 Quasi-geostrophic omega equation

Define $J(a,b)=a_x{b}_y-a_y{b}_x$ and $\zeta ={\nabla }_h^2\psi$. Combining quasi-geostrophic potential vorticity dynamics with buoyancy evolution eliminates the pressure tendency and givesThe beta term remains when the nonlinear terms are omitted. For a steady weak response to heating about rest, $w=R/N^2$ solves this equation using $\beta {\psi }_x=f_0{R}_z/N^2$ and suitable no-through-flow and decay conditions.