= Solution
Take $\beta>0$, a uniform eastward $U>0$, and the nondivergent <Barotropic Rossby wave> model. Its <dispersion relation> is $\omega=Uk-\beta k/(k^2+l^2)$. A stationary wave with $k\ne0$ has
$$
\boxed{K^2=k^2+l^2=\frac\beta U,\qquad \lambda=\frac{2\pi}{K}=2\pi\sqrt{\frac U\beta}.}
$$
The directional component wavelengths are $2\pi/|k|$ and $2\pi/|l|$ when nonzero. Differentiating the full <dispersion relation> before imposing stationarity gives its ground-frame <group velocity>:
$$
\mathbf c_g=\left(U+\frac{\beta(k^2-l^2)}{K^4},\frac{2\beta kl}{K^4}\right)=\boxed{\frac{2U}{K^2}(k^2,kl)}.
$$
Writing $k=K\cos\theta$, $l=K\sin\theta$ shows $\mathbf c_g=U(1+\cos2\theta,\sin2\theta)$. Therefore the <stationary Rossby-wave ray envelope> at time $\tau$, for <wave packets> emitted from a localized mountain throughout $0<t<\tau$, is the disk
$$
\boxed{(x-U\tau)^2+y^2\le(U\tau)^2,\qquad x\ge0.}
$$
The circle is the front of the earliest emissions; later emissions fill ray segments back to the source. The sketch represents the geometrical <wave packet> envelope, not compact support of the full balanced near field. All propagating <wave packet> energy lies downstream in this model.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-333-stationary-rays.png]
{title=Downstream ray envelope of stationary barotropic Rossby waves emitted by a mountain between zero and the observation time}
If the finite-depth free-surface model of the preceding part is retained consistently, its basic surface slope adds $U/R_D^2$ to the potential-vorticity gradient. Stationarity still gives $K^2=\beta/U$ and the same wavelength, but the <group velocity> is reduced by $\gamma=K^2/(K^2+R_D^{-2})$; the disk center and radius become $\gamma U\tau$. This tends to the plotted nondivergent result as $R_D\to\infty$.
\b[A westward wind has no nontrivial stationary propagating planetary Rossby-wave train in this model:] $U<0$ would require $K^2=\beta/U<0$. Its advective phase motion and intrinsic westward <Rossby wave> phase motion cannot cancel. A local forced, evanescent response and time-dependent disturbances can still occur. The degenerate $k=0$ limit is not a propagating wave train from an isolated mountain.
Back to article page
