Rayleigh–Bénard convection

Consider a horizontal layer of depth d between plates whose temperatures differ by positive $\Delta T$, with the lower plate warmer. The motionless reference state has a linear temperature profile if material properties are constant and internal heating is absent. In the Boussinesq approximation, density is held constant in inertia and incompressibility, while its small temperature dependence is retained in buoyancy. Thermal expansion coefficient $\alpha_T$, kinematic viscosity $\nu$, and thermal diffusivity $\kappa$ are treated as constants.

A small upward velocity carries relatively warm fluid into a cooler environment. The fluid is therefore buoyant and can accelerate upward further. The feedback fails if temperature diffuses away too quickly or viscous stresses damp the motion strongly enough. Conversely, displacement of colder material downward can reinforce a descending branch. The closed circulation must also satisfy incompressibility and the plate conditions. This is why inspecting the sign of a local density anomaly is insufficient to determine the threshold of a whole layer.

Linearize momentum and the temperature equation about the conducting state. An upward velocity produces a positive temperature disturbance because the base temperature decreases with height. The temperature disturbance drives vertical momentum through buoyancy. Diffusion acts on both variables. Search over horizontal wavelengths for a nondecaying disturbance, while applying the mechanical and thermal conditions at both plates. The lowest Rayleigh number admitting one is the onset threshold in this ideal linear calculation.

For an infinite horizontal layer with two rigid no-slip plates at fixed temperatures, the classical minimum is $\mathrm{Ra}_c\approx1707.76$. If both boundaries instead are impermeable and stress-free, still at fixed temperatures, the ideal threshold is $27\pi^4/4\approx657.51$. The stress-free neutral curve below has a minimum at dimensionless horizontal wavenumber $a=\pi/\sqrt2$. Do not transplant either number to finite-aspect-ratio containers, mixed thermal boundaries, free surfaces with additional physics, rotation, or strongly temperature-dependent properties.

Just above onset, a small-amplitude analysis can describe patterns and their slow evolution, but linear growth cannot determine their eventual amplitude. Further above onset, plumes, boundary layers, and unsteady motion may appear. Heat transport is often summarized by the Nusselt number, the total vertical heat flux divided by the conducting flux for the same imposed temperature difference. A value above one indicates enhanced transport. A stability threshold does not, by itself, supply a law for that enhancement at arbitrary Ra.

An experimental comparison needs more than a thermometer difference. Check plate uniformity, layer depth, sidewall effects, material properties at an appropriate reference temperature, and whether density variations remain small enough for Boussinesq reasoning. Since depth enters cubically, an error in depth can dominate the inferred driving parameter. A negative observation is informative only after accounting for observation time: a mode very close to onset can grow slowly enough that a short experiment misses its early amplification.