Potential-vorticity inversion ($[{\nabla }_h^2+(f_0^2/N^2){\partial }_z^2]\psi =q-\beta y$)

Potential-vorticity inversion recovers the balanced quasi-geostrophic streamfunction from interior potential vorticity and suitable boundary buoyancy or pressure data by solving an elliptic boundary-value problem. Boundary anomalies can drive nonzero flow even when the interior potential-vorticity anomaly is zero. Boundary conditions and any pressure gauge or compatibility condition are essential to uniqueness.