Nonlinear effects in buoyancy-driven variable density turbulence

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.