The Linear Mean Gradient Model for Two-Particle Turbulent Diffusion

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  • 1 National Center for Atmospheric Research, Boulder Colo. 80303
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Abstract

The linear mean gradient model (or K-theory closure) is applied to the diffusion of a pair of particles passively advected by an inhomogeneous, nonstationary ensemble of velocity fields. The central result is a differential equation satisfied by a model joint probability density function of the particle positions, approximating that of the real ensemble. Phenomenologically determined coefficients appear in the equation. One of these is the one-particle effective eddy diffusivity tensor while the other is a tensor quantity representative of the correlation between the motions of the particles. Conditions an the tensor coefficients are found to ensure that the one-particle model is properly embedded in the two-particle model. The separation into a center-of-mass diffusion model and a relative diffusion model is accomplished. Corresponding diffusivity tensors are derived and specialized to the case of homogeneous turbulence. An application to the problem of showing the initial appearance of small-scale structures resulting from turbulent mixing is presented. A method is suggested for determining the required diffusivity tensors from observations of the statistical properties of tracer trajectories.

Abstract

The linear mean gradient model (or K-theory closure) is applied to the diffusion of a pair of particles passively advected by an inhomogeneous, nonstationary ensemble of velocity fields. The central result is a differential equation satisfied by a model joint probability density function of the particle positions, approximating that of the real ensemble. Phenomenologically determined coefficients appear in the equation. One of these is the one-particle effective eddy diffusivity tensor while the other is a tensor quantity representative of the correlation between the motions of the particles. Conditions an the tensor coefficients are found to ensure that the one-particle model is properly embedded in the two-particle model. The separation into a center-of-mass diffusion model and a relative diffusion model is accomplished. Corresponding diffusivity tensors are derived and specialized to the case of homogeneous turbulence. An application to the problem of showing the initial appearance of small-scale structures resulting from turbulent mixing is presented. A method is suggested for determining the required diffusivity tensors from observations of the statistical properties of tracer trajectories.

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