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We consider the problem of horizontal convection in which non-uniform buoyancy, $b_{rm s}(x,y)$, is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $mathbf{J}$, defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $overline{mathbf{J}cdotmathbf{ abla}b_{rm s}}=-kappalangle|boldsymbol{ abla}b|^2rangle$; overbar denotes a space-time average over the top surface, angle brackets denote a volume-time average and $kappa$ is the molecular diffusivity of buoyancy $b$. This connection between $mathbf{J}$ and $kappalangle|boldsymbol{ abla}b|^2rangle$ justifies the definition of the horizontal-convective Nusselt number, $Nu$, as the ratio of $kappa langle|boldsymbol{ abla}b|^2rangle$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontal-convective Nusselt number currently in use. We investigate transient effects and show that $kappa langle|boldsymbol{ abla}b|^2rangle$ equilibrates more rapidly than other global averages, such as the domain averaged kinetic energy and bottom buoyancy. We show that $kappalangle|boldsymbol{ abla} b|^2rangle$ is essentially the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux of entropy 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 $b_{rm s}(x,y)$. In experiments it is likely easier to evaluate the surface entropy flux, rather than the volume integral of $|mathbf{ abla}b|^2$ demanded by $kappalangle|mathbf{ abla}b|^2rangle$.
Three dimensional roll-type double-diffusive convection in a horizontally infinite layer of an uncompressible liquid is considered in the neighborhood of Hopf bifurcation points. A system of amplitude equations for the variations of convective rolls
Multi-fluid models have recently been proposed as an approach to improving the representation of convection in weather and climate models. This is an attractive framework as it is fundamentally dynamical, removing some of the assumptions of mass-flux
Simulations of strongly stratified turbulence often exhibit coherent large-scale structures called vertically sheared horizontal flows (VSHFs). VSHFs emerge in both two-dimensional (2D) and three-dimensional (3D) stratified turbulence with similar ve
In a range of physical systems, the first instability in Rayleigh-Bernard convection between nearly thermally insulating horizontal plates is large scale. This holds for thermal convection of fluids saturating porous media. Large-scale thermal convec
We analyse the nonlinear dynamics of the large scale flow in Rayleigh-Benard convection in a two-dimensional, rectangular geometry of aspect ratio $Gamma$. We impose periodic and free-slip boundary conditions in the streamwise and spanwise directions