Past exam of the geophysical fluid dynamics course of the University of Oxford / 2024 / 1 / d / Solution

Hold $l,m,f_0,N$ fixed when differentiating the dispersion relation of Rossby waves, so that ${\alpha }^2\ge 0$ is independent of $k$. For $k\ne 0$, stationary zonal phase requiresThe zonal group velocity measured relative to the ground isUsing the stationarity condition,Here $\beta =2\Omega \cos \varphi /a>0$ on the ordinary midlatitude beta plane, so the stationary Rossby wave requires an eastward background current and carries its wave packet eastward. If $\alpha >0$, a real nonzero stationary $k$ exists only for $0<U<\beta /{\alpha }^2$. The restriction $k\ne 0$ matters: a zonally uniform disturbance with $k=0$ has $\omega =0$ identically, but its zonal phase velocity is undefined, so it is not a counterexample within the stated meaning of stationary waves.