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For a nonzero zonal wavenumber $k$, stationarity requiresA real propagating solution therefore requires a westerly $U>0$. For $0<|k|<K_s$ there are two meridional wavenumbers:The source's assertion of two stationary solutions needs these qualifications: at $|k|=K_s$ they coincide, and at $|k|>K_s$ there is no real meridionally propagating solution. For example $k=2\sqrt{\beta /U}$ gives $l^2=-3\beta /U$. Modes with $k=0$ are a separate zonally invariant degeneracy and do not have a defined zonal phase velocity.

Adding $U$ to the zonal group velocity and imposing stationarity givesThus a localized midlatitude disturbance radiates stationary Rossby wave activity downstream, along one poleward and one equatorward branch. For $k>0$ in the Northern Hemisphere, $l_{+}$ gives the northeastward branch and $l_{-}$ the southeastward branch. With strictly constant $U,\beta$, these wave rays are straight. Changing latitude cannot bend them within that constant-coefficient model.

To discuss a planet, extend it locally to slowly varying $\beta (\varphi )=2\Omega \cos \varphi /a_p$ and possibly $U(\varphi )$. The conserved zonal wavenumber in a locally zonally uniform medium then hasFor constant $U$ and northward propagation in the Northern Hemisphere, decreasing $\beta$ reduces $l$ until $l=0$: the wave ray becomes zonal at a turning latitude and can refract equatorward. The equatorward branch initially encounters larger allowed $K_s$ and becomes more meridional. This local midlatitude description eventually needs replacement near the equator or wherever the background varies rapidly.

Orographic forcing by a mountain range or persistent spatially localized heating can supply such a disturbance. Its steady forcing selects the ground-stationary part of the response. The corresponding local total wavelength isFor fixed $U$ it increases toward the pole as $\beta$ decreases. Locally the zonal integer scale has upper bound $n_{\max }\simeq a_p\cos \varphi K_s=\sqrt{2\Omega a_p{\cos }^3\varphi /U}$, so fewer planetary-scale zonal waves fit around higher-latitude circles. This is not an unconditional trend in actual weather: changing $U$, its curvature, stratification and the forcing spectrum can change the dominant Rossby wave wavelengths.