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The Invariant Measure and the Probability Density Function

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Book cover The Kolmogorov-Obukhov Theory of Turbulence

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Abstract

The invariant measure of the stochastic Navier–Stokes equation determines all the one-point statistics of turbulence, or the statistics of quantities defined at one point x in the flow. This quantity determines all the statistical properties of the turbulent velocity field, see [56], and in distinction to the nonlinear Navier–Stokes equation, the invariant measure satisfies a linear but a functional differential equation; see [56]. In fact Hopf [29] found a linear equation for the characteristic function (Fourier transform) of the invariant measure in 1952, but at that time methods for solving such an equation were not available. In Hopf’s equation the noise for fully developed turbulence was missing, but in Kolmogorov’s equation for the invariant measure the noise is always supplied. Since only the linearized Navier–Stokes equation (2.38) appears below in the Kolmogorov–Hopf equation for the invariant measure, we will think about the linearized Navier–Stokes equation as the infinite-dimensional Ito process, whose generator gives the Kolmogorov–Hopf equation. Thus associated with such an Ito process is a diffusion equation, a linear functional differential equation, that is the Kolmogorov–Hopf equation determining the invariant measure. We will now derive this equation. This will make clear how to compute the coefficients in the Kolmogorov–Hopf equation.

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References

  1. R. A. Adams. Sobolev Spaces. Academic Press, New York, 1975.

    MATH  Google Scholar 

  2. F. Anselmet, Y. Gagne, E. J. Hopfinger, and R. A. Antonia. High-order velocity structure function sin turbulent shear flows. J. Fluid Mech., 14:63–89, 1984.

    Article  Google Scholar 

  3. A. Babin, A. Mahalov, and B. Nicolaenko. Long-time averaged Euler and Navier-Stokes equations for rotation fluids. In Structure and Dynamics of non-linear waves in Fluids, 1994 IUTAM Conference, K. Kirehgassner and A. Mielke (eds), World Scientific, page 145–157, 1995.

    Google Scholar 

  4. A. Babin, A. Mahalov, and B. Nicolaenko. Global splitting, integrability and regularity of 3d Euler and Navier-Stokes equation for uniformly rotation fluids. Eur. J. Mech. B/Fluids, 15(2):08312, 1996.

    Google Scholar 

  5. A. V. Babin and M. I Vishik. Attractors of Evolution Equations. Studies in Appl. Math and its Applic. vol. 25, North Holland Amsterdam, 1992.

    Google Scholar 

  6. O. E. Barndorff-Nielsen. Exponentially decreasing distributions for the logarithm of the particle size. Proc. R. Soc. London, A 353:401–419, 1977.

    Google Scholar 

  7. O. E. Barndorff-Nielsen. Processes of normal inverse Gaussian type. Finance and Stochastics, 2:41–68, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  8. O. E. Barndorff-Nielsen, P. Blaesild, and Jurgen Schmiegel. A parsimonious and universal description of turbulent velocity increments. Eur. Phys. J. B, 41:345–363, 2004.

    Google Scholar 

  9. G. K. Batchelor. The Theory of Homogenous Turbulence. Cambridge Univ. Press, New York, 1953.

    Google Scholar 

  10. P. S. Bernard and J. M. Wallace. Turbulent Flow. John Wiley & Sons, Hoboken, NJ, 2002.

    MATH  Google Scholar 

  11. R. Betchov and W. O. Criminale. Stability of Parallel Flows. Academic Press, New York, 1967.

    Google Scholar 

  12. R. Bhattacharya and E. C. Waymire. Stochastic Processes with Application. John Wiley, New York, 1990.

    Google Scholar 

  13. R. Bhattacharya and E. C. Waymire. A Basic Course in Probability Theory. Springer, New York, 2007.

    MATH  Google Scholar 

  14. P. Billingsley. Probability and Measure. John Wiley, New York, 1995.

    MATH  Google Scholar 

  15. B. Birnir. Turbulence of a unidirectional flow. Proceedings of the Conference on Probability, Geometry and Integrable Systems, MSRI, Dec. 2005 MSRI Publications, Cambridge Univ. Press, 55, 2007. Available at http://repositories.cdlib.org/cnls/.

  16. B. Birnir. Turbulent Rivers. Quarterly of Applied Mathematics, 66:565–594, 2008.

    MathSciNet  MATH  Google Scholar 

  17. B. Birnir. The Existence and Uniqueness and Statistical Theory of Turbulent Solution of the Stochastic Navier-Stokes Equation in three dimensions, an overview. Banach J. Math. Anal., 4(1):53–86, 2010. Available at http://repositories.cdlib.org/cnls/.

  18. B. Birnir. The Kolmogorov-Obukhov statistical theory of turbulence. To appear in Journal of Nonlinear Science, 2013. Available at http://repositories.cdlib.org/cnls/.

  19. S. Y. Chen, B. Dhruva, S. Kurien, K. R. Sreenivasan, and M. A. Taylor. Anomalous scaling of low-order structure functions of turbulent velocity. Journ. of Fluid Mech., 533:183–192, 2005.

    MathSciNet  MATH  Google Scholar 

  20. P. A. Davidson, Y. Kaneda, K. Moffatt, and K. R. Sreenivasan. A Voyage Through Turbulence. Cambridge Univ. Press, New York, 2012.

    Google Scholar 

  21. A. Debussche and C. Odasso. Markov solutions for the 3 d stochastic Navier-Stokes equations with state dependent noise. Journal of Evolution Equations, 6(2):305–324, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Dhruva. An experimental study of high-Reynolds-number turbulence in the atmosphere. Ph.D. Thesis Yale University, New Haven, CT, 2000.

    Google Scholar 

  23. B. Dubrulle. Intermittency in fully developed turbulence: in log-Poisson statistics and generalized scale covariance. Phys. Rev. Letters, 73(7):959–962, 1994.

    Article  Google Scholar 

  24. F. Flandoli and G. Gatarek. Martingale and stationary solutions for stochastic Navier-Stokes equations. Prob. Theory Rel. Fields, 102:367–391, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  25. C. Foias, O. Manley, R. Rosa, and R. Temam. Navier-Stokes Equations and Turbulence. Cambridge Univ. Press, Cambridge UK, 2001.

    Book  MATH  Google Scholar 

  26. U. Frisch. Turbulence. Cambridge Univ. Press, Cambridge, 1995.

    MATH  Google Scholar 

  27. M. Hairer and J. Mattingly. Ergodic properties of highly degenerate 2d stochastic Navier-Stokes equations. Comptes Rendus Mathématique. Académie des Sciences. Paris, 339(12):879–882, 2004.

    MathSciNet  MATH  Google Scholar 

  28. M. Hairer, J. Mattingly, and É. Pardoux. Malliavin calculus for highly degenerate 2d stochastic Navier-Stokes equations. Comptes Rendus Mathématique. Académie des Sciences. Paris, 339(11):793–796, 2004.

    MathSciNet  MATH  Google Scholar 

  29. E. Hopf. Statistical hydrodynamics and functional calculus. J. Rat. Mech. Anal., 1(1):87–123, 1953.

    MathSciNet  Google Scholar 

  30. S. Hou, B. Birnir, and N. Wellander. Derivation of the viscous Moore-Greitzer equation for aeroengine flow. Journ. Math. Phys., 48:065209, 2007.

    Article  MathSciNet  Google Scholar 

  31. B.R. Hunt, T. Sauer, and J.A. Yorke. Prevalence: A translation-invariant “almost every” on infinite-dimensional spaces. Bull. of the Am. Math. Soc., 27(2):217–238, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Kato. Perturbation Theory for Linear Operators. Springer, New York, 1976.

    Book  MATH  Google Scholar 

  33. J. F. C. Kingman. Poisson Processes. Clarendon Press, Oxford, 1993.

    MATH  Google Scholar 

  34. A. N. Kolmogorov. Dissipation of energy under locally isotropic turbulence. Dokl. Akad. Nauk SSSR, 32:16–18, 1941.

    MATH  Google Scholar 

  35. A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR, 30:9–13, 1941.

    Google Scholar 

  36. A. N. Kolmogorov. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech., 13:82–85, 1962.

    Article  MathSciNet  MATH  Google Scholar 

  37. R. H. Kraichnan. The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech., 5:497–543, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  38. R. H. Kraichnan. Lagrangian-history closure approximation for turbulence. Phys. Fluids, 8:575–598, 1965.

    Article  MathSciNet  Google Scholar 

  39. R. H. Kraichnan. Inertial ranges in two dimensional turbulence. Phys. Fluids, 10:1417–1423, 1967.

    Article  Google Scholar 

  40. R. H. Kraichnan. Turbulent cascade and intermittency growth. In Turbulence and Stochastic Processes, eds. J. C. R. Hunt, O. M. Phillips and D. Williams, Royal Society, pages 65–78, 1991.

    Google Scholar 

  41. S. Kuksin and A. Shirikyan. A coupling approach to randomly forced nonlinear pdes. Comm. Math. Phys., 221:351–366, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 63(3):193–248, 1934.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Lunch and J. Sethuraman. Large deviations for processes with independent increments. The Annals of Probability, 15(2):610–627, 1987.

    Article  MathSciNet  Google Scholar 

  44. H. P. McKean. Turbulence without pressure: Existence of the invariant measure. Methods and Applications of Analysis, 9(3):463–468, 2002.

    MathSciNet  MATH  Google Scholar 

  45. J. Milnor. On the concept of attractor. Communications in Mathematical Physics, 99:177–195, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  46. A. S. Momin and A. M. Yaglom. Statistical Fluid Mechanics, volume 1. MIT Press, Cambridge, MA, 1971.

    Google Scholar 

  47. A. S. Momin and A. M. Yaglom. Statistical Fluid Mechanics, volume 2. MIT Press, Cambridge, MA, 1975.

    Google Scholar 

  48. M. Nelkin. Turbulence in fluids. Am. J. Phys., 68(4):310–318, 2000.

    Article  Google Scholar 

  49. A. M. Obukhov. On the distribution of energy in the spectrum of turbulent flow. Dokl. Akad. Nauk SSSR, 32:19, 1941.

    Google Scholar 

  50. A. M. Obukhov. Some specific features of atmospheric turbulence. J. Fluid Mech., 13:77–81, 1962.

    Article  MathSciNet  Google Scholar 

  51. B. Oksendal. Stochastic Differential Equations. Springer, New York, 1998.

    Book  Google Scholar 

  52. B. Oksendal and A. Sulem. Applied Stochastic Control of Jump Diffusions. Springer, New York, 2005.

    Google Scholar 

  53. L. Onsager. The distribution of energy in turbulence. Phys. Rev., 68:285, 1945.

    Google Scholar 

  54. L. Onsager. Statistical hydrodynamics. Nuovo Cimento., 6(2):279–287, 1945.

    MathSciNet  Google Scholar 

  55. S. B. Pope. Turbulent Flows. Cambridge Univ. Press, Cambridge UK, 2000.

    Book  MATH  Google Scholar 

  56. G. Da Prato. An Introduction of Infinite-Dimensional Analysis. Springer Verlag, New York, 2006.

    Google Scholar 

  57. G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge UK, 1992.

    Book  MATH  Google Scholar 

  58. G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge UK, 1996.

    Book  MATH  Google Scholar 

  59. R. Renzi, S. Ciliberto, C. Baudet, F. Massaioli, R. Tripiccione, and S. Succi. Extended self-similarity in turbulent flow. Phys. Rev. E, 48(29):401–417, 1993.

    Google Scholar 

  60. O. Reynolds. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law resistance in parallel channels. Phil. Trans. Roy. soc. Lond., 174(11):935–982, 1883.

    MATH  Google Scholar 

  61. R.Temam. “Infinite-Dimensional Dynamical Systems in Mechanics and Physics”. Springer New York, 1988.

    Google Scholar 

  62. D. Ruelle. “Large volume limit of distribution of characteristic exponents in turbulence”. Comm. Math. Phys., 87:287–302, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  63. D. Ruelle. “Characteristic exponents for a viscous fluid subjected to time-dependent forces”. Comm. Math. Phys., 92:285–300, 1984.

    Article  MathSciNet  Google Scholar 

  64. Z-S She and E. Leveque. Universal scaling laws in fully developed turbulence. Phys. Rev. Letters, 72(3):336–339, 1994.

    Google Scholar 

  65. Z-S She and E. Waymire. Quantized energy cascade and log-poisson statistics in fully developed turbulence. Phys. Rev. Letters, 74(2):262–265, 1995.

    Google Scholar 

  66. Z-S She and Zhi-Xiong Zhang. Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems. Acta Mech Sin, 25:279–294, 2009.

    Article  Google Scholar 

  67. Y. Sinai. Burgers equation driven by a periodic stochastic flow. Ito’s Stochastic Calculus and Probability Theory, Springer New York, pages 347–353, 1996.

    Google Scholar 

  68. K. R. Sreenivasan and R. A. Antonia. The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech., 29:435–472, 1997.

    Article  MathSciNet  Google Scholar 

  69. K. R. Sreenivasan and B. Dhruva. Is there scaling in high- Reynolds-number turbulence? Prog. Theor. Phys. Suppl., 103–120, 1998.

    Google Scholar 

  70. G. I. Taylor. Statistical theory of turbulence. Proc. Royal Soc. London, 151:421–444, 1935.

    Article  MATH  Google Scholar 

  71. A. A. Townsend. The passage of turbulence through wire gauzes. Quart. J. Mech. Appl. Math., 4:308–320, 1951.

    Article  MATH  Google Scholar 

  72. A. A. Townsend. The Structure of Turbulent Flow. Cambridge Univ. Press, New York, 1976.

    MATH  Google Scholar 

  73. S. R. S. Varadhan. Large Deviations and Applications. SIAM, Philadelphia, PA, 1884.

    Google Scholar 

  74. M. I. Vishik and A. V. Fursikov. Mathematical Problems of Statistical Hydrodynamics. Kluwer, Dordrecht, Netherlands, 1988.

    Book  Google Scholar 

  75. J. B. Walsh. An Introduction to Stochastic Differential Equations. Springer Lecture Notes, eds. A. Dold and B. Eckmann, Springer, New York, 1984.

    Google Scholar 

  76. M. Wilczek. Statistical and Numerical Investigations of Fluid Turbulence. PhD Thesis, Westfälische Wilhelms Universität, Münster, Germany, 2010.

    Google Scholar 

  77. M. Wilczek, A. Daitche, and R. Friedrich. On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity. J. Fluid Mech., 676:191–217, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  78. H. Xu, N. T. Ouellette, and E. Bodenschatz. Multifractal dimension of Lagrangian turbulence. Phys. Rev. Letters, 96:114503, 2006.

    Article  Google Scholar 

  79. Y. Yang and D. I. Pullin. Geometric study of Lagrangian and Eulerian structures in turbulent channel flow. J. Fluid Mech., 674:6792, 2011.

    Article  MathSciNet  Google Scholar 

  80. Y. Yang, D. I. Pullin, and I. Bermejo-Moreno. Multi-scale geometric analysis of Lagrangian structures in isotropic turbulence. J. Fluid Mech., 654:233270, 2010.

    Article  MathSciNet  Google Scholar 

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    Björn Birnir

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Birnir, B. (2013). The Invariant Measure and the Probability Density Function. In: The Kolmogorov-Obukhov Theory of Turbulence. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6262-0_3

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