Three-dimensional quasi-geostrophic potential vorticity (3D QG potential vorticity)

For constant buoyancy frequency $N$ on a beta plane, three-dimensional quasi-geostrophic potential vorticity can be written

A quasi-geostrophic vertical mode separates horizontal and vertical dependence of the streamfunction. With rigid horizontal boundaries and constant stratification, the vertical eigenfunctions are $\cos (n\pi z/H)$.

An isopycnal displacement is the vertical displacement $\xi$ of a constant-density surface. In linear stratified quasi-geostrophic flow,