the west-minus-east height difference would be

The northward volume transport per unit meridional distance isThusThis is the basin-integrated form of Sverdrup balance.

A potential vorticity budget has the formThe material derivative follows the fluid motion and $S_q$ collects forcing or dissipation. With $S_q=0$ it becomes potential-vorticity conservation. For a quasi-geostrophic streamfunction, horizontal advection is $J(\psi ,q)={\psi }_x{q}_y-{\psi }_y{q}_x$.

The shallow-water quasi-geostrophic potential vorticity relation is a modified Helmholtz equation for the quasi-geostrophic streamfunction. Its inverse with specified boundary conditions recovers horizontal geostrophic flow and surface elevation $\eta =f_0\psi /g$. The Rossby deformation radius $L_D=\sqrt{gH}/|f_0|$ is the range of the balanced response: for a horizontal Fourier component, $\hat{\psi }=-\hat{q}/(|{\mathbf{k}}_h{|}^2+L_D^{-2})$, after subtracting background potential vorticity.

The three-dimensional quasi-geostrophic potential vorticity relation defines an anisotropic elliptic boundary value problem. Interior potential vorticity and boundary buoyancy perturbation $b=f_0{\psi }_z$, together with lateral and global conditions, determine the quasi-geostrophic streamfunction up to its allowed constant. If two solutions have the same data, their difference has zero elliptic source and homogeneous boundary data. Integration by parts gives zero integral of $|{\nabla }_h\psi {|}^2+(f_0^2/N^2)|{\psi }_z{|}^2$, proving uniqueness modulo a constant in a connected domain when the boundary terms vanish. Eady edge waves show that zero interior source alone is insufficient.