Rayleigh-Bénard Convection — Onset & Stability

A fluid layer heated from below undergoes convection when the Rayleigh number Ra = gαΔTd³/(νκ) exceeds Ra_c ≈ 1708. The marginal stability curve σ(k) = 0 separates stable diffusion from unstable convection rolls. The critical wavenumber k_c ≈ 3.12/d selects the preferred roll wavelength.

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Rayleigh-Bénard theory
Ra_c = 1707.76 (classical)
k_c = π/√2 ≈ 3.12
λ_c = 2π/k_c ≈ 2.016d
Growth rate: σ ~ (Ra−Ra_c)^β
Nusselt: Nu ~ Ra^(2/7) turbulent