Probability Distributions of Passive Tracers in Randomly Moving Media

  • A. I. Saichev
  • W. A. Woyczynski
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 85)


Statistical properties of fluctuations of the density of passive tracer and related random fields in a randomly moving medium are discussed. Diffusion approximation and a Gaussian velocity field model is used. We find probability distributions of density fields and of Jacobians in a chaotically compressible medium. Formulas connecting statistical characteristics of random fields in Lagrangian and in Eulerian coordinates are provided. For an incompressible medium, we analyze statistical properties of the passive scalar field’s gradient, and also statistics of the total gradient and the length of a contour carried in the chaotic flow of an incompressible fluid.


Random Field Stochastic Differential Equation Diffusion Approximation Density Field Passive Tracer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Avellaneda M., Majda A. (1992), Approximate and exact renormalization theories for a model for turbulent transport, Phys. Fluids A 4, 41–57.CrossRefGoogle Scholar
  2. [2]
    Batchelor G.K. (1959), Small-scale variation of convected quantities like temperature in turbulent fluid, J. Fluid Mech. 5, 113.CrossRefGoogle Scholar
  3. [3]
    Careta A., Sagues F., Ramirez-Piscina L., Sancho J.M. (1993), Effective diffusion in a stochastic velocity field, J. Stat. Phys. 71, 235–313.CrossRefGoogle Scholar
  4. [4]
    Crisanti A., Vulpiani A. (1993), On the effects of noise and drift on diffusion in fluids, J. Stat. Phys. 70, 197–211.CrossRefGoogle Scholar
  5. [5]
    Csanady G.T. (1980), Turbulent Diffusion in the Environment, Reidel, Boston.Google Scholar
  6. [6]
    Davis R.E. (1982), On relating Eulerian and Lagrangian velocity statistics: single particles in homogeneous flow, J. Fluid Mech. 74, 1–26.CrossRefGoogle Scholar
  7. [7]
    Fradkin L. (1991), Comparison of Lagrangian and Eulerian approaches to turbulent diffusion, Plasma Physics and Control]. Fusion, 685.Google Scholar
  8. [8]
    Gurbatov S.N., Malakhov A., Saichev A.I. (1991), Non-linear Random Waves and Turbulence in Non-dipersive Media: Waves, Rays and Particles, Manchester U Press.Google Scholar
  9. [9]
    Gurbatov S.N., Saichev A.I. (1993), Inertial nonlinearity and chaotic motion of particle fluxes, Chaos 3, 333–358.CrossRefGoogle Scholar
  10. [10]
    Herring J.R., Kerr R.M., Rotunno R. (1994), Ertel’s potential vorticity in unstratified turbulence, J. Atmospheric Sciences 51, 35–47.CrossRefGoogle Scholar
  11. [11]
    Kesten H., Papanicolaou G.C. (1979), A limit theorem for turbulent diffusion, Comm. Math. Phys. 65, 97–128.CrossRefGoogle Scholar
  12. [12]
    Klyatskin V. (1986), Method of Imbedding in the Theory of Wave Propagation, Moscow, Nauka.Google Scholar
  13. [13]
    Klyatskin V., Saichev A.I., (1992), Statistical and dynamical localization of plave waves in randomly layered media, Soviet Physics Usp. 35 (3), 231–247.CrossRefGoogle Scholar
  14. [14]
    Klyatskin V., Woyczynski W.A., D. Gurarie (1996), Diffusing passive tracers in random incompressible flows: statistical topography aspects, J. Stat. Phys. 84, 797–836.CrossRefGoogle Scholar
  15. [15]
    Klyatskin V., Woyczynski W.A. (1994), Dynamical and statistical characteristics of geophysical fields and waves, and related boundary-value problems, this volume.Google Scholar
  16. [16]
    Kraichnan R.H. (1970), Diffusion by a random velocity field, Phys. Fluids 13, 22–31.CrossRefGoogle Scholar
  17. [17]
    Lipscomb T.C., Frenkel A.L., Ter Haar D. (1970), On the convection of a passive scalar by a turbulent Gaussian velocity field, J. Stat. Phys. 63, 305–313.CrossRefGoogle Scholar
  18. [18]
    Majda A. (1993), Explicit inertial range renormalization theory in a model for turbulent diffusion, J. Stat. Physics 73, 515–542.CrossRefGoogle Scholar
  19. [19]
    Papanicolaou G.C. (1971), Wave propagation in one-dimensional random medium,, SIAM J. Appl. Math. 21, 13–18.Google Scholar
  20. [20]
    Piterbarg L. (1994), Short-correlation approximation in models of turbulent diffusion, this volume.Google Scholar
  21. [21]
    Saichev A.I., Woyczynski W.A. (1995a), Model description of passive tracer density fields in the framework of Burgers’ turbulence, in Non-linear Stochastic PDE’s: Burgers Turbulence and Hydrodynamic Limit, IMA Volume 77, Springer-Verlag, 167–192.Google Scholar
  22. [22]
    Saichev A.I., Woyczynski W.A. (1996), Density fields in Burgers’ and KdV-Burgers’ turbulence, SIAM J. Appl. Math. 56 (1), 1–36.Google Scholar
  23. [23]
    Zirbel C.L. (1993), Stochastic flows: dispersion of a mass distribution and Lagrangian observations of a random field, Princeton U Ph.D. Dissertation, 197–211.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1997

Authors and Affiliations

  • A. I. Saichev
    • 1
  • W. A. Woyczynski
    • 2
  1. 1.Radio Physics DepartmentNizhny Novgorod UniversityNizhny NovgorodRussia
  2. 2.Department of Statistics and Center for Stochastic and Chaotic Processes in Science and TechnologyCase Western Reserve UniversityClevelandUSA

Personalised recommendations