Rayleigh number ($\mathrm{Ra}$)

The Rayleigh number compares buoyancy driving with viscous and thermal diffusion. For a layer of thickness $h$, temperature drop $\Delta T$, density $\rho$, thermal expansion coefficient ${\alpha }_T$, gravitational acceleration $g$, thermal diffusivity $\kappa$, and dynamic viscosity $\mu$,Thermal convection begins when this exceeds a geometry-dependent critical value.

A local Rayleigh-number closure estimates the thickness $\delta$ of a turbulent thermal boundary layer by settingwhere ${\mathrm{Ra}}_c$ is an order-one or geometry-dependent critical value. Combining this with Fourier's law converts the boundary-layer temperature and density differences into a heat flux law.

For constant viscosity and thermal properties, marginal local Rayleigh number gives $\rho g\alpha \theta {\delta }^3/(\eta \kappa )\sim R_c$. Conduction through the resulting thermal boundary layer gives $E\sim k\theta /\delta \propto {\theta }^{4/3}$. This determines exponents, not an exact prefactor. The closure applies when the layer is thin compared with the convecting interior and its temperature contrast is positive.

Penetrative convection occurs when an unstable convecting region extends into, or is capped by, a stably stratified region. Water near its temperature of maximum density provides a common example: a cold stable layer can lie above a warmer turbulent convecting layer.

Fresh water reaches maximum mass density near $4{}^{\circ }\mathrm{C}$ at atmospheric pressure. A useful local equation of state isso water becomes less dense when its temperature moves either above or below $T_m$.

In an ice-covered freshwater lake, a stagnant layer colder than the density-maximum temperature can cap a warmer turbulent region. Maximizing the boundary-layer heat flux obtained from a Local Rayleigh-number closure gives an interface temperature $T_h$ satisfyingand a maximum upward heat flux proportional to $(T_w-T_m{)}^{5/3}$.