Lagrangian approach, turbulent diffusion

Later Taylor began to search for more suitable means for the the description of turbulence [165]. The statistical approach to the study of turbulence was initiated by a paper by Taylor [158]. In the work of Taylor [158] on turbulent transport the important role of the Lagrangian correlation function (i.e., the one point time correlation) of the velocity field was first demonstrated. Taylor showed that the turbulent diffusion of particles starting from a point depends on the correlation between the velocity of a fluid particle at any instant and that of the same particle after a certain correlation time interval. 

Having demonstrated that exact solution for the mean concentrations (c, (x, t j) even of inert species in a turbulent fluid is not possible in general by either the Eulerian or Lagrangian approaches, we now consider what assumptions and approximations can be invoked to obtain practical descriptions of atmospheric diffusion. In Section 18.4 we shall proceed from the two basic equations for (c,), (18.4) and (18.8), to obtain the equations commonly used for atmospheric diffusion. A particularly important aspect is the delineation of the assumptions and limitations inherent in each description. 

The Lagrangian approach to turbulent diffusion is concerned with the behavior of representative fluid particles. We therefore begin by considering a single particle that is at location x at time t in a turbulent fluid. The subsequent motion of the particle can be described by its trajectory, X[x, r], that is, its position at any later time t. Let... 

The approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

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