The Nusselt numbers of horizontal convection

In the problem of horizontal convection a non-uniform buoyancy, , is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, , defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that ; the overbar denotes a space-Time average over the top surface, angle brackets denote a volume-Time average and is the molecular diffusivity of buoyancyÂ. This connection between and justifies the definition of the horizontal-convective Nusselt number, , as the ratio of to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of over other definitions of horizontal-convective Nusselt number. We investigate transient effects and show that equilibrates more rapidly than other global averages, such as the averaged kinetic energy and bottom buoyancy. We show that is the volume-Averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux through the top surface. This leads to an equivalent 'surface Nusselt number', defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy. In experimental situations it is easier to evaluate the surface entropy flux, rather than the volume integral of demanded by.