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Layer and forcing
Raise r = Ra/Ra_c from below 1 (still fluid) through steady rolls to chaos.
Layers
Model checks
The science
The model
Expanding the stream function and temperature of a 2D, free-slip Boussinesq layer in the lowest three Fourier modes (Saltzman 1962, Lorenz 1963) gives dX/dτ = σ(Y - X), dY/dτ = rX - Y - XZ, dZ/dτ = XY - bZ with b = 4/(1+a²). Here r = Ra/Rac, σ is the Prandtl number and τ is time in units of H²/(π²(1+a²)κ). The picture is rebuilt from the modes: ψ ∝ X sin(πax/H) sin(πz/H) and the temperature departure from the linear profile ∝ Y cos(πax/H) sin(πz/H) - Z sin(2πz/H).
Onset and chaos
For r < 1 the fluid stays at rest and heat only conducts. At r = 1 steady rolls appear with X = ±√(b(r-1)), Z = r-1. The steady rolls lose stability at rH = σ(σ+b+3)/(σ-b-1) (24.74 for σ = 10, b = 8/3) and the flow becomes chaotic: the roll direction flips irregularly (the Lorenz butterfly in the X-Z plane). The critical Rayleigh number of this free-slip model is Rac = π&sup4;(1+a²)³/a², minimum 27π&sup4;/4 = 657.5 at a² = 1/2. (For rigid plates the textbook value is 1708.)
Honest simplifications: three modes only, so real turbulent convection (plumes, boundary layers, higher Nusselt numbers at large Ra) is not captured; the Nusselt proxy 1+2Z/r is the model's own heat-transport estimate; fluid properties enter only through σ, a and the time scale (water diffusivity 1.43×10-7 m²/s).
What to try
- "Conduction": the roll amplitude decays to zero and the colours stay in horizontal layers.
- "Steady rolls": X settles on the fixed point marked in the phase plane.
- "Chaos at the classic r = 28": the trajectory loops around both fixed points and the roll direction reverses unpredictably.
- Set sigma = 2 to see how a low-Prandtl fluid behaves differently.