  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
$\overline{\mathbf{J}\cdot\mathbf{\nabla}b_{\rm
s}}=-\kappa\langle|\boldsymbol{\nabla}b|^2\rangle$; 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
$\kappa\langle|\boldsymbol{\nabla}b|^2\rangle$ justifies the definition of the
horizontal-convective Nusselt number, $Nu$, as the ratio of $\kappa
\langle|\boldsymbol{\nabla}b|^2\rangle$ 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 $\kappa
\langle|\boldsymbol{\nabla}b|^2\rangle$ equilibrates more rapidly than other
global averages, such as the domain averaged kinetic energy and bottom
buoyancy. We show that $\kappa\langle|\boldsymbol{\nabla} b|^2\rangle$ 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 $|\mathbf{\nabla}b|^2$
demanded by $\kappa\langle|\mathbf{\nabla}b|^2\rangle$.
