Table Of ContentDRAFT for IUTAM symposium 1999 Version of February 8, 2008
Anomalous Scaling in Passive Scalar Advection and Lagrangian Shape Dynamics
Itai Arad and Itamar Procaccia
Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
Theproblemofanomalousscalinginpassivescalaradvection,especially withδ-correlatedveloc-
ityfield(theKraichnan model) hasattracted alot ofinterest sincetheexponentscan becomputed
analytically in certain limiting cases. In this paper we focus, rather than on the evaluation of the
exponents, on elucidating the physical mechanism responsible for the anomaly. We show that the
anomalous exponentsζn stem from theLagrangian dynamics of shapes which characterize configu-
0
rations of n pointsin space. Usingtheshape-to-shapetransition probability,we definean operator
0
whose eigenvalues determine the anomalous exponents for all n, in all the sectors of the SO(3)
0
2 symmetry group.
n
a
J
0
3
I. INTRODUCTION
]
D In the lecture at the IUTAM symposium the work of our groupon the consequences of anisotropyon the universal
C statistics of turbulence has been reviewed. This material is available in print, and the interested reader can find it in
. [1–6]. Ashortreviewisavailableintheproceedingsof“DynamicsDaysAsia”[7]. Inthispaperwereviewsomerecent
n
work aimed at understanding the physical mechanism responsible for the anomalous exponents that characterize the
i
l statistics of passive scalars advected by turbulent velocity fields. We will consider isotropic advecting velocity fields,
n
butwillallowanisotropyintheforcingofthepassivescalar. Insuchcasethestatisticalobjectslikestructurefunctions
[
and correlation functions are not isotropic. Instead, they are composed of an isotropic and non-isotropic parts. We
1 overcome this complication by characterizing these functions in terms of the SO(3) irreducible representations. Any
v
such function can be written as a linear combination of parts which belong to a given irreducible representation of
7
SO(3). We will show that each part is characterizedby a set of universal scaling exponents. The weight of each part
6
however will turn out to be non-universal, set by the boundary conditions.
0
1 The SO(3) classification will appear to be natural once we focus on the physics of Lagrangian trajectories in the
0 flow. Wewillseethatonecanofferasatisfactoryunderstandingofthephysicsofanomalousscalingbyconnectingthe
0 thestatisticsofthepassivescalartotheLagrangiantrajectories. Thisconnectionprovidesaveryclearunderstanding
0 of the physical origin of the anomalous exponents, relating them to the dynamics in the space of shapes of groups of
/
n Lagrangianparticles.
i
l
n
: II. THE KRAICHNAN MODEL OF PASSIVE SCALAR ADVECTION
v
i
X
The model of passive scalar advection with rapidly decorrelating velocity field was introduced by R.H. Kraichnan
r [8] already in 1968. In recent years [9–14] it was shown to be a fruitful case model for understanding multiscaling in
a
the statistical description of turbulent fields. The basic dynamical equation in this model is for a scalar field T(r,t)
advected by a random velocity field u(r,t):
∂ κ 2+u(r,t) ∇ T(r,t)=f(r,t) . (2.1)
t 0
− ∇ ·
(cid:2) (cid:3)
In this equation f(r,t) is the forcing and κ is the molecular diffusivity. In Kraichnan’s model the advecting field
0
u(r,t) aswellasthe forcingfieldf(r,t)aretakento be Gaussian,time andspace homogeneous,anddelta-correlated
in time:
(uα(r,t) uα(r′,t))(uβ(r,t′) uβ(r′,t′)) =hαβ(r r′)δ(t t′) , (2.2)
− − u − −
(cid:10) (cid:11)
where the “eddy-diffusivity” tensor hαβ(r) is defined by
r ξ ξ rαrβ
hαβ(r)= (δαβ ) , η r Λ . (2.3)
(cid:16)Λ(cid:17) − d 1+ξ r2 ≪ ≪
−
1
Hereη andΛ arethe inner andouterscaleforthe velocityfields,andthe coefficientsarechosensuchthat∂ hαβ =0.
α
The averaging ... is done with respect to the realizations of the velocity field.
u
h i
The forcing f is also taken white in time and Gaussian:
f(r,t)f(r′,t′) =Ξ(r r′)δ(t t′) . (2.4)
h if − −
Heretheaverageisdonewithrespecttorealizationsoftheforcing. Theforcingistakentoactonlyonthelargescales,
ofthe orderofL (with acompactsupportinFourier space). This means that the function Ξ(r) is nearlyconstantfor
r L but is decaying rapidly for r >L.
≪
¿From the point of view of the statistical theory one is interested mostly in the scaling exponents characterizing
the structure functions
S (r ,r )= (T(r ,t) T(r ,t))2n . (2.5)
2n 1 2 1 − 2 u,f
(cid:10) (cid:11)
For isotropic forcing one expects S to depend only on the distance R r r such that in the scaling regime
2n 1 2
≡| − |
L δ2n
S (R) Rζ2n [S (R)]n . (2.6)
2n 2
∝ ∝ (cid:18)R(cid:19)
In this equation we introduced the “normal” (nζ ) and the anomalous (δ ) parts of the scaling exponents ζ =
2 2n 2n
nζ δ . The firstpartcanbe obtainedfromdimensionalconsiderations,but the anomalouspartcannotbe guessed
2 2n
−
from simple arguments.
When the forcing is anisotropic, the structure functions depend on the vector distance R = r r . In this case
1 2
−
we can represent them in terms of spherical harmonics,
S (R)= a (R)Y (Rˆ) , (2.7)
2n ℓ,m ℓ,m
Xℓ,m
where Rˆ R/R. This is a case in which the statistical object is a scalar function of one vector, and the appropriate
≡
irreduciblerepresentationofthe SO(3)symmetrygroupareobvious. We aregoingto explaininthe nextsectionthat
(ℓ)
the coefficients a (R) are expected to scale with a universal leading scaling exponent ζ . The exponent will turn
ℓ,m 2n
out to be ℓ dependent but not m dependent.
Theoreticallyitisnaturaltoconsidercorrelationfunctionsratherthanstructurefunctions. The2n-ordercorrelation
functions are defined as
F (r ,..., r ) T(r )T(r )...T(r ) . (2.8)
2n 1 2n ≡h 1 2 2n iu,f
For separations r 0 the correlation functions converges to T2n , whereas for r L decorrelation leads to
ij → f ij →
convergence to T n . For all r O(r) L one expects a beh(cid:10)aviou(cid:11)r according to
h iu,f ij ≈ ≪
F (r ,..., r )=Ln(2−ξ)(c + +c (r/L)ζ2nF˜ (r˜ ,...,r˜ )+ ) , (2.9)
2n 1 2n 0 k 2n 1 2n
··· ···
where F˜ is a scaling function depending on r˜ which denote a set of dimensionless coordinates describing the
2n i
configuration of the 2n points. The exponents and scaling functions are expected to be universal, but not the c
coefficients, which depend on the details of forcing.
Ithasbeenshown[11]thattheanomalousexponentsζ canbeobtainedbysolvingforthezeromodesoftheexact
2n
differential equations which are satisfied by F . The equations for the zero modes read
2n
κ 2+ ˆ (r ,r ,...,r )=0 . (2.10)
− ∇i B2n F2n 1 2 2n
(cid:2) Xi (cid:3)
The operator ˆ 2n ˆ , and ˆ are defined by
B2n ≡ i>jBij Bij
P
ˆ ˆ(r ,r )=hαβ(r r )∂2/∂rα∂rβ . (2.11)
Bij ≡B i j i− j i j
2
III. LAGRANGIAN TRAJECTORIES, CORRELATION FUNCTIONS AND SHAPE DYNAMICS
An elegant approachto the correlationfunctions is furnished by Lagrangiandynamics [7,15–18]. In this formalism
one recognizes that the actual value of the scalar at position r at time t is determined by the action of the forcing
along the Lagrangiantrajectory from t= to t:
−∞
t0
T(r ,t )= dt f(r(t),t) , (3.1)
0 0 Z h iη
−∞
with the trajectory r(t) obeying
r(t )=r ,
0 0
∂ r(t)=u(r(t),t)+√2κη(t) , (3.2)
t
andη is a vectorofzero-meanindependent Gaussianwhite randomvariables, ηα(t)ηβ(t′) =δαβδ(t t′). With this
−
in mind, we can rewrite F2n by substituting each factor of T(ri) by its represe(cid:10)ntation(3.1(cid:11)). Performingthe averages
over the random forces, we end up with
t0
F (r ,...,r ,t )= dt dt Ξ(r (t ) r (t )) (3.3)
2n 1 2n 0 1 n 1 1 2 1
DZ−∞ ··· h − ···
Ξ(r (t ) r (t ))+permutations , (3.4)
2n−1 n 2n n
× − iEu,{ηi}
To understand the averaging procedure recall that each of the trajectories r obeys an equation of the form (3.2),
i
where u as wellas η 2n areindependent stochastic variableswhose correlationsare givenabove. Alternatively, we
{ i}i=1
refer the reader to section II of [18], where the above analysis is carried out in detail.
In considering Lagrangian trajectories of groups of particles, we should note that every initial configuration is
characterizedby a center of mass, say R, a scale s (say the radius of gyrationof the cluster of particles) and a shape
Z. In “shape” we mean here all the degrees of freedom other than the scale and R: as many angles as are needed to
fully determine a shape, in addition to the Euler angles that fix the shape orientationwith respect to a chosen frame
of coordinates. Thus a group of 2n positions r will be sometime denoted below as R,s,Z .
i
{ } { }
One component in the evolution of an initial configuration is a rescaling of all the distances which increase on the
average like t1/ζ2; this rescaling is analogous to Richardson diffusion. The exponent ζ2 which determines the scale
increase is also the characteristic exponent of the second order structure function [8]. This has been related to the
exponent ξ of (2.3) according to ζ = 2 ξ. After factoring out this overall expansion we are left with a normalized
2
−
‘shape’. It is the evolution of this shape that determines the anomalous exponents.
ConsiderafinalshapeZ withanoverallscales whichisrealizedatt=0. Thisshapehasevolvedduringnegative
0 0
times. We fixa scales>s andexaminethe shapewhenthe configurationreachesthe scalesforthe lasttime before
0
reaching the scale s . Since the trajectories are random the shape Z which is realized at this time is taken from a
0
distribution γ(Z;Z ,s s ). As long as the advecting velocity field is scale invariant, this distribution can depend
0 0
→
only on the ratio s/s .
0
Next, we use the shape-to-shape transition probability to define an operator γˆ(s/s ) on the space of functions Ψ(Z)
0
according to
[γˆ(s/s )Ψ](Z )= dZγ(Z;Z ,s s )Ψ(Z) (3.5)
0 0 0 0
Z →
Wewillbeinterestedintheeigenfunctionandeigenvaluesofthisoperator. Thisoperatorhastwoimportantproperties.
First,foranisotropicstatisticsofthe velocityfieldtheoperatorisisotropic. Thismeansthatthis operatorcommutes
with all rotation operators on the space of functions Ψ(Z). In other words, if is the rotation operator that takes
Λ
O
the function Ψ(Z) to the new function Ψ(Λ−1Z), then
γˆ =γˆ . (3.6)
Λ Λ
O O
ThispropertyfollowsfromtheobvioussymmetryoftheKernelγ(Z;Z ,s s )torotatingZ andZ simultaneously.
0 0 0
→
Accordingly the eigenfunctions of γˆ can be classified according to the irreducible representations of SO(3) symmetry
group. We will denote these eigenfunctions as B (Z). Here ℓ=0,1,2,..., m= ℓ, ℓ+1,...ℓ and q stands for a
qℓm
− −
running index if there is more than one representation with the same ℓ,m. The fact that the B (Z) are classified
qℓm
accordingtothe irreduciblerepresentationsofSO(3)inmanifestedinthe actionofthe rotationoperatorsuponthem:
3
OΛBqℓm = Dm(ℓ)′m(Λ)Bqℓm′ (3.7)
Xm′
where D(ℓ) (Λ) is the SO(3) ℓ ℓ irreducible matrix representation.
m′m ×
Thesecondimportantpropertyofγˆ followsfromtheδ-correlationintimeofthevelocityfield. Physicallythismeans
thatthefuturetrajectoriesofnparticlesarestatisticallyindependentoftheirtrajectoriesinthepast. Mathematically,
it implies for the kernel that
γ(Z;Z ,s s )= dZ γ(Z;Z ,s s )γ(Z ;Z ,s s ) , s>s >s (3.8)
0 0 1 1 1 1 0 1 0 1 0
→ Z → →
and in turn, for the operator, that
γˆ(s/s )=γˆ(s/s )γˆ(s /s ) . (3.9)
0 1 1 0
Accordingly, by a successive application of γˆ(s/s ) to an arbitrary eigenfunction, we get that the eigenvalues of γˆ
0
have to be of the form αq,ℓ =(s/s0)ζ2(qn,ℓ):
( s )ζ2(qn,ℓ)Bqℓm(Z0)= dZγ(Z;Z0,s s0)Bqℓm(Z) (3.10)
s Z →
0
Notice thatthe eigenvalues arenot a function ofm. This follows fromSchur’slemmas [19],but canbe also explained
from the fact that the rotation operator mixes the different m’s (3.7): Take an eigenfunction B (Z), and act on it
qℓm
oncewiththe operator γˆ(s/s )andoncewiththeoperatorγˆ(s/s ) . Byvirtueof(3.6)weshouldgetthatsame
Λ 0 0 Λ
O O
result, but this is only possible if all the eigenfunctions with the same ℓ and the same q share the same eigenvalue.
To proceed we want to introduce into the averaging process in (3.4) by averaging over Lagrangian trajectories of
the 2n particles. This willallowus to connectthe shape dynamics to the statisticalobjects. To this aimconsiderany
set of Lagrangiantrajectories that started at t= and end up at time t=0 in a configurationcharacterizedby a
−∞
scale s and center of mass R =0. A full measure of these have evolved through the scale L or larger. Accordingly
0 0
they must have passed, during their evolution from time t = through a configuration of scale s > s at least
0
−∞
once. Denote now
µ (t,R,Z;s s ,Z )dtdRdZ (3.11)
2n 0 0
→
as the probability that this set of 2n trajectories crossedthe scale s for the last time before reaching s ,Z , between
0 0
t and t+dt, with a center of mass between R and R+dR and with a shape between Z and Z+dZ.
In terms of this probability we can rewrite Eq.(3.4) (displaying, for clarity, R =0 and t=0) as
0
0
F (R =0,s ,Z ,t=0)= dZ dt dRµ (t,R,Z;s s ,Z )
2n 0 0 0 2n 0 0
Z Z Z →
−∞
0
dt dt Ξ(r (t ) r (t )) Ξ(r (t ) r (t ))+perms (s;R,Z,t) . (3.12)
1 n 1 1 2 1 2n−1 n 2n n
×(cid:28)Z−∞ ··· h − ··· − i(cid:12)(cid:12) (cid:29)u,ηi
(cid:12)
The meaningofthe conditionalaveragingisanaveragingoverallthe realizationsofthe velocityfieldandthe random
η for which Lagrangiantrajectories that ended up at time t=0 in R=0,s ,Z passed through R,s,Z at time t.
i 0 0
Next, the time integrations in Eq.(3.12) are split to the interval [ ,t] and [t,0] giving rise to 2n different contri-
−∞
butions:
t t 0 t t
dt dt + dt dt dt +... (3.13)
1 n 1 2 n
Z ···Z Z Z ···Z
−∞ −∞ t −∞ −∞
Consider firstthe contributionwith n integrals inthe domain [ ,t]. It follows fromthe delta-correlationin time of
−∞
the velocity field, that we can write
t
dt dt Ξ(r (t ) r (t )) Ξ(r (t ) r (t ))+perms (s;R,Z,t)
1 n 1 1 2 1 2n−1 n 2n n
(cid:28)Z−∞ ··· h − ··· − i(cid:12)(cid:12) (cid:29)u,ηi
t (cid:12)
= dt dt Ξ(r (t ) r (t )) Ξ(r (t ) r (t ))+perms
1 n 1 1 2 1 2n−1 n 2n n
(cid:28)Z−∞ ··· h − ··· − i(cid:29)u,η
i
=F (R,s,Z,t)=F (s,Z) . (3.14)
2n 2n
4
The last equality follows from translational invariance in space-time. Accordingly the contribution with n integrals
in the domain [ ,t] can be written as
−∞
0
dZF (s,Z) dt dR µ (t,R,Z;s s ,Z ) . (3.15)
2n 2n 0 0
Z Z Z →
−∞
We identify the shape-to-shape transition probability:
0
γ(Z;Z ,s s )= dt dR µ (t,R,Z;s s ,Z ) . (3.16)
0 0 2n 0 0
→ Z Z →
−∞
Finally, putting all this added wisdom back in Eq.(3.12) we end up with
F (s ,Z )=I + dZγ(Z;Z ,s s )F (s,Z) . (3.17)
2n 0 0 0 0 2n
Z →
Here I represents all the contributions with one or more time integrals in the domain [t,0]. The key point now is
that only the term with n integralsin the domain[ ,t] containsinformation aboutthe evolutionof 2n Lagrangian
−∞
trajectories that probed the forcing scale L. Accordingly, the term denoted by I cannot contain information about
the leading anomalous scaling exponent belonging to F , but only of lower order exponents. The anomalous scaling
2n
dependence of the LHS of Eq.(3.17) has to cancel against the integral containing F without the intervention of I.
2n
Representing now
F (s ,Z )= a (s )B (Z ) ,
2n 0 0 q,ℓm 0 qℓm 0
Xqℓm
F (s,Z)= a (s)B (Z) ,
2n q,ℓm qℓm
Xqℓm
I = I B (Z ) (3.18)
qℓm qℓm 0
Xqℓm
and substituting on both sides of Eq.(3.17) and using Eq.(3.10) we find, due to the linear independence of the
eigenfunctions B
qℓm
s ζ2(qn,ℓ)
a (s )=I + a (s) (3.19)
q,ℓm 0 qℓm q,ℓm
(cid:18)s (cid:19)
0
To leading order the contribution of I is neglected, leading to the conclusion that the spectrum of anomalous
qℓm
exponents of the correlation functions is determined by the eigenvalues of the shape-to-shape transition probability
operator. Calculations show that the leading exponent in the isotropic sector is always smaller than the leading
exponents in all other sectors. This gap between the leading exponent in the isotropic sector to the rest of the
exponents determines the rate of decay of anisotropy upon decreasing the scale of observation.
IV. CONCLUDING REMARKS
Thederivationpresentedabovehasusedexplicitlythepropertiesoftheadvectingfield,inparticulartheδ-correlation
in time. Accordingly it cannot be immediately generalized to more generic situations in which there exist time
correlations. Nevertheless we find it pleasing that at least in the present case we can trace the physical origin of
the exponents anomaly, and connect it to the underlying dynamics. In more generic cases the mechanisms may be
more complicated, but one should still keep the lesson in mind - higher order correlation functions depend on many
coordinates,andthese define a configurationin space. The scalingproperties of suchfunctions may verywell depend
on how such configurations are reached by the dynamics. Focusing on static objects like structure functions of one
variable may be insufficient for the understanding of the physics of anomalous scaling.
5
ACKNOWLEDGMENTS
This work has been supported in part by the German-Israeli Foundation, The European Commission under the
TMR programand the Naftali and Anna Backenroth-BronickiFund for Research in Chaos and Complexity.
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6
