 Solution

Write $u=U+u^{\prime }$, $v=v^{\prime }$, and $\xi ={\xi }^{\prime }$, with $U=\bar{u}$ constant. Discarding products of perturbations in the vorticity equation givesIntroduce the streamfunction conventionIt enforces incompressibility identically. The linear partial differential equation isFor the plane wave ${\psi }^{\prime }=\mathrm{Re}\{\hat{\psi }\exp [i(kx+ly-\omega t)]\}$, put $K^2=k^2+l^2>0$. Substitution yieldsThe first term is the Doppler shift by the background flow; the second is the intrinsic Rossby wave frequency. The source's convention $\omega \ge 0$ selects representatives of real waves, rather than making this expression nonnegative for arbitrary $k,l$. For $U=0$ and $\beta >0$, nonzero positive frequency requires $k<0$; the conjugate mode has opposite signs of $k,l,\omega$ and describes the same real field.