= Solution
{c}
Take a Northern-Hemisphere cyclonic <vortex>, with positive <relative vorticity>. For a fixed-depth incompressible layer, conservation of <absolute vorticity> gives
$$
\frac D{Dt}(\xi+f)=0,\qquad \frac{D\xi}{Dt}=-\beta v.
$$
The anticlockwise flow on the east side moves parcels northward, increasing their <planetary vorticity> and creating negative <relative vorticity> anomalies. On the west side it moves parcels southward and creates positive <relative vorticity> anomalies. Initially,
$$
\partial_t\xi\simeq-\beta v,
$$
so the positive anomaly grows on the western flank while the eastern flank loses positive <vorticity>. The cyclonic pattern consequently shifts westward. The induced anomalies themselves drive meridional flow and further <relative vorticity> anomalies, transmitting a <Rossby wave> pattern through the surrounding fluid. The restoring process is this exchange between <planetary vorticity> and <relative vorticity>, rather than a parcel executing a permanently westward translation.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2021-rossby-vorticity-mechanism.png]
{title=<Relative vorticity> generated by a Northern-Hemisphere cyclonic <vortex> on a <beta plane>}
The sketch shows the initial tendency, with the generated signs separated from the initial positive core. An isolated disturbance contains many <wavevectors> and disperses; westward <phase velocity> does not imply that all its <wave energy> travels westward. A Southern-Hemisphere cyclonic <vortex> reverses the signs of the initial and generated <relative vorticities>, while the same <beta plane> mechanism remains.
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