Consider an element of air containing a concentration of a passive
pollutant (passive means that it doesn't react and is neutrally buoyant, i.e.
it doesn't settle).

If the flow is also incompressible, then the volume of the fluid element
remains constant and so the concentration remains constant. In mathematical
terms

It will be sufficient for our purposes to take the air to be incompressible so that

Hence an alternative form of Eq. (1) is

Suppose now that there is a source or sink of the pollutant within the
element of air. This could be because the pollutant is created or destroyed
by chemical reactions or because there is an outflow from, say, a chimney.
The equation for would then become

where represents the source term (in kgms). Examples could be:

(i) (5)

This represents decay of by, for example, chemical decomposition or radioactive decay.

(ii) , (6)

where is a delta function at . This represents a continuous point source (e.g. emission from a chimney).

(iii) . (7)

This represents an instantaneous point source occurring at time at (e.g. accidental release of a radioactive substance).

Another type of source/sink term is caused by diffusion. The pollutant
may diffuse in or out of our element. The equation for is then

is the molecular diffusivity. has dimensions . Typical values for pollutants in the atmosphere are 5-50 ms. We are usually concerned with dispersion on the scale of hundreds of metres to hundreds of kilometres. Consider m. Then the time-scale is

which is infinite for all practical purposes. Hence

Atmospheric flows are almost always turbulent. Turbulence occurs when the Reynolds number is high. It is characterised by eddy motions on a wide range of scales. When describing the dispersion of pollutants, we are usually interested in dispersion on scales much larger than many, if not all, of the eddies. In other words, we are interested in averages over length or time scales large compared to the turbulence.

In order to analyse dispersion in this way, we assume that it is possible to
divide the flow into a ``mean'' flow which is slowly varying in time and a
rapidly fluctuating, or ``turbulent'' part. We could perform this separation
by defining an average as follows:

The average period should be chosen to be long compared to the turbulence time-scales. Then we put

represents the turbulent part of . It follows (to a reasonable approximation at least) that

Let us perform this separation for all the variables in the concentration
equation, Eq. (3).

Therefore

We now average this equation. Consider, for example

since . Similar results hold for and , giving

The right-hand side represents the average effect of turbulent eddies on the concentration. Molecular diffusion is caused by the random motion of molecules, whereas the effect here is caused by the random eddy motions. By

, and are analogous to the molecular diffusivity . They must be measured experimentally. They differ from in that (i) , and need not be equal, (ii) In general, , and are not constant, (iii) .

Using Eq. (13),

This equation is the basis of much of the modelling of atmospheric dispersion.