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Generalization of K Theory for Turbulent Diffusion. Part I: Spectral Turbulent Diffusivity Concept

Ruwim BerkowiczDanish Meteorological Institute, Air Pollution Section, Lyngbyvej 100, DK-2100 Copenhagen, Denmark

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Lars P. PrahmDanish Air Pollution Laboratory, National Agency of Environmental Protection, Risø National Laboratory, DK-4000 Roskilde, Denmark

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Abstract

The gradient transfer theory for turbulent diffusion is reformulated in order to obtain an improved method for applied dispersion studies. The basic innovation is that diffusivity of single Fourier components of the concentration field is treated separately, i.e., spectral turbulent diffusivity coefficients are introduced. The value of the diffusivity decreases with increasing wave vector k of the concentration spectrum. The rate of growth of an expanding cloud of material thus becomes dependent on the stage of growth. This is in qualitative agreement with the statistical dispersion theory. It is shown that the assumption of k-dependent diffusivity leads to a nonlocal flux-gradient relation. A new function, the turbulent diffusivity transfer function, is introduced. The turbulent diffusive flux depends on concentration gradients at all points in the space. The diffusion equation is written in terms of the turbulent diffusivity transfer function. The width of the turbulent diffusivity transfer function is shown to determine the validity of the traditional gradient transfer theory formulation. The turbulent dispersion can be considered as Gaussian when the size of the cloud is considerably larger than the size of the most energetic turbulent eddies. These eddies determine the width of the turbulent diffusivity transfer function.

In general, it is shown that the shape of the cloud is non-Gaussian and the width, computed in terms of spectral turbulent diffusivity coefficients, is smaller than in a Gaussian distribution. This deviation decreases with increasing size of the cloud.

The present theory reveals properties in agreement with experiments and Lagrangian statistical dispersion theory and has the advantage of being an Eulerian method which can be used for air pollution dispersion models treating multiple, interacting sources.

Abstract

The gradient transfer theory for turbulent diffusion is reformulated in order to obtain an improved method for applied dispersion studies. The basic innovation is that diffusivity of single Fourier components of the concentration field is treated separately, i.e., spectral turbulent diffusivity coefficients are introduced. The value of the diffusivity decreases with increasing wave vector k of the concentration spectrum. The rate of growth of an expanding cloud of material thus becomes dependent on the stage of growth. This is in qualitative agreement with the statistical dispersion theory. It is shown that the assumption of k-dependent diffusivity leads to a nonlocal flux-gradient relation. A new function, the turbulent diffusivity transfer function, is introduced. The turbulent diffusive flux depends on concentration gradients at all points in the space. The diffusion equation is written in terms of the turbulent diffusivity transfer function. The width of the turbulent diffusivity transfer function is shown to determine the validity of the traditional gradient transfer theory formulation. The turbulent dispersion can be considered as Gaussian when the size of the cloud is considerably larger than the size of the most energetic turbulent eddies. These eddies determine the width of the turbulent diffusivity transfer function.

In general, it is shown that the shape of the cloud is non-Gaussian and the width, computed in terms of spectral turbulent diffusivity coefficients, is smaller than in a Gaussian distribution. This deviation decreases with increasing size of the cloud.

The present theory reveals properties in agreement with experiments and Lagrangian statistical dispersion theory and has the advantage of being an Eulerian method which can be used for air pollution dispersion models treating multiple, interacting sources.

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