Theoretical Model for the Kramers-Moyal Description of Turbulence Cascades

We derive the Kramers-Moyal equation for the conditional probability density of velocity increments from the theoretical model recently proposed by V.Yakhot [Phys.Rev.E {\bf 57}, 1737 (1998)] in the limit of high Reynolds number limit. We show that the higher order (n>=3) Kramers-Moyal coefficients tends to zero and the velocity increments are evolved by the Fokker-Planck operator. Our result is compatible with the phenomenological descriptions by R.Friedrich and J.Peinke [Phys.Rev.Lett. {\bf 78}, 863 (1997)].

[1]  V. Yakhot.  PROBABILITY DENSITY AND SCALING EXPONENTS OF THE MOMENTS OF LONGITUDINAL VELOCITY DIFFERENCE IN STRONG TURBULENCE , 1997, chao-dyn/9708016.

[2]  Joachim Peinke,et al.  FOKKER-PLANCK EQUATION FOR THE ENERGY CASCADE IN TURBULENCE , 1997 .

[3]  M. R. R. Tabar,et al.  Exact two-point correlation functions of turbulence without pressure in three dimensions , 1997, hep-th/9707173.

[4]  Velocity-difference probability density functions for Burgers turbulence , 1997 .

[5]  J. Peinke,et al.  Description of a Turbulent Cascade by a Fokker-Planck Equation , 1997 .

[6]  M. Chertkov,et al.  INSTANTON FOR RANDOM ADVECTION , 1996, chao-dyn/9606011.

[7]  Bouchaud,et al.  Velocity fluctuations in forced Burgers turbulence. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Statistical dependency of eddies of different sizes in turbulence , 1996 .

[9]  Migdal,et al.  Instantons in the Burgers equation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  M. R. R. Tabar,et al.  The exact N-point generating function in Polyakov-Burgers turbulence , 1995, hep-th/9507165.

[11]  Polyakov.  Turbulence without pressure. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  Parisi,et al.  Scaling and intermittency in Burgers turbulence. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  Peinke,et al.  Transition toward developed turbulence. , 1994, Physical review letters.

[14]  Y. Sinai,et al.  Statistics of shocks in solutions of inviscid Burgers equation , 1992 .

[15]  E. Aurell,et al.  The inviscid Burgers equation with initial data of Brownian type , 1992 .

[16]  Yakhot,et al.  Limiting probability distributions of a passive scalar in a random velocity field. , 1989, Physical review letters.

[17]  S. Orszag,et al.  Renormalization group analysis of turbulence. I. Basic theory , 1986 .

[18]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[19]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[20]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[21]  Paul C. Martin,et al.  Energy spectra of certain randomly-stirred fluids , 1979 .

[22]  S. Pope.  The probability approach to the modelling of turbulent reacting flows , 1976 .