Rigid-lid constant-shear baroclinic model ($D(\eta )=1+{\eta }^2/4-\eta \coth \eta$)

For a uniform buoyancy frequency, constant Coriolis parameter and zonal shear $U=\Lambda z$ between rigid lids separated by $H$, a zero-interior-PV normal mode has ${\Psi }^{\prime \prime }-K^2\Psi =0$. Adiabatic no-normal-flow conditions give $c=\Lambda H/2\pm \Lambda H\sqrt{D(\eta )}/\eta$, $\eta =KH$. Growth requires $D<0$, or a negative squared phase speed relative to the mean flow, rather than a literal $c^2<0$ in an arbitrary moving frame. The cutoff is ${\eta }_c\simeq 2.39936$; for zero meridional wavenumber maximum growth occurs at $k{L}_D\simeq 1.60612$, with rate $0.30982\Lambda f_0/N$.