= Harmonic heating response of quasi-geostrophic flow
{title2=$\widehat\psi''-\lambda_\omega^2\widehat\psi=C_\omega e^{-\mu z}$}
For constant <buoyancy frequency>, <density> forcing $r_0e^{-\mu z}\cos kx\cos\omega_0t$ 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 $C_\omega=ig\mu r_0/(2\rho_0f_0\omega)$. 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 $-C_\omega z e^{-\mu z}/(2\mu)$.
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