# A Velocity–Dissipation Lagrangian Stochastic Model for Turbulent Dispersion in Atmospheric Boundary-Layer and Canopy Flows

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## Abstract

An extended Lagrangian stochastic dispersion model that includes time variations of the turbulent kinetic energy dissipation rate is proposed. The instantaneous dissipation rate is described by a log-normal distribution to account for rare and intense bursts of dissipation occurring over short durations. This behaviour of the instantaneous dissipation rate is consistent with field measurements inside a pine forest and with published dissipation rate measurements in the atmospheric surface layer. The extended model is also shown to satisfy the well-mixed condition even for the highly inhomogeneous case of canopy flow. Application of this model to atmospheric boundary-layer and canopy flows reveals two types of motion that cannot be predicted by conventional dispersion models: a strong sweeping motion of particles towards the ground, and strong intermittent ejections of particles from the surface or canopy layer, which allows these particles to escape low-velocity regions to a high-velocity zone in the free air above. This ejective phenomenon increases the probability of marked fluid particles to reach far regions, creating a heavy tail in the mean concentration far from the scalar source.

## Keywords

Canopy Dispersion Dissipation Lagrangian stochastic model## Notes

### Acknowledgments

This work was supported by Research Grant Award No. IS-4374-11C from BARD, the United States—Israel Binational Agricultural Research and Development Fund.

## References

- Anand M, Pope S, Mongia H (1993) Pdf calculations for swirling flows. In: Proceedings of 31st aerospace sciences meeting and exhibition, Reno, NV. AIAA Paper 93–0106Google Scholar
- Antonia R (1973) Some small scale properties of boundary layer turbulence. Phys Fluids 16:1198–1206CrossRefGoogle Scholar
- Baldocchi D (1997) Flux footprints within and over forest canopies. Boundary-Layer Meteorol 85(2):273–292CrossRefGoogle Scholar
- Cassiani M, Franzese P, Giostra U (2005a) A PDF micromixing model of dispersion for atmospheric flow. Part I: development of the model, application to homogeneous turbulence and to neutral boundary layer. Atmos Environ 39(8):1457–1469CrossRefGoogle Scholar
- Cassiani M, Radicchi A, Giostra U (2005b) Probability density function modelling of concentration fluctuation in and above a canopy layer. Agric For Meteorol 133:153–165CrossRefGoogle Scholar
- Chen W (1971) Lognormality of small-scale structure of turbulence. Phys Fluids 14:1639–1642CrossRefGoogle Scholar
- Finnigan J (2000) Turbulence in plant canopies. Annu Rev Fluid Mech 32:519–571CrossRefGoogle Scholar
- Flesch T, Wilson J (1992) A two-dimensional trajectory-simulation model for non-Gaussian, inhomogeneous turbulence within plant canopies. Boundary-Layer Meteorol 61:349–374CrossRefGoogle Scholar
- Flesch TK, Wilson JD, Yee E (1995) Backward-time Lagrangian stochastic dispersion models and their application to estimate gaseous emissions. J Appl Meteorol 34(6):1320–1332CrossRefGoogle Scholar
- Freytag C (1978) Statistical properties of energy dissipation. Boundary-Layer Meteorol 14(2):183–198CrossRefGoogle Scholar
- Frisch U (1996) Turbulence: the legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge, 296 ppGoogle Scholar
- Hsieh C, Katu G, Chi T (2000) An approximate analytical model for footprint estimation of scalar fluxes in thermally stratified atmospheric flows. Adv Water Resour 23(7):765–772CrossRefGoogle Scholar
- Hsieh CI, Katul G (2009) The Lagrangian stochastic model for estimating footprint and water vapor fluxes over inhomogeneous surfaces. Int J Bioclimatol Biometeorol 53(1):87–100CrossRefGoogle Scholar
- Juang J, Katul G, Siqueira M, Stoy P, Palmroth S, McCarthy HR, Kim H, Oren R (2006) Modeling nighttime ecosystem respiration from measured \(\text{ CO }_2\) concentration and air temperature profiles using inverse methods. J Geophys Res 111:D08S05Google Scholar
- Juang J, Katul G, Siqueira M, McCarthy H (2008) Investigating a hierarchy of eulerian closure models for scalar transfer inside forested canopies. Boundary-Layer Meteorol 128:1–32CrossRefGoogle Scholar
- Kaimal J, Finnigan J (1994) Atmospheric boundary layer flows: their structure and measurement. Oxford University Press, New York, 289 ppGoogle Scholar
- Katul G, Albertson J (1998) An investigation of higher-order closure models for a forested canopy. Boundary-Layer Meteorol 89(1):47–74CrossRefGoogle Scholar
- Kljun N, Rotach N, Schmid H (2002) A three-dimensional backward Lagrangian footprint model for a wide range of boundary-layer stratifications. Boundary-Layer Meteorol 103:205–226CrossRefGoogle Scholar
- Kolmogorov A (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J Fluid Mech 13:82–85CrossRefGoogle Scholar
- Kurbanmuradov O, Sabelfeld K (2000) Lagrangian stochastic models for turbulent dispersion in the atmospheric boundary layer. Boundary-Layer Meteorol 97(2):191–218CrossRefGoogle Scholar
- Leuning R, Denmead O, Miyata A, Kim J (2000) Source/sink distributions of heat, water vapor, carbon dioxide, and methane in a rice canopy estimated using lagrangian dispersion analysis. Agric For Meteorol 103:233–249CrossRefGoogle Scholar
- Li P, Taylor P (2005) Three-dimensional Lagrangian simulation of suspended particles in the neutrally stratified atmospheric surface layer. Boundary-Layer Meteorol 116(2):301–311CrossRefGoogle Scholar
- Minier J, Pozorski J (1999) Wall-boundary conditions in probability density function methods and application to a turbulent channel flow. Phys Fluids 11:2632–2644CrossRefGoogle Scholar
- Monin A, Yanglom A (1975) Statistical fluid mechanics: mechanics of turbulence, vol 2. MIT Press, Cambridge, 874 ppGoogle Scholar
- Nathan R, Katul G, Horn H, Thomas S, Oren R, Avissar R, Pacala S, Levin S (2002) Mechanisms of long-distance dispersal of seeds by wind. Nature 418(6896):409–413CrossRefGoogle Scholar
- Nemitz E, Sutton M, Gut A, San-José R, Husted S, Schjoerring J (2000) Sources and sinks of ammonia within an oilseed rape canopy. Agric For Meteorol 105:385–404CrossRefGoogle Scholar
- Novikov E (1969) Relation between the Lagrangian and Eulerian descriptions of turbulence. J Appl Math Mech 33(5):862–864CrossRefGoogle Scholar
- Novikov E (1986) The Lagrangian–Eulerian probability relations and the random force method for nonhomogeneous turbulence. Phys Fluids 29(12):3907–3909CrossRefGoogle Scholar
- Obukhov A (1962) Some specific features of atmospheric turbulence. J Geophys Res 67:3011–3014CrossRefGoogle Scholar
- Poggi D, Katul G, Albertson J (2004) Momentum transfer and turbulent kinetic energy budgets within a dense model canopy. Boundary-Layer Meteorol 111:589–614CrossRefGoogle Scholar
- Poggi D, Katul G, Albertson J (2006) Scalar dispersion within a model canopy: measurements and three-dimensional Lagrangian models. Adv Water Resourc 29(2):326–335CrossRefGoogle Scholar
- Pope S (1991) Application of the velocity–dissipation probability density function model to inhomogeneous turbulent flows. Phys Fluids A 3:1947–1957CrossRefGoogle Scholar
- Pope S (2000) Turbulent flows. Cambridge University Press, Cambridge, 771 ppGoogle Scholar
- Pope S, Chen Y (1990) The velocity–dissipation probability density function model for turbulent flows. Phys Fluids A 2:1437–1449CrossRefGoogle Scholar
- Porta AL, Voth GA, Crawford AM, Alexander J, Bodenschatz E (2001) Fluid particle accelerations in fully developed turbulence. Boundary-Layer Meteorol 409:1017–1019Google Scholar
- Rannik Ü, Aubinet M, Kurbanmuradov O, Sabelfeld K, Markkanen T, Vesala T (2000) Footprint analysis for measurements over a heterogeneous forest. Boundary-Layer Meteorol 97(1):137–166CrossRefGoogle Scholar
- Raupach M (1989) Applying Lagrangian fluid-mechanics to infer scalar source distributions from concentration profiles in plant canopies. Agric For Meteorol 47:85–108Google Scholar
- Raupach M, Thom A (1981) Turbulence in and above plant canopies. Annu Rev Fluid Mech 13:97–129CrossRefGoogle Scholar
- Rodean H (1996) Stochastic Lagrangian models of turbulent diffusion. Meteorological monographs, vol 26, no. 48. American Meteorological Society, Boston, 84 ppGoogle Scholar
- Simon E, Lehmann B, Ammann C, Ganzeveld L, Rummel U, Meixner F, Nobre A, Araujo A, Kesselmeier J (2005) Lagrangian dispersion of \(^{222}\)Rn, \(\text{ H }_2\)O and \(\text{ CO }_2\) within Amazonian rain forest. Agric For Meteorol 132:286–304CrossRefGoogle Scholar
- Siqueira M, Lai C, Katul G (2000) Estimating scalar sources, sinks, and fluxes in a forest canopy using Lagrangian, Eulerian, and hybrid inverse models. J Geophys Res 105:29475–29488CrossRefGoogle Scholar
- Siqueira M, Katul G, Lai C (2002) Quantifying net ecosystem exchange by multilevel ecophysiological and turbulent transport models. Adv Water Resour 25:1357–1366CrossRefGoogle Scholar
- Siqueira M, Leuning R, Kolle O, Kelliher F, Katul G (2003) Modeling sources and sinks of \(\text{ CO }_2, \text{ H }_2\)O and heat within a Siberian pine forest using three inverse methods. Q J R Meteorol Soc 129:1373–1393CrossRefGoogle Scholar
- Siqueira M, Katul G, Tanny J (2012) The effect of the screen on the mass, momentum, and energy exchange rates of a uniform crop situated in an extensive screenhouse. Boundary-Layer Meteorol 142(3):339–363CrossRefGoogle Scholar
- Taylor G (1921) Diffusion by continuous movements. Proc Lond Math Soc 20:196–211Google Scholar
- Thomson D (1987) Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J Fluid Mech 180:529–556CrossRefGoogle Scholar
- Tiwary A, Fuentes J, Barr J, Wang D, Colls J (2007) Inferring the source strength of isoprene from ambient concentrations. Environ Modell Softw 22:1281–1293CrossRefGoogle Scholar
- Vesala T, Kljun N, Rannik U, Rinne J, Sogachev A, Markkanen T, Sabelfeld K, Foken T, Leclerc M (2008) Flux and concentration footprint modelling: state of the art. Environ Pollut 152(3):653–666CrossRefGoogle Scholar
- Wilson J (1988) A second order closure model for flow through vegetation. Boundary-Layer Meteorol 42: 371–392Google Scholar
- Wilson JD, Sawford BL (1996) Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Boundary-Layer Meteorol 78:191–210CrossRefGoogle Scholar
- Yeung P, Pope S (1989) Lagrangian statistics from direct numerical simulations of isotropic turbulence. J Fluid Mech 207:581–586CrossRefGoogle Scholar