= Rossby wave in a log-pressure atmosphere
{c}
For constant <buoyancy frequency> $N$, the <log-pressure quasi-geostrophic potential vorticity> equation has modes $\psi=e^{z/(2H)}e^{i(kx+mz-\omega t)}$ with
$$
\omega=-\frac{\beta k}{k^2+(f_0^2/N^2)[m^2+1/(4H^2)]}.
$$
The prefactor compensates the density decrease in the energy weight. For $\beta,k>0$, vertical propagation occurs for $-\beta k/[k^2+f_0^2/(4N^2H^2)]<\omega<0$. An outgoing <radiation condition> selects the root with positive vertical <group velocity>; bounded weighted amplitude alone does not select between two propagating roots.
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