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Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere

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Abstract

We review the theoretical basis for, and the advantages of, random flight models for the trajectories of tracer particles in turbulence. We then survey their application to calculate dispersion in the principal types of atmospheric turbulence (stratified, vertically-inhomogeneous, Gaussian or non-Gaussian turbulence in the surface layer and above), and show that they are especially suitable for some problems (e.g., quantifying ground emissions).

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References

  • Anand M. S. and Pope, S. B.: 1983, ‘Diffusion Behind a Line Source in Grid Turbulence’, in L. J. S. Bradbury, F. Durst, B. E. Launder, F. W. Schmidt and J. H. Whitelaw (eds.), Turbulent Shear Flows 4, Springer, pp. 46–52.

  • de Baas, A. F., Van Dop, H., and Nieuwstadt, F. T. M.: 1986, ‘An Application of the Langevin Equation for Inhomogeneous Conditions to Dispersion in a Convective Boundary Layer’, Quart. J. Roy. Meteorol. Soc. 112, 165–180.

    Google Scholar 

  • Baerentsen, J. H. and Berkowicz, R.: 1984, ‘Monte Carlo Simulation of Plume Dispersion in the Convective Boundary Layer’, Atmos. Environ. 18, 701–712.

    Google Scholar 

  • Borgas, M. S. and Sawford, B. L.: 1991, ‘The Small-Scale Structure of Acceleration Correlations and Its Role in the Statistical Theory of Turbulent Dispersion’, J. Fluid Mech. 228, 295–320.

    Google Scholar 

  • Borgas, M. S. and Sawford, B. L.: 1994a, ‘Stochastic Equations with Multifractal Random Increments for Modelling Turbulent Dispersion’, Phys. Fluids 6, 618–633.

    Google Scholar 

  • Borgas, M. S. and Sawford, B. L.: 1994b, ‘A Family of Stochastic Models for Two-Particle Dispersion in Isotropic Homogeneous Stationary Turbulence’, J. Fluid Mech. 279, 69–99.

    Google Scholar 

  • Borgas, M. S., Flesch, T. K., and Sawford, B. L.: 1995, ‘Turbulent Dispersion with Broken Reflexional Symmetry’, J. Fluid Mech. (Submitted).

  • Coppin, P. A., Raupach, M. R., and Legg, B. J.: 1986, ‘Experiments on Scalar Dispersion Within a Model Plant Canopy, Part II: An Elevated Plane Source’, Boundary-Layer Meteorol. 35, 167–191.

    Google Scholar 

  • Corrsin, S.: 1974, ‘Limitations of Gradient Transport Models’, Adv. Geophys. 18A, 25–60.

    Google Scholar 

  • Davis, P. A.: 1983, ‘Markov Chain Simulations of Vertical Dispersion from Elevated Sources into the Neutral Planetary Boundary Layer’, Boundary Layer Meteorol. 26, 355–376.

    Google Scholar 

  • Deardorff, J. W.: 1978, ‘Closure of Second and Third Moment Rate Equations for Diffusion in Homogeneous Turbulence’, Phys. Fluids 21, 525–530.

    Google Scholar 

  • Denmead, O. T. and Bradley, E. F.: 1985, ‘Flux-Gradient Relationships in a Forest Canopy’, in B. A. Hutchison and B. B. Hicks (eds.), The Forest-Atmosphere Interaction, D. Reidel Publ. Co., ISBN 90-277-1936-5.

  • van Dop, H., Nieuwstadt, F. T. M., and Hunt, J. C. R.: 1985, ‘Random Walk Models for Particle Displacements in Inhomogeneous Unsteady Turbulent Flows’, Phys. Fluids 28, 1639–1653.

    Google Scholar 

  • Du, S., Wilson, J. D., and Yee, E.: 1994a, ‘Probability Density Functions for Velocity in the Convective Boundary Layer, and Implied Trajectory Models’, Atmos. Environ. 28, 1211–1217.

    Google Scholar 

  • Du, S., Wilson, J. D., and Yee, E.: 1994b, ‘On the Moments Approximation Method for Constructing a Lagrangian Stochastic Model’, Boundary-Layer Meteorol. 70, 273–292.

    Google Scholar 

  • Du, S., Sawford, B. L., Wilson, J. D., and Wilson, D. J.: 1995, ‘A Determination of the Kolmogorov Constant (C 0) for the Lagrangian Velocity Structure Function, Using a Second-Order Lagrangian Stochastic Model for Decaying Homogeneous, Isotropic Turbulence’, Phys. Fluids. 1, 3083–3090.

    Google Scholar 

  • Durbin, P. A.: 1983, ‘Stochastic Differential Equations and Turbulent Dispersion’, NASA Reference Publication 1103.

  • Flesch, T. K.: 1996, ‘The Footprint for Flux Measurements, from Backward Lagrangian Stochastic Models’, Boundary-Layer Meteorol. (in press).

  • Flesch, T. K. and Wilson, J. D.: 1992, ‘A Two-Dimensional Trajectory-Simulation Model for Non-Gaussian, Inhomogeneous Turbulence Within Plant Canopies’, Boundary-Layer Meteorol. 61, 349–374.

    Google Scholar 

  • Flesch, T. K., and Wilson, J. D.: 1995, ‘Backward-Time Lagrangian Stochastic Dispersion Models, and Their Application to Estimate Gaseous Emissions’, J. Appl. Meteorol. 34, 1320–1332.

    Google Scholar 

  • Gardiner, C. W.: 1983, Handbook of Stochastic Methods for Physics Chemistry and the Natural Sciences, Springer-Verlag, Berlin.

    Google Scholar 

  • Hurley, P. and Physick, W.: 1993, ‘A Skewed Homogeneous Lagrangian Particle Model for Convective Conditions’, Atmos. Environ. 27A, 619–624.

    Google Scholar 

  • Hurley, P. J.: 1994, ‘PARTPUFF — A Lagrangian Particle-Puff Approach for Plume Dispersion Modeling Applications’, J. Appl. Meteorol. 33, 285–294.

    Google Scholar 

  • Jaynes, E. T.: 1957, ‘Information Theory and Statistical Mechanics’, Phys. Rev. 106, 620–630.

    Google Scholar 

  • Legg, B. J. and Raupach, M. R.: 1982, ‘Markov Chain Simulations of Particle Dispersion in Inhomogeneous Flows: The Mean Drift Velocity Induced by a Gradient in Eulerian Mean Velocity’, Boundary-Layer Meteorol. 24, 3–13.

    Google Scholar 

  • Legg, B. J.: 1983, ‘Turbulent Dispersion from an Elevated Line Source: Markov Chain Simulations of Concentration and Flux Profiles’, Quart. J. Roy. Meteorol. Soc. 109, 645–660.

    Google Scholar 

  • Legg, B. J., Raupach, M. R., and Coppin, P. A.: 1986, ‘Experiments on Scalar Dispersion Within a Model Plant Canopy, Part III: An Elevated Line Source’, Boundary-Layer Meteorol. 35, 277–302.

    Google Scholar 

  • Luhar, A. K. and Britter, R. E.: 1989, ‘A Random Walk Model for Dispersion in Inhomogeneous Turbulence in a Convective Boundary Layer’, Atmos. Environ. 23, 1911–1924.

    Google Scholar 

  • Luhar, A. L. and Rao, K. S.: 1993, ‘Random-Walk Model Studies of the Transport and Diffusion of Pollutants in Katabatic Flows’, Boundary-Layer Meteorol. 66, 395–412.

    Google Scholar 

  • Manins, P. C.: 1995, ‘Selected Papers on LADM, CSIRO's Lagrangian Atmospheric Dispersion Model’, Environmental Consulting and Research Unit, CSIRO Division of Atmospheric Research.

  • McNider, R. T., Moran, M. D., and Pielke, R. A.: 1988, ‘Influence of Diurnal and Inertial Boundary-Layer Oscillations on Long-Range Dispersion’, Atmos. Environ. 22, 2445–2462.

    Google Scholar 

  • Monin, A. S. and Yaglom, A. M.: 1975, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2, MIT Press, Cambridge MA.

    Google Scholar 

  • Nappo, C. J. and Rao, K. S.: 1987, ‘A Model Study of Pure Katabatic Flows’, Tellus 39A, 61–71.

    Google Scholar 

  • Physick, W. L., Noonan, J. A., McGregor, J. L., Hurley, P. J., Abbs, D. J., and Manins, P. C.: 1993, ‘LADM: A Lagrangian Atmospheric Dispersion Model’. CSIRO Division of Atmospheric Research Technical Report No. 24.

  • Pielke, R. A., Cotton, W. R., Walko, R. L., Tremback, C. J., Lyons, W. A., Grasso, L. D., Nicholls, M. E., Moran, M. D., Wesley, D. A., Lee, T. J., and Copeland, J. H.: 1992, ‘A Comprehensive Meteorological Monitoring System — RAMS’, Meteorol. Atmos. Phys. 49, 69–91.

    Google Scholar 

  • Pope, S. B.: 1994, ‘On the Relationship Between Stochastic Lagrangian Models of Turbulence and Second-Moment Closures’, Phys. Fluids 6, 973–985.

    Google Scholar 

  • Raupach, M. R.: 1987, ‘A Lagrangian Analysis of Scalar Transfer in Vegetation Canopies’, Quart. J. Roy. Meteorol. Soc. 113, 107–120.

    Google Scholar 

  • Rodean, H. C.: 1996, ‘Stochastic Lagrangian Models of Turbulent Diffusion’, Am. Meteorol. Soc. monograph, in press.

  • Sawford, B. L.: 1985, ‘Lagrangian Statistical Simulation of Concentration Mean and Fluctuation Fields’, J. Clim. Appl. Meteorol. 24, 1152–1166.

    Google Scholar 

  • Sawford, B. L.: 1993, ‘Recent Developments in the Lagrangian Stochastic Theory of Turbulent Dispersion’, Boundary-Layer Meteorol. 62, 197–215.

    Google Scholar 

  • Sawford, B. L. and Guest, F. M.: 1987, ‘Lagrangian Stochastic Analysis of Flux-Gradient Relationships in the Convective Boundary Layer’, J. Atmos. Sci. 44, 1152–1165.

    Google Scholar 

  • Sawford, B. L. and Guest, F. M.: 1988, ‘Uniqueness and Universality of Lagrangian Stochastic Models of Turbulent Dispersion’, Preprints of 8th Symp. Turb. Diff., AMS, San Diego, CA, pp. 96–99.

    Google Scholar 

  • Sawford, B. L. and Tivendale, C. M.: 1992, ‘Measurements of Concentration Statistics Downstream of a Line Source in Grid Turbulence’, in Proc. 11th Australasian Fluid Mechanics Conf., Hobart, Dec.14–18, 1992, pp. 945–948, University of Tasmania.

  • Sawford, B. L. and Borgas, M. S.: 1994, ‘On the Continuity of Stochastic Models for the Lagrangian Velocity in Turbulence’, Physica D 76, 297–311.

    Google Scholar 

  • Shaw, R. H., Tavangar, J., and Ward, D. P.: 1983, ‘Structure of the Reynolds Stress in a Canopy Layer’, J. Clim. Appl. Meteorol. 22, 1922–1931.

    Google Scholar 

  • Smith, F. B.: 1982, ‘The Integral Equation of Diffusion’, in Proc. 13th NATO/CCMS Conf. on Air Pollution Modelling and its Application, Plenum, pp. 23–34.

  • Smith, F. B. and Thomson, D. J.: 1984, ‘Solutions of the Integral Equation of Diffusion and the Random Walk Model for Continuous Plumes and Instantaneous Puffs in the Atmospheric Boundary Layer’, Boundary-Layer Meteorol. 30, 143–157.

    Google Scholar 

  • Stapountzis, H., Sawford, B. L., Hunt, J. C. R., and Britter, R. E.: 1986, ‘Structure of the Temperature Field Downstream of a Line Source in Grid Turbulence’, J. Fluid Mech. 165, 401–421.

    Google Scholar 

  • Taylor, G. I.: 1921: ‘Diffusion by Continuous Movements’, Proc. London Math. Soc. Series 2 20, 196–212.

    Google Scholar 

  • Tennekes, H. and Lumley, J. L.: 1972, A First Course in Turbulence, MIT Press, Cambridge MA.

    Google Scholar 

  • Tennekes, H.: 1979, ‘The Exponential Lagrangian Correlation Function and Turbulent Diffusion in the Inertial Subrange’, Atmos. Environ. 13, 1565–1567.

    Google Scholar 

  • Thomson, D. J.: 1984, ‘Random Walk Modelling of Diffusion in Inhomogeneous Turbulence’, Quart. J. Roy. Meteorol. Soc. 110, 1107–1120.

    Google Scholar 

  • Thomson, D. J.: 1987, ‘Criteria for the Selection of Stochastic Models of Particle Trajectories in Turbulent Flows’, J. Fluid Mech. 180, 529–556.

    Google Scholar 

  • Uliasz, M.: 1993, ‘The Atmospheric Mesoscale Dispersion Modeling System’, J. Appl. Meteorol. 32, 139–149.

    Google Scholar 

  • Underwood, B. Y.: 1991, ‘Deposition Velocity and the Collision Model of Atmospheric Dispersion — 1. Framework and Application to Cases with Constant Turbulent Velocity Scale’, Atmos. Environ. 25A, 2749–2759.

    Google Scholar 

  • Wang, L-P. and Stock, D. E.: 1992, ‘Stochastic Trajectory Models for Turbulent Diffusion: Monte Carlo Process Versus Markov Chains’, Atmos. Environ. 26A, 1599–1607.

    Google Scholar 

  • Warhaft, Z.: 1984, ‘The interference of thermal fields from line sources in grid turbulence’, J. Fluid Mech. 144, 363–387.

    Google Scholar 

  • Weil, J. C.: 1990, ‘A diagnosis of the asymmetry in top-down and bottom-up diffusion using a Lagrangian stochastic model’, J. Atmos. Sci. 47, 501–515.

    Google Scholar 

  • Willis, G. E. and Deardorff, J. W.: 1976, ‘A Laboratory Model of Diffusion into the Convective Planetary Boundary Layer’, Quart. J. Roy. Meteorol. Soc. 102, 427–445.

    Google Scholar 

  • Willis, G. and Deardorff, J. W.: 1978, ‘A Laboratory Study of Dispersion from an Elevated Source Within a Modeled Convective Planetary Boundary Layer’, Atmos. Environ. 12, 1305–1311.

    Google Scholar 

  • Willis, G. E. and Deardorff, J. W.: 1981, ‘A Laboratory Study of Dispersion from a Source in the Middle of the Convective Mixed Layer’, Atmos. Environ. 15, 109–117.

    Google Scholar 

  • Wilson, J. D.: 1989, ‘Turbulent Transport Within the Plant Canopy’, in Estimation of Areal Evapotranspiration, Intl. Assoc. Hydro. Sci., Publ. No. 177.

  • Wilson, J. D., Thurtell, G. W., and Kidd, G. E.: 1981a, ‘Numerical Simulation of Particle Trajectories in Inhomogeneous Turbulence, III. Comparison of Predictions with Experimental Data for the Atmospheric Surface-Layer’, Boundary-Layer Meteorol. 21, 443–463.

    Google Scholar 

  • Wilson, J. D., Thurtell, G. W., and Kidd, G. E.: 1981b, ‘Numerical Simulation of Particle Trajectories in Inhomogeneous Turbulence. II. Systems with Variable Turbulent Velocity Scale’, Boundary-Layer Meteorol. 21, 423–441.

    Google Scholar 

  • Wilson, J. D., Thurtell, G. W., Kidd, G. E., and Beauchamp, E. G.: 1982, ‘Estimation of the Rate of Gaseous Mass Transfer from a Surface Source Plot to the Atmosphere’, Atmos. Environ. 16, 1861–1868.

    Google Scholar 

  • Wilson, J. D., Legg, B. J., and Thomson, D. J.: 1983, ‘Calculation of Particle Trajectories in the Presence of a Gradient in Turbulent-Velocity Variance’, Boundary-Layer Meteorol. 27, 163–169.

    Google Scholar 

  • Wilson, J. D., Ferrandino, F. J., and Thurtell, G. W.: 1989, ‘A Relationship Between Deposition Velocity and Trajectory Reflection Probability for Use in Stochastic Lagrangian Dispersion Models’, Agric. Forest Meteorol. 47, 139–154.

    Google Scholar 

  • Wilson, J. D. and Flesch, T. K.: 1993, ‘Flow Boundaries in Random-Flight Dispersion Models: Enforcing the Well-Mixed Condition’, J. Appl. Meteorol. 32, 1695–1707.

    Google Scholar 

  • Wilson, J. D., Flesch,T. K., and Swaters, G. E.: 1993, ‘Dispersion in Sheared Gaussian Homogeneous Turbulence’, Boundary-Layer Meteorol. 62, 281–290.

    Google Scholar 

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Wilson, J.D., Sawford, B.L. Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Boundary-Layer Meteorol 78, 191–210 (1996). https://doi.org/10.1007/BF00122492

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