Table Of ContentKolmogorov scaling from random force fields
Mogens H. Jensen 1, Kim Sneppen 1 and Luiza Angheluta 2∗
1 Niels Bohr Institute, Blegdamsvej 17, Dk 2100, Copenhagen, Denmark
2 Center for Physics of Geological Processes, Univ. of Oslo, Oslo, Norway†
(Dated: February 3, 2008)
We show that the classical Kolmogorov and Richardson scaling laws in fully developed turbu-
lence are consistent with a random Gaussian force field. Numerical simulations of a shell model
approximation to the Navier-Stokes equations suggest that the fluctuations in the force (acceler-
ation) field are scale independent throughout the inertial regime. We conjecture that Lagrangian
statisticsoftherelativevelocityinaturbulentflowisdeterminedbythetypicalforcefield,whereas
themultiscaling is associated to extremeeventsin theforce field fluctuations.
8
0 In studies of fully developed turbulence, two discov- ∆r(t)= r (t) − r (t)
0 1 2
eries are highly noticeable as fundamental and seminal.
2
OneregardsL.R.Richardson’sstudyoftheenhanceddis-
n
persionofparticlesadvectedbyaturbulentflow[1]. The
a
v (t)
J other result is Kolmogorov’sfundamental derivation, es- 1
sentially based on dimensional arguments, of the energy
4 v (t)
2 spectruminfullydevelopedturbulence[2]. Boththeories 2
employ the energy cascade,from the integralscale down
] tothedissipationscale,astheparadigmaticphysicalpic- t=0
D
tureoftheenergydissipationflow. Indeed,pair-particles
C passively advected by turbulence exhibit a superdiffu-
. sive behavior with their relative distance given by the
n
li Rpeicrshiaornd,stohne’smscoatiloinngof[3a],s[i4n]g.leInpcaorntitcrlaesitstdoettherempianierddbisy- d ∆ r = v(r (t)) − v(r (t )) = ∆v (t)
n dt 1 2
[ the correlationtime ofthe underlyingvelocityfield, such d ∆ v
= F(r (t)) − F(r (t))
that it is transported ballistically for times smaller than dt 1 2
1
thecorrelationtimeanddiffusesnormallyforlargertime
v
7 scales [5]. In the cascade scenario, the pair-particle su- FIG.1: Twoparticlesbeingadvectedinarandomforcefield,
1 perdiffusionisduetothelargeenergyjumpsbetweenthe generated by theGOY shell model.
7 eddies containing each particle. However,in the velocity
3 space no superdiffusive behavior is needed to substanti-
1. ate this jumpy motion. r1(t) andr2(t), and∆F(t)=F1(t)−F2(t) isthe relative
0 In this Letter, we show that the velocity increments force. The prefactor 4 instead of the usual factor 2 ap-
8 generated by a white-noise force field are sufficient to pearingintheforcecorrelationisduetotheparametriza-
0 generate the superdiffusive behavior, as well as the Kol- tion of the relative dispersion in terms of the diffusion
v: mogorov energy spectrum. To put it in very simple constant ǫ∗ for a single particle in the velocity space.
i terms: integrating ’up’ from the random acceleration The δ−function may have, in principle, a finite width
X field to the velocity field and subsequently to the dis- givenbythe time correlationofthe relativerandomfield
r placementisenoughtoreproducethewell-knownscaling alongthetwotrajectories. Thiswidthwillbedetermined
a
laws. both by the time it takes to pass a correlationlength for
a givenforce realization,and the time it takes to change
To clarify the underlying physicalpicture, we consider
the force in a certain point of the system.
asimplestochasticmodelofrelativedispersioninawhite
In this set up, the relative velocity field, ∆v(t) =
noise acceleration field given by
t
∆F(s)ds, is a Wiener process with a Gaussian dis-
0
d∆v Rtribution, namely
= ∆F(t) (1)
dt
∗
h∆F(t′)∆F(t”)i = 4ǫ∗ δ(t′−t”), (2) h∆v(t1)∆v(t2)i=4ǫ min(t1,t2), (3)
while the relative separation is described by a non-
where ∆v(t)=v1(t)−v2(t) is the velocity difference be- Gaussiandistributionwiththesecondmomentsatisfying
tweenthetwoparticlesmovingalongthetwotrajectories
the Richardson’s scaling, that is
∗
4ǫ
h∆r2(t)i= t3. (4)
3
∗Electronicaddress: [email protected],[email protected],[email protected]
†URL:http://cmol.nbi.dk Byeliminatingthetimedependenceoftherelativeveloc-
2
model [8, 9]. This model proposed originally by Glen-
zer,YamadaandOhkitani[10,11]providesadescription
of the turbulent motion embodied in the Navier-Stokes
equations. TheGOYmodelisformulatedonaN-discrete
setofwavenumbers,k =2n,withtheassociatedFourier
n
modes u evolving according to
n
d b
(dt +νkn2) un = ikn(anu∗n+1u∗n+2 + 2nu∗n−1u∗n+1 +
c
n ∗ ∗
4 un−1un−2) + fδn,1, (7)
for n = 1···N. The coefficients of the non-linear
terms are constrained by two conservation laws, namely
the total energy, E = |u |2, and the helicity (for
n n
3d), H = n(−1)nkn|uPn|, or the enstrophy (for 2d),
FIG. 2: The squared relative acceleration ∆F∆F and its in- Z = nkn2|Pun|2,intheinviscidlimit,i.e. f =ν =0[12].
finite moment versus ∆r2. The thin lines are for the La- TherPefore, they may be expressed in terms of a free pa-
grangian trajectories where distances and accelerations are rameter only δ ∈ [0,2], an = 1, bn+1 = −δ, cn+2 =
−(1 − δ). As observed by Kadanoff [13], one obtains
parameterized by the time of advection. The squares repre-
sent the corresponding Eulerian measures of the same quan- the canonical value ǫ = 1/2, when the 3d-helicity is
tities. conserved. The set (7) of N-coupled ordinary differen-
tialequationscanbe numericallyintegratedbystandard
techniques [14]. We have used standard parameters in
ityanddistance,weobtaintheexactKolmogorovscaling, this paper N =19,ν =10−6,k0 =2·10−4,f =5·10−3.
The GOY model is defined in k-space but we study
h∆v2(t)i=481/3ǫ∗2/3h∆r2(t)i1/3, (5) particledispersionindirectspaceobtainedbyaninverse
Fourier transform [8] of the form
in the Lagrangian framework (for the higher moments
see [6]). Thus,Kolmogorovscalingisconsistentwiththe N
assumption that the dispersion is driven by sufficiently ~v(~r,t)= ~c [u (t)ei~kn·~r+c.c.]. (8)
n n
random and uncorrelated acceleration fields. nX=1
InderivingEq.(5)weassumedthattherelativeveloc-
ity is obtained by following the Lagrangian trajectories, Here the wavevectors are~kn = kn~en where~en is a unit
vector in a randomdirection, for each shell n and~c are
whichinarealturbulentflowmaydifferfromthetypical n
velocityincrementsseparatedbythedistancer(Eulerian unit vectors in random directions. We ensure that the
velocityfieldisincompressible,∇·~v =0,byconstraining
measurement of the velocity differences) [7].
∗ ~c ·~e = 0, ∀n. In our numerical computations we con-
Eq. 4 implies that 2ǫ is the diffusion constant for the n n
siderthevectors~c and~e quenchedintimebutaveraged
relative velocity. For a Lagrangian stochastic flow gen- n n
∗ over many different realizations of these.
erated by the white noise acceleration field, ǫ can be
Asanexampleofthe motioninthisfield, Fig.1shows
estimated as
the trajectories of two passively advected particles. As
t the relative distance diverges in time, the two particles
h∆~v(t)∆F~(t)i = h ds∆F~(s)·∆F~(t)i
Z experience different force fields, which in turn typically
0
increase the difference in the relative velocities of the
t
= dsh∆F~(s)·∆F~(t)i two particles. The figure shows the individual particles
Z
0 as they are advected, first together and later diverging
t awayfrom eachother when they are encased in different
∗
= 12ǫ δ(t−s)ds=12ǫ , (6)
Z eddies.
0
Fig. 2 examines the noise in the effective force field
wheretheadditionalfactor3comparedtoeq.4isrelated h∆F∆Fifortherelativemotionofthetwoadvectedpar-
to the 3D system. From dimensional considerations, ǫ∗ ticles. In Fig 2a) we use viscosity ν =10−6, with a Kol-
has the same units [length2/time3] as the standard en- mogorov scale ∆r ∼ 1.0·10−4. The noise amplitude is
ergydissipationrateǫ characterizingthe turbulence cas- plotted versus the average square distance between the
cade. particles h∆r2i = h(r1(t) −r2(t))2i , with the time as
To examine how the Lagrangian white-noise accelera- parametrization of the curves, as in eq. 5. The average
tion relates to the anomalous scaling laws in a more re- is over independent trials of the two advected particles.
alistic turbulent field, we consider the kinematics of pair One observes that both the typical noise and the max-
particlesadvectedbythehomogeneousturbulentflowob- imum value at any distance is constant throughout the
tained by a real-space transformation of the GOY shell inertial range, i.e. above the Kolmogorov scale.
3
0
10
0
10
> >
v 10-2 v
∆ ∆ 10-1
v F
∆ ∆
< <
-4
10
-2
10
-9 -6 -3 0 -9 -6 -3 0
10 10 10 10 10 10 10 10
<∆r ∆r > <∆r ∆r >
FIG. 3: The squared relative velocity (∆v)2 and its infinite FIG. 4: A measure of an effective diffusion constant in ve-
moment versus (∆r)2 (ie a representation of the structure locity: h∆v∆Fi versus ∆r2. As in the previous figures the
function). The thin lines are for the Lagrangian trajectories thin line describes Lagrangian trajectories parametriced by
where distances and velocities are parametriced by the time the time of advection. The filled squares correspond to the
ofadvection. Thefilledsquaresrepresentarethecorrespond- Eulerian case.
ing Eulerian measures. The straight line represents standard
Kolmogorov scaling h∆v2i∝h∆r2i1/3.
structure functions shown in Fig. 3.
Using Eq. 6 we estimate h∆v∆Fi ∼ 0.1 through-
We conclude thatthe forcefield is equivalentto Gaus- out the inertial range in the GOY model simulations,
sian white noise, and therefore the structure function of see Fig. 4. This value of the effective velocity diffusion
this turbulent field should be close to the one predicted constant is larger than the averageenergy dissipation at
byEq. (5). ThisisconfirmedinFig. 3whereweshowthe the Kolmogorov scale, estimated from the energy input
∗
deviationsinvelocityasafunctionofthesquaredistance Rehu ·fi=0.001 in the GOY model. This discrepancy
1
between the particles. One indeed sees that h∆v2(t)i in the effective diffusion terms we attribute to the huge
versus h∆r2(t)i scales with an exponent close to 1/3 in contributions from the spikes in the acceleration which
agreement with our expectations. For completeness, we areabsentinthe simplewhite noisecalculationofEq. 6.
alsoshowtheinfinitemomentofthevelocity,andremark Thesespikesalsogivesrisetomultidiffusion,asdiscussed
thatthishighermomentscaleswithanexponentcloseto above.
0.23. This signals multidiffusion [15] where extreme ve- We believe that the accelerationfield as shown in Fig.
locitydifferencessometimes,butrarely,arereachedafter 2 should be experimentally accessible either by particle
short separations. In the current context, we see these tracking in a 3-D flow [16] or from probe measurements
extreme deviations as a measure of very unlikely and in- in channel flows employing the Taylor hypothesis. In
termittent events which only addlittle to the typicalbe- the first case the acceleration is easily estimated from
havior of the flow. Indeed also the Eulerian statistics the temporal variations in the velocity field of the 3-D
shows clear multiscaling as expected [8]. advected particles.
While our intuition has been basedon the Lagrangian Overallwehaveseenthattheforcefieldreachesanav-
picture of advected particles, it is remarkable that erage value that is independent on the distance between
the corresponding Eulerian quantities behaves similarly. the advected points in the turbulent fluid. Already at
Thisisdemonstratedinsimulationswherewenowfixthe distances slightly above the Kolmogorov scale the two
distance between two points, and then calculate respec- particlesoften receiverandom“kicks”whichareas large
tively the difference in velocity and acceleration. The at small scales as they are at the integral scale. Thus,
continuous curves in Figs. 2 and 3 show how h∆F2(r)i hugeaccelerationsareassociatedtotheverysmallscales,
and h∆v2(r)i vary with the square relative distance be- presumablytothe coreofeddies atthe vergeoftheirde-
tween the investigated points. From Fig. 2 we see that struction by dissipation. The acceleration between two
thevalueoftheplateaufortherandomforcefieldisadi- particles are primarily dependent on how close each of
rect consequence of its random expectation at any large them are to the center of an eddy. When examining the
distance. Therefore, there is nothing special about the distribution of the accelerations at a fixed distance we
selection of advected points in the Lagrangian case. In observe a broad power law like behavior with a cutoff
facttheonsetoftheplateauisslightlysharperintheEu- which is independent of the distance (as demonstrated
leriancase,presumablyreflectingaveragingassociatedto by the constant max norm). The size of the cutoff is
the underlying time parameter in the Lagrangianadvec- determined by the size of the forcing and the scale at
tion. Similarly, there is no significant difference for the which this forcing is acting (in our simulations, the scale
4
is ∆r =1). the Kolmogorovscale.
In conclusion, the motion associated to the relatively
slowturnoverdynamicsofthelargeeddiesisnotneeded
for obtaining Richardson or Kolmogorov statistics. We are grateful to Hiizu Nakanishi, Simone Pigolotti
These two seminal laws are primarily a consequence of and Yves Pomeau for valuable discussions. We thank
the randomforce field that fluctuates with an amplitude the Danish National Research Foundation for support
set by the system size andwith a correlationtime set by through the Center for Models of Life.
[1] L.F. Richardson, Proceedings of the Royal Society of [9] T. Bohr and M.H. Jensen and G. Paladin and A. Vulpi-
London.Series A,110, 756, pp.709-737, 1926 ani, Cambridge University Press, Cambridge, 1998
[2] A.N. Kolmogorov, C.R. Acad. Sci. USSR 30, 301; ibid [10] E. B. Gledzer, Sov.Phys. Dokl. 18, 216 (1973).
32, 16 (1941). [11] M. Yamada and K. Ohkitani, J. Phys. Soc. Japan 56,
[3] G. Boffetta and I.M. Sokolov, Physics of fluids, 14, 9, 4210(1987); Prog. Theor. Phys.79,1265(1988).
2002 [12] L. Biferale, and R.M. Kerr, Phys.Rev. E 52, 6 (1995).
[4] G. Boffetta and I.M. Sokolov, PRL, 88, 9, 2002 [13] L. Kadanoff, D. Lohse, J. Wang, and R. Benzi, Phys.
[5] G. Falkovich and K. Gawedzki and M. Vergassola, Re- Fluids 7, 617 (1995).
views of Modern Physics, 73, 2001 [14] D. Pisarenko, L. Biferale, D. Courvoisier, U. Frisch, and
[6] Using Wick’s theorem we obtain the 2n-th moment M. Vergassola, Phys.Fluids A5, 10 (1993).
of the velocity difference at time t: h∆v2n(t)i = [15] K. Sneppen and M.H. Jensen, Phys.Rev.E 49, 919
(2n)!(48ǫ∗2)n/3h∆r2(t)in/3 (1994).
2nn!
[7] N.MordantandJ.DelourandE.L´evequeandO.Michel [16] B. Lu¨thi and J. Berg and S. Ott and J. Mann, Physics
andA.Arn´eodoandJ.-F.Pinton,J.ofStat.Phys.,113, of Fluids 19, 2007
5/6, 2003
[8] M.H. Jensen, Phys.Rev.Lett. 83, 76 (1999).
