  We consider the time-dependence of a hierarchy of scaled $L^{2m}$-norms
$D_{m,\omega}$ and $D_{m,\theta}$ of the vorticity $\boldsymbol {\omega} =
\boldsymbol{\nabla} \times {\mathbf u}$ and the density gradient
$\boldsymbol{\nabla} \theta$, where $\theta=\log (\rho^*/\rho^*_0)$, in a
buoyancy-driven turbulent flow as simulated by \cite{LR2007}. $\rho^*({\mathbf
x},\,t) $ is the composition density of a mixture of two incompressible
miscible fluids with fluid densities $\rho^*_2 > \rho^*_1$ and $\rho^*_{0}$ is
a reference normalisation density. Using data from the publicly available Johns
Hopkins Turbulence Database we present evidence that the $L^{2}$-spatial
average of the density gradient $\boldsymbol{\nabla} \theta$ can reach
extremely large values, even in flows with low Atwood number $At = (\rho^*_{2}
- \rho^*_{1})/(\rho^*_{2} + \rho^*_{1}) = 0.05$, implying that very strong
mixing of the density field at small scales can arise in buoyancy-driven
turbulence. This large growth raises the possibility that the density gradient
$\boldsymbol{\nabla} \theta$ might blow up in a finite time.
