Buoyancy-induced turbulent mixing in a narrow tilted tank

We describe a series of experiments in which a constant buoyancy flux $B_s$ of dyed salty water of density $\rho_s$ is introduced at the top of a long narrow tank of square cross-section tilted at an angle $\theta$ from the vertical. The tank is initially filled with fresh clear water of density $\rho_0<\rho_s$, and we investigate the resulting buoyancy-driven high Reynolds number turbulent mixing at various tilt angles $\theta$ using a light-attenuation method. When $\theta>0^{\circ}$, the ensemble averaged reduced gravity develops a statically stable gradient normal to the walls of the tank, and this induces a counterflow. We model the evolution of the cross-tank and ensemble averaged reduced gravity $\left<\overline{g'}\right>_e$ as a diffusive process using Prandtl's mixing length theory, building on the model of van Sommeren \textit{et al.} (\textit{JFM} \textbf{701}, 2012) who considered vertical tanks. We show that the counterflow acts to enhance the effective along-tank turbulent diffusivity, and from experiments, we find that the mixing length increases approximately linearly with $\theta$, and that both the along-tank and cross-tank turbulent diffusivities are proportional to $\left(\partial\left<\overline{g'}\right>_e/\partial z \right)^{1/2}$.