 Buoyancy-driven turbulent density diffusion

In a narrow tube with unstable mean density gradient $q={\partial }_z\bar{\rho }>0$, width-limited turbulent eddies have velocity scale $u_{\ast }\sim W\sqrt{gq/{\rho }_0}$. A mixing-length diffusivity gives downward density flux $F_{\rho }=K{W}^2\sqrt{g/{\rho }_0}{q}^{3/2}$ per unit area. With upward coordinate $z$, conservation is ${\bar{\rho }}_t={\partial }_z{F}_{\rho }$ and $q_t=K{W}^2\sqrt{g/{\rho }_0}{\partial }_z^2(q^{3/2})$. The closure presupposes established high-Reynolds number turbulence and requires boundary fluxes and finite available potential energy to be respected.