Forced Eady edge wave ($\dot{A}+i\omega A=C{e}^{-i{\omega }_{0}t}$)

With uniform thermal wind $U=\Lambda z$ above a plane boundary, write the zero-interior-potential vorticity response as ${\psi }^{\prime }=\mathrm{Re}\{A{e}^{-\mu z+i(kx+ly)}\}$, where $\mu =N\sqrt{k^2+l^2}/f$ and $f>0$. Boundary vertical velocity $ϵ\cos (kx+ly-{\omega }_{0}t)$ gives $C=N^2ϵ/(f\mu )$ and natural angular frequency $\omega =\Lambda k/\mu$. An initially zero response has $A=C(e^{-i{\omega }_{0}t}-e^{-i\omega t})/[i(\omega -{\omega }_0)]$. Its continuous resonant limit is $Ct{e}^{-i\omega t}$: forcing in phase with the boundary mode supplies energy every cycle, and linear undamped theory predicts secular growth.