Random Flight Models for Turbulent Trajectories
A "Lagrangian stochastic" (LS; or "Random Flight") model describes the paths of particles in a turbulent flow, given a knowledge (ie. statistical description) of the random velocity field. It is the natural and most powerful means to describe many interesting atmospheric processes (eg. the dispersion of pollen, or of air pollutants), and with the aid of such models we can expect eventually to develop better strategies for (eg.) the application of aerial sprays.
The LS method offers many advantages over the classical treatment of turbulent dispersion, ie. solution of the mass conservation equation. Particularly, one can exploit all given statistical information on the wind field within which diffusion is occurring: modern LS models are derived rigorously from the probability density function (pdf) of the turbulent velocity vector, and thus account (eg.) for velocity skewness, a crucial factor in the daytime atmospheric boundary layer (ABL) where thermal plumes in a gently subsiding ambient flow correspond to a wind field having negatively skewed vertical velocity (consequence: plume centreline from an elevated source declines in height with increasing distance downwind).
Early work in this field, eg. Wilson et al. (1981; Boundary-Layer Meteorol. 21, 443-463), showed that the LS model provides an excellent description of turbulent dispersion in the atmospheric surface layer (ASL), over the entire range of thermal stratification.
- Equations of LS model for particle trajectory in Atmospheric Surface Layer
- Simulation of the plume of a point source (of SO2) emitting continuously [Q, kg/s] at height z=0.5 m into the atmosphere: the following diagrams compare measured ("Project Prairie Grass") and modelled profiles (of suitably non-dimensionalised "cross-wind integrated" concentration) at distance x=100 metres downwind from the source. Atmospheric stability is indicated by the value of L, the "Monin-Obukhov length." Negative values connote unstable (daytime) stratification.
In 1987, D.J. Thomson (J. Fluid Mech. Vol. 180) provided rigorous criteria for LS models (a simple and obvious necessity, yet one whose mathematical implication is powerful: a well-mixed tracer should remain well-mixed, in the velocity-position phase space). Our (Wilson/Thurtell/Kidd) early model was shown to be the uniquely correct ("well-mixed") LS model for inhomogeneous turbulence (statistics height-dependent) having Gaussian velocity statistics.
My work subsequent to Thomson's criteria for these models has included:
- J.D. Wilson and E. Yee, 2007: A Critical Examination of the Random Displacement Model of Turbulent Dispersion. Boundary-Layer Meteorology, 125, 399 - 416. DOI 10.1007/s10546-007-9201-x.
- Wilson, J.D., 2007: Turbulent velocity distributions and implied trajectory models.'' Boundary-Layer Meteorology, 125, 39-47. DOI 10.1007/s10546-007-9188-3
- Bouvet, T., B. Loubet, J.D. Wilson and A. Tuzet, 2007: Filtering of windborne particles by a natural windbreak. Boundary-Layer Meteorology, 123, 481-509.
- Yee, E., and J.D. Wilson, 2007: On the Presence of Instability in Lagrangian Stochastic Trajectory Models, and a Method for its Cure. Boundary-Layer Meteorology, 122, 243-261.
- Shadwick, E.H., J.D. Wilson and T.K. Flesch, 2007: Forward Lagrangian stochastic simulation of a transient source in the atmospheric surface layer. Research Note, Boundary-Layer Meteorology, 122, 263-272.
- Bouvet, T., J.D. Wilson and A. Tuzet, 2006: Observations and modeling of particle deposition in a windbreak flow. Journal of Applied Meteorology and Climatology, 45, 1332-1349.
- Taylor, P.A., P.Y. Li and J.D. Wilson, 2002: Lagrangian simulation of suspended particles in the neutrally stratified atmospheric boundary layer. J. Geophys. Res. (Atmospheres), AAC 7-1 to AAC 7-11.
- Wilson, J.D., T.K. Flesch and R. d'Amours, 2001, Surface delays for gases dispersing in the atmosphere. J. Applied Meteorol., . Applied Meteorol., 40, 1422-1430.
- Wilson, J.D., 2000, Trajectory models for heavy particles in atmospheric turbulence: Comparison with observations. J. Applied Meteorol., 39, 1894-1912. More
- Wilson, J.D., and T.K. Flesch, 1997: Trajectory curvature as a
selection criterion for valid Lagrangian stochastic models. Boundary Layer Meteorol. 84, 411-425.
- Wilson, J.D., and B.L.Sawford, 1996: Lagrangian stochastic models for trajectories in the turbulent atmosphere. Boundary Layer Meteorol. 78, 191-210. Review article.
- Du, S., and J.D. Wilson, 1995: The effect of turbulence on the collection of cloud droplets. J. Atmos. Sci. 52, 3849-3856.
- Du, S., B.L. Sawford, J.D. Wilson, and D.J. Wilson, 1995: A determination of the Kolmogorov constant (C0) for the Lagrangian velocity structure function, using a second-order Lagrangian stochastic model for decaying homogeneous, isotropic turbulence. Physics of Fluids 7, 3083-3090.
- Flesch, T.K., J.D. Wilson, and E. Yee, 1995: Backward-time Lagrangian stochastic dispersion models, and their application to estimate gaseous emissions. J. Appl. Meteorology 34,1320-1332.
- Du, S., J.D. Wilson, and E. Yee, 1994: On the moments approximation method for constructing a Lagrangian stochastic model. Boundary Layer Meteorol. 70, 273-292.
- Du, S., J.D. Wilson, J.D. and E Yee, 1994: Probability density functions for velocity in the convective boundary layer, and implied trajectory models. Atmos. Environ. 28, 1211-1217. More
- Wilson, J.D., and T.K. Flesch, 1993: Flow boundaries in random flight dispersion models: enforcing the well-mixed condition. J. Appl. Meteorol. 32, 1695-1707. Re- formulation of Thomson's criteria to recognise that realisable models use a finite time step (Thomson's treatment assumed infinitesmal time increments), and application to deduce the proper means of treatment of boundaries. More
- Mooney, C.J., and J.D. Wilson, 1993: Disagreements between gradient-diffusion and Lagrangian stochastic dispersion models, even for sources near ground. Boundary Layer. Meteorol. 64, 291-296. (Research Note).
- Wilson, J.D., T. K. Flesch, and G.E. Swaters, 1993: Dispersion in sheared Gaussian homogeneous turbulence. Boundary Layer Meteorol. 62, 281-290. An analytical LS model for sheared homogeneous turbulence. More
- Flesch, T.K., and J.D. Wilson, 1992: A two-dimensional trajectory-simulation model for non-Gaussian, inhomogeneous turbulence within plant canopies. Boundary Layer Meteorol. 61, 349-374. A well-mixed model for the highly skewed and inhomogeneous turbulence of a plant canopy. More.
- Wilson, J.D. and Y.Zhuang, 1989: Restriction on the timestep to be used in stochastic Lagrangian models of turbulent dispersion, Boundary Layer Meteorol. 49, 309-316, (Research Note)
- Wilson, J.D., F.J.Ferrandino, and G.W.Thurtell, 1989: A relationship between deposition velocity and trajectory reflection probability for use in stochastic Lagrangian dispersion models. Agric. Forest Meteorol. 47, 139-154.
- Wilson, J.D., E.P. Lozowski, and Y. Zhuang, 1988: Comments on a relationship between fluid and immersed-particle velocity fluctuations proposed by Walklate(1987). Boundary-Layer Meteorol. 43, 93-98, (Research note).
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Last Modified: 25 Sept., 2008
