Three-dimensional quasi-geostrophic potential vorticity
= Three-dimensional quasi-geostrophic potential vorticity
{title2=3D QG potential vorticity}
For constant buoyancy frequency $N$ on a beta plane, three-dimensional quasi-geostrophic potential vorticity can be written
$$
q=\psi_{xx}+\psi_{yy}
+\frac{\partial}{\partial z}
\left(\frac{f_0^2}{N^2}\psi_z\right)+\beta y.
$$