Physics
Ra = gαΔT d³ / (νκ)
g: gravity, α: thermal expansion
ν: viscosity, κ: diffusivity
Critical: Ra_c = 1707.76
(free-slip) or 1708 (no-slip)
k_c d = π/√2 ≈ 2.22
Nu ≈ 0.149 Ra^0.281
(Nusselt number, heat flux)
Above onset: rolls → chaos
Below Ra_c ≈ 1708, heat flows by conduction — a linear temperature gradient. Above this critical value, the conduction state becomes linearly unstable and convection rolls spontaneously form. The wavelength selected is 2d√2 (d = fluid depth). Nusselt number measures enhanced heat transfer.