A thin fluid layer is held hot at the bottom and cold at the top. When the temperature difference is small, heat simply diffuses upward by conduction and nothing moves. Once the Rayleigh number exceeds the critical value Rac ≈ 1708, buoyancy overwhelms viscous damping and the fluid spontaneously organises into rotating convection rolls that carry heat far more efficiently.
Ra = g·β·ΔT·H³ / (ν·κ) — Rayleigh number
∂T/∂t + (u·∇)T = κ ∇²T — temperature advection-diffusion
∂ω/∂t + (u·∇)ω = ν ∇²ω + g·β·∂T/∂x — vorticity with buoyancy
∇²ψ = -ω, u = ∂ψ/∂y, v = -∂ψ/∂x — stream function
The hexagonal and roll patterns you see here also paint the surface of the Sun: each bright "granule" is the top of a convection cell roughly the size of a continent, lasting only a few minutes before sinking back down.
Rayleigh-Bénard convection is the buoyancy-driven motion that develops in a thin horizontal fluid layer heated uniformly from below and cooled from above. When the heating is strong enough, the still fluid becomes unstable and breaks into regular, rotating convection cells called rolls.
The Rayleigh number Ra is a dimensionless ratio comparing the strength of buoyancy (which drives convection) to the damping effects of viscosity and thermal diffusion. Ra = (g·β·ΔT·H³)/(ν·κ). Higher Ra means buoyancy dominates and convection is more vigorous.
For a fluid layer between two rigid, no-slip plates, linear stability theory predicts that convection first sets in at Ra_c ≈ 1708. Below this value, heat is carried only by conduction and the fluid stays still; above it, infinitesimal disturbances grow into convection rolls.
The Boussinesq approximation treats the fluid density as constant everywhere except in the buoyancy (gravity) term, where small temperature-induced density changes are kept. This greatly simplifies the equations while capturing the essential physics of thermal convection.
Among all possible disturbances, the one with a particular horizontal wavelength (about twice the layer depth) grows fastest. This mode wins the competition and organises the flow into evenly spaced counter-rotating rolls, the lowest-energy way to transport the imposed heat.
Conduction moves heat molecule by molecule, slowly. Convection physically carries hot parcels of fluid upward and cold parcels downward, advecting thermal energy in bulk. Above Ra_c the convective heat flux quickly exceeds the conductive flux, measured by the Nusselt number.
The Nusselt number Nu is the ratio of total heat transport to the heat that pure conduction alone would carry. Nu = 1 means no convection; Nu > 1 means convection is enhancing heat transfer. It rises with the Rayleigh number.
It appears in the Sun's surface granulation, in the Earth's mantle and outer core, in the atmosphere as cloud streets and hexagonal cells, in the oceans, and even in a pot of heated soup or oil. It is one of the most studied examples of pattern formation.
The Prandtl number Pr = ν/κ compares momentum diffusion to thermal diffusion. Low Pr fluids (like liquid metals) develop large-scale, inertia-dominated flows, while high Pr fluids (like oils) form smoother, more viscous rolls. It shapes the texture of the convection.
It solves a 2D Boussinesq model on a lattice grid: temperature is advected and diffused, the temperature difference produces a buoyancy force, and a stream-function / vorticity update enforces an incompressible velocity field. Colour shows temperature and arrows show velocity.