= Solution
{c}
For $U=0$ and fixed $\omega>0$, completing the square in the <dispersion relation> gives
$$
\boxed{\left(k+\frac\beta{2\omega}\right)^2+l^2=\left(\frac\beta{2\omega}\right)^2.}
$$
These <Rossby-wave isofrequency circles> lie in the $k\leq0$ half-plane and touch the origin, where the <dispersion relation> is undefined. Their radii decrease as <frequency> increases. The zero-<frequency> line is $k=0,l\ne0$.
The <group velocity> is normal to the constant-<frequency> curves, toward increasing <frequency>:
$$
\boxed{c_{gx}=\frac{\beta(k^2-l^2)}{(k^2+l^2)^2},\qquad
c_{gy}=\frac{2\beta kl}{(k^2+l^2)^2}.}
$$
At $l=0$, the <group velocity> is eastward; close to $k=0$ it is westward. For the positive-<frequency> circles its arrows point inward toward the circle centre. Its meridional component is southward in $k<0,l>0$ and northward in $k<0,l<0$.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2021-rossby-dispersion.png]
{title=Positive-<frequency> <Rossby-wave isofrequency circles> and <group velocity> directions in dimensionless <wavenumber> coordinates}
For $k\ne0$, the zonal <phase velocity>, meaning propagation at fixed $y$, is
$$
\boxed{c_{px}=\frac\omega k=-\frac\beta{k^2+l^2}<0.}
$$
It is westward everywhere it is defined. The zonal <group velocity> is eastward for $|k|>|l|$, westward for $|k|<|l|$, and zero on $|k|=|l|$. There is no region of eastward zonal <phase velocity>. At $k=0$, $\omega/k$ is undefined, though its continuous limiting expression is westward and $c_{gx}=-\beta/l^2$. Using instead the normal-to-wavecrest <phase velocity> vector gives $\mathbf c_p=\omega(k,l)/(k^2+l^2)$; its zonal component is also westward for $k\ne0$, but is not the same scalar as $\omega/k$.
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