Fastest growth in infinite-Prandtl convection (${\lambda }_{\infty }=R{k}^2/(k^2+{\pi }^2{)}^2-(k^2+{\pi }^2)$)

For a stress-free, fixed-temperature layer, the infinite-Prandtl number growth rate of the fundamental vertical normal mode is ${\lambda }_{\infty }=R{k}^2/(k^2+{\pi }^2{)}^2-(k^2+{\pi }^2)$. For large Rayleigh number, its maximum has $k^2={\pi }^2-8{\pi }^6/R+O(R^{-2})$ and ${\lambda }_{\max }=R/(4{\pi }^2)-2{\pi }^2+O(R^{-1})$. 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.