= Buoyancy-driven turbulent density diffusion
{title2=$F_\rho=KW^2\sqrt{g/\rho_0}\,(\partial_z\bar\rho)^{3/2}$}
In a narrow tube with unstable mean density gradient $q=\partial_z\bar\rho>0$, width-limited turbulent eddies have velocity scale $u_*\sim W\sqrt{gq/\rho_0}$. A mixing-length diffusivity gives downward density flux $F_\rho=KW^2\sqrt{g/\rho_0}\,q^{3/2}$ per unit area. With upward coordinate $z$, conservation is $\bar\rho_t=\partial_zF_\rho$ and $q_t=KW^2\sqrt{g/\rho_0}\,\partial_z^2(q^{3/2})$. The closure presupposes established high-<Reynolds number> turbulence and requires boundary fluxes and finite available potential energy to be respected.
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