P H Y S I C A L R E V I E W L E T T E R S Theoretical Model for the Kramers-Moyal Description of Turbulence Cascades
Jahanshah Davoudi1,3 and M. Reza Rahimi Tabar 1,2 1Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5531, Tehran, Iran 2Department of Physics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran 3Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran 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 57, 1737 (1998)] in the limit
of the high Reynolds number. We show that the higher order (n $ 3 ) Kramers-Moyal coefficients
tend to zero and the velocity increments are evolved by the Fokker-Planck operator. Our results are
compatible with the phenomenological description, developed for explaining recent experiments by
R. Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)].
PACS numbers: 47.27.Ak, 05.40. – a, 47.27.Eq, 47.27.Gs The problem of scaling behavior of longitudinal velocity locity increments to be a Markovian process in terms of difference U u͑x1͒ 2 u͑x2͒ in turbulence and the prob- length scales. By fitting the observational data they have ability density function of U, i.e., P͑U͒, attracts a great succeeded in finding the different Kramers-Moyal (KM) deal of attention [1 – 7]. Statistical theory of turbulence coefficients, and they find that the approximations of the has been brought forward by Kolmogorov [8] and further third and fourth order coefficients tend to zero, whereas developed by others [9 – 12]. The approach is to model the first and second coefficients have well-defined limits.
turbulence using stochastic partial differential equations.
Then by addressing the implications dictated by [23] theo- Kolmogorov conjectured that the scaling exponents are rem they have gotten a Fokker-Planck evolution operator.
universal, independent of the statistics of large-scale fluc- As an evolution equation for the probability density func- tuations and the mechanism of the viscous damping, when tion of velocity increments, the Fokker-Planck equation the Reynolds number is sufficiently large. However, re- has been used to give information on the changing shape cently it has been found that there is a relation between of the distribution as a function of the length scale. By the probability distribution function (PDF) of velocity and using this strategy the information on the observed inter- those of the external force (see [13] for more details). In mittency of the turbulent cascade is verified. In their de- this direction, Polyakov [1] has recently offered a field the- scription and based on simplified assumptions on the drift oretic method to derive the probability distribution or den- and diffusion coefficients, they have considered two pos- sity of states in ͑1 1 1͒ dimensions in the problem of the sible scenarios in order to indicate that both the Kol- randomly driven Burgers equation [14,15]. In one dimen- mogorov 41 and 62 scalings are recovered as possible sion, turbulence without pressure is described by the Burg- behaviors in their phenomenological theory.
ers equation [see also [16] concerning the relation between In this paper we derive the Kramers-Moyal equation the Burgers equation and the Kardar-Parisi-Zhang (KPZ) from the Navier-Stokes equation and show how the higher equation]. In the limit of the high Reynolds number, using order (n $ 3) Kramers-Moyal coefficients tend to zero in the operator product expansion (OPE), Polyakov reduces the high Reynolds number limit. Therefore, we find the the problem of computation of correlation functions in the Fokker-Planck equation from first principles. We show inertial subrange, to the solution of a certain partial dif- that the breakdown of the Galilean invariance is respon- ferential equation [17,18]. Yakhot recently [13,19] gen- sible for the scale dependence of the Kramers-Moyal co- eralized the Polyakov approach in three dimensions and efficients. Finally, using the path-integral expression for found a closed differential equation for the two-point gen- the PDF we show how small-scale statistics is affected by erating function of the “longitudinal” velocity difference PDF’s in the large scale and this is confirmed by Lan- in the strong turbulence (see also [20] about the closed dau’s remark that the large-scale fluctuations of turbulence equation for the PDF of the velocity difference for two production in the integral range can invalidate the Kol- and three-dimensional turbulence without pressure). On the other hand, recently [21,22] from a detailed analysis Our starting point is the Navier-Stokes equations, of experimental data of a turbulent free jet, Friedrich and Pienke have been able to obtain a phenomenological de- vt 1 ͑v ? =͒v ෇ n=2v 2
1 f͑x, t͒ ,
scription of the statistical properties of a turbulent cascade using a Fokker-Planck equation. In other words, they have seen that the conditional probability density of velocityincrements satisfies the Chapman-Kolmogorov equation.
for the Eulerian velocity v͑x, t͒ and the pressure p with
Mathematically this is a necessary condition for the ve- viscosity n, in N dimensions. The force f͑x, t͒ is the
P H Y S I C A L R E V I E W L E T T E R S external stirring force, which injects energy into the a nontrivial effect in the dynamics of the NS equation.
system on a length scale L. More specifically, one can Proceeding to find a closed equation for the generating take, for instance, a Gaussian distributed random force, function of the longitudinal velocity difference, ˆ dissipation and pressure terms in Eq. (1) give contribu-tions, and the longitudinal part of the dissipation term ͗fx, t͒fx0, t0͒͘ ෇ k͑0͒d͑t 2 tkmn͑x 2 x0͒ , (2)
renormalizes the coefficient in front of O͑ 1 ͒ in the equa- Z [13]. Also, it generates a term with the order of and ͗fx, t͒͘ ෇ 0, where m, n ෇ x1, x2, . . . , xN . The
O͑U͒ which can be written in terms of ˆZ as l mn ͑r ͒ is normalized to unity at the origin and decays rapidly enough where r becomes larger into account all the possible terms and using the symme- or equal to the integral scale L.
try of the PDF, i.e., P͑U, r͒ ෇ P͑2U, 2r͒, the following The force-free NS equation is invariant under space-time translation, parity, and scaling transformation. Also it is invariant under the Galilean transformation, x ! x 1 Vt and y ! y 1 V , where V is the constant velocity of the moving frame. Both boundary conditions and forcing can the theory. Also we suppose that k violate some or all of the symmetries of the force-free NS ͑ri,j͒m͑ri,j͒n
equation. However, it is usually assumed that in the high kmn͑ri,j͒ ෇ k͑0͒ ͓1 2
Reynolds number flow all symmetries of the NS equa- 1 and ri,j xi 2 xj.
tion are restored in the limit r ! 0 and r ¿ h, where “single-point” probability density fixes the value of the h is the dissipation scale where the viscous effects be- coefficient C urms and the C term corresponds to the come important. This means that in this limit the root- breakdown of G invariance in the limited Polyakov’s mean square velocity fluctuations u are not invariant under the constant shift V , cannot en- In the limit r ! 0 the equation for the probability ter the relations describing moments of velocity difference.
Therefore, the effective equations for the inertial-range ve- locity correlation functions must have the symmetries of the original NS equation. For many years this assump- tion was the basis of turbulence theories. But based onthe recent understanding of turbulence, some of the con- Using the exact results S3 ෇ 2 er in the small scale straints on the allowed turbulence theories can be relaxed (e is the mean energy dissipation rate) one finds A ෇ [13]. Polyakov’s theory of the large-scale random force 31B , where B ෇ 2B driven Burgers turbulence [1] was based on the assump- Eq. (4) can be written as ≠r P ෇ ͑2≠UU 2 B0͒21 3 tion that weak small-scale velocity difference fluctuations ͓2͑A͞r͒≠UU 1 ͑urms͞L͒≠2UU͔P, and so its solution can (i.e., jy͑x 1 r͒ 2 y͑x͒j ø urms and r ø L), where L is obviously be written as a scalar-ordered exponential [23], the integral scale of the system, obey the G-invariant dy- Rr dr0LKM͑U,r0͒ namic equation, meaning that the integral scale and the P͑U, r͒ ෇ T ͓e 0 single-point urms induced by random forcing cannot enterthe resulting expression for the probability density. Ac- where LKM can be obtained formally by computing the in- cording to [13] it has been shown how the u verse operator. Using the properties of scale-ordered ex- the equation for the PDF and therefore breaks the G in- ponentials the conditional probability density will satisfy variance in the limited Polyakov’s sense. We are inter- the Chapman-Kolmogorov equation. Equivalently we de- ested in the scaling of the longitudinal structure function rive that the probability density and, as a result, the con- ditional probability density of velocity increments satisfy q ෇ ͓͗u͑x 1 r ͒ 2 u͑x͔͒q͘ ෇ ͗Uq͘, where u͑x͒ is the x component of the three-dimensional velocity field, and r is the displacement in the direction of the x axis. Let us de- ͓D͑n͒͑r, U͒P͔ , Z ෇ ͗elU͘. According to [13] in the spherical co- ordinates the advective term in Eq. (1) involves the terms where D͑n͒͑r, U͒ ෇ an Un 1 b O͑ ≠2 ˆZ ͒, O͑ ≠ ˆZ ͒, O͑ ≠ ˆZ ͒, O͑ ˆZ ͒ [20]. It is noted that the that the coefficients an and bn depend on A, B, advection contributions are accurately accounted for in the urms, and the integral length scale L which are given Z, but it is not closed due to the dissipation and pressure terms. Using Polyakov’s OPE approach, Yakhot bn ෇ ͑ urms ͒ m͑m21͒ has shown that the dissipation term can be treated easily while the pressure term has an additional difficulty. The pressure contribution leads to effective energy redistribu- mic length scale l ෇ ln͑ L ͒ which varies from zero tion between components of the velocity field, and it has to infinity as r decreases from L to h.
P H Y S I C A L R E V I E W L E T T E R S U, r͒ ෇ 2͑ A ͒ ˜ how the large scale l ! 0 Gaussian probability density can change its shape when going to small scales l ! The drift and diffusion coefficients for various scales of ` and consequently give rise to intermittent behavior.
l, determined in the theory of Yakhot, show the same Instead of working with the probability functional of functional form as the calculated coefficients from the velocity increments, the formal solution of Fokker-Planck equation, as a scale-ordered exponential [24], can be In comparison with the phenomenological theory of converted to an integral representation for the probability Friedrich and Pienke we are able to construct a KM equa- tion for velocity increments that is analytically derived from the Yakhot theory which is based on just general un- derlying symmetries and OPE conjecture. Furthermore, es2͓͒͞4g͑l͔͒f͑ ˜ this viewpoint on Eq. (4) gives the expressions for scale dependence of the coefficients in the KM equation. The important result is that scale dependent KM coefficients are ͓2a1͑l0͒ 1 2a2͑l0͔͒ dl0 and g1͑l͒ ෇ between the breakdown of G invariance and the scale de- U͒ is the probability measure in the integral length pendence of the KM coefficients in the equivalent theory.
scales ͑l ! 0͒ . We consider the Gaussian distribution The two unknown parameters A and B in the theory are U͒ Х e2m ˜U2 in the integral scale which is a reasonable reduced to 1 by fitting the j3 ෇ 1, so all the scaling ex- choice (experimental data show that up to third moments ponents and D͑n͒’s are described by one parameter, B.
the PDF in the integral scale is consistent with the Gauss- Considering the results in [13,21] on which the value ian distribution [13]), and we derive the dependence of of B is obtained, we have used the value B Х 20 and the variance of the probability density on the scale in the have calculated the numerical values of the KM coeffi- limit when the original distribution satisfies the condition cients. Ratios of the first three coefficients an and bn The result shows an exponential dependence are a3͞a2 ෇ 0.04, a4͞a2 ෇ 0.001, b3͞b2 ෇ 0.04, and such as m ! me2z , where z ෇ 3a tent picture with the shape change of probability measure ues of higher order coefficients we find that the series under the scale is that when l grows, the width decreases can be cut safely after the second term, and a good ap- and vice versa. Moreover, we should emphasize that the proximation for the evolution operator of velocity incre- shape change is somehow complex which gives some corrections in order O͑m2 ˜ the value of the parameter, B Х 20 is calculated numeri- limit, i.e., m ø 1. Starting with a Gaussian measure at cally in the limit of infinite Reynolds numbers. Using integral scales and using the calculated scale indepen- this value for the calculation of the numerical values of dent Fokker-Planck coefficients, we have numerically D͑2͒ we find that the contribution of scale depen- calculated the PDF’s for fully developed turbulence and dent terms is essentially negligible. As it is well known, Burgers turbulence in different length scales from which the Fokker-Planck description of probability measure is their plots in Figs. 1 and 2 are completely compatible equivalent with the Langevin description written as [23], with experimental and simulation results [13,21,22]. The ෇ ˜D͑1͒͑ ˜U, l͒ 1 ˜D͑2͒͑ ˜U, l͒ h͑l͒, where h͑l͒ is a extreme case of Burgers problem (i.e., B Х 0) shows the white noise and the diffusion term acts as a multiplica- ever localizing behavior as if in the limit of l ! ` goes tive noise. By considering the Ito prescription and using to a Dirac delta function which again is consistent with the path-integral representation of the Fokker-Planck equa- our knowledge about Burgers problem [6,13]. Clearly tion, we can give an expression for all the possible paths Eqs. (4) and (5) give the same result for the multifractal in the configuration space of velocity differences and thus exponent of structure function, i.e., Sn͑r͒ Х Anrjn is demonstrate the change of the measure under the change In summary, we have constructed a theoretical bridge between two recent theories involving the statistics of lon- gitudinal velocity increment in fully developed turbulence.
On the basis of the recent theory proposed by Yakhot weshowed that the probability density of longitudinal veloc- When calculating, the measure of the path integral is ity components satisfies a Kramers-Moyal equation which meaningful when some form of discretization is chosen encodes the Markovian property of these fluctuations in [23], but we have written it in a formal way.
a necessary way. We are able to give the exact form of Kramers-Moyal coefficients in terms of a basic parameter scale independent ones in the infinite Reynolds number in the Yakhot theory B. The qualitative behavior of drift limit, one can easily see that the transition functional can and diffusion terms are consistent with the experimental P H Y S I C A L R E V I E W L E T T E R S lieve that it would be possible to derive the Kramers-Moyaldescription for the statistics of energy dissipation [28].
We thank A. Aghamohamadi, B. Davoudi, R. Ejtehadi, M. Khorrami, A. Langari, and S. Rouhani for helpful [1] A. Polyakov, Phys. Rev. E 52, 6183 (1995).
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Schematic view of the logarithm of PDF in terms of different length scales. These graphs are numerically obtained [6] J. Bouchaud, M. Mezard, and G. Parisi, Phys. Rev. E 52,
from the integral representation of PDF at the Fokker-Planck approximation. The curves correspond with the scales L͞r ෇ [8] A. N. Kolmogorov, C. R. Acad. Sci. USSR 30, 301 (1941).
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[13] V. Yakhot, Phys. Rev. E 57, 1737 (1998).
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equation is not restricted to Polyakov’s specific approach.
[15] Z. She, E. Aurell, and U. Frisch, Commun. Math. Phys.
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might be generalized to give a theoretical basis for the Mar- [18] S. Boldyrev, Phys. Rev. E 55, 6907 (1997); hep-th/
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[26] J. Davoudi and M. R. Rahimi Tabar (to be published).
[27] R. Friedrich, K. Marzinzik, and A. Schmigel, in A Perspective Look at Nonlinear Media, edited by Jurgen Schematic view of the logarithm of PDF in the Parisi, Stefan C. Muller, and Walter Zimmermann, Lecture Burgers turbulence (B Х 0), in terms of different length Notes in Physics Vol. 503 (Springer-Verlag, Berlin, 1997), scales. These graphs are numerically obtained from the integral representation of PDF at the Fokker-Planck approximation.
[28] A. Naert, R. Friedrich, and J. Peinke, Phys. Rev. E 56,
The scales are L͞r ෇ 1.5, 2, 5, 10, and 20.


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