= Stationary Rossby-wave vertical propagation
{title2=$m^2=(N^2/f_0^2)[\beta/U-k^2-l^2]$}
For constant <buoyancy frequency> and uniform <zonal flow>, the <three-dimensional quasi-geostrophic potential vorticity> equation gives $\omega=Uk-\beta k/[k^2+l^2+(f_0^2/N^2)m^2]$. Set $\omega=0$ with $k\ne0$ to obtain the displayed formula. For $\beta>0$, oscillatory vertical structure requires $0<U<\beta/(k^2+l^2)$. Differentiation gives $c_{g,z}=2\beta k(f_0^2/N^2)m/[k^2+l^2+(f_0^2/N^2)m^2]^2$, so an upward outgoing <radiation condition> selects $km>0$. The <Coriolis parameter> enters squared: the criterion does not reverse between hemispheres. A density-scale-height correction belongs to a different, compressible <log-pressure quasi-geostrophic potential vorticity> model.
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