{c}
{title2=$\widetilde\alpha=\alpha N^2/(f_0\Lambda)$}

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 is
$$
C^2-\left(1-a\frac{\coth M}{M}\right)C+(1-a)\left(\frac{\coth M}{M}-\frac1{M^2}\right)=0.
$$
For $a>1$ 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.


 Eady model with a sloping lower boundary

ID: eady-model-with-a-sloping-lower-boundary