Journal of Statistical Physics

, Volume 110, Issue 1–2, pp 87–136 | Cite as

Two Different Rapid Decorrelation in Time Limits for Turbulent Diffusion

  • Peter R. Kramer


A turbulent diffusion model in which the velocity field is Gaussian and rapidly decorrelating in time (GRDT) has been widely used recently in an endeavor to understand the emergence of anomalous scaling behavior of physical fields in fluid mechanics from the underlying stochastic partial differential equations. The utility of the GRDT model is the fact that correlation functions of the passive scalar field solve closed partial differential equations; the usual moment closure obstacle is averted. We study here the sense in which the GRDT model describes turbulent diffusion by a general, non-Gaussian velocity field with nontrivial temporal structure in the limit in which the correlation time of the velocity field is taken to zero. When the velocity field is rescaled in a particular manner in this rapid decorrelation limit, then a limit theorem of Khas'minskii indeed shows that the passive scalar statistics are described asymptotically by the GRDT Model for a broad class of velocity field models. We provide, however, an explicit example of a “Poisson blob model” velocity field which has two different well-defined rapid decorrelation in time limits. In one, the passive scalar correlation functions converge to those of the GRDT Model, and in the other, they converge to a distinct nontrivial limit in which the correlation functions do not solve closed PDE's. We provide both mathematical and heuristic explanations for the differences between these two limits. The conclusion is that the GRDT Model provides a universal description of the rapid decorrelation in time limit of general non-Gaussian velocity field models only when the velocity field is rescaled in a particular manner during the limit process.

turbulent diffusion Kraichnan model Poisson process convergence of probability measures Levy–Khinchine theorem Feynman–Kac formula 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Lesieur, Turbulence in Fluids, Chap. 7, pp. 161-163, No. 1 in Fluid Mechanics and Its Applications (Kluwer, Dordrecht, (1990), 2nd revised edn.Google Scholar
  2. 2.
    W. D. McComb, The physics of fluid turbulence, Oxford Engineering Science Series, Vol. 25, Chap. 2.2.1 (Clarendon Press, New York, 1991).Google Scholar
  3. 3.
    R. A. Antonia and B. R. Pearson, Scaling exponents for turbulent velocity and temperature increments, Europhys. Lett. 40(2):123-128 (1997).Google Scholar
  4. 4.
    S. Grossman, D. Lohse, and A. Reeh, Different intermittency for longitudinal and transversal turbulent fluctuations, Phys. Fluids 9(12):3817-3825 (1997).Google Scholar
  5. 5.
    K. R. Sreenivasan and R. A. Antonia, The phenomenology of small-scale turbulence, in Annual review of fluid mechanics, Vol. 29, Vol. 29 of Annu. Rev. Fluid Mech., pp. 435-472 (Annual Reviews, Palo Alto, California, 1997).Google Scholar
  6. 6.
    R. H. Kraichnan, Small-scale structure of a scalar field convected by turbulence, Phys. Fluids 11(5):945-953 (1968).Google Scholar
  7. 7.
    A. P. Kazantsev, Enhancement of a magnetic field by a conducting fluid, Sov. Phys. JETP 26:1031(1968).Google Scholar
  8. 8.
    S. A. Molchanov, Ideas in the theory of random media, Acta Applicandae Math. 22:139-282 (1991).Google Scholar
  9. 9.
    S. A. Molchanov, A. A. Ruzmaikin, and D. D. Sokoloff, Dynamo equations in a random short-term correlated velocity field, Magnitnaja Gidrodinamika 4:67-73 (1983) [in Russian].Google Scholar
  10. 10.
    A. J. Majda, Explicit inertial range renormalization theory in a model for turbulent diffusion, J. Statist. Phys. 73:515-542 (1993).Google Scholar
  11. 11.
    R. H. Kraichnan, Anomalous scaling of a randomly advected passive scalar, Phys. Rev. Lett. 72(7):1016-1019 (1994).Google Scholar
  12. 12.
    D. Bernard, K. Gawęedzki, and A. Kupiainen, Slow modes in passive advection, J. Statist. Phys. 90(3/4):519-569 (1998).Google Scholar
  13. 13.
    M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive scalar, Phys. Rev. E 52(5):4924-4941 (1995).Google Scholar
  14. 14.
    A. L. Fairhall, B. Galanti, V. S. L'vov, and I. Procaccia, Direct numerical simulations of the Kraichnan model: Scaling exponents and fusion rules, Phys. Rev. Lett. 79(21)Google Scholar
  15. 15.
    U. Frisch, A. Mazzino, and M. Vergassola, Intermittency in passive scalar advection, Phys. Rev. Lett. 80 (25): 5532-535 (1998).Google Scholar
  16. 16.
    O. Gat and R. Zeitak, Multiscaling in passive scalar advection as stochastic shape dynamics, Phys. Rev. E 57(5):5511-5519 (1998).Google Scholar
  17. 17.
    A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling and physical phenomena, Phys. Rep. 314(4-5):237-574 (1999).Google Scholar
  18. 18.
    J. C. Bronski and R. M. McLaughlin, Scalar intermittency and the ground state of periodic Schrödinger equations, Phys. Fluids 9 (1):181-190 (1997).Google Scholar
  19. 19.
    J. C. Bronski and R. M. McLaughlin, The problem of moments and the Majda model for scalar intermittency, Phys. Lett. A 265:257-263 (2000).Google Scholar
  20. 20.
    M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Statistics of a passive scalar advected by a large-scale two-dimensional velocity field: Analytic solution, Phys. Rev. E 51 (6):5609-5627 (1995).Google Scholar
  21. 21.
    A. J. Majda, The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions, Phys. Fluids A 5(8):1963-1970 (1993).Google Scholar
  22. 22.
    R. M. McLaughlin and A. J. Majda, An explicit example with non-Gaussian probability distribution for nontrivial scalar mean and fluctuation, Phys. Fluids 8(2):536(1996).Google Scholar
  23. 23.
    B. I. Shraiman and E. D. Siggia, Lagrangian path integrals and fluctuations in random flow, Phys. Rev. E 49(4):2912-2927 (1994).Google Scholar
  24. 24.
    V. I. Klyatskin, W. A. Woyczynski, and D. Gurarie, Short-time correlation approximations for diffusing tracers in random velocity fields: A functional approach, in Stochastic Modelling in Physical Oceanography, Vol. 39 of Progr. Probab., pp. 221-269 (Birkhäuser Boston, Boston, 1996).Google Scholar
  25. 25.
    A. I. Saichev and W. A. Woyczynski, Probability distributions of passive tracers in randomly moving media, in Stochastic models in geosystems, S. A. Molchanov, ed., IMA Volumes in Mathematics and Its Applications (Springer-Verlag, Berlin, 1996).Google Scholar
  26. 26.
    L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil'ev, Renormalization group, operator product expansion, and anomalous scaling in a model of advected passive scalar, Phys. Rev. E (3) 58(2, part A):1823-1835 (1998).Google Scholar
  27. 27.
    A. L. Fairhall, O. Gat, V. L'vov, and I. Procaccia, Anomalous scaling in a model of passive scalar advection: Exact results, Phys. Rev. E 53(4A):3518-3535 (1996).Google Scholar
  28. 28.
    K. Gawęedzki and A. Kupiainen, Universality in turbulence: An exactly solvable model, in Low-Dimensional Models in Statistical Physics and Quantum Field Theory (Schladming, 1995), Lecture Notes in Phys., Vol. 469, pp. 71-105 (Springer, Berlin, 1996).Google Scholar
  29. 29.
    T. C. Lipscombe, A. L. Frenkel, and D. ter Haar, On the convection of a passive scalar by a turbulent Gaussian velocity field, J. Statist. Phys. 63(1/2):305-313 (1991).Google Scholar
  30. 30.
    H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, No. 24 (Cambridge University Press, Cambridge, United Kingdom, 1990).Google Scholar
  31. 31.
    P. R. Kramer, Passive Scalar Scaling Regimes in a Rapidly Decorrelating Turbulent Flow, Ph.D. thesis (Princeton University, 1997).Google Scholar
  32. 32.
    C. L. Zirbel and E. ÇCinlar, Mass transport by Brownian flows, in Stochastic Models in Geosystems, S. A. Molchanov, ed., IMA Volumes in Mathematics and Its Applications (Springer-Verlag, Berlin, 1996).Google Scholar
  33. 33.
    A. J. Majda, Random shearing direction models for isotropic turbulent diffusion, J. Statist. Phys. 25(5/6):1153-1165 (1994).Google Scholar
  34. 34.
    M. Avellaneda and A. J. Majda, Mathematical models with exact renormalization for turbulent transport, II: Fractal interfaces, non-Gaussian statistics and the sweeping effect, Comm. Pure Appl. Math. 146:139-204 (1992).Google Scholar
  35. 35.
    M. Avellaneda and A. J. Majda, Mathematical models with exact renormalization for turbulent transport, Comm. Pure Appl. Math. 131:381-429 (1990).Google Scholar
  36. 36.
    L. Ts. Adzhemyan and N. V. Antonov, Renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow, Phys. Rev. E (3) 58 (6, part A):7381-7396 (1998).Google Scholar
  37. 37.
    D. Bernard, K. Gawęedzki, and A. Kupiainen, Anomalous scaling in the N-point functions of passive scalar, Phys. Rev. E 54(3):2564-2572 (1996).Google Scholar
  38. 38.
    M. Chertkov and G. Falkovich, Anomalous scaling exponents of a white-advected passive scalar, Phys. Rev. Lett. 76(15):2706-2709 (1996).Google Scholar
  39. 39.
    U. Frisch, A. Mazzino, A. Noullez, and M. Vergassola, Lagrangian method for multiple correlations in passive scalar advection, Phys. Fluids 11(8): 2178-2186 (1999), The International Conference on Turbulence (Los Alamos, New Mexico, 1998).Google Scholar
  40. 40.
    K. Gawęedzki and M. Vergassola, Phase transition in the passive scalar advection, Phys. D 138(1/2):63-90 (2000).Google Scholar
  41. 41.
    A. Pumir, B. I. Shraiman, and E. D. Siggia, Perturbation theory for the δ‐correlated model of passive scalar advection near the Batchelor limit, Phys. Rev. E 55(2):R1263-R1266 (1997).Google Scholar
  42. 42.
    A. C. Fannjiang, Phase diagram for turbulent transport: Sampling drift, eddy diffusivity and variational principles, Phys. D 136(1/2):145-174 (2000).Google Scholar
  43. 43.
    H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Comm. Math. Phys. 65(2):97-128 (1979).Google Scholar
  44. 44.
    I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Chap. 17 (Wolters-Noordhoff Publishing, Groningen, The Netherlands, 1971).Google Scholar
  45. 45.
    I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Section 1.1, 2nd edn. (Springer-Verlag, New York, 1991).Google Scholar
  46. 46.
    T. Fujiwara and H. Kunita, Limit theorems for stochastic difference-differential equations, Nagoya Math. J. 127:83-116 (1992).Google Scholar
  47. 47.
    R. Z. Khas'minskii, On stochastic processes defined by differential equations with a small parameter, Theor. Probability Appl. 11(2):211-228 (1966).Google Scholar
  48. 48.
    B. Øksendal, Stochastic Differential Equations, Universitext (Springer-Verlag, Berlin, 1998), fifth edn., An introduction with applications.Google Scholar
  49. 49.
    S. A. Molchanov and L. I. Piterbarg, Averaging in turbulent diffusion problems, in Probability Theory and Random Processes, pp. 35-47 (Kijev, Naukova Dumka, 1987) [in Russian].Google Scholar
  50. 50.
    R. A. Carmona and J. P. Fouque, Diffusion-approximation for the advection-diffusion of a passive scalar by a space-time Gaussian velocity field, in Seminar on stochastic analysis, random fields and applications, E. Bolthausen, M. Dozzi, and F. Russo, eds., Progress in Probability, Vol. 36, pp. 37-49 (Basel, 1995) (Centro Stefano Franscini, Birkhäuser Verlag).Google Scholar
  51. 51.
    S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Section 16.1 (Academic Press, Boston, 1981).Google Scholar
  52. 52.
    A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results (Springer-Verlag, Berlin, 1987).Google Scholar
  53. 53.
    P. Billingsley, Probability and Measure, 3rd edn. (Wiley, New York/London/Sydney, 1995).Google Scholar
  54. 54.
    A. Friedman, Partial Differential Equations of Parabolic Type, Chapter 1 (Prentice–Hall, Englewood Cliffs, New Jersey, 1964).Google Scholar
  55. 55.
    O. A. Ladyžzhenskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Vol. 23 of Translations of Mathematical Monographs, Chap. IV (American Mathematical Society, Providence, Rhode Iland, 1968).Google Scholar
  56. 56.
    A. Friedman, Stochastic Differential Equations and Applications, Vol. 2 (Academic Press, New York, 1976).Google Scholar
  57. 57.
    M. Chertkov, Instanton for random advection, Phys. Rev. E 55(3):2722-2735 (1997).Google Scholar
  58. 58.
    E. M. Stein, Harmonic Analysis—Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Chap. 6 (Princeton University Press, 1993).Google Scholar
  59. 59.
    D. W. Stroock, Probability Theory, an Analytic View (Cambridge University Press, Cambridge, United Kingdom, 1993).Google Scholar
  60. 60.
    P. Billingsley, Convergence of Probability Measures (Wiley, New York/London/Sydney, 1968).Google Scholar
  61. 61.
    H. L. Royden, Real Analysis, 3rd edn. (MacMillan, New York, 1988).Google Scholar
  62. 62.
    I. M. Gel'fand and N. Ya. Vilenkin, Generalized Functions. Applications of Harmonic Analysis, Vol. 4 (Academic Press, New York, 1964).Google Scholar
  63. 63.
    A. Friedman, Stochastic Differential Equations and Applications, Vol. 1 (Academic Press, New York, 1975).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Peter R. Kramer
    • 1
  1. 1.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroy

Personalised recommendations