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).

Figure 1. Top: the critical Rayleigh-Benard roll eigenmode

\theta\propto\sin(\pi y)\cos(k x)

, its amplitude evolving as

e^{\sigma t}

(growing above

Ra_c

, decaying below). Bottom: the free-free neutral curve

Ra(k)=(k^2+\pi^2)^3/k^2

with the exact critical point

(k_c,\,27\pi^4/4)

and the live operating point. Method: closed-form linear stability of the free-free Boussinesq layer; the discrete critical Rayleigh number and growth-rate sign come from the gate-tested shared engine, converging to

27\pi^4/4

with resolution.

Ra (units of Ra_c)2.00 Ra_c

wavenumber k2.22

Prandtl Pr1.0

WHAT TO TRY

Push Ra above the critical value Ra_c: the motionless conducting layer breaks into counter-rotating rolls and the state readout flips to convecting. Below Ra_c any perturbation just decays.
Tune the wavenumber k: the growth rate sigma is largest near k = pi/sqrt(2), the most unstable mode that sets the natural roll width. The marginal-stability curve marks where sigma crosses zero.
Change the Prandtl number Pr: it reshapes how momentum and heat diffuse, shifting the roll structure while the onset threshold Ra_c stays put for these free-free boundaries.