where Mi is the molecular weight, g is gravitational acceleration, T is atmospheric temperature, and R is the universal gas constant. We see that when the molecular density structure follows that given by the stratification structure we have zero Brownian diffusive flux.
Similarly, the atmospheric scale height when we have complete mixing is given by
1 Mg 1 dT , where M is the atmospheric mean molecular weight and dn n
We can now express the Brownian diffusive flux in terms of the mixing ratio fi = n-/n by
For turbulent or eddy diffusion we need to correct the diffusive flux to remove the apparent flux of molecules i that arises from the stratification in the total number density dz \ dz ^
where the second term is the fraction of the apparent total turbulent flux that corresponds to molecules i. For a well-mixed atmosphere we can now write fc)'"* <™>
The eddy diffusive flux then reduces to
The total diffusive flux can then be written in terms of the mixing ratio fi as df-
where the buoyancy term f3i is given by
and we have also introduced a correction factor 7i. In the lower atmosphere, where eddy diffusion maintains a well-mixed atmosphere, the buoyancy term needs to be corrected as the stratification of light species is destroyed by turbulent mixing with the result that the scale height H* needs to be replaced by H. Thus, the correction factor y is equal to zero near the Earth's surface increasing to 1 in the upper atmosphere. A possible variation of 7 for the present atmosphere is
where the altitude z is in km so that by 100 km y = 1.
Another component of diffusion is thermal diffusion with the flux given by cf>iT = (7.33)
where ai is the thermal diffusion factor. When this is included we get
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