Instability for negative slopes in the sloping-boundary Eady model

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.