= Rigid-lid constant-shear baroclinic model
{title2=$D(\eta)=1+\eta^2/4-\eta\coth\eta$}
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 $\Psi''-K^2\Psi=0$. 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 $\eta_c\simeq2.39936$; for zero meridional wavenumber maximum growth occurs at $kL_D\simeq1.60612$, with rate $0.30982\Lambda f_0/N$.
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