= Solution
The <turbulent Schmidt number> compares turbulent momentum and concentration diffusivities:
$$
\boxed{\mathrm{Sc}_T=\frac{\nu_T}{\kappa_T}=2.}
$$
The mean fluid has no vertical <velocity>. Taking $z$ positive upward and $W_s>0$, the settling flux is $-W_s\phi$, while the given turbulent particle flux is $-\kappa_T\phi_z$. Local particle-volume conservation therefore gives
$$
\phi_t+\partial_zJ=0,\qquad J=-W_s\phi-\kappa_T\phi_z.
$$
Consequently the evolution equation is
$$
\boxed{\phi_t=\partial_z\left(W_s\phi+\kappa_T\phi_z\right).}
$$
For constant settling speed this is $\phi_t=W_s\phi_z+\partial_z(\kappa_T\phi_z)$. In the logarithmic region, $\kappa_T=Cz/2$ from the preceding part. The sign is consistent with downward settling: without diffusion, a profile evolves as $\phi(z,t)=\phi(z+W_st,0)$. The dilute suspension approximation neglects feedback on the mean flow and particle interactions; deposition or erosion must be imposed through the bed boundary flux.
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