 Exponential-profile stationary stratified waves

For positive background velocity $U=U_0{e}^{Mz}$ and buoyancy frequency $N=N_0{e}^{Mz}$, a stationary nonzero-horizontal-wavenumber Taylor–Goldstein equation reduces to the displayed constant-coefficient equation. Its structures are oscillatory, affine or exponential according to the sign of $N_0^2/U_0^2-M^2-k^2$. The disturbance Laplacian is a constant times its streamfunction, but its buoyancy anomaly is $N_0^2{e}^{Mz}\varphi /U_0$. Therefore vorticity self-advection vanishes while buoyancy self-advection produces a nonzero second harmonic for $M\ne 0$. These nontrivial real linear modes are not exact finite-amplitude solutions without additional fields. For $M=0$, both polarizations are constant and these nonlinear self-interactions vanish.