Statistical Theory of Turbulent Diffusion

In most applications of the Lagrangian formulas, the dependences of Oy and a on x are determined empirically rather than as indicated in (17.82). Thus the main purpose of the formulas in Table 17.1 is to provide a comparison between the two approaches to atmospheric diffusion theory. 

TABLE 18.1 Expressions for the Mean Concentration from a Continuous Point Source in an Infinite Fluid in Stationary, Homogeneous Turbulence 

Up to this point in this chapter we have developed the common theories of turbulent diffusion in a purely formal manner. We have done this so that the relationship of the approximate models for turbulent diffusion, such as the K theory and the Gaussian formulas, to the basic underlying theory is clearly evident. When such relationships are clear, the limitations inherent in each model can be appreciated. We have in a few cases applied the models obtained to the prediction of the mean concentration resulting from an instantaneous or continuous source in idealized stationary, homogeneous turbulence. In Section 18.7.1 we explore further the physical processes responsible for the dispersion of a puff or plume of material. Section 18.7.2 can be omitted on a first reading of this chapter that section goes more deeply into the statistical properties of atmospheric dispersion, such as the variances a (r), which are needed in the actual use of the Gaussian dispersion formulas. 

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The definition of X incorporates the fact that the Lagrangian integral time scale is of the order of Zilw,. The statistical theory of turbulent diffusion outlined in the beginning of Section VIII,B can be used to estimate the functional dependence of g as... 

The statistical theory of turbulent diffusion (Section VIII,B) predicts that the mean square displacement of a fluid particle in, say, the y direction manifests the following behavior ... 

This result is identical to (18.25) if b = TL 1, which provides a nice connection between the statistical theory of turbulent diffusion and the basic example considered earlier. 

In this connection it is important to be able to ascertain the smallest material ball attainable and to estimate the homogenization time, which is thereby required. Mixing or stirring power has to be expended to decrease the diffusion length or decrease the size of the segregated liquid balls. According to the statistical theory of turbulence due to Kolmogorov [143, 289], see Section 1.4.2, the size of the liquid balls can be estimated ... 

Taylor GI (1935) Statistical Theory of Turbulence, IV. Diffusion in a tirrbulent air stream. Proc Roy Soc London A151(874) 465-478... 

Mesoscopic statistical theories of turbulence laminar and turbulent transport thermal conductivity, diffusivity effective transport coefficients... 

S A study of turbulent diffusion of gas clouds over several terrains, Report OSRD No. 6185, 84 pages, by Harold Johnston, Robert Merrill, and Robert Mills. 1945. Part I. An Empirical Approach to the Effect of Turbulent Diffusion on Gas Clouds over Several Terrains, pages 1-23. Part II. A Critical Examination of the British Statistical Diffusion Theories, pages 24-46. 

In Moffat s review [1] an excellent account is given of the relation between the elementary approach and modern statistical methods in turbulence theory. In particular, by analogy with kinetic theory we easily derive for the diffusion coefficient of a scalar admixture the expression... 

The theory of mixing of a passive scalar concentration field subject to advection and diffusion in a high Reynolds number turbulent flow is based on the works of Obukhov (1949) and Corrsin (1951). Consider a statistically stationary state with a large-scale source of scalar fluctuations in the case when both Pe and Re are large. The... 

Dyson-type equations have been used extensively in quantum electrodynamics, quantum field theory, statistical mechanics, hydrodynamic instability and turbulent diffusion studies, and in investigations of electromagnetic wave propagation in a medium having a random refractive index (Tatarski, 1961). Also, this technique has recently been employed to study laser light scattering from a macromolecular solution in an electric field. 

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