Thermal convection

Thermal convection is primarily controlled by the balance between buoyancy forces and the resisting effects of viscous deformation and thermal diffusion. This balance is commonly quantified by the Rayleigh number, a dimensionless measure of the relative importance of buoyancy forces compared to viscous resistance and thermal diffusion. Larger Rayleigh numbers correspond to stronger buoyancy forces relative to viscous and diffusive effects and therefore promote increasingly vigorous convective flow.

For convection driven by an imposed temperature difference between the lower and upper boundaries, the appropriate control parameter is the basal Rayleigh number

\[\begin{equation} Ra_b = \frac{\rho_0 g \alpha \Delta T H^3}{\eta \kappa}, \end{equation}\]

where $\rho_0$ is the reference density, $g$ the gravitational acceleration, $\alpha$ the coefficient of thermal expansion, $\Delta T$ the imposed temperature difference across a layer of thickness $H$, $\eta$ the dynamic viscosity, and $\kappa$ the thermal diffusivity.

If buoyancy is instead generated by uniform volumetric heat production, the relevant control parameter is the internal-heating Rayleigh number

\[\begin{equation} Ra_Q = \frac{\rho_0 g \alpha Q H^5}{k \eta \kappa}, \end{equation}\]

where $Q$ denotes the volumetric heat-production rate and $k$ is the thermal conductivity.

Mixed-heated convection combines both buoyancy sources and is therefore characterized by both $Ra_b$ and $Ra_Q$. Throughout the following examples, the appropriate Rayleigh number(s) are reported for each heating configuration.