Near-field dispersion from instantaneous sources in the surface layer

Abstract

This paper considers the near-field dispersion of an ensemble of tracer particles released instantaneously from an elevated source into an adiabatic surface layer. By modelling the Lagrangian vertical velocity as a Markov process which obeys the Langevin equation, we show analytically that the mean vertical drift velocity w(t) is w(τ)=bu _*(1−e ^−τ(1+τ)), where Τ is time since release (nondimensionalized with the Lagrangian time scale at the source), b Batchelor's constant, and u _*, the friction velocity. Hence, the mean height and mean depth of the ensemble are calculated. Although the derivation is formally valid only when Τ ≪ 1, the predictions for w, mean height and mean depth are consistent in the downstream limit (Τ ≫ 1) with surface-layer Lagrangian similarity theory and with the diffusion equation. By comparing the analytical predictions with numerical, randomflight solutions of the Langevin equation, the analytical predictions are shown to be good approximations at all times, both near-field and far-field.