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  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  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  for more details). In
mittency of the turbulent cascade is verified. In their de-
this direction, Polyakov  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  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  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
͗fm͑x, t͒fn͑x0, t0͒͘ k͑0͒d͑t 2 t0͒kmn͑x 2 x0͒ , (2)
renormalizes the coefficient in front of O͑ 1 ͒ in the equa-
Z . Also, it generates a term with the order of
and ͗fm͑x, 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
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
. Polyakov’s theory of the large-scale random force
31B , where B 2B
driven Burgers turbulence  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 ,
the integral scale of the system, obey the G-invariant dy-
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  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  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 ͒ . 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 , 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,
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 ), 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 ,
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
, 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 .
We thank A. Aghamohamadi, B. Davoudi, R. Ejtehadi,
M. Khorrami, A. Langari, and S. Rouhani for helpful
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Schematic view of the logarithm of PDF in terms of
different length scales. These graphs are numerically obtained
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from the integral representation of PDF at the Fokker-Planck
approximation. The curves correspond with the scales L͞r
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we could find the form of path probability functional of
the velocity increments in scale which naturally encodes
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the scale dependence of probability density. This gives a
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equation is not restricted to Polyakov’s specific approach.
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conditional averaging methods [25,26]. A clearly analytic
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form of the KM coefficients D͑n͒ can be estimated numeri-
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cally but analytic derivation is not possible . Our work
Lett. A 212, 60 (1996).
might be generalized to give a theoretical basis for the Mar-
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kovian fluctuations of the moments of height difference in
the surface growth problems like KPZ [16,27], and we be-
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The scales are L͞r 1.5, 2, 5, 10, and 20.
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___________________Joseph P Walton, PhD____________________ CURRICULUM VITAE Joseph P. Walton, Ph.D. Personal information Address: Department of Communication Sciences and Disorders University of South Florida PCD 1007 Tampa, FL 33647 Phone (813) 974-6473 E-mail [email protected] Birth place Education 1976 B.A. (Hearing Science) Employment 9/84 to 8/90 Assistant Pr