a Statistical Study of Turbulent Diffusion.

The dispersion of particles in a turbulent fluid flow is studied by means of the transition probability distribution functions. Applying the propagator method, we derive the evolution equations of the transition functions in the form of a series of correlations. Based upon the discussion of the behavior of the series, a low order truncation is made to close the equations. The eddy transport properties are found as integral operators that contain the memory effect. The formal transport equations are used to investigate the dispersion of fluid particles. The transition function of a single particle and the relative transition function of a pair of particles are found to obey non-linear integro -differential equations. The general properties of these equations agree with observations. In comparison with other theories, our results yield better short time behavior for the one-particle dispersion, and the equation for the relative dispersion of a pair of particles seems more reasonable and more convenient for use in analysis. Based upon the Kolmogoroff spectrum, an analytical proof is given for the empirical 4/3 power law of relative diffusion. A numerical calculation based upon the von Karman spectrum agrees with the analysis. The diffusion of a puff of a passive quantity is studied by means of the dispersion of particles. For the absolute diffusion, the eddy diffusivity is given in tensorial form to account for anisotropy and the mean velocity gradient. With a locally isotropic approximation, the relative diffusion undergoes different stages because of the changes in the relative importance of shear and buoyancy turbulence. A prediction based on the theory and upon an empirical spectrum estimated from turbulent wind data is compared with observations of diffusion made under conditions similar to those when the wind data were obtained.