A Lagrangian analysis of turbulent diffusion

Abstract

This is an analysis of diffusion of a scalar field by molecular transport and isotropic turbulence. Existing results are surveyed, and some new results are advanced. The discussion is supported with oceanographic and atmospheric observations of dispersion and diffusion. The existing results were originally obtained using a variety of mathematical techniques. However, all results are derived here using an approximate solution of the Lagrangian form of the advection‐diffusion equation. The approximation is equivalent to neglecting the spatial dependence of the transformation factors in the Lagrangian representation of the molecular flux divergence. Examinations of the diffusive subranges show the approximation to justified: infinitesimal line stretching is either controlled by relatively large scale shears (viscous‐diffusive subrange at large Prandtl number) or else is negligible during the diffusion process (inertia‐diffusive subrange at small Prandtl number). Estimation of scalar mean fields, total variances, and wave number spectra requires, in general, joint statistics of infinitesimal line stretching and either single particle displacement or particle pair separation. Normality is assumed for displacement statistics; separation statistics are determined from the Richardson‐Kraichnan equation. A simple derivation of that equation is presented here. Joint stretching‐separation statistics are modeled by a uniform shear flow, with time‐dependent amplitudes described by the Wiener process (white noise). With the possible exception of this random process, the only mathematics required here is elementary calculus, so details have been kept to a minimum. In the diffusion problems considered here, the turbulence is isotropic. However, both the approximate solution of the advection‐diffusion equation and the equations for joint displacements are equally valid for inhomogeneous turbulence.