Abstract
Turbulent dispersion proceeds not only much faster but also in a qualitatively different manner than molecular diffusion. Yet, the majority of hydraulic, oceanic and atmospheric models rely on the concept of an eddy diffusivity. It is shown here that an alternative model can be developed to exhibit observed behavior. The new term in the diffusion equation, which is non-local, may be interpreted in terms of the probability density function (pdf) of the turbulent velocity. Different assumptions about this distribution lead to a family of models, one of which is the model proposed here and another, the classical Fickian model of diffusion. A connection is also made with models using fractional calculus.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bakunin OG (2004) Correlation effects and turbulent diffusion scalings. Rep Prog Phys 67: 965–1032
Batchelor GK, Townsend AA (1956) Turbulent diffusion. In: Batchelor GK, Davies RM (eds). Surveys in mechanics. Cambridge University Press, pp 352–399
Boussinesq J (1877) Essai sur la thèorie des eaux courantes [Essay on the theory of flowing waters]. Mem Acadèmie des Sciences, vol 23. Inst France, Paris, pp 252–260
Clark JF, Schlosser P, Stute M, Simpson HJ (1996) SF6 − 3He tracer release experiment: a new method of determining longitudinal dispersion coefficients in large rivers. Environ Sci Technol 30: 1527–1532
Cushman-Roisin B, Jenkins AD (2006) On a non-local parameterization for shear turbulence and the uniqueness of its solutions. Boundary-Layer Meteor 118: 69–82
Fick A (1855) On liquid diffusion. Philos Mag J Sci 10: 31–19
Gifford FA (1957) Relative atmospheric diffusion of smoke puffs. J Meteor 14: 410–414
Klafter J, Blumen A, Schlesinger MF (1987) Stochastic pathway to anomalous diffusion. Phys Rev A 35: 3081–3085
Meerschaert MM, Mortensen J, Wheatcraft SW (2006) Fractional vector calculus for fractional advection-dispersion. Physica A 367: 181–190
Min IA, Abernathy RN, Lundblad HL (2002) Measurement and analysis of puff dispersion above the atmospheric boundary layer using quantitative imagery. J Appl Meteor 41: 1027–1041
Obukhov AM (1941) Spectral energy distribution in turbulent flow. Izv Akad Nauk SSSR 5: 453–566
Okubo A (1962) A review of theoretical models for turbulent diffusion in the sea. J Oceanogr Soc Jpn 20th Anniversary Volume:286–320
Okubo A (1974) Some speculations on oceanic diffusion diagrams. Rapports et Procès-verbaux des Rèunions du Conseil Permanent International pour l‘Exploration de la Mer 167: 77–85
Ouellette NT, Xu H, Bourgoin M, Bodenschatz E (2006) An experimental study of turbulent relative dispersion models. New J Phys 8. doi:10.1088/1367-2630/8/6/109.
Richardson LF (1926) Atmospheric diffusion shown on a distance-neighbour graph. Proc R Soc London A 110: 709–737
Richardson LF, Stommel H (1948) A note on eddy diffusion in the sea. J Meteor 5: 238–240
Stommel H (1949) Horizontal diffusion due to oceanic turbulence. J Mar Res 8: 199–225
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cushman-Roisin, B. Beyond eddy diffusivity: an alternative model for turbulent dispersion. Environ Fluid Mech 8, 543–549 (2008). https://doi.org/10.1007/s10652-008-9082-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10652-008-9082-7