= Stratified quasi-geostrophic inversion
{title2=$[\nabla_h^2+\partial_z(f_0^2N^{-2}\partial_z)]\psi=q-\beta y$}
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.
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