 Log-pressure quasi-geostrophic potential vorticity

In a log-pressure coordinate, quasi-geostrophic potential vorticity uses density-weighted vertical stretching. For stratification $N^2=RS/H$ and geostrophic flow ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x,0)$, the inviscid equation is $D_{g}q/Dt=0$. Combining horizontal vorticity balance with the thermal equation eliminates ageostrophic flow and gives this conservation law.