# American Institute of Mathematical Sciences

September  2014, 13(5): 1737-1757. doi: 10.3934/cpaa.2014.13.1737

## The Kolmogorov-Obukhov-She-Leveque scaling in turbulence

 1 Department of Mathematics, University of California, Santa Barbara

Received  September 2013 Revised  December 2013 Published  June 2014

We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced by She-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.
Citation: Björn Birnir. The Kolmogorov-Obukhov-She-Leveque scaling in turbulence. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1737-1757. doi: 10.3934/cpaa.2014.13.1737
##### References:
 [1] F. Anselmet, Y. Gagne, E. J. Hopfinger and R. A. Antonia, High-order velocity structure function sin turbulent shear flows, J. Fluid Mech., 14 (1984), 63-89. [2] 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. Kirehg¨assner and A. Mielke (eds), World Scientific, page 145-–157, 1995. [3] A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3d Euler and Navier-Stokes equation for uniformely rotation fluids, Eur. J. Mech. B/Fluids, 15 (1996), 08312. [4] O. E. Barndorff-Nilsen, Exponentially decreasing distributions for the logarithm of the particle size, Proc. R. Soc. London A, 353, (1977), 401-419. [5] O. E. Barndorff-Nilsen, Processes of normal inverse Gaussian type, Finance and Stochastics, 2 (1998), 41-68. doi: 10.1007/s007800050032. [6] O. E. Barndorff-Nilsen, P. Blaesild and Jurgen Schmiegel, A parsimonious and universal description of turbulent velocity increments, Eur. Phys. J. B, 41 (2004), 345-363. [7] R. Benzi, S. Ciliberto, C. Baudet, F. Massaioli, R. Tripiccione and S. Succi, Extended self-similarity in turbulent flow, Phys. Rev. E, 48 (1993), 401-417. [8] 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. [9] 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 (2010), 53-86. [10] B. Birnir, The Kolmogorov-Obukhov statistical theory of turbulence, J. Nonlinear Sci., 2013. DOI 10.1007/s00332-012-9164-z. [11] B. Birnir, The Kolmogorov-Obukhov Theory of Turbulence, Springer, New York, 2013. doi: 10.1007/978-1-4614-6262-0. [12] B. Dubrulle, Intermittency in fully developed turbulence: in log-Poisson statistics and generalized scale covariance, Phys. Rev. Letters, 73 (1994), 959-962. [13] U. Frisch, Turbulence Cambridge Univ. Press, Cambridge, 1995. [14] E. Hopf, Statistical hydrodynamics and functional calculus, J. Rat. Mech. Anal., 1 (1953), 87-123. [15] A. N. Kolmogorov, Dissipation of energy under locally istotropic turbulence, Dokl. Akad. Nauk SSSR, 32 (1941), 16-18. [16] A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Dokl. Akad. Nauk SSSR, 30 (1941), 9-13. doi: 10.1098/rspa.1991.0075. [17] 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 (1962), 82-85. [18] 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. doi: 10.1098/rspa.1991.0080. [19] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. [20] H. P. McKean, Turbulence without pressure: Existence of the invariant measure, Methods and Applications of Analysis, 9 (2002), 463-468. doi: 10.4310/MAA.2002.v9.n3.a10. [21] A. M. Obukhov, On the distribution of energy in the spectrum of turbulent flow, Dokl. Akad. Nauk SSSR, 32 (1941). [22] A. M. Obukhov, Some specific features of atmospheric turbulence, J. Fluid Mech., 13 (1962), 77-81. [23] B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, New York, 2005. [24] L. Onsager, Statistical hydrodynamics, Nuovo Cimento., 6 (1945), 279-287. [25] G. Da Prato, An Introduction of Infinite-Dimensional Analysis, Springer Verlag, New York, 2006. [26] Z-S She and E. Leveque, Universal scaling laws in fully developed turbulence, Phys. Rev. Letters, 72 (1994), 336-339. [27] Z-S She and E. Waymire, Quantized energy cascade and log-poisson statistics in fully developed turbulence, Phys. Rev. Letters, 74 (1995), 262-265. [28] Z-S She and Zhi-Xiong Zhang, Universal hierarchial symmetry for turbulence and general multi-scale fluctuation systems, Acta Mech Sin, 25 (2009), 279-294. [29] J. B. Walsh, An Introduction to Stochastic Differential Equations, Springer Lecture Notes, eds. A. Dold and B. Eckmann, Springer, New York, 1984.

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##### References:
 [1] F. Anselmet, Y. Gagne, E. J. Hopfinger and R. A. Antonia, High-order velocity structure function sin turbulent shear flows, J. Fluid Mech., 14 (1984), 63-89. [2] 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. Kirehg¨assner and A. Mielke (eds), World Scientific, page 145-–157, 1995. [3] A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3d Euler and Navier-Stokes equation for uniformely rotation fluids, Eur. J. Mech. B/Fluids, 15 (1996), 08312. [4] O. E. Barndorff-Nilsen, Exponentially decreasing distributions for the logarithm of the particle size, Proc. R. Soc. London A, 353, (1977), 401-419. [5] O. E. Barndorff-Nilsen, Processes of normal inverse Gaussian type, Finance and Stochastics, 2 (1998), 41-68. doi: 10.1007/s007800050032. [6] O. E. Barndorff-Nilsen, P. Blaesild and Jurgen Schmiegel, A parsimonious and universal description of turbulent velocity increments, Eur. Phys. J. B, 41 (2004), 345-363. [7] R. Benzi, S. Ciliberto, C. Baudet, F. Massaioli, R. Tripiccione and S. Succi, Extended self-similarity in turbulent flow, Phys. Rev. E, 48 (1993), 401-417. [8] 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. [9] 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 (2010), 53-86. [10] B. Birnir, The Kolmogorov-Obukhov statistical theory of turbulence, J. Nonlinear Sci., 2013. DOI 10.1007/s00332-012-9164-z. [11] B. Birnir, The Kolmogorov-Obukhov Theory of Turbulence, Springer, New York, 2013. doi: 10.1007/978-1-4614-6262-0. [12] B. Dubrulle, Intermittency in fully developed turbulence: in log-Poisson statistics and generalized scale covariance, Phys. Rev. Letters, 73 (1994), 959-962. [13] U. Frisch, Turbulence Cambridge Univ. Press, Cambridge, 1995. [14] E. Hopf, Statistical hydrodynamics and functional calculus, J. Rat. Mech. Anal., 1 (1953), 87-123. [15] A. N. Kolmogorov, Dissipation of energy under locally istotropic turbulence, Dokl. Akad. Nauk SSSR, 32 (1941), 16-18. [16] A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Dokl. Akad. Nauk SSSR, 30 (1941), 9-13. doi: 10.1098/rspa.1991.0075. [17] 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 (1962), 82-85. [18] 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. doi: 10.1098/rspa.1991.0080. [19] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. [20] H. P. McKean, Turbulence without pressure: Existence of the invariant measure, Methods and Applications of Analysis, 9 (2002), 463-468. doi: 10.4310/MAA.2002.v9.n3.a10. [21] A. M. Obukhov, On the distribution of energy in the spectrum of turbulent flow, Dokl. Akad. Nauk SSSR, 32 (1941). [22] A. M. Obukhov, Some specific features of atmospheric turbulence, J. Fluid Mech., 13 (1962), 77-81. [23] B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, New York, 2005. [24] L. Onsager, Statistical hydrodynamics, Nuovo Cimento., 6 (1945), 279-287. [25] G. Da Prato, An Introduction of Infinite-Dimensional Analysis, Springer Verlag, New York, 2006. [26] Z-S She and E. Leveque, Universal scaling laws in fully developed turbulence, Phys. Rev. Letters, 72 (1994), 336-339. [27] Z-S She and E. Waymire, Quantized energy cascade and log-poisson statistics in fully developed turbulence, Phys. Rev. Letters, 74 (1995), 262-265. [28] Z-S She and Zhi-Xiong Zhang, Universal hierarchial symmetry for turbulence and general multi-scale fluctuation systems, Acta Mech Sin, 25 (2009), 279-294. [29] J. B. Walsh, An Introduction to Stochastic Differential Equations, Springer Lecture Notes, eds. A. Dold and B. Eckmann, Springer, New York, 1984.
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