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The turbulent Schmidt number compares turbulent momentum and concentration diffusivities:The mean fluid has no vertical velocity. Taking $z$ positive upward and $W_s>0$, the settling flux is $-W_s\varphi$, while the given turbulent particle flux is $-{\kappa }_T{\varphi }_z$. Local particle-volume conservation therefore givesConsequently the evolution equation isFor constant settling speed this is ${\varphi }_t=W_s{\varphi }_z+{\partial }_z({\kappa }_T{\varphi }_z)$. 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 $\varphi (z,t)=\varphi (z+W_{s}t,0)$. The dilute suspension approximation neglects feedback on the mean flow and particle interactions; deposition or erosion must be imposed through the bed boundary flux.

The molecular Schmidt number is the ratio of kinematic viscosity to scalar diffusivity. A large value allows a passive scalar to develop structure below the viscous smoothing scale of the velocity field. It is distinct from the turbulent Schmidt number, which compares effective turbulent transport coefficients, and from the Schmidt number used for quantum entanglement.