Rayleigh-Benard Convection: Onset of Instability

A fluid layer heated from below is motionless until the Rayleigh number $Ra=g\alpha\Delta T d^3/(\nu\kappa)$ crosses a sharp threshold, then it breaks into counter-rotating convection rolls. For stress-free, perfectly conducting plates (the free-free case) the linear theory is exact: a normal mode $\propto\sin(\pi y)\,e^{ikx}$ is neutrally stable on the curve $Ra(k)=(k^2+\pi^2)^3/k^2$, minimised at $k_c=\pi/\sqrt2$, giving the closed-form critical value $Ra_c=\tfrac{27\pi^4}{4}\approx 657.51$. The top panel is the critical roll eigenmode growing ($\sigma\gt 0$, above the curve) or decaying ($\sigma\lt 0$, below it); the bottom panel is the neutral curve with the marked critical point and your operating point. The onset is independent of the Prandtl number (Chandrasekhar). The engine reproduces $Ra_c$ to better than 0.2% and the value converges with resolution (gate-tested).