= Quasi-geostrophic omega equation
{title2=$N^2\nabla_h^2w+f_0^2w_{zz}$}
Define $J(a,b)=a_xb_y-a_yb_x$ and $\zeta=\nabla_h^2\psi$. Combining <quasi-geostrophic potential vorticity> dynamics with buoyancy evolution eliminates the pressure tendency and gives
$$
N^2\nabla_h^2w+f_0^2w_{zz}=\nabla_h^2R+f_0\beta\psi_{xz}+f_0\left[\partial_zJ(\psi,\zeta)-\nabla_h^2J(\psi,\psi_z)\right].
$$
The 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 $\beta\psi_x=f_0R_z/N^2$ and suitable no-through-flow and decay conditions.
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