Trajectory end point distribution of a test particle in the atmosphere

The classic meteorological law of diffusion in the atmosphere was given experimentally, by Richardson in 1926, whose result that the mean squared distance <R^2>=cT^3, the time cubed, is in accord with the scaling theory of Komogorov [ Obukhov (1941)]. In some cases it might be important to have more information than that provided by Richardson's law. An example would be the distribution of pollutants in time by turbulent flow. Here small amounts of material reaching relatively large distances are of importance. This motivates our interest in the full distribution of the location of particles swept by the fluid as a function of time. The distribution depends on the distance through the dimensionless quantity X^2=R^2/<R^2(T)> . Using the Kolmogorov picture, we find that for small X, the distribution is proportional to exp(-aX^2) and exp(-bX^4/3) at its tail when X is large.