Fundamental Equations of Turbulent Diffusion

The basis of the description of turbulent diffusion is the species conservation equation (Bird et al., 1960)  

We will generally not be concerned here with chemical reactions, so the Ri term can be omitted, in which case the subscript i denoting species is no longer required. In addition, for the turbulent flows of interest the molecular diffusion term in Eq. (2.1) may be neglected. [Although for the spatial scales of interest to us the molecular diffusion term may be neglected, molecular and turbulent diffusion are not independent, linearly additive, physical processes (Saffman, I960).] As a result of the above two simplifications, Eq. (2.1) becomes 

Equation (2.2) can be considered as the fundamental governing equation for the concentration of an inert constituent in a turbulent flow. Because the flow in the atmosphere is turbulent, the velocity vector u is a random function of location and time. Consequently, the concentration c is also a random fimction of location and time. Thus, the dispersion of a pollutant (or tracer) in the atmosphere essentiaUy involves the propagation of the species molecules through a random medium. Even if the strength and spatial distribution of the source 5 are assumed to be known precisely, the concentration of tracer resulting from that source is a random quantity. The instantaneous, random concentration, c(x, y, z, t), of an inert tracer in a turbulent fluid with random velocity field u( c, y, z, t) resulting from a source distribution S x, y, z, t) is described by Eq. (2.2). 

What is really desired, of course, is not the random concentration resulting from one realization of the flow, but the expected (mean) value resulting from an entire ensemble of flows with identical macroscopic conditions. Letting (c(x, y, z, /)) = E c x, y, z, r) and then taking the expected value of Eq. (2.3) leads to 

Equation (2.5) can be interpreted physically as follows. E G is just the probability density p that a particle released at location (x , y, z ) at time / will be at location (jt, y, z) at time t, the transition probability density, 

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