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.

For a polar gamma-plane Rossby wave, the drift fixes ${\alpha }^2=K^2+L_d^{-2}$. It does not separately fix radial wavenumber $K$ and Rossby deformation radius $L_d$. Scaling an eigenfunction changes its amplitude without changing either wavenumber, so a meander amplitude cannot supply the missing radial wavelength. A radial boundary condition or a separately justified envelope approximation is needed to determine $L_d$.