Incorporating terminal velocities into Lagrangian stochastic models of particle dispersal in the atmospheric boundary layer

Lagrangian stochastic models for simulation of tracer-particle trajectories in turbulent flows can be adapted for simulation of particle trajectories. This is conventionally done by replacing the zero-mean fall speed of a tracer-particle with the terminal speed of the particle. Such models have been used widely to predict spore and pollen dispersal. Here I show that this modification predicts that particles become uniformly distributed throughout the air column, which is at variance with the seminal experimental studies of Hirst et al. (1967) that demonstrated spore concentrations (and pollen concentrations) declined exponentially with height in unstable air. This discrepancy arises because the terminal speed, which is a Lagrangian property of a particle, has always been treated as if it were an Eulerian property of an ensemble of particles. In this study models are formulated correctly. I show that the mean acceleration of a tracer-particle should be replaced by the mean acceleration of a particle. Model predictions for aerial density profiles then agreed with the observations of Hirst et al. (1967) and with observations of ground-level concentrations but differed significantly from predictions obtained using conventional models. In accordance with the results of numerical simulations, the models also predict that particles are moving downwind marginally more slowly than the wind itself. Finally, the new modelling approach can be extended to predict the dispersal of small insects with active flight behaviours.