= Finite-depth Eady dispersion relation
{title2=$\left(c-\Lambda D/2\right)^2=(\Lambda/m)^2[(mD)^2/4-(mD)\coth(mD)+1]$}
For constant shear $U=\Lambda z$ between impermeable horizontal walls at $0,D$, with constant <buoyancy frequency> and no interior <potential-vorticity gradient>, define $m=NK/|f_0|$. The boundary conditions yield
$$
\left(c-\frac{\Lambda D}{2}\right)^2=\frac{\Lambda^2}{m^2}\left[\frac{(mD)^2}{4}-(mD)\coth(mD)+1\right].
$$
The bracket is negative for $0<mD<2.399\ldots$, giving <baroclinic instability> for nonzero zonal wavenumber. For large separation the boundary waves decouple; the lower-wave speed tends to $\Lambda/m$, matching the <dispersion relation of a semi-infinite Eady edge wave>.
Back to article page
