Let $\epsilon=\Omega/(\nu k^2)\ll1$ and $s=\epsilon\Omega s_1+\cdots$. Since $D=\Omega/\epsilon+O(\epsilon\Omega)$, the leading terms of
$$
s(D^2+\varpi^2)+n^2\varpi^2D=0
$$
give
$$
\boxed{s_1=-n^2\frac{k_z^2}{k^2}},
\qquad
\boxed{s=-\frac{N^2k_z^2}{\nu k^4}+\cdots}.
$$
The mode is unstable for every $\boxed{N^2<0}$, a weaker criterion than the inviscid condition. Strong viscosity damps the velocity response and removes the rapid epicyclic restoration, permitting a slow <viscous-convective instability>; its <growth rate> nevertheless tends to zero on very small scales.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 3 / f / Solution

Viscous-convective instability is slow growth in a rotating fluid with an adverse buoyancy gradient when viscosity weakens epicyclic stabilization.


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