= Fastest growth in infinite-Prandtl convection
{title2=$\lambda_\infty=Rk^2/(k^2+\pi^2)^2-(k^2+\pi^2)$}
For a stress-free, fixed-temperature layer, the infinite-<Prandtl number> growth rate of the fundamental vertical <normal mode> is $\lambda_\infty=Rk^2/(k^2+\pi^2)^2-(k^2+\pi^2)$. For large <Rayleigh number>, its maximum has $k^2=\pi^2-8\pi^6/R+O(R^{-2})$ and $\lambda_{\rm max}=R/(4\pi^2)-2\pi^2+O(R^{-1})$. The infinite-<Prandtl number> limit must be taken first, or inertia must remain small on the resulting growth-time scale. This optimum differs from the <free-slip convection neutral curve> minimum, which selects onset rather than fastest growth far above it.
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