= Shallow-water quasi-geostrophic inversion
{title2=$[\nabla_h^2-L_D^{-2}]\psi=q-\beta y$}
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, $\widehat\psi=-\widehat q/(|\mathbf k_h|^2+L_D^{-2})$, after subtracting background <potential vorticity>.
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