Turbulent Diffusion in Three Dimensions

Abstract

In this chapter, we review, and in some cases expand, the material in Thomson (1987) on the probability density function (PDF) for turbulent flow, the Langevin and Fokker-Planck equations for stochastic models of turbulent diffusion, the well-mixed criterion, and solutions for the “drift” and “diffusion” coefficients in the stochastic differential equations. He began with general forms of the Lagrangian and Eulerian stochastic differential equations [Eqs. (4) and (5), respectively, in his text] and ended with the “simplest” of the possible solutions for the nonstationary, three-dimensional Langevin equation [see his Eqs. (24), (32), and the line following (32)]. There are few, if any, hints as to how his Eq. (32) was derived. The main purpose of this chapter is to follow the mathematical processes that lead from Eq. (4) toward Eq. (32), based on suggestions from Brian Sawford (1992, unpublished lecture; 1993, private communications). Our procedure is similar to that in section 7.2 for the one-dimensional case. In addition, we derive the three-dimensional random displacement model.