= Instability for negative slopes in the sloping-boundary Eady model
The discriminant of the <Eady model with a sloping lower boundary> is
$$
\Delta=\left[1-(2-a)\frac{\coth M}{M}\right]^2-4(1-a)\frac{\operatorname{csch}^2M}{M^2}.
$$
For 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.
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