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.

For constant stratification and $U=U_0+\Lambda z$, a decaying zero-interior-PV mode has vertical structure $e^{-Kz}$. Its topographic boundary condition gives $A^{\prime }+ik(U_0+\Lambda /K)A=ikϵN^2{U}_0/(Kf)$. Resonance requires $K{U}_0+\Lambda =0$ and nonzero forcing $kϵU_0$. With zero initial disturbance, $A$ grows linearly and is imaginary relative to cosine topography. Its meridional velocity is therefore opposite in phase to the relief for positive $f,U_0,ϵ$. A zero zonal wavenumber or zero boundary current removes the forcing and invalidates an unconditional claim of secular growth.