Quasi-geostrophic wave at a stratification interface

Two vertically unbounded regions with buoyancy frequencies $N_1,N_2$ and thermal wind shears ${\Lambda }_1,{\Lambda }_2$ support a localized wave if pressure and normal velocity are continuous at their common material interface. With decays ${\varphi }_1=A{e}^{{\mu }_{1}z}$ below and ${\varphi }_2=A{e}^{-{\mu }_{2}z}$ above, ${\mu }_j=N_{j}K/|f_0|$, matching givesWhen $N_1/N_2\to \infty$ with bounded shear, the lower layer becomes effectively rigid and the upper-layer Eady edge wave speed ${\Lambda }_2/{\mu }_2$ is recovered.