= Solution
{c}
For a nonzero zonal <wavenumber> $k$, stationarity requires
$$
0=\frac\omega k=U-\frac\beta{k^2+l^2},\qquad K_s^2=\frac\beta U.
$$
A real propagating solution therefore requires a westerly $U>0$. For $0<|k|<K_s$ there are two meridional <wavenumbers>:
$$
\boxed{l_\pm=\pm\sqrt{\frac\beta U-k^2}.}
$$
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 gives
$$
\boxed{\mathbf c_g=\frac{2U k}{K_s^2}(k,l),\qquad
\frac{dy}{dx}=\frac{c_{gy}}{c_{gx}}=\frac lk.}
$$
Thus 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 $\beta(\phi)=2\Omega\cos\phi/a_p$ and possibly $U(\phi)$. The conserved zonal <wavenumber> in a locally zonally uniform medium then has
$$
l^2(\phi)=\frac{\beta(\phi)}{U(\phi)}-k^2.
$$
For 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> is
$$
\boxed{\Lambda_s=\frac{2\pi}{K_s}=2\pi\sqrt{\frac U\beta}.}
$$
For fixed $U$ it increases toward the pole as $\beta$ decreases. Locally the zonal integer scale has upper bound $n_{\max}\simeq a_p\cos\phi\,K_s=\sqrt{2\Omega a_p\cos^3\phi/U}$, 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>.
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