Harmonic heating response of quasi-geostrophic flow (${\hat{\psi }}^{\prime \prime }-{\lambda }_{\omega }^2\hat{\psi }=C_{\omega }{e}^{-\mu z}$)

For constant buoyancy frequency, density forcing $r_0{e}^{-\mu z}\cos kx\cos {\omega }_{0}t$ gives two frequencies $\omega =\pm {\omega }_0$. The quasi-geostrophic streamfunction satisfies ${\lambda }_{\omega }^2=(N^2/f_0^2)(k^2+\beta k/\omega )$ and $C_{\omega }=ig\mu r_0/(2{\rho }_0{f}_0\omega )$. With zero streamfunction at the bottom, a positive ${\lambda }_{\omega }^2$ selects the decaying solution $C_{\omega }(e^{-\mu z}-e^{-{\lambda }_{\omega }z})/({\mu }^2-{\lambda }_{\omega }^2)$. For $\beta ,k>0$, the westward branch propagates vertically when ${\omega }_0<\beta /k$; a radiation condition selects positive vertical group velocity, not merely bounded amplitude. Coincident exponential exponents are handled by the finite limit $-C_{\omega }z{e}^{-\mu z}/(2\mu )$.