Advection, Turbulent Flux, and Molecular Diffusion

Let us consider a fluid in which a tracer i is mixed. A flux of the tracer within the fluid can be brought 

If variations in fluid density and diffusivity can be neglected, we have  

In most situations, a fluid would be turbulent, implying that the velocity vector, as well as the concentration c exhibits considerable variability on time-scales smaller than those of prime interest. This situation can be described by writing these quantities as the sum of an average quantity (normally a time average) and a perturbation  

From equation (32), the transport flux Fn then becomes  

Note that the averages of V and c are equal to zero. The continuity equation can now be written as  

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