Baroclinic Rossby wave

A baroclinic Rossby wave has nontrivial vertical structure. Constant stratification and rigid boundaries add a positive vertical-mode contribution to the denominator of its dispersion relation.

For fixed meridional wavenumber and deformation radius, differentiating $\omega =Uk-\beta k/(k^2+l^2+L_d^{-2})$ gives $c_{gx}=U+\beta (k^2-l^2-L_d^{-2})/(k^2+l^2+L_d^{-2}{)}^2$. A localized packet uses this velocity rather than $\omega /k$. For $l=0$ its intrinsic sign changes at $|k|L_d=1$.

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.

For dispersion relation $\omega =Uk-\beta k/(k^2+{\alpha }^2)$ with fixed ${\alpha }^2\ge 0$, nonzero zonal wavenumber and stationary phase imply $U=\beta /(k^2+{\alpha }^2)$. Differentiating at fixed $\alpha$ yields $c_{g,x}=U+\beta (k^2-{\alpha }^2)/(k^2+{\alpha }^2{)}^2$, which reduces to the displayed positive value for $\beta >0$. Thus a stationary Rossby wave can transport energy eastward even though its crests stay fixed.

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.