We consider the time dependence of a hierarchy of scaled L²ᵐ-norms D_m,ω and D_m,θ of the vorticity ω =∇ x u and the density gradient ∇θ, where θ = log.(ρ*/ ρ*₀), in a buoyancy-driven turbulent flow as simulated by Livescu & Ristorcelli (J. Fluid Mech., vol. 591, 2007, pp. 43–71). Here, ρ* (x,t) is the composition density of a mixture of two incompressible miscible fluids with fluid densities ρ*₂ > ρ*₁, and ρ*₀ is a reference normalization density. Using data from the publicly available Johns Hopkins turbulence database, we present evidence that the L²-spatial average of the density gradient can reach extremely large values at intermediate times, even in flows with low Atwood number At = (ρ*₂ - ρ*₁)/(ρ*₂ + ρ*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 ∇θ might blow up in a finite time.We acknowledge, with thanks, the staff of IPAM UCLA where this collaboration began in the Autumn of 2014 on the programme ‘Mathematics of Turbulence’. We would also like to thank C. Doering and D. Livescu for useful discussions. All of the numerical data used are from the JHTDB (Livescu et al. 2014), a publicly available DNS database. For more information, please see http://turbulence.pha.jhu.edu/. We also thank the referees for suggesting substantial improvements
