  In an important study, Maffioli et al. (J. Fluid Mech., Vol. 794 , 2016) used
a scaling analysis to predict that in the weakly stratified flow regime
$Fr_h\gg1$ ($Fr_h$ is the horizontal Froude number), the mixing coefficient
$\Gamma$ (defined as the ratio of the dissipation rates of potential to kinetic
energy) scales as $\Gamma\sim O(Fr_h^{-2})$. Direct numerical simulations
confirmed this result, and also indicated that for the strongly stratified
regime $Fr_h\ll 1$, $\Gamma\sim O(1)$. Furthermore, the study argued that
$\Gamma$ does not depend on the buoyancy Reynolds number $Re_b$, but only on
$Fr_h$. We present an asymptotic analysis to predict theoretically how $\Gamma$
should behave for $Fr_h\ll1$ and $Fr_h\gg1$ in the limit $Re_b\to\infty$. To
correctly handle the singular limit $Re_b\to\infty$ we perform the asymptotic
analysis on the filtered Boussinesq-Navier-Stokes equations, and demonstrate
the precise sense in which the inviscid scaling analysis of Billant \& Chomaz
(Phys. Fluids, vol. 13, 1645-1651, 2001) applies to viscous flows with
$Re_b\to\infty$. The analysis yields $\Gamma\sim O(Fr_h^{-2}(1+Fr_h^{-2}))$ for
$Fr_h\gg1$ and $\Gamma\sim O(1+Fr_h^{2})$ for $Fr_h\ll 1$, providing a
theoretical basis for the numerical observation made by Maffioli et al, as well
as predicting the sub-leading behavior. Our analysis also shows that the
Ozmidov scale $L_O$ does not describe the scale below which buoyancy forces are
sub-leading, which is instead given by $O(Fr_h^{1/2} L_O)$, and that the
condition for there to be an inertial sub-range when $Fr_h\ll 1$ is not
$Re_b\gg1$, but the more restrictive condition $Re_b\gg Fr_h^{-4/3}$.
