Eady model with a sloping lower boundary ($\tilde{\alpha }=\alpha N^2/(f_0\Lambda )$)

A gently sloping lower boundary modifies the lower Boundary Rossby wave of the Eady model, while a horizontal upper boundary retains the usual upper wave. With $M=N|k|D/|f_0|$, $a=\alpha N^2/(f_0\Lambda )$ and $C=c/(\Lambda D)$, the two-boundary dispersion relation isFor $a>1$ both speeds are real for every nonzero wavenumber. For every $a<1$ there is a band of exponentially growing modes; this includes negative slopes. The case $a=1$ has no exponential instability but can have a defective neutral mode at the crossing of the two uncoupled speeds.

The discriminant of the Eady model with a sloping lower boundary isFor any $a<1$, choose the unique $M>0$ satisfying $M\tanh M=2-a$. Then $\Delta <0$, proving existence of an unstable band even when $a<0$. For large negative $a$, resonance occurs at large $M$ and the exponentially weak boundary-wave coupling makes the unstable band narrow.