A Lagrangian statistical analysis of diffusion from a ground‐level source in a turbulent boundary layer

A Lagrangian statistical analysis is proposed for the motions of a single particle and the growth of a small cloud released from the ground in a turbulent boundary layer, essentially by considering the fluctuating as well as the mean vertical Lagrangian velocity, by noting that its integral time scale is much smaller than the travel time of a particle from the source and by estimating (quite conventionally) the Lagrangian statistics in terms of the measured Eulerian statistics of atmospheric surface layer turbulence, in neutrally stable conditions. Comparing the results with a 1968 analysis by P. C. Chatwin based on a diffusion equation, the same value is obtained for the dispersion of the vertical displacement of a particle as a function of its mean travel time; however, a small but physically significant difference is predicted for the growth of a cloud released at ground level. The validated diffusion equation is then used to relate the concentration of a cloud to that of a continuous plume. It is shown that the index, s , used in the exponential formula exp(–Az s ) to describe observations of the concentration in a plume, varies slowly with distance downwind of a source. Diffusion measurements in the atmospheric boundary layer and in wind tunnels are shown by the theory to be consistent with turbulence measurements. The theory suggests the kind of further turbulence and diffusion measurements that ought to be made.