= Free-slip convection neutral curve
{title2=$R_0(k)=(k^2+\pi^2)^3/k^2$}
For nonrotating convection under the <Boussinesq approximation> with fixed-temperature impermeable <stress-free boundary conditions>, a marginal vertical mode $\sin(n\pi z)$ has neutral <Rayleigh number> $(k^2+n^2\pi^2)^3/k^2$. The minimum is at $n=1$, $k=\pi/\sqrt2$, and $R_c=27\pi^4/4$. These follow by eliminating the <temperature> from $(D^2-k^2)^2w=Rk^2\theta$ and $(D^2-k^2)\theta=-w$, then minimizing over $k$.
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