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An adverse radial specific entropy gradient can drive an oscillatory instability through the finite lag supplied by thermal conduction or relaxation. This differs from a viscous-convective instability, in which viscosity changes rotational stabilization.

Let $ϵ=\Omega /(\nu k^2)\ll 1$ and $s=ϵ\Omega s_1+\cdots$. Since $D=\Omega /ϵ+O(ϵ\Omega )$, the leading terms ofgiveThe 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.