= Solution
Write $q=\partial_z\bar\rho>0$ for the unstable, horizontally averaged <density gradient>. A turbulent displacement over a <mixing length> of order $W$ encounters a density contrast $\Delta\rho\sim Wq$. Balancing its <buoyancy> acceleration with the inertial acceleration gives
$$
\frac{u_*^2}{W}\sim g\frac{Wq}{\bar\rho},\qquad \boxed{u_*\sim W\sqrt{\frac{gq}{\bar\rho}}.}
$$
A <turbulent diffusivity> for density is therefore $D_T=K u_*W=KW^2\sqrt{gq/\bar\rho}$, absorbing dimensionless factors into $K>0$. The upward density flux is $J_\rho=\langle w'\rho'\rangle=-D_Tq$: turbulent exchange carries denser water downwards and lighter water upwards. Defining $F_\rho=-J_\rho$ as the positive downward flux yields the <buoyancy-driven turbulent density diffusion> law
$$
\boxed{F_\rho=KW^2\sqrt{\frac g{\bar\rho}}\left(\partial_z\bar\rho\right)^{3/2}.}
$$
This is a flux per unit horizontal area, with units of mass per area per time. The total density transport across a horizontal section is $W^2F_\rho$.
The assumptions are established high-<Reynolds number> <turbulence>, width-limited eddies, an instantaneous local balance between <buoyancy> and inertia, approximately constant $K$, negligible molecular diffusion, and zero mean vertical volume flux. The last condition follows from <incompressibility> and impermeable sidewalls in a closed tube. Density variations and the eddy statistics must also vary slowly enough for a local <mixing-length closure> to be meaningful.
Horizontal averaging of <mass conservation>, with upward $z$, gives $\bar\rho_t=-\partial_zJ_\rho=\partial_zF_\rho$. Differentiating once more supplies the requested evolution of the <density gradient>:
$$
\boxed{\partial_tq=\partial_z^2\left[KW^2\sqrt{\frac g{\bar\rho}}\,q^{3/2}\right].}
$$
Under the <Boussinesq approximation>, replace $\bar\rho$ in the inertial prefactor by the constant reference density $\rho_0$; retain the changing <density gradient> inside the flux.
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