using standard syste - Weizmann Institute of Science

PHYSICAL REVIEW E

VOLUME 62, NUMBER 4

OCTOBER 2000

Anomalous scaling in the anisotropic sectors of the Kraichnan model of passive scalar advection Itai Arad,1 Victor S. L’vov,1,2 Evgenii Podivilov,1,2 and Itamar Procaccia1 1

2

Departments of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel Institute of Automation and Electrometry, Academy of Sciences of Russia, Novosibirsk 630090, Russia 共Received 12 July 1999兲

Kraichnan’s model of passive scalar advection in which the driving 共Gaussian兲 velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields. We show here that the solutions of the Kraichnan equation for the n-order correlation functions foliate into sectors that are classified by the irreducible representations of the SO(d) symmetry group. We find a discrete spectrum of universal anomalous exponents, with a different exponent characterizing the scaling behavior in every sector. Generically the correlation functions and structure functions appear as sums over all these contributions, with nonuniversal amplitudes that are determined by the anisotropic boundary conditions. The isotropic sector is always characterized by the smallest exponent, and therefore for sufficiently small scales local isotropy is always restored. The calculation of the anomalous exponents is done in two complementary ways. In the first they are obtained from the analysis of the correlation functions of gradient fields. The theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result. In the second method we compute the exponents from the zero modes of the Kraichnan equation for the correlation functions of the scalar field itself. In this case the renormalization scale is the outer scale. The two approaches lead to the same scaling exponents for the same statistical objects, illuminating the relative role of the outer and inner scales as renormalization scales. In addition we derive exact fusion rules, which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions. PACS number共s兲: 47.27.Gs, 47.27.Jv, 05.40.⫺a

I. INTRODUCTION

The aim of this paper is twofold. On the one hand, we are interested in the effects of anisotropy on the universal aspects of scaling behavior in turbulent systems. To this aim we present below a theory of the anomalous scaling of the Kraichnan model of turbulent advection 关1兴 in anisotropic sectors that are classified by the irreducible representations of the SO(d) symmetry group. On the other hand, we are interested in clarifying the relationship between ultraviolet and infrared anomalies in turbulent systems. Again, it turns out that the Kraichnan model is an excellent case model in which this relationship can be exposed with complete clarity. As is well known by now, the Kraichnan model describes the advection of a passive scalar by a velocity field that is random, Gaussian, and ␦ correlated in time. The correlation functions of the field scale in their spatial dependence, and the main question is what are the scaling 共or homogeneity兲 exponents of the statistical objects of the scalar field that are induced by the given value of the scaling exponent of the advecting velocity field. The two issues discussed in this paper have an importance that transcends the particular example that we treat in detail in this paper. The first is the role of anisotropy in the observed scaling properties in turbulence. We have shown recently that in the presence of anisotropic effects 共which are ubiquitous in realistic turbulent systems兲 one needs to carefully disentangle the various universal scaling contributions. Even at the largest available Reynolds numbers the observed scaling behavior is not simple, being composed of several contributions with different scaling exponents. The statistical 1063-651X/2000/62共4兲/4904共16兲/$15.00

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objects like structure functions and correlations functions are characterized by one scaling 共or homogeneity兲 exponent only in the idealized case of full isotropy, or infinite Reynolds numbers when the scaling regime is of infinite extent. Anisotropy results in mixing in various contributions to the statistical objects, each of which is characterized by one universal exponent, but the total is a sum of such contributions that appears not to ‘‘scale’’ in standard log-log plots. By realizing that the correlation functions have natural projections on the irreducible representations of the SO(3) symmetry group we could offer methods of data analysis that allow one to measure the universal scaling exponents in each sector separately. In this paper we show that this foliation is an exact property of the statistical objects that arise in the context of the Kraichnan model. The second issue that transcends the particular example of the Kraichnan model is the identification of the renormalization scales that are associated with anomalous exponents. As has been already explained before, the renormalization scale that appears in the correlation functions of the passively advected scalar field is the outer scale of turbulence L. On the other hand, the theory of correlations of gradients of the field expose in the inner 共or dissipative兲 scale an additional renormalization scale. Having below a theory of anomalous scaling in various sectors of the symmetry group allows us to explain clearly the relationship between the two renormalization scales and the anomalous exponents that are implied by their existence. Since we expect that Kolmogorov-type theories, which assume that no renormalization scale appears in the theory, are generally invalidated by the appearance of both the outer and the inner scales as renormalization scales, 4904

©2000 The American Physical Society

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ANOMALOUS SCALING IN THE ANISOTROPIC . . .

the clarification of the relation between the two is important also for other cases of turbulent statistics. The central quantitative result of this paper is the expression for the scaling exponent ␨ (l) n , which is associated with the scaling behavior of the n-order correlation function 共or structure function兲 of the scalar field in the lth sector of the symmetry group. In other words, this is the scaling exponent of the projection of the correlation function on the lth irreducible representation of the SO(d) symmetry group, with n and l taking on even values only, n⫽0,2,... and l⫽0,2,...,

␨ 共nl 兲 ⫽n⫺ ⑀

冋

册

n 共 n⫹d 兲 共 d⫹1 兲 l 共 l⫹d⫺2 兲 ⫺ ⫹O 共 ⑀ 2 兲 . 2 共 d⫹2 兲 2 共 d⫹2 兲共 d⫺1 兲

4905

II. KRAICHNAN’S MODEL OF TURBULENT ADVECTION AND THE STATISTICAL OBJECTS

The model of passive scalar advection with rapidly decorrelating velocity field was introduced by Kraichnan 关1兴 already in 1968. In recent years 关2–7兴 it was shown to be a fruitful case model for understanding multiscaling in 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): 关 ⳵ t ⫺ ␬ 0 ⵜ 2 ⫹u共 r,t 兲 •“ 兴 T 共 r,t 兲 ⫽ f 共 r,t 兲 .

共1.1兲

The result is valid for any even l⭐n, and to O( ⑀ ) where ⑀ is the scaling exponent of the eddy diffusivity in the Kraichnan model 共and see below for details兲. In the isotropic sector (l ⫽0) we recover the well known result of 关2兴. It is noteworthy that for higher values of l the discrete spectrum is a strictly increasing function of l. This is important, since it shows that for diminishing scales the higher order scaling exponents become irrelevant, and for sufficiently small scales only the isotropic contribution survives. As the scaling exponent appears in power laws of the type (R/⌳) ␨ , with L being some typical outer scale and RⰆ⌳, the larger the exponent is, the faster is the decay of the contribution as the scale R diminishes. This is precisely how the isotropization of the small scales takes place, and the higher order exponents describe the rate of isotropization. Nevertheless for intermediate scales or for finite values of the Reynolds and Peclet numbers the lower lying scaling exponents will appear in measured quantities, and understanding their role and disentangling the various contributions cannot be avoided. The organization of this paper is as follows: In Sec. II we recall the Kraichnan model of passive scalar advection, and introduce the statistical objects of interest. In Sec. III we set up the calculation of the correlation functions of gradients of the field. It turns out that it is most straightforward to compute the fully fused correlation functions of gradient field, as these objects depend only on the ratio of the outer and inner scales. We compute these quantities and their exponents to first order in ⑀. We introduce the appropriate irreducible representations of the SO(d) symmetry group and evaluate the scaling exponents in all its sectors. In Sec. IV we turn to the correlation functions of the passive scalar field itself, and compute the scaling exponents of the structure functions in the presence of anisotropy, again correct to first order in ⑀. To this aim we compute the zero modes in all the sectors of the symmetry group. One of the interesting points of this paper is that the results of this calculation and the calculation via the correlations of the gradient fields gives the same results for the scaling exponents if one accepts the fusion rules. To clarify the issue we prove the fusion rules here in all the sectors of the symmetry group by a direct computation of the fusion of the zero modes. In Sec. V we offer a summary and a discussion.

共2.1兲

In this equation f (r,t) is the forcing. In Kraichnan’s model the advecting field u(r,t) as well as the forcing field f (r,t) are taken to be Gaussian, time and space homogeneous, and ␦ correlated in time: f 共 r,t 兲 f 共 r⬘ ,t ⬘ 兲 ⫽⌽ 共 r⫺r⬘ 兲 ␦ 共 t⫺t ⬘ 兲 ,

共2.2兲

具 u ␣ 共 r,t 兲 u ␤ 共 r⬘ ,t ⬘ 兲 典 ⫽W␣␤ 共 r⫺r⬘ 兲 ␦ 共 t⫺t ⬘ 兲 .

共2.3兲

Here the symbols ¯ and 具¯典 stand for independent ensemble averages with respect to the statistics of f and u, which are given a priori. We will study this model in the limit of large Peclet 共Pe兲 number, Pe⬅U ⌳ ⌳/ ␬ 0 , where U ⌳ is the typical size of the velocity fluctuations on the outer scale ⌳ of the velocity field. We stress that the forcing is not assumed isotropic, and actually the main goal of this paper is to study the statistic of the scalar field under anisotropic forcing. The correlation function of the advecting velocity needs further discussion. It is customary to introduce W␣␤ (R) via its k representation: W␣␤ 共 R兲 ⫽

⑀D ⍀d

冕

d d p ␣␤ d⫹ ⑀ P 共 p 兲 exp共 ⫺ip•R 兲 , 共2.4兲 ⌳ ⫺1 p ␭ ⫺1

冋

P ␣␤ 共 p兲 ⫽ ␦ ␣␤ ⫺

册

p␣p␤ , p2

共2.5兲

were P ␣␤ (p) is the transversal projector, ⍀ d ⫽(d ⫺1)⍀(d)/d, and ⍀(d) is the volume of the sphere in d dimensions 关i.e., ⍀(2)⫽2 ␲ , ⍀(3)⫽4 ␲ 兴: Equation 共2.4兲 introduces the four important parameters that determine the statistics of the driving velocity field: ⌳ and ␭ are the outer and inner scales of the driving velocity field, respectively. The scaling exponent ⑀ characterizes the correlation functions of the velocity field, lying in the interval 共0,2兲. The factor D is related to the correlation function 共2.3兲 as follows: W␣␤ 共 0 兲 ⫽D ␦ ␣␤ 共 ⌳ ⑀ ⫺␭ ⑀ 兲 .

共2.6兲

The most important property of the driving velocity field from the point of view of the scaling properties of the passive scalar is the tensor of ‘‘eddy diffusivity’’ 关1兴

␬ T␣␤ 共 R兲 ⬅2 关 W␣␤ 共 0 兲 ⫺W␣␤ 共 R兲兴 .

共2.7兲

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ARAD, L’VOV, PODIVILOV, AND PROCACCIA

The scaling properties of the scalar depend sensitively on the scaling exponent ⑀ that characterizes the R dependence of ␬ T␣␤ (R):

␬ T␣␤ 共 R兲 ⬀ 关 ⌳ ⑀ ⫺␭ ⑀ 兴 ␦ ␣␤ , for RⰇ⌳,

冋

␣

␤

册

⑀ R R ␬ T␣␤ 共 R兲 ⬀R ⑀ ␦ ␣␤ ⫺ , ␭ⰆRⰆ⌳. d⫺1⫹ ⑀ R 2

where 兵 ␣ m 其 is a set of even n vector indices 兵 ␣ m 其 ⫽ ␣ 1 , ␣ 2 ,..., ␣ n . We introduce also one-point correlations, which in the space homogeneous case are independent of the space coordinates: 兵␣m其

Hn

¯1 ,...rn ,r ¯n 兲 ⬅ 具 关 T 共 r1 ,t 兲 ⫺T 共¯r1 ,t 兲兴关 T 共 r2 ,t 兲 S n 共 r1 ,r ⫺T 共¯r2 ,t 兲兴 ... 关 T 共 rn ,t 兲 ⫺T 共¯rn ,t 兲兴 典 , 共2.9兲

冋

⫺␬0

冉 冊

共2.10兲

1/⑀

共2.12兲

共2兲 In addition to structure functions we are also interested in the simultaneous nth-order correlation functions of the temperature field, which is time independent in stationary statistics, Fn 共 兵 rm 其 兲 ⬅ 具 T 共 r1 ,t 兲 T 共 r2 ,t 兲 ,...,T 共 rn ,t 兲 典 ,

冓兿 n

Hn

共 兵 rm 其 兲 ⬅

i⫽1

␣i

冔

关 ⵜ T 共 ri ,t 兲兴 ,

兺

1 ⌽ 共 ri ⫺r j 兲 Fn⫺2 共 兵 rm 其 m⫽i, j 兲 , 2 兵 i⫽ j 其 ⫽1

兺

冋

共2.14兲

共2.16兲

兺

i⫽1

册

n

n

⫺␬

ⵜ 2i ⫹

兺

W␣␤ 共 ri ⫺r j 兲 ⵜ ␣i ⵜ ␤j Fn 共 兵 rm 其 兲

兵 i⫽ j 其 ⫽1

n

1 ⫽ ⌽ 共 ri ⫺r j 兲 Fn⫺2 共 兵 rm 其 m⫽i, j 兲 , 2 兵 i⫽ j 其 ⫽1

兺

共2.17兲

where

␬ ⫽ ␬ 0 ⫹D 关 ⌳ ⑀ ⫺␭ ⑀ 兴 .

共2.18兲

n Here we used that in the space-homogeneous case 兺 i⫽1 “i ⫽0 and therefore

冏兺 冏 n

i⫽1

2

n

n

“i ⫽

兺

i⫽1

ⵜ 2i ⫹

兺

兵 i⫽ j 其 ⫽1

ⵜ ␣i ⵜ ␤j ⫽0.

Consider the k-Fourier transform of Fn , which is defined as

共2.13兲

where we used the shorthand notation 兵 rm 其 for the whole set of arguments of nth-order correlation function Fn ,r1 ,r2 ,...,rn . 共3兲 Finally, we are interested in correlation functions of the gradient field “T. There can be a number of these, and we denote 兵␣m其

1 ␬ ␣␤ 共 r ⫺r 兲 ⵜ ␣ ⵜ ␤ F 共 兵 r 其 兲 2 i, j⫽1 T i j i j n m

where 兵 rm 其 m⫽i, j is the set off all rm with m from 1 to n, except of m⫽i and m⫽ j. Substituting ␬ T␣␤ (r) from Eqs. 共2.6兲, 共2.7兲 one gets

共2.11兲

.

兺

i⫽1

ⵜ 2i ⫹ n

characterize their R dependence in the limit of large Pe, when R is in the ‘‘inertial’’ interval of scales. This range is ␭, ␩ ⰆRⰆ⌳, L, where ␩ is the dissipative scale of the scalar field,

␬0 ␩ ⫽⌳ D

共2.15兲

册

n

n

⫽

In writing this definition we used the fact that the stationary and space-homogeneous statistics of the velocity and the forcing fields lead to a stationary and space-homogeneous ensemble of the scalar T. If the statistics is also isotropic, then S n becomes a function of R only, independent of the direction of R. The ‘‘isotropic scaling exponents’’ ␨ n of the structure functions S n 共 R 兲 ⬀R ␨ n ,

共 兵 rm ⫽r其 兲 .

The tensor Hn m ( 兵 rm 其 ) can be contracted in various ways. For example, binary contractions ␣ 1 ⫽ ␣ 2 , ␣ 3 ⫽ ␣ 4 , etc. with r1 ⫽r2 , r3 ⫽r4 , etc. produce the correlation functions of the dissipation field 兩 “T 兩 2 . When the ensemble is not isotropic we need to take into account the angular dependence of R, and the scaling behavior consists of multiple contributions arising from anisotropic effects. The formalism to describe this is set up in Appendix A and in the forthcoming sections. The correlation functions Fn satisfy the equation 关1兴

and in particular the standard structure functions are S n 共 R兲 ⬅ 具 关 T 共 r⫹R,t 兲 ⫺T 共 r,t 兲兴 典 .

␣ ␣ 2 ... ␣ n

⬅Hn 1

兵␣ 其

共2.8兲

We are interested in the scaling properties of the scalar field. By this we mean the power laws characterizing the R dependence of the various correlation and response functions of T(r,t) and its gradients. We will focus on three types of quantities: 共1兲 ‘‘Unfused’’ structure functions are defined as

n

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冉兺 冊 冕冋兿 n

共 2 ␲ 兲d␦

s⫽1

ks F n 共 兵 km 其 兲

n

⫽

m⫽1

册

drm exp共 ikm •rm 兲 Fn 共 兵 rm 其 兲 . 共2.19兲

Here the ␦ function applies to a homogeneous ensemble in which Fn ( 兵 rm 其 ) depends only on differences of coordinates. For F n ( 兵 km 其 ) Eq. 共2.17兲 yields:

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ANOMALOUS SCALING IN THE ANISOTROPIC . . .

␬ F n 共 兵 km 其 兲 兺 k 2i ⫹ i

⑀D ⍀d

冕

d d p ␣␤ d⫹ ⑀ P 共 p 兲 ⌳ ⫺1 p ␭ ⫺1

兵␣m其

Hn

⫻

k ␣i k ␤j F n 共 ki ⫹p,k j ⫺p, 兵 km 其 m⫽i, j 兲 兺 兵 i⫽ j 其 ⫽1

˜ n 共 兵 km 其 兲 , ⫽⌽

⫻

共2.20兲 兵␣m其

兺

冕

dR exp共 ik•R兲 ⌽ 共 R兲 .

⌿n

共2.22兲

共 2 ␲ 兲 共 n⫺1 兲 d

冉兺 冊 n

␦

s⫽1

ks

n ␣ ␤ d d p P ␣␤ 共 p兲 兺 兵 i⫽ j 其 ⫽1 k i k j n d⫹ ⑀ 兺 s⫽1 k s2 ⌳ ⫺1 p ␭ ⫺1

兵␣m其

n

˜ 共 k兲 ⫽ ⌽

冕

␣

n 兿 s⫽1 k s sd dk s

⫻F n 共 ki ⫹p,k j ⫺p, 兵 km 其 m⫽i, j 兲 ⫹⌿ n

共 2␲ 兲d ˜ n 共 兵 km 其 兲 ⬅ ⌽ 共 ki 兲 ␦ 共 ki ⫹k j 兲 ⌽ 2 兵 i⫽ j 其 ⫽1

⫻F n⫺2 共 兵 km 其 m⫽i, j 兲 , for n⬎2, 共2.21兲

冕

⑀D ⫽⫺ ␬⍀d

n

4907

⬅

冕

␣ n ˜ n 共 兵 km 其 兲 兿 s⫽1 k s sd dk s ⌽ ␦ n k s2 共 2 ␲ 兲 共 n⫺1 兲 d k 兺 s⫽1

,

冉兺 冊 n

s⫽1

ks . 共3.3兲

Shifting the dummy variables ki ⫺p→ki and k j ⫹p→k j we have another representation of this equation: 兵␣m其

Hn

冕

⑀D ⫽⫺ ␬⍀d

共2␲兲

共 n⫺1 兲 d

␣

冉兺 冊

兺 冕 兵 i⫽ j 其 ⫽1 ⌳

n

n 兿 s⫽1 d dk s

␦

s⫽1

n

ks

␭ ⫺1 ⫺1

␣

␣ ␤ ␣ ␣ ␣␤ d d p 共 k i i ⫺ p i 兲共 k j j ⫹ p j 兲 P 共 p兲 k i k j ⫻ d⫹ ⑀ n p 2 p 2 ⫹2p• 共 k j ⫺ki 兲 ⫹ 兺 s⫽1 k s2

˜ (k) is the Fourier transform of ⌽(R) and ⌽ ˜ 2 (k) Here ⌽ ⫽⌽(k). Equation 共2.20兲 may be rewritten as

n

⑀D F n 共 兵 km 其 兲 ⫽⫺ ␬⍀d

冕

⫻

兺 兵ni⫽ j 其 ⫽1 k ␣i k ␤j d d p ␣␤ P p 兲 共 n d⫹ ⑀ 兺 s⫽1 k s2 ⌳ ⫺1 p ␭ ⫺1

⫻F n 共 ki ⫹p,k j ⫺p, 兵 km 其 m⫽i, j 兲 ⫹

˜ n 共 兵 km 其 兲 ⌽ n ␬ 兺 s⫽1 k s2

.

共2.23兲

兿

␣

s⫽1,s⫽i, j

兵␣m其

k s s F n 共 兵 k m 其 兲 ⫹⌿ n

共3.4兲

.

In order to analyze this equation further we choose to nondimensionalize all the wave vectors by ⌳. We write ˜ks ⫽⌳ks , ˜p⫽⌳p, etc., and for simplicity drop the tilde signs at the end. We simplify the appearance of the equation further by introducing the definition of the dimensionless function ␣␣

This equation will serve as the basis for future analysis in Sec. III.

2 A ␤ i , ␤j 共 兵 k m 其 s⫽i, j ki ,k j ,p兲 i

j

⫽⫺

III. SCALING OF THE TEMPERATURE GRADIENT FIELDS

冕

␣

␣

dpˆ共 k i i ⫺ p ␣ i 兲共 k j j ⫹ p ␣ j 兲 P ␤ i ␤ j 共 pˆ兲 n ⍀ d 2 p 2 ⫹2p• 共 k j ⫺ki 兲 ⫹ 兺 s⫽1 k s2

, 共3.5兲

A. Basic equations in k representation

It appears that Eq. 共2.23兲 is as difficult to solve as Eq. 共2.16兲. In fact, very important information about scaling behavior may be extracted from Eq. 共2.23兲 for small ⑀ 关8兴. In order to develop our method we will analyze first the simultaneous, n-point correlation functions of the gradient fields 兵␣ 其 兵␣ 其 Hn m ( 兵 rm 其 ) and H n m of Eqs. 共2.14兲, 共2.15兲: These objects are expressed in terms of F n ( 兵 km 其 ) as follows:

冕兿

where dpˆ stands for integrating over all the angles of the unit vector pˆ⬅p/p. The resulting equation is 兵␣m其

Hn

⫽g

冕

⫻

n

兵␣m其

Hn

共 兵 rm 其 兲 ⫽ 共 2 ␲ 兲

共 1⫺n 兲 d

i⫽1

关 d d k i k ␣ i exp共 iki •ri 兲兴

冉兺 冊

共2␲兲

冉兺 冊 n

n 兿 s⫽1 d dk s 共 n⫺1 兲 d

␦

s⫽1

ks

兺 冕 兵 i⫽ j 其 ⫽1 1 n

⌳/␭

⑀ d p ␣i␣ j 2 A k ,k ,p兲 k ␤ i k ␤ j 共 兵k 其 p 1⫹ ⑀ ␤ i ␤ j m s⫽i, j i j n

⫻

␣ 兵␣ k s F n 共 兵 k m 其 兲 ⫹⌿ n 兿 s⫽1,s⫽i, j s

m其

,

共3.6兲

n

⫻F n 共 兵 km 其 兲 ␦

冕兿

s⫽1

n

兵␣m其

Hn

⫽ 共 2 ␲ 兲 共 1⫺n 兲 d

i⫽1

共3.1兲

ks ,

冉兺 冊 n

关 d d k i k ␣ i 兴 F n 共 兵 km 其 兲 ␦

s⫽1

ks . 共3.2兲

From this and Eq. 共2.23兲 one gets

where the dimensionless factor g is g⬅

D⌳ ⑀ . ␬ 0 ⫹D 共 ⌳ ⑀ ⫺␭ ⑀ 兲

共3.7兲

In fact, we should recognize that the natural expansion parameter is actually not g, but ˜g , where ˜g ⬅g

冕

⌳/␭

1

⑀dp . p 1⫹ ⑀

共3.8兲

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ARAD, L’VOV, PODIVILOV, AND PROCACCIA

Evaluating the integral we find

兵␣m其

H n,1 ⫽g ˜ D 关 ⌳ ⑀ ⫺␭ ⑀ 兴 ˜g ⫽ . ␬ 0 ⫹D 共 ⌳ ⑀ ⫺␭ ⑀ 兲

共3.9兲

We will see below that ˜g can take on very different values in different limiting cases. In particular it can be of O( ⑀ ) or of O(1) depending on the order of limits. The relevant limit for the physics at hand will be discussed below. At this point we 兵␣ 其 perform a calculation of Hn m to first order in ˜g in all the sectors of the symmetry group. B. Theory to first order in ˜g

The theory for F n and Hn can be formulated iteratively, resulting in the following series: ⬁

F n 共 兵 rm 其 兲 ⫽

兺

q⫽0

兵␣m其

F n,q 共 兵 rm 其 兲 , H n

⬁

兺

⫽

q⫽0

⫻

冕

n

兵␣ 其

˜ H n,1m ⫽g

H n,q .

␣ ␤

⫻F n 共 ki ⫹p,k j ⫺p, 兵 km 其 m⫽i, j 兲 ⫹F n,0共 兵 km 其 兲 , 共3.11兲 ˜ n,0共 兵 km 其 兲 ⌽ n k 兺 s⫽1 k s2

,

n

˜ n,0共 兵 km 其 兲 ⬅ ⌽

共 2␲ 兲d ⌽ 共 ki 兲 ␦ 共 ki ⫹k j 兲 2 兵 i⫽ j 其 ⫽1

兺

⫻F n⫺2,0共 兵 km 其 m⫽i, j 兲 . Thus we are interested in calculating

s⫽1

ks

兺 冕 兵 i⫽ j 其 ⫽1 1

⌳/␭

⑀ d p ␣i␣ j 2 A k ,k ,p兲 k ␤ i k ␤ j 共 兵k 其 p 1⫹ ⑀ ␤ i ␤ j m s⫽i, j i j

兿

␣

k s s F n,0共 兵 km 其 兲 .

s⫽1,s⫽i, j

共3.13兲

冕

冉兺 冊 n

n 兿 s⫽1 d dk s

共2␲兲

共 n⫺1 兲 d ␦

s⫽1

ks

兺 冕 兵 i⫽ j 其 ⫽1 1 n

⌳/␭

⑀ dp p 1⫹ ⑀

n

兺 兵 i⫽ j 其 ⫽1 k i k j d d p ␣␤ n d⫹ ⑀ P 共 p 兲 ⫺1 p 兺 s⫽1 k s2 ⌳

F n,0共 兵 km 其 兲 ⫽

共 n⫺1 兲 d ␦

n

Recall that the function ⌽(k) is constant for kLⰆ1 and it vanishes for kLⰇ1. Therefore the leading contribution to the integrals 共3.13兲 over ki comes from the region k i L⭐1. In integral 共3.13兲 p⬎1/⌳ and in all our approaches we consider LⰇ⌳. Therefore in 共3.13兲 pⰇk j and this equation may be simplified up to:

兵␣m其

Here F n,q is the result of the q step of the iteration procedure ˜ q . There are two contributions for each of Eq. 共2.23兲, F n,q ⬀g q term of order ˜g . One arises from substituting F n,q⫺1 into the integral on the right-hand side 共RHS兲 of 共2.23兲, and the ˜ n according second arises from F n⫺2,q , which appears in ⌽ to 共2.21兲. Correspondingly, also H n,q has two contributions. One is obtained from Eq. 共3.6兲 when we substitute F n,q⫺1 in the integral on the RHS, and the second when we substitute F n⫺2,q in the term denoted as ⌿ n . In this section we compute explicitly H n,1 . Analyzing the relative importance of these two contributions to H n,1 we found that the second contribution is negligible compared to the first when ⌳ⰆL. In other words, we can disregard the contribution to H n,1 that arises from F n⫺2,q . This means that ˆ n in for the sake of our iteration procedure we can replace ⌽ 0 ˜ ˜ ). This means 共2.23兲 by the quantity ⌽ n,0 , which is of O(g that instead of Eq. 共2.23兲 we iterate ␭ ⫺1

共2␲兲

冉兺 冊 n

n 兿 s⫽1 d dk s

n

⫻

␣␣ ⫻A ␤ i ␤ j k ␤ i k ␤ j i j

共3.10兲

⑀D F n 共 兵 km 其 兲 ⫽⫺ ␬⍀d

冕

PRE 62

共3.12兲

兿

␣

s⫽1,s⫽i, j

k s s F n,0共 兵 km 其 兲 ,

共3.14兲

where now ␣␣

A ␤i␤ j⬅ i j

1 2⍀ d

冕

dpˆpˆ ␣ i pˆ ␣ j P ␤ i ␤ j 共 pˆ 兲 ,

共3.15兲

is the constant tensor that obtains from the tensor function 共3.5兲 when all k s Ⰶ p. Performing the all the wave-vector integrals we observe that the explicit ⑀ is canceled by the integral over p. Accordingly n

兵␣m其

H n,1 ⫽g ˜

兺

兵 i⫽ j 其 ⫽1

␣␣

␤ ␤ 兵 ␣ m 其 m⫽i, j

A ␤ i ␤ j H n,0i j i j

.

共3.16兲

An actual integration in 共3.15兲 yields ␣␣

A ␤i␤ j⫽ i j

␦ ␣ i ␣ j 共 d⫹1 兲 ⫺ ␦ ␣ i ␤ i ␦ ␣ j ␤ j ⫺ ␦ ␣ i ␤ j ␦ ␣ j ␤ i 2 共 d⫹2 兲共 d⫺1 兲

. 共3.17兲

C. Analysis in all the anisotropic sectors

We note that Eqs. 共2.17兲 contain only isotropic operators. On the other hand ⌽(ri ⫺r j ) can be anisotropic, depending on the direction of the vector ri ⫺r j . Since the equations are linear, we can expand all the objects in terms of the irreducible representations of the SO(d) group of all rotations, and be guaranteed that the solutions foliate in the sense that different irreducible representations cannot be mixed. This considerations are valid for all the equations in this theory, including Eq. 共3.16兲. To know which irreducible representations we need to use in every case one has to consult Appendix A. After doing so one notes that for any order 兵␣ 其 q, the tensors H n,qm are constant tensors, fully symmetric in all their indices. Using the exposition of Appendix A we know that the projections on the irreducible representations of the SO(d) symmetry group must be of the form 兵␣ 其

兵␣ 其

m ⫽␭ 共ql 兲 B l, ␴m,n . H n,q,l

共3.18兲

PRE 62

ANOMALOUS SCALING IN THE ANISOTROPIC . . .

Our first order calculation is aimed at finding the ratio (l) ␭ (l) 1 /␭ 0 . Substituting 共3.17兲, 共3.18兲 in 共3.16兲 we find 兵␣ 其

m H n,1,l ⫽

冋

兺

␤ ␤ 兵 ␣ m 其 m⫽i, j ⫺

册 冋

H共nl 兲 ⫽

␣ ␤ ␣ ␣ 关␦␤ ␦␤ ⫹␦␤ ␦␤ 兴 兺 i⫽ j i

i

i

j

i

j

j

i

␤ ␤ 兵␣m其

⫻␭ 共0l 兲 B l, ␴i ,nj ⫽

If we expect that H(l) n is a scale invariant function of ⌳/␭ we can interpret Eq. 共3.23兲 as the beginning of an expansion that can be resummed into a power law

˜g ␦ ␣ i ␣ j ␦ ␤ i ␤ j ,␭ 共0l 兲 共 d⫹1 兲 2 共 d⫹2 兲共 d⫺1 兲 i⫽ j ⫻B l, ␴i ,nj

册

˜g 共 d⫹1 兲 z n,l 兵␣ 其 ⫺n 共 n⫺1 兲 ␭ 共0l 兲 B l, ␴m,n . 2 共 d⫹2 兲共 d⫺1 兲

Defining now A (l) n via the relation l兲 l兲 H共n,1 ⫽g ˜ A 共nl 兲 H共n,0 ,

共3.19兲

we conclude that A 共nl 兲 ⫽

2n 共 d⫹n 兲 共 d⫹1 兲 l 共 l⫹d⫺2 兲 ⫺ . d⫹2 2 共 d⫺1 兲共 d⫹2 兲

共3.20兲

To interpret the result 共3.19兲–共3.20兲 we should observe that the nature of the theory that we develop depends on the order of the limits that we take. We should recognize that the 兵␣ 其 quantity H n m does not exist without an inner 共ultraviolet兲 cutoff. We are thus interested in limiting values of ˜g subject to the condition that ␩ is finite. Thus one order of limits that makes sense is ␭→0 first 共corresponding to the Reynolds number going to infinity first兲, and then ⑀ going to zero second, but keeping ␩ fixed 关for example by controlling ␬ 0 in Eq. 共2.12兲兴. Another order of limits is ⑀ →0 first 共still keeping ␩ fixed, but very small兲 with ␭ being fixed and larger than ␩. If we have ␭→0 first, and then when ⑀ →0 second we find that the expansion parameter is close to unity:

冉冊 ␩

⑀

⌳

, for ⑀ ln

冉冊 ␩

⌳

Ⰶ1.

共3.21兲

Thus we cannot stop at 共3.19兲, and we are forced to consider higher order terms in the expansion in ˜g and appropriate resummations. This is done in Sec. III F. On the other hand, if ⑀ →0 first, we find an apparently ‘‘small’’ expansion parameter that is proportional to ⑀: ˜g ⬇ ⑀ ln

冉冊

冉冊

⌳ ␩ , for ⑀ ln Ⰷ1. ␭ ⌳

冉冊 ⌳ ␭

⑀ A 共nl 兲

l兲 H共n,0 .

共3.22兲

共3.24兲

Of course, this is hardly justified just by examining the O( ⑀ ) term, since one can have more than one branch of scaling exponents proportional to ⑀. If we have m branches only the analysis up to O( ⑀ m ) can reveal this. We return to this issue in the next subsection. An additional issue is the magnitude of ␭ that can be arbitrarily small, making any reexponentiation even more dubious. To overcome this problem one usually invokes the renormalization group equations to justify the exponentiation. We shortly present this method next. In doing so we want to argue that for the case in question there is nothing more in this approach than direct reexponentiation as long as higher order terms in ⑀ are not included. Within the renormalization group method 关8兴 one consid(l) ers Eq. 共3.23兲 as the ‘‘bare’’ value of H(l) n , Hn,B . One then seeks a multiplicative renormalization group by defining a renormalized function l兲 l兲 H 共n,R 共 ␮ ,⌳,... 兲 ⫽Z H 共 ␮ ,⌳,␭ 兲 H 共n,B 共 ⌳,␭ 兲 .

D. Interpretation of the result

˜g ⬇1⫺

4909

共3.25兲

Here ␮ is a ‘‘running length,’’ and the only ␮ dependence of the RHS is through the Z H function. Defining Z H so that it eliminates the dependence of the left-hand side 共LHS兲 on ␭ (l) we get and setting the initial condition Hn,R (⌳,⌳,...)⫽Hn,0 Z H 共 ␮ ,␭ 兲 ⫽1⫹ ⑀ A 共nl 兲 ln共 ␭/ ␮ 兲 .

共3.26兲

From Eq. 共3.25兲 we get l兲 d ln关 H 共n,R 共 ␮ ,⌳,... 兲兴

d ln ␮

⫽␥H ,

共3.27兲

where

␥ H ⫽⫺ ⑀ A 共nl 兲 ⫹O 共 ⑀ 2 兲 .

共3.28兲

Solving the differential equation 共3.27兲 we get l兲 H共n,R 共 ␮ ,⌳,... 兲 ⫽

冉冊 ⌳ ␮

⑀ A 共nl 兲

Hn,0 .

共3.29兲

(l) Exponentiating 共3.26兲 and solving Eq. 共3.25兲 in favor of Hn,B we recover Eq. 共3.24兲. Note that the inner scale in this case is ␭, since ␭⬎ ␩ . This fact casts an additional doubt on this limit of the theory, since it misses altogether the existence of the Betchelor regime 关9兴 between ␭ and ␩. For all these reasons we tend to disqualify 共3.24兲 despite the relative simplicity of its derivation. We turn next to the other limiting procedure.

E. Exponentiating using renormalization group equations

Using Eq. 共3.22兲 in Eq. 共3.19兲 we get

冋

冉冊

F. Theory for ␭\0 first

册

⌳ l兲 ⫹O 共 ⑀ 2 兲 H共n,0 . H共nl 兲 ⫽ 1⫹ ⑀ A 共nl 兲 ln ␭

共3.23兲

Considering the limit ␭→0 first, Eq. 共3.19兲 is still valid, but now ˜g is of order unity, and we cannot justify reexponentiation. The small parameter ⑀ seems to have disappeared.

4910

ARAD, L’VOV, PODIVILOV, AND PROCACCIA

PRE 62

FIG. 1. The graphic representation of Eq. 共3.3兲.

This forces us to consider all the higher order terms in ˜g to understand how to resum them. We will see that at the end ⑀ reappears. The resummation of the ˜g dependence is aided significantly by some graphic representations of the relevant equations and their perturbative solutions. In Fig. 1 we represent 兵␣ 其 graphically the definition 共3.2兲 of H n m in terms of F n ( 兵 km 其 ). This helps us to introduce the basic diagrammatic notations. A solid long rectangle with n ␣ m wavy lines stands for 兵␣ 其 H n m , while the elongated solid ellipse represents F n . The ks wave vectors are denoted by arrows, and the dots represent multiplications. The vertical line connecting all the arrow heads stands for the integration with a ␦ function over the sum of all the k vectors. Figure 2 is a graphic notation of Eq. 共3.6兲. The little ellipse with a vertical arrow designated ␣␣ by p stands for the tensor function A ␤ i ␤ j ( 兵 ks 其 s⫽i, j ki ,k j ,p). i j

This ellipse is involved in the integration over the vector p in the loop to the left of the ellipse with a weight consisting of the sum of squares of the k vectors. The thin elongated ellipse in the second term on the RHS stands for the zeroth order term F n,0 , cf., Eq. 共3.12兲. The actual values of the wave vectors are indicated in this diagram. In later diagrams, Fig. 3, we drop this obvious notation. In Fig. 3 we display the perturbative solution, which results from the iteration procedure in Eq. 共3.11兲, the result of which is substituted in Eq. 共3.3兲. (l) , whereas the second The first diagram on the RHS is Hn,0 (l) is the first order term Hn,1 , Eq. 共3.19兲. The last diagram ␣␣ (l) shown in Hn,2 . One should note that when the A ␤ i ␤ j ellipse

FIG. 3. The graphic representation of the iteration scheme.

appears once we have a sum over all pairs i,j indices with i ⫽ j. When it appears twice there is a double sum, with respect to the pairs i⫽ j and l⫽m. In analyzing such diagrams one needs to identify three distinct possibilities. These are denoted as case 共a兲, i⫽l, j⫽m, case 共b兲 i⫽l but j⫽m, and case 共c兲 where all the indices are different. There are two integrals over p1 and p2 in the loops to the left of the corresponding ellipse. We refer to the functions A of Eq. 共3.5兲 that appear in these integrals as A1 and A2 , respectively. Analyzing the integrals it is useful to separate the discussion to region I, in which p 1 ⬎ p 2 , and region II, in which p 2 ⬎ p 1 . In region I we can neglect the contribution of p2 and all ks with respect to p1 . Accordingly we have two independent integrals and sums, and the result is therefore 1 l兲 l兲 ⫽ 关 ˜g A 共nl 兲 兴 2 H共n,0 H共n,2 共 region I兲 , 2

共3.30兲

where the factor 1/2 stems from the fact that the volume of region I is a half of the whole volume of (p1 ,p2 ) space. In region II we should distinguish between the cases 共a兲, 共b兲, and 共c兲, for which the evaluation of A2 will be different. In case 共c兲 Eq. 共3.5兲 shows that A2 is of the order of p 22 / p 21 , which is small. In case 共b兲 A2 is of the order of p 2 / p 1 , which is still small. Only case 共a兲, in which the loops appear as two rungs on the same ladder, we have A2 of the order of unity. The actual calculation of this integral is presented in Appendix B, with the final result

i j

1 l兲 l兲 ⫽ ˜g 2 A 共nl 兲 H共n,0 H共n,2 2

关 region II, case 共a兲兴 .

共3.31兲

(l) is Together the second order result for Hn,2

1 l兲 l兲 H共n,2 ⫽ ˜g 2 关 A 共nl 兲 ⫹ 共 A 共nl 兲 兲 2 兴 H共n,0 . 2

FIG. 2. The graphic representation of Eq. 共3.8兲.

共3.32兲

Our aim is to find the fully resummed form, correct to all (l) order in ˜g and A (l) n , of Hn . We can express it in the form

PRE 62

ANOMALOUS SCALING IN THE ANISOTROPIC . . . l兲 H共nl 兲 ⬅K 共 ˜g ,A 共nl 兲 兲 H共n,0 ,

共3.33兲

where the function K(g ˜ ,A (l) n ) is represented as the double infinite sum ⬁

K 共 ˜g ,A 兲 ⫽1⫹

⬁

兺

兺

Am

m⫽1

s⫽m

D m,s˜g s .

共3.34兲

Up to now we have information about D 1,1⫽1 and D 1,2 ⫽D 2,2⫽1/2. In Appendix C we derive the following recurrent relation for the higher order terms s⫺1

1 1 D 1,s ⫽ , D m⫹1,s ⫽ s s

兺

q⫽m

D q,m .

共3.35兲

Using 共3.35兲 in Eq. 共3.34兲 we find the contribution proportional to A: ⬁

K 1 共 ˜g ,A 兲 ⫽A

兺

s⫽1

˜g s ⫽⫺A ln共 1⫺g ˜ 兲. s

共3.36兲

Considering all the terms quadratic in A, and using the recurrent relations to determine D 2,s we find ⬁

K 2 共 ˜g ,A 兲 ⫽A

2

兺

s⫺1

s⫽2

˜g s 1 . s q⫽1 q

兺

共3.37兲

This double sum is computed in Appendix C with the result 1 ˜ 兲兴 2 . K 2 共 ˜g ,A 兲 ⫽ 关 ⫺A ln共 1⫺g 2

共3.38兲

The general result can be derived using similar techniques with the result K m 共 ˜g ,A 兲 ⫽

1 ˜ 兲兴 m . 关 ⫺A ln共 1⫺g m!

共3.39兲

˜ ,A): Accordingly we conclude with the series for K(g ⬁

K 共 ˜g ,A 兲 ⫽

兺 m⫽0

˜ 兲兴 关 ⫺A ln共 1⫺g ⫽exp关 ⫺A ln共 1⫺g ˜ 兲兴 . m! 共3.40兲

Using now Eq. 共3.21兲 one finds

冉冊 ⌳

␩

⑀A

.

共3.41兲

Finally, using Eq. 共3.33兲 we have the final result H共nl 兲 ⫽

冉冊 ⌳

␩

⑀ A 共nl 兲

l兲 H共n,0 .

⬇1兲, while neglecting terms of O( ⑀ ). However, the sums ˜ ⫺1)⬀ ⑀ . In 共3.38兲–共3.39兲 result in expressions containing (g other words, we end up with terms that appear of the same order as those neglected during the procedure. In order to justify the results 共3.42兲 we must go back and analyze contributions of O( ⑀ ). These appear, for example, in contributions in which the ‘‘rungs’’ appear on adjacent ladders, like case 共b兲 in region II. The two ‘‘simple’’ rungs appearing in two adjacent ladders can be now considered as a single compounded rung. We focus now on the infinite set of diagrams in which this compounded rung repeats many times. The sum of such diagrams will again generate results containing ˜ ⫺1), and in the end will be responsible for terms of (g O( ⑀ 2 ) in the scaling exponent. The reason for this phenomenon is the structure of the iterative solution. Sums of terms of order ˜g have cancellations, leading eventually to a result of O( ⑀ ). The sum of terms of O( ⑀ ) have a very similar structure, just we a redefined ‘‘rung.’’ Therefore they automatically generate another factor of ⑀ by resummation. This phenomenon repeats in higher orders, again by redefining what do mean by a ‘‘rung.’’ Thus Eq. 共3.42兲 can be considered as the final result for the scaling of the fused correlation function of gradient fields with the exponent correct to O( ⑀ ). In the next section we turn to the calculation of the scaling exponent of the unfused correlation function of the scalar field itself. We will show that the exponents computed in both methods agree when the same objects are evaluated. This agreement is connected in the final section with the existence of fusion rules that control the asymptotic properties of unfused correlation functions when some coordinates are fused together. IV. ZERO MODES IN THE ANISOTROPIC SECTORS A. Calculation of the correlation functions

In this section we consider the zero modes of Eq. 共2.16兲. In other words we seek solutions Z n ( 兵 rm 其 ) that in the inertial interval solve the homogeneous equation n

m

K 共 ˜g ,A 兲 ⫽

4911

兺

i⫽ j⫽1

␬ T␣␤ 共 ri ⫺r j 兲 ⵜ ␣i ⵜ ␤j Z n 共 兵 rm 其 兲 ⫽0.

We allow anisotropy on the large scales. Since all the operators here are isotropic and the equation is linear, the solution space foliates into sectors 兵l,␴其 corresponding to the irreducible representations of the SO(d) symmetry group. Accordingly we write the wanted solution in the form Z n 共 兵 rm 其 兲 ⫽

共3.42兲

We are pleased to find that the inner scale is now ␩, in agreement with our expectation. The exponentiation was achieved naturally in the present case. In assessing this result, we need to return to a delicate point in the derivation of Eq. 共3.42兲. The procedure involved summing all the terms of the order of unity 共powers of ˜g

共4.1兲

Z n,l, ␴ 共 兵 rm 其 兲 , 兺 l, ␴

共4.2兲

where Z n,l, ␴ is composed of functions that transform according to the 共l,␴兲 irreducible representations of SO(d). Each of these components is now expanded in ⑀. In other words, we write, in the notation of Ref. 关2兴, Z n,l, ␴ ⫽E n,l, ␴ ⫹ ⑀ G n,l, ␴ ⫹O 共 ⑀ 2 兲 . For ⑀ ⫽0, Eq. 共4.1兲 simplifies to

共4.3兲

4912

ARAD, L’VOV, PODIVILOV, AND PROCACCIA n

兺 ⵜ 2i E n,l, ␴共 兵 rm 其 兲 ⫽0, i⫽1

共4.4兲

n

兺 ⵜ 2i G n,l, ␴共 兵 rm 其 兲 ⫽V n E n,l, ␴共 兵 rm 其 兲 ,

i⫽1

共4.5兲

where ⑀ V n is the first order term in the expansion of the operator in 共4.1兲, V n⬅

兺

j⫽k⫽1

冋

␦

␣␤

ln共 r jk 兲 ⫺

r ␣jk r ␤jk 共 d⫺1 兲 r 2jk

册

ⵜ ␣j ⵜ ␤k ,

共4.6兲

where r jk ⬅r j ⫺rk . In solving Eq. 共4.4兲 we are led by the following considerations: we want scale invariant solutions, which are powers of r jk . We want analytic solutions, and thus we are limited to polynomials. Finally we want solutions that involve all the n coordinates for the function E n,l, ␴ ; solutions with fewer coordinates do not contribute to the structure functions 共2.9兲. To see this, note that the unfused structure function is a linear combination of correlation functions. This linear combination can be represented in terms of the difference operator ␦ j (r,r⬘ ) defined by

␦ j 共 r,r⬘ 兲 F共 兵 rm 其 兲 ⬅F共 兵 rm 其 兲 兩 r j ⫽r⫺F共 兵 rm 其 兲 兩 r j ⫽r⬘ . 共4.7兲 Then, Sn 共 r1 ,r⬘1 ,...,rn r⬘n 兲 ⫽

兿j ␦ j 共 rj ,r⬘j 兲 F共 兵 rm其 兲 .

共4.8兲

Accordingly, if F( 兵 rm 其 ) does not depend on rk , then ␦ k (rk ,rk⬘ )F( 兵 rm 其 )⫽0 identically. Since the difference operators commute, we can have no contribution to the structure functions from parts of F that depend on less than n coordinates. Finally we want the minimal polynomial because higher order ones are negligible in the limit r jk Ⰶ⌳. Accordingly, E n,l, ␴ with l⭐n is a polynomial of order n. Consulting Appendix A for the irreducible representations of the SO(d) symmetry group, we can write the most general form of E n,l, ␴ , up to an arbitrary factor, as ␣

␣

␣ ,..., ␣ n

1 E n,l, ␴ ⫽r 1 1 ...r n n B n,l, ␴

兺 H l,jk␴共 兵 rm 其 兲 ln共 r jk 兲 ⫹H l, ␴共 兵 rm 其 兲 ,

j⫽k

共4.10兲

for any value of l, ␴. Next we expand the operator in Eq. 共4.1兲 in ⑀ and collect the terms of O( ⑀ ):

n

G n,l ␴ 共 兵 rm 其 兲 ⫽

PRE 62

⫹关¯兴,

共4.9兲

where 关¯兴 stands for all the terms that contain less than n coordinates; these do not appear in the structure functions, but maintain the translational invariance of our quantities. ␣ 1 ... ␣ n of Appendix A is jusThe appearance of the tensor B n,l, ␴ tified by the fact that E n,l, ␴ must be symmetric to permutations of any pair of coordinates on the one hand, and it has to belong to the l, ␴ sector on the other hand. This requires the appearance of the fully symmetric tensor 共A5兲. In light of Eqs. 共4.5兲–共4.6兲 we seek solution for G (l) n ( 兵 rm 其 ) of the form

where H l,jk␴ ( 兵 rm 其 ) and H l, ␴ ( 兵 rm 其 ) are polynomials of degree n. The latter is fully symmetric in the coordinates. The former is symmetric in r j ,r k and separately in all the other 兵 r m 其 m⫽i, j . Substituting Eq. 共4.10兲 into Eq. 共4.5兲 and collecting terms of the same type yields three equations:

兺i ⵜ 2i H l,jk␴ ⫽ⵜ j •ⵜ k E 2n,l, ␴ , 2 关 d⫺2⫹r jk • 共 “ j ⫺“ k 兲兴 H l,jk␴ ⫹

r ␣jk r ␤jk ⵜ ␣j ⵜ k␤ d⫺1

共4.11兲

E 2n,l, ␴

⫽⫺r 2jk K l,jk␴ ,

共4.12兲

兺i ⵜ 2i H l, ␴ ⫽ j⫽k 兺 K l,jk␴ .

共4.13兲

Here K l,jk␴ are polynomials of degree n⫺2, which are separately symmetric in the j,k coordinates and in all the other coordinates except j,k. In Ref. 关2兴 it was proven that for l ⫽0 these equations posses a unique solution. The proof follows through unchanged for any l⫽0, and we thus proceed to finding the solution. By symmetry we can specialize the discussion to j⫽1,k ⫽2. In light of Eq. 共4.12兲 we see that H l,12␴ must have at least a quadratic contribution in r 12 . This guarantees that 共4.10兲 is nonsingular in the limit r 12→0. The only part of H l,12␴ that will contribute to structure functions must contain r3 ¯rn at least once. Since H l,12␴ has to be a polynomial of degree n in the coordinates, it must be of the form ␣

␣

␣

␣

H l,12␴ ⫽r 121 r 122 r 3 3 ¯r n n C ␣ 1 ␣ 2 ,..., ␣ n ⫹ 关 ¯ 兴 1,2 ,

共4.14兲

where 关 ¯ 兴 1,2 contains terms with higher powers of r 12 and therefore do not contain some of the other coordinates r 3 ¯r n . Obviously such terms are unimportant for the structure functions. Since H l,12␴ has to be symmetric in r1 , r2 and r3 ¯rn separately, and it has to belong to an l, ␴ sector, we conclude that the constant tensor C must have the same symmetry and to belong to the same sector. Consulting Appendix A, the most general form of C is ␣ , ␣ ,..., ␣ n ␣ 3 , ␣ 4 ,..., ␣ n ⫹b ␦ ␣ 1 ␣ 2 B 2n⫺2,l, ␴

1 2 C ␣ 1 ␣ 2 ... ␣ n ⫽aB 2n,l, ␴

⫹c

兺

i⫽ j⬎2

␣ , ␣ ,..., ␣ n .

3 4 ␦ ␣ 1 ␣ i ␦ ␣ 2 ␣ j B 2n⫺4,l, ␴

共4.15兲

Substituting in Eq. 共4.12兲 one finds ␣ ␣ ␣ ␣ r 121 r 122 r 3 3 ...r 2nn 12 共 d⫹2 兲 H l, ␴ ⫹

2d⫺2

␣ ,..., ␣ n

1 B 2n,l, ␴

1 ␣ ␣ ⫹ r 121 r 122 ␦ ␣ 1 ␣ 2 K l,1,2␴ ⫽ 关 ¯ 兴 1,2 . 2

共4.16兲

PRE 62

ANOMALOUS SCALING IN THE ANISOTROPIC . . .

Substituting Eq. 共4.14兲 and demanding that coefficients of ␣ ␣ the term r 1 1 ...r n n will sum up to zero, we obtain ⫺2 共 d⫹2 兲 a⫺

⬅ 共 r i ⫺r j 兲 ␣ i 共 r i ⫺r j 兲 ␣ j ln兩 ri ⫺r j 兩 ⫹ 共¯r i ⫺r ¯ j 兲␣i ⫻ 共¯r i ⫺r ¯ j 兲 ␣ j ln兩¯ri ⫺r ¯ j 兩 ⫺ 共 r i ⫺r ¯ j 兲 ␣ i 共 r i ⫺r ¯ j 兲␣i ⫻ln兩 ri ⫺r ¯ j ⫺ 共¯r i ⫺r j 兲 ␣ i 共¯r i ⫺r j 兲 ␣ j ln兩¯ri ⫺r j 兩 .

␣ ,..., ␣

共4.23兲

共4.17兲

The coefficient b is not determined from this equation due to possible contributions from the unknown last term. We determine the coefficient b from Eq. 共4.11兲. After substituting the forms we find ␣

f ␣ i ␣ j 共 ri ,r ¯i ,r j ,r ¯j兲

2 ⫽0, 2d⫺2

⫺2 共 d⫹2 兲 c⫽0; ⇒c⫽0.

␣

4913

␣ , ␣ ,..., ␣ n

␣1␣2 1 n 3 4 B 2n⫺2,l, 4 ␦ ␣ 1 ␣ 2 r 3 3 ¯r 2nn 关 aB 2n,l, ␴ ⫹b ␦ ␴

The scaling exponent of Sn,l, ␴ can be found by multiplying all its coordinates by ␮. A direct calculation yields:

兴

␣ ,..., ␣

1 n ⫽ ␦ ␣ 1 ␣ 2 r ␣ 3 ¯r ␣ n B 2n,l, ␴ ⫹ 关 ¯ 兴 1,2 .

共4.18兲

Recalling the identity 共A6兲 we obtain b⫽

z n,l 关 1⫺4a 兴 . 4d

共4.19兲

Finally we find that a is n,l independent, a⫽⫺

1 , 2 共 d⫹2 兲共 d⫺1 兲

共4.20兲

whereas b does depend on n and l, and we therefore denote it as b n,l b n,l ⫽

共 d⫹1 兲 z . 4 共 d⫹2 兲共 d⫺1 兲 n,l

Using 共A8兲, we find that

共4.21兲

In the next subsection we compute from these results the scaling exponents in the sectors of the SO(d) symmetry group with l⭐n.

and therefore, we finally obtain Sn 共 ␮ r1 , ␮¯r1 ;... 兲 ⫽ ␮ n 兵 1⫺2 ⑀ 关 n 共 n⫺1 兲 a⫹b n,l 兴 ln ␮ 其

B. The scaling exponents of the structure functions

We now wish to show that the solution for the zero modes of the correlation functions Fn 共i.e., Z n 兲 result in homogeneous structure functions Sn . In every sector l⭐n, ␴ we compute the scaling exponents, and show that they are independent of ␴. Accordingly the scaling exponents are denoted ␨ ln , and we compute them to first order in ⑀. Using 共4.7兲, 共4.8兲, the structure function is given by

⫻Sn 共 r1 ,r ¯1 ;... 兲 ⫹O 共 ⑀ 兲 2 共l兲

⫽ ␮ ␨ n Sn 共 r1 ,r ¯1 ;... 兲 ⫹O 共 ⑀ 2 兲 . The result of the scaling exponent is now evident:

冋

␨ 共nl 兲 ⫽n⫺2 ⑀ ⫺

n 共 n⫺1 兲 共 d⫹1 兲 ⫹ z 2 共 d⫹2 兲共 d⫺1 兲 4 共 d⫹2 兲共 d⫺1 兲 n,l

⫹O 共 ⑀ 2 兲 ⫽n⫺ ⑀

冋

n 共 n⫹d 兲 共 d⫹1 兲 l 共 l⫹d⫺2 兲 ⫺ 2 共 d⫹2 兲 2 共 d⫹2 兲共 d⫺1 兲

⫹O 共 ⑀ 2 兲 .

共4.22兲 ␣

␣

␣

¯ i i , and the function f is defined as: where ⌬ i i ⬅r i i ⫺r

册

册 共4.24兲

For l⫽0 this result coincides with 关2兴. This is the final result of this calculation. It is noteworthy that this result is in full agreement with 共3.42兲 and 共3.20兲, even though the scaling exponents that appear in these results refer to different quantities. The way to understand this is the fusion rules that are discussed next.

4914

ARAD, L’VOV, PODIVILOV, AND PROCACCIA C. Fusion rules

The fusion rules address the asymptotic properties of the fully unfused structure functions when two or more of the coordinates are approaching each other, whereas the rest of the coordinates remain separated by much larger scales. A full discussion of the fusion rules for the Navier-Stokes and the Kraichnan model can be found in 关7,10兴. In this section we wish to derive the fusion rules directly from the zero modes that were computed to O( ⑀ ), in all the sectors of the symmetry group. In other words, we are after the dependence ¯1 ;...) on its first p pairs of of the structure function Sn (r1 ,r ¯1 ;...;rp ,r ¯ p in the case where these points coordinates r1 ,r are very close to each other compared to their distance from the other n⫺p pairs of coordinates. Explicitly, we consider ¯1 ;...;rp ,r ¯ p Ⰶrp⫹1 ,r ¯ p⫹1 ;...;rn ,r ¯n . 共We the case where r1 ,r have used here the property of translational invariance to put the center of mass of the first 2p coordinates at the origin.兲 The calculation is presented in Appendix D, with the final result 关to O( ⑀ )兴

PRE 62

l-dependent scaling exponents, and is of course a power law. The result shows that the l-dependent part is n independent. This means that the rate of isotropization of all the moments of the distribution function of field differences across a given scale is the same. This is a demonstration of the fact that the distribution function itself tends towards a locally isotropic distribution function at the same rate. We note in passing that to first order in ⑀ the l-dependent part is also identical for ␨ 2 , a quantity whose isotropic value is not anomalous. For all l ⬎1 also ␨ (l) 2 is anomalous, and in agreement with the n⫽1 value of Eq. 共1.1兲. Significantly, for ␨ 2 we have a nonperturbative result that was derived in 关7兴, namely,

␨ 共2l 兲 ⫽

冋

1 2⫺d⫺ ⑀ 2 ⫹

冑

共 2⫺d⫺ ⑀ 兲 2 ⫹

共5.1兲

Sn,l, ␴ 共 r1 ,r ¯1 ;...;rn ,r ¯n 兲 p

⫽

兺 兺 ␺j, ␴ ⬘S p, j, ␴ ⬘共 r1 ,r¯1 ;...;rp¯rp 兲 .

j⫽ j max ␴ ⬘

共4.25兲 In this expression the quantity ␺ j, ␴ ⬘ is a function of all the coordinates that remain separated by large distances, and j max⫽max兵 0,p⫹l⫺n 其 ,

册

4 共 d⫹ ⑀ ⫺1 兲 l 共 d⫹l⫺2 兲 , d⫺1

l⭐n.

共4.26兲

We have shown that the LHS has a homogeneity exponent ␨ (l) n . The RHS is a product of functions with homogeneity exponents ␨ (pj) and the functions ␺ j, ␴ ⬘ . Using the linear independence of the functions Sp, j, ␴ ⬘ we conclude that ␺ j, ␴ ⬘ ( j) must have homogeneity exponent ␨ (l) n ⫺ ␨ p . This is precisely the prediction of the fusion rules, but in each sector separately. One should stress the intuitive meaning of the fusion rules. The result shows that when p coordinates approach each other, the homogeneity exponent corresponding to these coordinates becomes simply ␨ (pj) as if we were considering a p-order correlation function. The meaning of this result is that p field amplitudes measured at p close by coordinates in the presence of n⫺p field amplitudes determined far away behave scalingwise, like p field amplitudes in the presence of anisotropic boundary conditions. In closing, we note that the tensor functions ␺ j, ␴ ⬘ do not necessarily belong to the j, ␴⬘ sector of SO(d).

valid for all values of ⑀ in the interval 共0,2兲 and for all l ⭓2. This exact result agrees after expanding to O( ⑀ ) with 共4.24兲 for n⫽1 and l⫽2. Our second motivation was to expose the correspondence between the scaling exponents of the zero modes in the inertial interval and the corresponding scaling exponents of the gradient fields. The latter do not depend on any inertial l scales, and the exponent appears in the combination (⌳/ ␰ ) ␨ n , where ␰ is the appropriate ultraviolet inner cutoff, either ␭ or ␩, depending on the limiting process. We found exact agreement with the exponents of the zero modes in all the sectors of the symmetry group and for all values of n. The reason behind this agreement is the linearity of the fundamental equation of the passive scalar 共2.1兲. This translates to the fact that the viscous cutoff ␩, Eq. 共2.12兲, is n and l independent, and also does not depend on the inertial separations in the unfused correlation functions. This point has been discussed in detail in 关7,11兴. In the case of Navier-Stokes statistics we expect this ‘‘trivial’’ correspondence to fail, but nevertheless the ‘‘bridge relations’’ that connect these two families of exponents has been presented in 关10兴 for the isotropic sector. Finally we note that in the present case we have displayed the fusion rules in all the l sectors, using the O( ⑀ ) explicit form of the zero modes. We expect the fusion rules to have a nonperturbative validity for any value of ⑀. It would be interesting to explore similar results for the Navier-Stokes case.

V. SUMMARY AND DISCUSSION

One motivation in this paper was to understand the scaling properties of the statistical objects under anisotropic boundary conditions. The scaling exponents were found for all l⭐n, cf., Eq. 共1.1兲. We found a discrete and strictly increasing spectrum of exponents as a function of l. This means that for higher l the anisotropic contributions to the statistical objects decay faster upon decreasing scales. In other words, the statistical objects tend towards locally isotropic statistics upon decreasing the scale. The rate of isotropization is determined by the difference between the

ACKNOWLEDGMENTS

We want to thank Yoram Cohen for numerous useful discussions. This work has been supported in part by the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities, the German-Israeli Foundation, The European Commission under the TMR program, the Henri Gutwirth Fund for Research, and the Naftali and Anna Backenroth-Bronicki Fund for Research in Chaos and Complexity.

PRE 62

ANOMALOUS SCALING IN THE ANISOTROPIC . . .

4915

APPENDIX A: ANISOTROPY IN d DIMENSIONS

u 2 ⳵ ␣ ⳵ ␣ Y l, ␴ 共 uˆ 兲 ⫽⫺l 共 l⫹d⫺2 兲 Y l, ␴ 共 uˆ 兲 .

To deal with anisotropy in d dimensions we need classify the irreducible representations of the group of all d-dimensional rotations, SO(d) 关12兴, and then to find a proper basis for these representations. The main linear space that we work in 共the carrier space兲 is the space of constant tensors with n indices. This space possesses a natural representation of SO(d), given by the well known transformation of tensors under d-dimensional rotation. The traditional method to find a basis for the irreducible representations of SO(d) in this space is using the Young tableaux machinery on the subspace of traceless tensors 关12兴. It turns out that in the context of the present paper, we do not need the explicit structure of these tensors. Instead, all that matters are some relations among them. A convenient way to derive these relations is to construct the basis tensors from functions on the unit d-dimensional sphere that belong to a specific irreducible representation. Here also, the explicit form of these functions in unimportant. All that matters for our calculations is the action of the Laplacian operator on these functions. Let us therefore consider first the space Sd of functions over the unit d-dimensional sphere. The representation of SO(d) over this space is naturally defined by

One can easily check that for d⫽3, 共A2兲 gives the factor l(l⫹1), well known from the theory of angular momentum in quantum mechanics. To prove this identity for any d, note that

OR⌿ 共 uˆ 兲 ⬅⌿ 共 R⫺1 uˆ 兲 ,

共A1兲

where ⌿(uˆ ) is any function on the d-dimensional sphere, and R is a d-dimensional rotation. Sd can be spanned by polynomials of the unit vector uˆ . Obviously 共A1兲 does not change the degree of a polynomial, and therefore each irreducible representation in this space can be characterized by an integer l⫽0, 1, 2,..., specifying the degree of the polynomials that span this representation. At this point, we cannot rule out the possibility that some other integers are needed to fully specify all irreducible representations in Sd and therefore we will need below another set of indices to complete the specification. We can now choose a basis of polynomials 兵 Y l, ␴ (uˆ ) 其 that span all the irreducible representations of SO(d) over Sd . The index ␴ counts all integers other than l needed to fully specify all irreducible representations, and in addition, it labels the different functions within each irreducible representation. Let us demonstrate this construction in two and three dimensions. In two dimensions ␴ is not needed since all the irreducible representations are one dimensional and are spanned by Y l (uˆ )⫽e il ␾ with ␾ being the angle between uˆ and the the vector eˆ 1 ⬅(1,0). Any rotation of the coordinates in an angle ␾ 0 results in a multiplicative factor e i ␾ 0 . It is clear that Y l (uˆ ) is a polynomial in uˆ since Y l (uˆ )⫽ 关 uˆ •pˆ 兴 l , where pˆ ⬅(1,i). In three dimensions ␴ ⫽m, where m takes on 2l⫹1 values m⫽⫺l,⫺l⫹1,...,l. Here Y l,m ⬀e im ␾ P m l (cos ␪), where ␾ and ␪ are the usual spherical coordinates, and P m l is the associated Legendre polynomial of degree l⫺m. Obviously we again have a polynomial in uˆ of degree l. We now wish to calculate the action of the Laplacian operator with respect to u on the Y l, ␴ (uˆ ). We prove the following identity:

兩 u 兩 2⫺l ⳵ 2 兩 u 兩 l Y l, ␴ 共 uˆ 兲 ⫽0.

共A2兲

共A3兲

This follows from the fact that the Laplacian is an isotropic operator, and therefore is diagonal in the Y l, ␴ . The same is true for the operator 兩 u 兩 2⫺l ⳵ 2 兩 u 兩 l . But this operator results in a polynomial in uˆ of degree l⫺2, which is spanned by Y l ⬘ , ␴ ⬘ such that l ⬘ ⭐l⫺2. Therefore the RHS of 共A3兲 must vanish. Accordingly we write

⳵ 2 兩 u l 兩 Y l, ␴ 共 uˆ 兲 ⫹2 ⳵ ␣ 兩 u l 兩 ⳵ ␣ Y l, ␴ ⫹ 兩 u l 兩 ⳵ 2 Y l, ␴ 共 uˆ 兲 ⫽0.

共A4兲

The second term vanishes since it contains a radial derivative u ␣ ⳵ ␣ operating on Y l, ␴ (uˆ ), which depends on uˆ only. The first and third terms, upon elementary manipulations, lead to 共A2兲. Having the Y l, ␴ (uˆ ) we can now construct the irreducible representations in the space of constant tensors. The method is based on acting on the Y l, ␴ (uˆ ) with the isotropic operators u ␣ , ⳵ ␣ , and ␦ ␣␤ . Due to the isotropy of the above operators, the behavior of the resulting expressions under rotations is similar to the behavior of the scalar function we started with. For example, the tensor fields ␦ ␣␤ Y l, ␴ (uˆ ), ⳵ ␣ ⳵ ␤ Y l, ␴ (uˆ ) transform under rotations according to the (l, ␴ ) sector of SO(d). Next, we wish to find the basis for the irreducible representations of the space of constant and fully symmetric tensors with n indices. We form the basis ␣ ,..., ␣ n ⬅ ⳵ ␣1¯ ⳵ ␣nu nY

B l, ␴1 ,n

ˆ 兲, l, ␴ 共 u

l⭐n.

共A5兲

Note that when l and n are even 共as is the case invariably in ␣ ,..., ␣ this paper兲, B l, ␴1 ,n n no longer depends on uˆ , and is indeed fully symmetric by construction. Simple arguments can also prove that this basis is indeed complete, and spans all fully symmetric tensors with n indices. Other examples of this procedure for the other spaces are presented directly in the text. Finally let us introduce two identities involving the B n,l, ␴ that are used over and over through the paper. The first one is ␣ ,..., ␣ n ⫽z

␦ ␣ 1 ␣ 2 B l, ␴1 ,n

␣ 3 ,..., ␣ n n,l B l, ␴ ,n⫺2 ,

z n,l ⫽ 关 n 共 n⫹d⫺2 兲 ⫺l 共 l⫹d⫺2 兲兴 .

共A6兲 共A7兲

It is straightforward to derive this identity using 共A2兲. The second identity is ␣ 其 ,m⫽i, j ␣ ,..., ␣ ␦ ␣ ␣ B 兵l, ␴ ,n⫺2 ⫽B l, ␴ ,n , 兺 i⫽ j i j

m

1

n

l⭐n⫺2 .

共A8兲

This identity is proven by writing u n in 共A5兲 as u 2 u n⫺2 , and operating with the derivative on u 2 . The term obtained as u 2 ⳵ ␣ 1 ¯ ⳵ ␣ n u n⫺2 Y l, ␴ (uˆ ) vanishes because we have n derivatives on a polynomial of degree n⫺2. It is worthwhile no-

4916

ARAD, L’VOV, PODIVILOV, AND PROCACCIA

ticing that these identities connect tensors from two different spaces: The space of tensors with n indices and the space of tensors with n⫺2 indices. Nevertheless, in both spaces, the tensors belong to the same (l, ␴ ) sector of the SO(d) group. This is due to the isotropy of the contraction with ␦ ␣ 1 ␣ 2 in the first identity, and the contraction with ␦ ␣ i ␣ j in the second identity.

in a diagram with s rungs. After some combinatorial calculations of the weights one finds s⫺1

D m⫹1,s ⫽

冕

dpˆ 2 ␤ ␤ p i j 共 pˆ 2 兲 . ⍀d

冕

d⫺1 , d

共B1兲

兺

共C3兲

q ⫺1

1 2 1 1 , s q 2 ⫽2 q 2 q 1 ⫽1 q 1

兺

兺

q ⫺1

s⫺1

共B2兲

1 1 , s q⫽1 q

s⫺1

D 3,s ⫽

D 4,s ⫽

dpˆ ⫽⍀ 共 d 兲 , ⍀ d ⬅⍀ 共 d 兲

共C2兲

s⫺1

Using the identities

冕

兺

D 2,s ⫽

In case a of region II where p 1 ⬎p 2 ⬎k s the analytic expression for A2 can be simplified to 1 ␣i ␣ j ˆp pˆ 2 1 1

1 D . s q⫽m q,m

Together with 共C1兲 this gives

APPENDIX B: PROOF OF EQ. „3.32…

␣␣ A 2 ␤i ␤j ⫽ i j

PRE 62

共C4兲

q ⫺1

1 3 1 2 1 1 , etc. s q 3 ⫽3 q 3 q 2 ⫽2 q 2 q 1 ⫽1 q 1

兺

兺

兺

共C5兲

The general structure of D m,s now becomes obvious. 2. Higher order terms in A

␣ ␤

dpˆ pˆ pˆ ⫽ ␦ ␣␤ ⍀ 共 d 兲 ,

共B3兲

Consider the equation 共3.37兲 ⬁

1 ␣ ␣ ␣␣ A 2 ␤i ␤j ⫽ ˆp 1 i pˆ 1 j ␦ ␤ i ␤ j . 共B4兲 i j 2 Substituting this form into the double rung ladder diagram results, after contracting all the indices of A1 and A2 , in a form identical to Eq. 共3.15兲 for A in the one rung ladder diagram. This leads directly to the final equation, Eq. 共3.31兲.

s⫽2

冋兺 冉

冊册

兺

兺

共C6兲

Observing that s⫺1

兺 q⫽1

1 1 ⫽ q 2

s⫺1

q⫽1

1 1 ⫹ s⫺q q

,

冉

冊

1 1 1 1 ⫹ ⫽ , s q s⫺q q 共 s⫺q 兲

1. Calculation of D m,s

In this Appendix we discuss the calculation of the coefficients D m,s in Eq. 共3.34兲, and the actual resummation of that equation. First, we need to introduce rules to evaluate the rungs in the general ladder diagram that appear in the expansion. The rule is actually quite simple: every rung contributes a term proportional to gA (l) n if the p vector associated with this rung is the largest among all the p vectors associated with rungs appearing to the right of it. Otherwise the contribution is proportional to ˜g . The weight of the contribution is obtained as a factor c⭐1 which reflects the proportional fraction of the volume of (p1 ,p2 ,...) space in which the associated ordering of the p vectors is valid. For example, if the rung with the largest p vector is in the extreme right, then all the other rungs contribute terms proportional to ˜g . Thus a diagram with s rungs ordered in this manner contributes with a weight C⫽1/s. Therefore 共C1兲

The recurrence relation for D m,s with m⬎1 is derived by inserting an additional rung that is associated with the largest p vector in any one of the (s⫹1) possible positions available

共C7兲

and

APPENDIX C: DOUBLE RESUMMATION

1 D 1,s ⫽ . s

s⫺1

˜g s 1 . s q⫽1 q

K 2 共 g,A 兲 ⫽A 2

we compute

共C8兲

we end up with ⬁

⬁

1 K 2 共 ˜g ,A 兲 ⫽ A 2 2 q 1 ⫽1

兺 q兺⫽1 2

˜g q 1 ⫹q 2 , q 1q 2

共C9兲

where we relabeled q→q 1 and s⫺q→q 2 and changed correspondingly the limit of summation over q 2 . Thus 1 K 2 共 ˜g ,A 兲 ⫽ A 2 2

冋兺 册 ⬁

2

1 1 ⫽ 关 ⫺A ln共 1⫺g 兲兴 2 . q 2

q⫽1

共C10兲

The terms proportional to A 3 give ⬁

K 3 共 ˜g ,A 兲 ⫽A 3

s⫺1

q ⫺1

˜g n 1 2 1 . n q 2 ⫽2 q 2 q 1 ⫽1 q 1

兺

兺

s⫽3

兺

共C11兲

We can rearrange the sums by summing over q 3 ⫽n⫺q 1 ⫺q 2 instead of n. Using relationships similar to 共C7兲 and 共C8兲 we find ⬁

1 K 3⫽ A 2 6 q 1 ⫽1

⬁

⬁

兺 q兺⫽1 q兺⫽1 2

3

˜g q 1 ⫹q 2 ⫹q 3 . q 1q 2q 3

共C12兲

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4917

Obviously this leads to 1 K 3 共 ˜g ,A 兲 ⫽ A 3 6

冋兺 册 ⬁

q⫽1

3

˜g q 1 ⫽ 关 ⫺A ln共 1⫺g ˜ 兲兴 3 . q 3! 共C13兲

The general structure is now clear, leading to Eq. 共3.38兲. APPENDIX D: DERIVATION OF THE FUSION RULES

In this Appendix we derive the fusion rules 共4.25兲. Consider a fully unfused structure function with n coordinates, such that p of them are separated from each other by a typical distance r, whereas n⫺ p coordinates are separated from them and from each other by a typical distance R, and R Ⰷr. We want to compute the asymptotic properties of S n and show that to leading order in r/R we find Eq. 共4.25兲. For homogeneous ensembles we can shift the origin to the center of mass of the p coordinates. In this case we have r j Ⰶr i for every j⭐p and i⬎p. Our aim is to separate the dependence on the small distances from the dependence on the large distances. We will see that some of the terms in S n lend themselves naturally to such a separation, and some call for more work. We start from Eq. 共4.22兲, and compute to first order in r/R: ¯i ,r j ,r ¯j兲 f ␣ i ␣ j 共 ri ,r ⬅ 共 r i ⫺r j 兲 ␣ i 共 r i ⫺r j 兲 ␣ j ln兩 ri ⫺r j 兩 ␣i

␣j

⫹ 共¯r i ⫺r ¯ j 兲 共¯r i ⫺r ¯ j 兲 ln兩¯ri ⫺r ¯ j 兩 ⫺ 共 r i ⫺r ¯ j兲

␣i

⫻ 共 r i ⫺r ¯ j 兲 ␣ j ln兩 ri ⫺r ¯ j 兩 ⫺ 共¯r i ⫺r j 兲 ␣ i 共¯r i ⫺r j 兲 ␣ j ln兩¯ri ⫺r j 兩

冋

␣

␣

␣

␣j

⫽ ⫺2r i i ␦ ␤ j ln r i ⫺r i i r i ␣

␣j

⫹r ¯ i i¯r i

共¯r i 兲 ␤

¯r 2i

册

共 ri兲␤

r 2i

␣

We note that of these three terms only I 2 has a nontrivial mixing of small and large coordinates, and indeed it is the only term in which the expansion 共D1兲 was employed. Collecting terms we find

␣

⫹2r ¯ i i ␦ ␤ j ln ¯r i

⌬ ␤j ⬅g ␣ i ␣ j ␤ 共 ri ,r ¯i 兲 ⌬ ␤j ,

共D1兲

and so, if r j ,r ¯ j Ⰶr i ,r ¯ i for j⫽1,...,p, i⫽ p⫹1,...,n then the first order in ⑀ of Sn will contain three types of terms: where

˜B ␣ 1 ,..., ␣ p ⫽⌬ ␣ p⫹1 ,...,⌬ ␣ n B ␣ 1 ... ␣ n p p⫹1 n n,..., ␴ p

⫽

␣ , 兺 兺 c j, ␴ ⬘B ␣p, j,,..., ␴ j⫽max兵 0,p⫹l⫺n 其 ␴ 1

p

⬘

˜B ␣ 3 ... ␣ p ⫽⌬ ␣ p⫹1 ,...,⌬ ␣ n B ␣ 3 ,..., ␣ n p⫺2 p⫹1 n n⫺2,l, ␴ p⫺2

⫽

␣ ,..., ␣ d j, ␴ ⬘ B p⫺2,j, ␴ , 兺 兺 j⫽max兵 0,p⫹l⫺n 其 ␴ 1

⬘

p

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PRE 62

␣

␣

p⫹1 g 共 uˆ 兲 ⬅u ⫺p⫹2 关 ⌬ p⫹1 ,...,⌬ n n 兴 ⳵ p⫹1 ¯ ⳵ n u n⫺2 Y l, ␴ 共 uˆ 兲

⫽

兺

j, ␴ ⬘

d j, ␴ ⬘ Y j, ␴ ⬘ 共 uˆ 兲 .

To obtain 共D2兲, operate with u 2 ⳵ 2 on f (uˆ ). On the one hand, we get

u 2 ⳵ 2 f 共 uˆ 兲 ⫽

兺

j, ␴ ⬘

⫺ j 共 j⫹d⫺2 兲 c j, ␴ ⬘ Y j, ␴ ⬘ 共 uˆ 兲

but on the other hand, we have ˜ p is a fully symmetric In these expressions we use the fact B tensor and therefore can be again expanded in terms of the basis functions Bp, j, ␴ ⬘ with coefficients that depend on the large separations. The sums on the right-hand sides run between j⫽max兵0,p⫹l⫺n 其 and p because not all the basis functions can appear when p⫹l⫺n⬎0. This can be checked ˜ p with (n⫺l⫹2)/2 ␦ function. This conby contracting the B traction vanishes since it contains a factor z l,l . On the other hand, the contraction results in a tensor with p⫹l⫺n⫺2 indices, and therefore all corresponding coefficients of j⭐ p ⫹l⫺n⫺2 must vanish. To proceed we establish the following identity: z p, j c ⫽d j, ␴ ⬘ . z n,l j, ␴ ⬘

共D2兲

The identity is proven by the following calculations: ␣

␣

␣ ,..., ␣ n

␣

␣

p⫹1 ,...,⌬ n n 兴 ⳵ p⫹1 ¯ ⳵ n u n Y l, ␴ 共 uˆ 兲兴 u 2 ⳵ 2 f 共 uˆ 兲 ⫽u 2 ⳵ 2 关 u ⫺p 关 ⌬ p⫹1

⫽⫺p 共 ⫺ p⫹d⫺2 兲 f 共 uˆ 兲 ⫺2 p 2 f 共 uˆ 兲 ⫹z n,l g 共 uˆ 兲 ⫽⫺p 共 p⫹d⫺2 兲 f 共 uˆ 兲 ⫹z n,l g 共 uˆ 兲 . Equating the two expressions, and projecting over the ( j, ␴ ⬘ ) sector, we obtain ⫺ j 共 j⫹d⫺2 兲 c j, ␴ ⬘ ⫽⫺p 共 p⫹d⫺2 兲 c j, ␴ ⬘ ⫹z n,l d j, ␴ ⬘

d j, ␴ ⬘ ⫽

z p, j 关 p 共 p⫹d⫺2 兲 ⫺ j 共 j⫹2⫺2 兲兴 c j, ␴ ⬘ ⫽ c . z n,l z n,l j, ␴ ⬘

Recalling Eq. 共4.21兲, b p, j ⫽b n,l z p, j /z n,l and we may write to leading order in r/R:

p⫹1 1 ,...,⌬ n n B n,l, ⌬ p⫹1 ␴

␣

␣

p⫹1 ⫽ ⳵ ␣ 1 ¯ ⳵ ␣ p u p u ⫺p 关 ⌬ p⫹1 ,...,⌬ n n 兴 ⳵ ␣ p⫹1 ¯ ⳵ ␣ n u n Y l, ␴ 共 uˆ 兲

⫽ ⳵ ␣1¯ ⳵ ␣ pu p

兺 j, ␴

⬘

␣

␣

¯1 ;...;rn¯rn 兲 S n 共 r1 ,r p

c j, ␴ ⬘ Y j, ␴ ⬘ 共 uˆ 兲

⫽

␣ p p⫺2 ⫺p⫹2

⫽⳵ ¯⳵ u

u

⫹⑀

␣ p⫹1 ␣ ,...,⌬ n n 兴 关 ⌬ p⫹1,

兺

j, ␴ ⬘

1

␣

␣

⫽ ␣

⫽

兺

j, ␴ ⬘

c j, ␴ ⬘ Y j, ␴ ⬘ 共 uˆ 兲 ,

␣

␣

␣ ,..., ␣ p

⌬ 1 1 , . . . , ⌬ p p B p,1j, ␴

⬘

兺

册

⬘

1

i j

p

⫹

␣ ,..., ␣ p

⫻⌬ 1 1 ,...,⌬ p p B p,1j, ␴ p

p⫹1 ,...,⌬ n n 兴 ⳵ p⫹1 ¯ ⳵ n u n Y l, ␴ 共 uˆ 兲 f 共 uˆ 兲 ⬅u ⫺p 关 ⌬ p⫹1

p

␣ 1 ,... ␣ p ⫹b p, j ␦ ␣ i ␣ j B p⫺2,l, 兴 ␴⬘

d j, ␴ ⬘ Y j, ␴ ⬘ 共 uˆ 兲 .

Denote now ␣

冋

␣ ,..., ␣ ␣ ␣ ⌬ 1 , . . . , ⌬ p f ␣ ␣ 共 ri ,r ¯i ,r j ,r ¯ j 兲关 aB p,l, ␴ 兺 ⬘ i⫽ j no i, j

⫻ ⳵ ␣ p⫹1 ¯ ⳵ ␣ n u n⫺2 Y l, ␴ 共 uˆ 兲 ⫽ ⳵ ␣3¯ ⳵ ␣ pu p

兺 C j, ␴ ⬘

no i, j

␣ ,..., ␣

p⫹1 3 n ⌬ p⫹1 ,...,⌬ n n B n⫺2,l, ␴

␣3

兺

j⫽max兵 0,p⫹l⫺n 其 ␴ ⬘

兺

⫹O 共 ⑀ 2 兲

兺 共 c j, ␴ ⬘⫹ ⑀ e j, ␴ ⬘ 兲

j⫽max兵 0,p⫹l⫺n 其 ␴ ⬘

⫻S p, j, ␴ ⬘ 共 r1 ,r ¯1 ;...;rp ,r ¯ p 兲 ⫹O 共 ⑀ 2 兲 . From this follows Eq. 共4.25兲.

兺 ⑀ e j, ␴ ⬘

j⫽max兵 0,p⫹l⫺n 其 ␴ ⬘

p

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ANOMALOUS SCALING IN THE ANISOTROPIC . . .

关1兴 R. H. Kraichnan, Phys. Fluids 11, 945 共1968兲. 关2兴 D. Bernard, K. Gawedcki, and A. Kupiainen, Phys. Rev. E 54, 2564 共1996兲. 关3兴 R. H. Kraichnan, Phys. Rev. Lett. 72, 1016 共1994兲. 关4兴 V. S. L’vov, I. Procaccia, and A. L. Fairhall, Phys. Rev. E 50, 4684 共1994兲. 关5兴 K. Gawedzki and A. Kupiainen, Phys. Rev. Lett. 75, 3834 共1995兲. 关6兴 M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedeev, Phys. Rev. E 52, 4924 共1995兲. 关7兴 A. Fairhall, O. Gat, V. S. L’vov, and I. Procaccia, Phys. Rev.

4919

E 53, 3518 共1996兲. 关8兴 L. Ts. Adzhemyan, N. V. Antonov, and A. V. Vasil’ev, Phys. Rev. E 58, 1823 共1998兲. 关9兴 G. K. Betchelor, The Theory of Homogeneous Turbulence 共Cambridge University Press, Cambridge, 1953兲. 关10兴 V. S. L’vov and I. Procaccia, Phys. Rev. E 54, 6268 共1996兲. 关11兴 V. S. L’vov and I. Procaccia, Phys. Rev. Lett. 77, 3541 共1996兲. 关12兴 M. Hamermesh, Group Theory and its Applications to Physical Problems 共Addison-Wesley, Reading, MA, 1962兲. See Chap. 10 for discussion of the irreducible representation of SO(d).