Stationary Rossby-wave vertical propagation ($m^2=(N^2/f_0^2)[\beta /U-k^2-l^2]$)

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 $k\ne 0$ 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.