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

Andy M Reynolds

doi:10.1038/s41598-018-34924-4

Abstract

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.

Highlights

In his seminal work, Thomson[1] correctly noted that turbulent dispersal within complex, inhomogeneous turbulent flows, such as the atmospheric boundary-layer, is best predicted by Lagrangian stochastic models because other methods are either invalid or inappropriate
The one-dimensionality of the analysis mirrors the modelling of tracer-particle trajectories in boundary layers where Lagrangian stochastic models are frequently used to model vertical movements due to turbulence, and where turbulence is assumed to have a negligible effect on downwind movements which are dominated by the mean flow
Preferential sampling of turbulent flows is not significant when Stokes numbers, St = τ/tη, are much less than one[11], as is the case here where the focus is on spores and pollen that are dispersing in atmospheric boundary-layers

Summary

Introduction

Thomson[1] correctly noted that turbulent dispersal within complex, inhomogeneous turbulent flows, such as the atmospheric boundary-layer, is best predicted by Lagrangian stochastic models because other methods (e.g., diffusion equations and similarity theory) are either invalid or inappropriate. In common with the formulation of Lagrangian stochastic models for simulation of tracer-particle trajectories[1] I have assumed that the turbulent velocities experienced by particles along their trajectories can be represented as continuous Markov processes.

Results

Conclusion