Tensor Diffusivity of a Trace Constituent in a Stratified Boundary Layer

Burton E. Freeman Science Applications, Inc., La Jolla, Calif. 92031

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Abstract

Diffusion coefficients applicable to the calculation of the dispersion of an inert trace constituent in the atmospheric boundary layer are derived from a closure model for atmospheric turbulence. While the atmosphere is assumed to be horizontally homogeneous, the trace constituent is taken to be distributed in an arbitrary three-dimensional distribution, such as encountered in a plume or puff.

The closure assumptions and equations for turbulent variances of Mellor and Yamada are supplemented by equations for the concentration of a trace constituent and its covariances with velocity and temperature. These equations are solved in an approximation which leads to Fickian diffusion theory and results in algebraic, rather than differential, equations. In view of the fact that the vertical shear and turning of the wind field are retained, there are mechanisms for diffusion in directions oblique to the concentration gradient. All of the terms of the tensor diffusivity are nonzero in this approximation. They depend in a simple way on the components of the vertical derivative of horizontal wind as required by tensor convariance. In addition, they depend on the turbulent kinetic energy, the magnitude of the vertical shear of the wind and on the gradient Richardson number. A critical value of the Richardson number exists above which the diffusivity vanishes.

Values of the tensor diffusivity are calculated for typical conditions in the atmospheric boundary layer. The diagonal components display the qualitatively expected dependence on atmospheric stability. In addition, the off-diagonal terms are evaluated; they are found to be comparable in magnitude to the diagonal terms.

Abstract

Diffusion coefficients applicable to the calculation of the dispersion of an inert trace constituent in the atmospheric boundary layer are derived from a closure model for atmospheric turbulence. While the atmosphere is assumed to be horizontally homogeneous, the trace constituent is taken to be distributed in an arbitrary three-dimensional distribution, such as encountered in a plume or puff.

The closure assumptions and equations for turbulent variances of Mellor and Yamada are supplemented by equations for the concentration of a trace constituent and its covariances with velocity and temperature. These equations are solved in an approximation which leads to Fickian diffusion theory and results in algebraic, rather than differential, equations. In view of the fact that the vertical shear and turning of the wind field are retained, there are mechanisms for diffusion in directions oblique to the concentration gradient. All of the terms of the tensor diffusivity are nonzero in this approximation. They depend in a simple way on the components of the vertical derivative of horizontal wind as required by tensor convariance. In addition, they depend on the turbulent kinetic energy, the magnitude of the vertical shear of the wind and on the gradient Richardson number. A critical value of the Richardson number exists above which the diffusivity vanishes.

Values of the tensor diffusivity are calculated for typical conditions in the atmospheric boundary layer. The diagonal components display the qualitatively expected dependence on atmospheric stability. In addition, the off-diagonal terms are evaluated; they are found to be comparable in magnitude to the diagonal terms.

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