 Polar gamma-plane Rossby wave

Near a north pole, $f=2\Omega -\gamma r^2/2$ with $\gamma =2\Omega /a^2$. If a quasi-geostrophic streamfunction satisfies ${\nabla }_h^2\psi -\psi /L_d^2=-{\alpha }^2\psi$, then $Q=f-{\alpha }^2\psi$. Its nonlinear self-advection cancels because $J(\psi ,\psi )=0$, leaving $-{\alpha }^2{\psi }_t+\gamma {\psi }_{\varphi }=0$. An azimuthal mode $\cos (\omega t-n\varphi )$ therefore has $\omega =-n\gamma /{\alpha }^2$ at any amplitude consistent with the underlying balanced model.