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Diffusing passive tracers in random incompressible flows: Statistical topography aspects

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Abstract

The paper studies statistical characteristics of the passive tracer concentrations and of its spatial gradient, in random incompressible velocity fields from the viewpoint of statistical topography. The statistics of interest include mean values, probability distributions, as well as various functionals characterizing topographic features of tracers. The functional approach is used. We consider the influence of the mean flow (the linear shear flow) and the molecular diffusion coefficient on the statistics of the tracer. Most of our analysis is carried out in the framework of the delta-correlated (in time) approximation and conditions for its applicability are established. But we also consider the diffusion approximation scheme for finite correlation radius. The latter is applied to a diffusing passive tracer that undergoes sedimentation in a random velocity field.

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References

  1. A. S. Monin and A. M. Yaglom,Statistical Fluid Mechanics (MIT Press, Cambridge, Massachusetts, 1980).

    Google Scholar 

  2. G. T. Csanady,Turbulent Diffusion in the Environment (Reidel, Dordrecht, 1980).

    Google Scholar 

  3. A. Okubo,Diffusion and Ecological Problems: Mathematical Models (Springer-Verlag, New York, 1980).

    Google Scholar 

  4. M. Lesieur,Turbulence in Fluids (Kluwer, Boston, 1990).

    Google Scholar 

  5. W. McComb,The Physics of Fluid Turbulence (Clarendon Press, Oxford, 1990).

    Google Scholar 

  6. G. Dagan, Theory of solute transport by groundwater,Annu. Rev. Fluid Mech. 19:183 (1987).

    Google Scholar 

  7. S. F. Shandarin and Ya. B. Zel'dovich, Turbulence, intermittency, structures in a self-gravitating medium: The large scale structure of the universe,Rev. Mod. Phys 61:185 (1989).

    Google Scholar 

  8. S. Gurbatov, A. Malakhov, and A. Saichev,Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles (Manchester University Press, Cambridge, 1991).

    Google Scholar 

  9. G. K. Batchelor, Small-scale variation of convected quantities like temperature in turbulent fluid. 1. General discussion and the case of small conductivity,J. Fluid Mech. 5:113 (1959).

    Google Scholar 

  10. G. K. Batchelor, I. D. Howells, and A. A. Townsend, Small-scale variation of convected quantities like temperature in turbulent fluid. 2. The case of large conductivity,J. Fluid Mech. 5:134 (1959).

    Google Scholar 

  11. P. H. Roberts, Analytical theory of turbulent diffusion,J. Fluid Mech. 11:257 (1961).

    Google Scholar 

  12. R. H. Kraichnan, Small scale structure of scalar field convected by turbulence,Phys Fluids 11:945 (1968); Diffusion by a random velocity field,Phys. Fluids 13:22 (1970); Anomalous scaling of a randomly advected passive scalar,Phys. Rev. Lett. 72:1016 (1994).

    Google Scholar 

  13. P. G. Saffman, Application of the Wiener-Hermite expansion to the diffusion of passive scalar in a homogeneous turbulent flow,Phys. Fluids 12(9):1786 (1972).

    Google Scholar 

  14. D. McLaughlin, G. Papanicolaou, and O. R. Pironneau, Convection of microstructures and related problems,SIAM J. Appl. Math. 45:780 (1985).

    Google Scholar 

  15. S. Molchanov and L. Piterbarg, Heat propagation in random flows,Russ. J. Math. Phys. 1:18 (1992).

    Google Scholar 

  16. M. B. Isichenko, Percolation, statistical topography, and transport in random media,Rev. Mod. Phys. 64(4):961 (1992).

    Google Scholar 

  17. R. J. Adler,The Geometry of Random Fields (Wiley, New York, 1981).

    Google Scholar 

  18. C. L. Zirbel and E. Çinlar, Mass transport by Brownian motion, inStochastic Models in Geosystems S. A. Molchanov and W. A. Woyczynski, eds. (Springer-Verlag, New York, 1966).

    Google Scholar 

  19. S. D. Rice, Mathematical analysis of random noise,Bell. Syst. Tech. J. 23:282 (1944);24:46 (1945).

    Google Scholar 

  20. M. S. Longuet-Higgins, The statistical analysis of a random moving surface,Philos. Trans. R. Soc. Lond. A 249:321 (1957); Statistical properties of an isotropic random surface,Philos. Trans. R. Soc. Lond. A 250:157 (1957).

    Google Scholar 

  21. P. Swerling, Statistical properties of the countours of random surfaces,IRE Trans. Inf. Theory IT-8:315 (1962).

    Google Scholar 

  22. V. I. Klyatskin, Statistical description of the diffusion of a passive tracer in a random velocity field,Physics-Uspekhi 37(5):501 (1994).

    Google Scholar 

  23. K. Furutsu, On the statistical theory of electromagnetic waves in a fluctuating media,J. Res. NBS N-67:303 (1963).

    Google Scholar 

  24. E. A. Novikov, Functionals and the random-force method in turbulence theory,Sov. Phys. JETP 20(5):1290 (1964).

    Google Scholar 

  25. V. I. Klyatskin and A. I. Saichev, Statistical and dynamical localization of plane waves in randomly layered media.Sov. Phys. Usp. 35(3):231 (1992).

    Google Scholar 

  26. A. I. Saichev and W. A. Woyczynski, Probability distributions of passive tracers in randomly moving media, inStochastic Models in Geosystems, S. A. Molchanov and W. A. Woyczynski, eds. (Springer-Verlag, New York, 1996).

    Google Scholar 

  27. A. S. Gurvich and A. M. Yaglom, Breakdown of eddies and probability distributions for small-scale turbulence,Phys. Fluids Suppl. 10(9):559 (1967).

    Google Scholar 

  28. A. R. Kerstein and W. T. Ashurst, Lognormality of gradients of diffusive scalars in homogeneous, two-dimensional mixing systems,Phys. Fluids 27(12):2819 (1984).

    Google Scholar 

  29. W. J. A. Dahm and K. A. Buch, Lognormality of the scalar dissipation pdf in turbulent flows,Phys. Fluids A 1(7):1290 (1989).

    Google Scholar 

  30. R. Kraichnan, Convection of a passive scalar by a quasi-uniform random straining field,J. Fluid Mech. 64:737 (1974).

    Google Scholar 

  31. E. Zambianchi and A. Griffa, Effects of finite scales of turbulence on disperions estimates,J. Marine Res. 52:129 (1994).

    Google Scholar 

  32. V. I. Klyatskin and W. A. Woyczynski, Fluctuations of passive scalar with nonzero mean concentration gradient in random velocity fields,Zh. Eksp. Teor. Fiz. 96(10) (1995) [Phys. JETP69(10), 1995].

  33. A. Pumir, B. Shraiman, and E. Siggia, Exponential tails and random advection,Phys. Rev. Lett. 66(23):2984 (1991).

    Google Scholar 

  34. J. Gollub, J. Clarke, M. Gharib, B. Lane, and O. Mesquita, Fluctuations and transport in a stirred fluid with a mean gradient,Phys. Rev. Lett. 67(25):3507 (1991).

    Google Scholar 

  35. M. Holzer and A. Pumir, Simple models of non-Gaussian statistics for a turbulently advected passive scalar,Phys. Rev. E 47(1):202 (1993).

    Google Scholar 

  36. A. Pumir, A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient,Phys. Fluids A 6(6):2118 (1994).

    Google Scholar 

  37. M. Holzer and E. Siggia, Turbulent mixing of a passive scalar,Phys. Fluids 6(5):1820 (1994).

    Google Scholar 

  38. A. Kerstein and P. A. McMurtry, Mean-field theories of random advection,Phys. Rev. E 49(1):474 (1994).

    Google Scholar 

  39. B. I. Shraiman and E. D. Siggia, Lagrangian path integrals and fluctuations in random flow,Phys. Rev. E 49:2912 (1994).

    Google Scholar 

  40. V. P. Dokuchaev, Method of dispersion relations for mean concentration of passive admixture in the turbulent diffusion theory,Izv. RAN, Fiz. Atm. Okeana 31(2):275 (1995).

    Google Scholar 

  41. V. I. Klyatskin,Stochastic Equations and Waves in Random Media (Nauka, Moscow, 1980) [in Russian];Ondes et équations stochastiques dans les milieus aléatorement non homogènes (Editions de Physique, Besançon, France, 1985).

    Google Scholar 

  42. Ya. G. Sinai and V. Yakhot, Limiting probability distributions of a passive scalar in a random velocity field,Phys., Rev. Lett. 63:1962 (1989).

    Google Scholar 

  43. H. Chen, S. Chen, and R. H. Kraichnan, Probability distribution of a stochastically adverted scalar field,Phys. Rev. Lett. 63:2657 (1989).

    Google Scholar 

  44. Y. Kimura and R. H. Kraichnan, Statistics of an adverted passive scalarPhys. Fluids A 5:2264 (1993).

    Google Scholar 

  45. F. Gao, An analytical solution for the scalar probability density function in homogeneous turbulence,Phys. Fluids A 3:511 (1991).

    Google Scholar 

  46. M. Avellaneda and A. Majda, An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows,Commun. Math. Phys. 138:339 (1991).

    Google Scholar 

  47. A. J. Majda, Random shearing direction models for isotropic turbulent diffusion,J. Stat. Phys. 75(516):1153 (1994).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Atmospheric Physics, Russian Academy of Sciences, 109017, Moscow, Russia

    V. I. Klyatskin

  2. Pacific Oceanological Institute, Russian Academy of Sciences, 690041, Vladivostok, Russia

    V. I. Klyatskin

  3. Department of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, 44106, Cleveland, Ohio

    W. A. Woyczynski

  4. Department of Mathematics and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, 44106, Cleveland, Ohio

    D. Gurarie

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  1. V. I. Klyatskin
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  2. W. A. Woyczynski
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  3. D. Gurarie
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Klyatskin, V.I., Woyczynski, W.A. & Gurarie, D. Diffusing passive tracers in random incompressible flows: Statistical topography aspects. J Stat Phys 84, 797–836 (1996). https://doi.org/10.1007/BF02179658

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  • DOI: https://doi.org/10.1007/BF02179658

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