 Resonant topographic quasi-geostrophic wave

A stationary boundary corrugation can resonate with a Rossby wave whose intrinsic propagation cancels the background current. For constant $N$, rigid upper height $H$, and $m=\pi /H$, define ${\kappa }^2=k^2+{\ell }^2+f_0^2{m}^2/N^2$. When $U=\beta /{\kappa }^2$, a lower corrugation proportional to $e^{ikx}\sin (\ell y)$ admits a disturbance proportional to $e^{ikx}\sin (\ell y)[Bt\cos (mz)+C(z-H)\sin (mz)]$, with $B=2ik{f}_0{U}^2/\beta$ and $C=N^2/(f_{0}m)$. Its wave activity grows quadratically with time while the upward Eliassen–Palm flux grows linearly. The linear prediction applies only while the disturbance remains small.