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Let $D_g={\partial }_t+{\mathbf{u}}_g\cdot {\nabla }_h$, $\psi =\tilde{\Phi }/f_0$, and ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x,0)$. The hydrostatic and thermal equations combine to giveTaking the vertical curl of the horizontal momentum equations to leading quasi-geostrophic order givesThe log-pressure coordinate continuity equation gives ${\nabla }_h\cdot {\mathbf{u}}_{a,h}=-e^{z/H}{\partial }_z(e^{-z/H}w)$. Substitute $w=-f_0{D}_g{\psi }_z/N^2$. In commuting the vertical derivative with $D_g$, the extra term is ${\mathbf{u}}_{g,z}\cdot {\nabla }_h{\psi }_z=(-{\psi }_{yz},{\psi }_{xz})\cdot ({\psi }_{xz},{\psi }_{yz})=0$. Therefore the log-pressure quasi-geostrophic potential vorticity equation is

The required small parameters have distinct roles. The Rossby number ${ϵ}_R=U/(|f_0|L)$ makes inertial acceleration small relative to the leading geostrophic balance. The parameter ${ϵ}_{\beta }=\beta L/|f_0|$ lets that leading balance use a single $f_0$ while retaining the beta effect in the slower vorticity evolution. Finally, $\psi \sim UL$ and the thermal equation imply $w\sim |f_0|U^2/(N^{2}D)$. Density-weighted continuity has vertical derivative scale $1/\delta$, where $\delta =\min (D,H)$, so $|{\mathbf{u}}_{a,h}|\sim Lw/\delta$. The vertical-advection consistency criterion for log-pressure quasi-geostrophy is thusIt also controls the omitted vertical-advection ratio $wL/(UD)\sim {ϵ}_a\delta /D$. Small Rossby number alone would not ensure small ageostrophic flow for every choice of stratification and vertical scales.

For constant $N$, linearization about rest givesPutting $\psi =\mathrm{Re}[{\hat{\psi }}_c{e}^{z/(2H)}{e}^{i(kx+mz-\omega t)}]$ replaces the vertical operator by $-m^2-1/(4H^2)$. Consequently the Rossby wave in a log-pressure atmosphere has

For the forced half-space, substitution of the specified ansatz givesAssume $\beta >0$ and $k>0$, and define ${\omega }_{\ast }=-\beta k/[k^2+f_0^2/(4N^2{H}^2)]<0$.

If $\omega >0$ or $\omega <{\omega }_{\ast }$, then $m^2<0$. Boundedness excludes the growing solution and uniquely selects $\boxed{\hat{\chi }={\Psi }_b{e}^{-\sqrt{-m^2}z}}$. If $\omega ={\omega }_{\ast }$, boundedness excludes the linear-in-$z$ solution, leaving $\hat{\chi }={\Psi }_b$; this is the zero-vertical-wavenumber threshold.

If ${\omega }_{\ast }<\omega <0$, both oscillatory solutions are bounded. Boundedness alone does not give a unique propagating solution. With forcing at the bottom and no incident wave from above, impose an upward radiation condition. Sincepositive $m$ has upward group velocity. The outgoing solution is $\boxed{\hat{\chi }={\Psi }_b{e}^{imz},m=\sqrt{m^2}>0}$. The physical streamfunction includes the prescribed $e^{z/(2H)}$ density-weighting factor; it is the amplitude $\hat{\chi }$ that must remain bounded.

Thus the strict range for vertically propagating waves isAt $\omega =0$, the unforced steady interior equation requires $\beta {\psi }_x=0$, so a nonzero $k$ bottom streamfunction cannot have a time-independent solution of this ideal rest-state problem without additional dynamics.

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