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,

Evolution of trajectory correlations in steady random ows

Albert Fannjiang, Leonid Ryzhik, and George Papanicolaou

To P.Lax and L. Nirenberg.

Abstract. We analyze the behavior of the correlation for two nearby trajectories of motion in a randomincompressible ow with nonzero mean and small uctuations. We show that the Fourier transform of theRichardson function of a passive scalar advected by the ow satis�es, under certain conditions, a radiativetransport equation. We also study the stretching of curves advected by the ow and show that their lengthgrows algebraically in time, and not exponentially as it does for time dependent, zero mean random ows.

Contents

1. Introduction 12. Long time di�usive behavior 23. Flow correlations 44. The Richardson function and its evolution 55. The expectation of the Richardson function 86. Higher moment equations 97. The Wigner distribution and connections with radiative transport theory 108. Di�usion{transport duality 129. General scaling of the uctuations 1410. Evolution of the Jacobian matrix 1511. Application to two-dimensional ows 1812. Deformation of the length 1913. Summary and conclusions 2014. Appendix A. A limit theorem for turbulent di�usion 2015. Appendix B. Oscillatory functions and the Wigner distribution 21Acknowledgments 22References 22

1. Introduction

We consider motion in a steady, random, incompressible ow with constant drift u. The random part ofthe ow v(x), with r � v = 0, is a smooth spatially homogeneous random �eld with zero mean and rapidlydecaying correlations. The trajectory X"(t;x) advected by the ow satis�es the scaled equation

dX"

dt= u+

p"v

�X"

"

�(1.1)

X"(0;x) = x;

1991 Mathematics Subject Classi�cation. Primary 60F05, 76F05, 76R50; Secondary 58F25.

c 0000 American Mathematical Society0160-7634/00 $1.00 + $.25 per page

1

2 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

where x 2 Rd, d = 2; 3, is the starting position. The small scaling parameter " determines the size of therandom uctuations and their correlation length. We are interested in the behavior of the rescaled two-pointdi�erence function

Y"(t;y;x) =1

"fX"(t;x)�X"(t;x� "y)g(1.2)

in the limit of "! 0. The individual trajectories X"(t) are close to the trajectories of the deterministic owX = x + ut for �nite times t, when " is small, and thus random perturbations do not play an essential rolein their behavior. The rescaled trajectory di�erences Y" that start nearby are, however, a�ected and thatis what we study in this paper. We note that this is di�erent from the analysis of the long time correlationsof trajectories that are initially a �xed distance apart, independently of " [18, 21].

The paper is organized as follows. In Section 2 we review brie y the known results for the long time (oforder "�1) behavior of single trajectories, or groups of trajectories that are not close initially. In Section 3 weshow that the asymptotic behavior of Y" as "! 0 can be deduced from the limit theorems of [18, 21, 16],stated here in Appendix A. This allows us to derive in Sections 4 and 5 a singular di�usion equation forthe rescaled Richardson function. Higher moments of the Richardson function are studied in Section 6.In Section 7 we show how the di�usion equation for the Richardson function can be transformed into aradiative transport equation for the Wigner distribution associated with the trajectory di�erence functionY". We study the duality between the limit two-point di�usion process and its analog in Fourier space inSection 8. A scaling more general than 1.1 is treated in Section 9. In Section 10 we show that the Jacobianmatrix of the ow (1.1) also converges to a di�usion process in the limit " ! 0, and we analyze in detailits law in two-dimensional potential ows in Section 11. In Section 12 we apply these results to study thestretching of curves transformed by (1.1). We show that their length grows algebraically in this case andnot exponentially, as in the case of zero mean randomly time dependent ows [2, 8, 31].

2. Long time diffusive behavior

The long time di�usive behavior of trajectories of mean zero random velocity �elds was �rst studied byG.I.Taylor [25, 32]. He considered the trajectories of the system

dX(t) = v(t;X(t)) + �dW(t)(2.1)

X(0) = 0;

where v(t;x) is a mean zero random velocity �eld with rapidly decaying correlations, and W(t) is thestandard d-dimensional Brownian motion. He argued that in the long time limit the trajectories behave likethose of d-dimensional Brownian motion with covariance matrixZ 1

0

E fvi(t;X(t))vj(0; 0) + vj(t;X(t))vi(0; 0)gdt+ �2�ij :(2.2)

The �rst term in expression (2.2), which corresponds to the turbulent enhancement of the di�usion coe�cient,depends also on the molecular di�usivity � through the path X(t). It is convenient to scale equations (2.1)with a small parameter "

dX"(t) =1

"v(

t

"2;X"

") + �dW(t)(2.3)

X"(0) = 0

and analyze the behavior of the trajectories as " ! 0. When the molecular di�usivity � is positive andthe ow is incompressible, r � v = 0, it is shown by homogenization methods [27, 26, 1, 23, 13] that as" ! 0 the trajectories X" ! 0 converge weakly to Brownian motion with the covariance matrix given bythe homogenization formula (2.2) with X(t) the solution of (2.1) corresponding to " = 1 in (2.3). Whenthe molecular di�usivity � is positive, di�usive behavior holds for random velocity �elds that, in the time

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 3

independent case, are spatially homogeneous, incompressible and have a square integrable stream matrix[13]. Time dependent ows when � > 0 are analyzed in [12] and when � = 0 and there is rapid timedecorrelation of v in [17].

The weak uctuation scaling (described in detail in Appendix A) was intensively studied (with � = 0and � > 0 [16, 18, 21]). A typical scaling in this case, with � = 0, is

dX"

dt=

1

"v(

t

"2;X"(t)

" );(2.4)

X"(0) = x;

with 0 � < 1, so that now the random velocity �eld is varying on spatial scales large compared to ", butpossibly small on the overall scale if 6= 0. Then, as before, the process X"(t) converges weakly as "! 0 toa di�usion process. Its generator is given by

Lxf(x) =1

2

dXi;j=1

aij(x)@2f

@xi@xj:

This operator is self-adjoint because the velocity �eld v(x) is incompressible. The corresponding generatorfor a compressible ow has an extra drift term [18, 16, 21]. The di�usion matrix is given by

aij(x) =

Z 1

0E fvi(0;x)vj(t;x) + vj(0;x)vi(t;x)g :(2.5)

Expression (2.5) is called the Kubo formula. Of particular interest here is the system

dX"

dt=

1

"v(X"(t) +

t

"2u);(2.6)

X"(0) = x;

where v(x) is a space homogeneous, mean zero divergence free random �eld. Then X"(t;x) converges toBrownian motion with the covariance matrix given by the Kubo formula

cij(0) =

Z 1

�1

E fvi(tu)vj(0)gdt;(2.7)

This corresponds to the Taylor prediction (2.2) with the path X = ut being frozen and independent of therandom medium. We shall see in Section 3 why the path in (2.7) is deterministic. The joint law for two

trajectories of (2.6) starting at two points x;x0 was also obtained in [18, 21]. The joint process (X";X0")

converges weakly to a di�usion process (X;X0) with generator

~Lf(x;x0) = Lxf + Lx0f + L0f;

where the cross-term L0 is

L0f =1

2cij(x;x

0)@2f

@xi@x0j:

Here

cij(x� x0) =dX

i;j=1

Z 1

�1

E fvi(0)vj(x0 � x + tu)g dt:

Thus the di�erence function Y" = X" �X0" converges weakly to a di�usion process Y with generator

LYf = (cij(0) � cij(y)) @2f

@yi@yj:(2.8)

The behavior of the di�erence function for a stochastic ow on a compact manifold was considered in [3],where the properties of the two-point motion for various Lyapunov exponents were investigated.

4 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

The scaling in (1.1) is di�erent from both that of (2.3) and (2.4). As we noted above, the trajectories of(1.1) stay close to the deterministic trajectories X = x+ ut. We shall show, however, that after a rescalingthe dynamics of many quantities of interest can be reduced to a form similar to (2.4). For example, therescaled di�erence function (1.2) for the trajectories of (1.1) converges to a di�usion process with the samegenerator (2.8).

3. Flow correlations

Let X"(t;x) and X"(t;x � "y) be two trajectories of (1.1) that are "y apart initially. We consider therescaled di�erence variable

Y"(t;y;x) =X"(t;x)�X"(t;x� "y)

";

which is the scaled separation of the particles at time t. Then X"(t;x), Y"(t;y;x) satisfy

dX"

dt= u+

p"v

�X"

"

�; X"(0) = x(3.1)

dY"

dt=

1p"

�v

�X"

"

�� v

�X"

"�Y"

��; Y"(0) = y:

As we noted above the trajectory X"(t;x) is close to the deterministic trajectory X = x + ut for �nite t,and so we introduce its rescaled uctuations

Z"(t; x) =X"(t;x)� ut� x

":(3.2)

Then the system (3.1) becomes

dZ"

dt=

1p"v

�x+ ut

"+ Z"

�; Z"(0) = 0(3.3)

dY"

dt=

1p"

�v

�x+ ut

"+ Z"

�� v

�x + ut

"+ Z" �Y"

��; Y"(0) = y:

This system has the general form (14.1) to which the limit theorem of Appendix A applies. The fullstatement of this theorem and the necessary assumptions on the random velocity �eld in the general caseare recalled in Appendix A. We shall assume in particular that v(y) is a mean zero, divergence-free, spacehomogeneous and strongly mixing velocity �eld bounded in C3(Rd). Then the limit theorem implies thefollowing.

Theorem 3.1. The processes Z", and Y" converge weakly to the correlated di�usion processes Z andY whose joint generator is

Lf =1

2

dXi;j=1

�Z 1

�1

Rij(ut)dt@2f

@zi@zj+

Z 1

�1

(2Rij(ut)� Rij(ut+ y)� Rij(ut� y)) dt@2f

@yi@yj

+

Z 1

�1

(2Rij(ut)� 2Rij(ut� y)) dt@2f

@zi@yj

�:(3.4)

Here the covariance tensor Rij is de�ned by

E fvi(y)vj (y + h)g = Rij(h);(3.5)

The individual generators for Z and Y are:

LZf(z) =1

2

dXi;j=1

Z 1

�1

Rij(ut)dt@2f

@zi@zj;(3.6)

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 5

and

LY f(y) =dX

i;j=1

Z 1

�1

�Rsij(ut)�Rsij(y + ut)

�dt

@2f

@yi@yj;(3.7)

respectively, where

Rsij =Rij + Rji

2:

Thus the uctuations Z"(t) converge to Brownian motion with covariance matrix given by the Kuboformula (2.7). It is clear in this case that the Kubo formula (2.7) is the expression (2.2) predicted by Taylor.The path X(t) = x + ut is now the deterministic trajectory of (1.1) around which X"(t;x) is uctuatingbecause these two paths are close to each other. The generator (3.7) for the separation process Y(t;y)coincides with the one given by (2.8), as expected. We note that the di�usion coe�cient in (3.7) vanishesfor y = cu, so that if two particles start at two nearby positions on the same deterministic trajectory thentheir separation is not changed by the ow in the limit. The generator LY is asymptotically close to LZfor all y that have large component in the direction perpendicular to the mean ow u. The reason for thisis that when the two starting points are separated by a large distance in the direction normal to u, theirtrajectories are almost independent, and the rescaled di�erence trajectory behaves like the uctuations ofeach individual trajectory. The other end of the asymptotics, the small y behavior of LY , is related to theJacobian of the map x! X"(t;x), and is described in detail in Section 10.

4. The Richardson function and its evolution

The Richardson function of a scalar �(t;x) advected by a random ow

@�

@t+ v(x) � rx� = 0

�(0;x) = �0(x)

is de�ned [25] by

Q(t;x;y) = �(x)�(x+ y):

It was predicted by Richardson [29] that the expectation of this function satis�es the usual di�usion equationin y, in the long time limit. This problem was studied extensively by physicists (see [19] for an extensivereview and references), especially for �-correlated in time velocity �elds, but to the best of our knowledgethe only mathematical results are those in [18] and [24, 28]. Molchanov and Piterbarg [24] considered inparticular a convection-di�usion equation of the form

@T "

@t+

1

"v(

t

"2;x) � rxT " = ��T "(4.1)

T "(0;x) = T0(x):

They argued that the expectation E fQ"(x;y)g of the Richardson function has a limit Q(x;y) as "! 0. Ifthe initial density T0(x) is a space homogeneous isotropic random process with correlation function Q0(r),the limit Richardson function satis�es the di�usion equation

@Q

@t=

1

rd�1@

@rrd�1(2�+ F (r))

@Q

@r(4.2)

Q(0; r) = Q0(r):

Here

F (r) =

Z 1

�1

(RL(t; 0)�RL(t; r)) dt;

6 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

and RL(t; r) comes from the correlation matrix R(t;x) of the isotropic velocity �eld v(x):

Rij =

�RL(t; r) +

r

d� 1

@RL@r

��ij � xixj

r(d� 1)

@RL@r

:

Equation (4.2) is the generalization of (2.8) for � 6= 0. We note that the additional term due to non-zeromolecular di�usivity is additive. This is di�erent from the homogenization formulas of the type (2.2) whichcome from the homogenization scaling of (2.3).

We study here the rescaled Richardson function of oscillatory initial densities advected by the random ow. Let �"(x) be the solution of

@�"

@t+�u+

p"v�x"

� � � r�" = 0(4.3)

�"(0;x) = �"0(x);

where the initial density �"0(x) is "-oscillatory (see Appendix B for the precise de�nition of this notion). Itmay be either deterministic, or random but independent of v(x). An example of such random initial data is

�"0(x) = �0(x

");(4.4)

where �0(x) is a spatially homogeneous ergodic random process. Interesting deterministic initial data ofoscillatory form are the localized initial densities

�"0(x) =1

"d=2�0(

x

");(4.5)

where �0(x) is an L1(Rd) \ L2(Rd) function, and the inhomogeneous wave family

�"(x) = A(x)eiS(x)="(4.6)

where A(x) and S(x) are smooth functions, and S(x) is real valued. The latter family describes the dis-tribution of tracers which have the form of high frequency waves propagating in the direction rS(x) withamplitude A(x). A particular case of (4.6) is the high frequency plane wave

�"(x) = Aeix�p=":(4.7)

The rescaled Richardson function of the family �"(x) is de�ned by

W "(t;x;y) = Q(t;x� "y; "y) = �"(t;x� "y)�"�(t;x);(4.8)

so that its expectation is the correlation function of the �led �"(x) at two nearby points separated by "y. It

has a weak limit W (t;x;y) as "! 0 in the space A0 dual to (see Appendix B)

A =

�f(x;y) :

Zdy sup

x

jf(x;y)j <1�;

introduced in [22]. The Richardson function of the family (4.4) is

W (x;y) = R0(y);(4.9)

with probability one, by the ergodic theorem, and is independent of x. Here R0(x) is the covariance functionof the random process �0(x). The Richardson functions of the localized deterministic family (4.5) is

W (x;y) = �(x)

ZQ0(z;y)dz;(4.10)

where

Q0(x;y) = �0(x � y)��0(x):(4.11)

The limit Richardson function of the WKB family (4.6) is

W (x;y) = jA(x)j2e�irS(x)�y;(4.12)

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 7

which reduces to

W (x;y) = jAj2e�ip�y(4.13)

for the plane waves (4.7). The limits (4.10) and (4.12) correspond to localization in space and direction,respectively.

The rescaled Richardson function W "(t;x;y) of the oscillatory �eld �"(t;x) may be reduced to theform studied in [18, 24], when the initial data has the form (4.4) and �0(x) is Gaussian. In that case

�"(t;x) = "(x� ut

"; t), where "(t;x) satis�es an equation of the form (4.1) with � = 0

@ "

@t+

1

"v

�x+ u

t

"2

�� rx " = 0(4.14)

"(0;x) = �0(x):

The rescaled Richardson function W "(t;x;y) of �"(x) and the unscaled Richardson function Q"(t;x;y) of "(x) are related by

W "(t;x;y) = Q"�t;x � ut

";y

�:

When the initial data for (4.3) is random, space homogeneous of the form (4.4), both the scaled and unscaledRichardson functions are independent of x in the limit " ! 0, hence they coincide in this limit, and theanalysis for (4.3) can be reduced to the one in [24] and [18]. We consider here a more general class of"-oscillatory initial data, not necessarily random and Gaussian, and also a more general scaling in Section9. The following theorem shows that a version of (2.8) and (4.2) still holds.

Theorem 4.1. Let �"(t;x) satisfy (4.3) with the initial data �"0(x) being either

(i) random of the form (4.4) with �0(x) spatially homogeneous and independent of v(x), or(ii) deterministic of the form (4.5) with �0(x) 2 L2(Rd), or(iii) the plane wave (4.7).

Let W (t;x;y) be the weak limit of the expectation of the rescaled Richardson function W "(t;x;y) in A0as " ! 0, and W0(x;y) be the limit of the expectation of the scaled Richardson function for �"0(x). Then

W (t;x;y) satis�es the di�usion equation with drift:

@W

@t+ u � rxW = LY W(4.15)

W (0;x;y) = W0(x;y);

where the operator LY is given by (3.7). The initial data W0(x;y) in (4.15) are given by (4.9) in the randomcase, and by (4.10) and (4.13) in the two deterministic cases.

We shall give the proof of Theorem 4.1 for the deterministic case (ii) of a localized pulse, with themodi�cation for random initial data (4.4) and the plane waves (4.7) being routine. Our proof given in

Section 5 is in several steps. First, we obtain an equation for W "(t;x;y) in the form (14.1) but with initial

data W "(0;x;y) depending on ". In the second step we show that these initial data may be replaced by the

limit rescaled Richardson function W0(x;y). Finally we apply the limit theorem of Appendix A to obtain

equation (4.15) for W (t;x;y) = lim"!0

E fW "(t;x;y)g.We believe that this theorem is true for a much wider class of initial data than (4.4), (4.5) and (4.7).

The technical di�culty in generalizing this result arises in the second step of the proof, that is, replacingthe initial data W "

0 by the limit W0. The limit theorem of Appendix A does not provide any informationon the uniformity of the convergence to the limit process with respect to the starting point in all of Rn. Inparticular, we do not have uniform bounds for the probability to visit a �xed set, which are needed in the

8 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

general case when we cannot control the convergence of W "0 (x;y) to W0(x;y). This does not allow us to

include the physically important case of general wave data (4.6) in Theorem 4.1.

5. The expectation of the Richardson function

Let the initial data for (4.3) be of the form (4.5). Then W "0 (x;y) =

1

"dQ0(

x

";y), and equation (4.3)

implies that W " satis�es the initial value problem

@W "

@t+ u � rxW " +

p"v�x"

�� rxW " +

1p"

nv�x"

�� v

�x"� y

"

�o� ryW " = 0(5.1)

with

W "(0;x;y) =1

"dQ0(

x

";y);(5.2)

where Q0 is given by (4.11). The solution of (5.1) is given explicitly by

W "(t;x;y) = W "0 (x � ut+ "Z"(�t;x);Y"(�t;y;x)):

Let

W0"(t;x;y) = W0(x � ut+ "Z"(�t;x);Y"(�t;y;x))

be the solution of (5.2) with the initial data W "0 (x;y) replaced by W0(x;y), which is given by (4.10). We

claim that

EfW "(t;x;y)� W 0"(t;x;y)g ! 0(5.3)

weakly in A0 as " ! 0. Let f(x;y) 2 A be a continuous non-random test function of compact support.Then, since the ow is incompressible

E

�ZZdxdy(W "(t;x;y)� W `"(t;x;y))f(x;y)

�=

ZZdxdy(W "

0 (x;y)� W0(x;y))E ff(x + ut+ "Z"(t;x);Y"(t;y;x))g(5.4)

=

ZZdxdy(

1

"dQ0(

x

";y)� �(x)

ZdpQ0(p;y))E ff(x + ut+ "Z"(t;x);Y"(t;y;x))g

=

ZZdxdyQ0(x;y)E ff("x + ut+ "Z"(t; "x);Y"(t;y; "x))� f(ut + "Z"(t; 0);Y"(t;y; 0))g :

Theorem 3.1 implies that

E ff(ut + "Z"(t; 0);Y"(t;y; 0))g ! �f (0;y);(5.5)

strongly in the uniform norm on compact sets in y. Here the function �f (x;y) satis�es the di�usion equationwith a drift

@ �f

@t= u � rx �f + LY �f

�f (0;x;y) = f(x;y);

with the operator LY given by (3.7). The limit theorem of Appendix A applies to the system

d~Z"

dt=

1p"v

�ut

"+ ~Z"

�; ~Z"(0) = x

d ~Yeps

dt=

1p"

�v

�ut

"+ ~Z"

�� v

�ut

"+ ~Z" � ~Y"

��;

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 9

where ~Z"(t;x) = x+ Z"(t; "x), and ~Y"(t;y;x) = Y"(t;y; "x). It implies that the expectation

E ff("x + ut+ "Z"(t; "x);Y"(t;y; "x))ghas the same limit as in (5.5). Then by the dominated convergence theorem (5.5) vanishes in the limit "! 0

since Q0(x;y) 2 L1(Rd � Rd). Thus (5.3) holds, and we may replace the initial data W "0 by W0 in (5.1),

and

W (t;x;y) = lim"!0

EnW "(t;x;y)

o= lim

"!0EnW

0"(t;x;y)o:

The limit theorem applies to the weak form of the resulting system, that is, for any test function f(x;y) wehave ZZ

EnW

0"(t;x;y)of(x;y)dxdy =

ZZW0(x;y)E ff(x + ut+ "Z"(t;x);Y"(t;x;y))gdxdy

=

ZZdxdyQ0(x;y)E ff(ut + "Z"(t; 0);Y"(t; 0;y))gdxdy!

ZZdxdyQ0(x;y) �f(ut;y)dxdy

by the dominated convergence theorem. Thus the function W (t;x;y) is the weak solution of

@W

@t+ u � rxW = LY W(5.6)

W (0;x;y) = W0(x;y):

Here the operator LY is given by (3.7). This �nishes the proof of Theorem 4.1.

6. Higher moment equations

We show how the joint behavior of trajectories starting at several points may be studied using the limittheorem of Appendix A. Consider �rst two pairs of trajectories, X"(t;x1), X

"(t;x1 � "y1), and X"(t;x2),

X"(t;x2 � "y2). Then the corresponding processes Z"j and Y"j satisfy the following system:

dZ"1dt

=1p"v

�x1 + ut

"+ Z"1

�; Z"1(0) = 0(6.1)

dY"1

dt=

1p"

�v

�x1 + ut

"+ Z"1

�� v

�x1 + ut

"+ Z"1 �Y"

1

��; Y"

1(0) = y1

dZ"2dt

=1p"v

�x2 + ut

"+ Z"2

�; Z"2(0) = 0

dY"2

dt=

1p"

�v

�x2 + ut

"+ Z"2

�� v

�x2 + ut

"+ Z"2 �Y"

2

��; Y"

2(0) = y2:

The joint process (Z"1;Y"1;Z

"2;Y

"2) converge to the process (Z1;Y1;Z2;Y2) with generator of the form

L(2) = L1 + L2 + L12;

where the operators L1 and L2 are given by (3.4) in the variables (z1;y1) and (z2;y2), respectively. Thecross term L12 involves terms of the type

cij(x1;x2;y1;y2)@2

@zi1@yj2

; dij(x1;x2;y1;y2)@2

@yi1@yj2

;

and other similar ones. The coe�cients cij, dij, : : : are non-zero only for points x1, x2 which lie on the samedeterministic trajectory, that is, x1 � x2 = �u for some � 2 R. Thus when the points x1 and x2 do not lie

10 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

on the same deterministic trajectory, the pairs of the limit processes Z1, Y1 and Z2, Y2 are independentand their joint generator is the sum of the corresponding generators:

L(2) = L1 + L2:

This phenomenon occurs also for all higher moments.Let us consider now the case when the two trajectories start at the same point x1 = x2 = x, so that

Z"1 = Z"2. Then the system (6.1) reduces to

dZ"

dt=

1p"v

�x+ ut

"+ Z"

�; Z"(0) = 0(6.2)

dY"1

dt=

1p"

�v

�x+ ut

"+ Z"

�� v

�x + ut

"+ Z" �Y"

1

��; Y"

1(0) = y1

dY"2

dt=

1p"

�v

�x+ ut

"+ Z"

�� v

�x + ut

"+ Z" �Y"

2

��; Y"

2(0) = y2:

The joint generator for the limit processes Y1, Y2 has the form:

L(2)f(y1;y2) =1

2a��ij (y1;y2)

@2f

@y�i @y�j

;

where the Greek indices label the points and the Latin indices label the coordinates. The coe�cientsa��ij (y1;y2) are given by

a��ij (y1;y2) =

Z 1

�1

[Rij(ut)�Rij(ut� y�) �Rji(ut� y�) + Rij(ut� y� + y�)] dt:

Let G(t;x;y1;y2) be the second moment of the Richardson function for the solution of (4.3)

G(t;x;y1;y2) = lim"!0

EnW "(t;x;y1)W

"(t;x;y2)o

= lim"!0

E f�"(t;x� "y1)�"(t;x)�"(t;x� "y2)�

"(t;x)g :

Assume that the initial data is of the form (4.4) with �0(x) being Gaussian. Then G satis�es the initialvalue problem:

@G

@t= L(2)G;(6.3)

G(0;y1;y2) = 2R0(y1)R0(y2) + R0(y1 � y2)R0(0):

All the higher moment equations for the Richardson function can be derived in a similar way.

7. The Wigner distribution and connections with radiative transport theory

The Wigner distribution of an oscillatory family �"(x) is de�ned as the inverse Fourier transform of therescaled Richardson function

W "(t;x;k) =

Zdy

(2�)deik�y�"(t;x� "y)�"�(t;x):(7.1)

The Wigner transform has a weak limit W (t;x;k) as " ! 0, which is a non-negative measure [14] (seeAppendix B). The limit Wigner distribution may be interpreted as the limit energy (or particle) density ofan "-oscillatory family �"(x), travelling in the direction k at position x. In particular, we have

lim"!0

j�"(x)j2 =ZW (t;x;k)dk(7.2)

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 11

in the weak sense if and only if �"(x) is "-oscillatory. The limit Wigner distribution also determines thelimit values as " ! 0 of various others quantities of interest, for example, the correlation functions of thederivatives of �". The limit of the correlation matrix of the gradient

~W "ij(t;x;y) = "

@�"

@xi(x� "y)"

@��"

@xj(x)

is the Hessian in y of W (t;x;k):

~W "ij(t;x;y)!

@2W

@yi@yj:(7.3)

The limits of the higher order derivatives can be similarly expressed via W (t;x;k). Thus, in the highfrequency limit the Wigner distribution gives a complete description both of the �eld �"(t;x) itself and itsderivatives. Relations of the type (7.2) and (7.3) are the main reason for the recent studies of the Wignerdistribution for waves in random media [30].

Let us brie y recall how radiative transport equations for waves arise [30]. Let w"(t;x) 2 CN be thesolution of a symmetric hyperbolic system (this scaling corresponds to � = 1 in (4.3))n

A(x) +p"V

�x"

�o @w"

@t+Dj @w

"

@xj= 0(7.4)

w"(0;x) = w"0(x):

Here the matrix A(x) is positive de�nite and the matrices Dj are symmetric. The initial data w"0(x) is

assumed to be "-oscillatory and deterministic, and V (y) is a matrix valued, space homogeneous randomprocess. Then the N �N limit Wigner matrix W (t;x;k) has a special form

W (t;x;k) =X�;i;j

W�ij(t;x;k)b

�;ib��;j:(7.5)

Here the vectors b�;j 2 CN form a basis for the eigenspace of the dispersion matrix

L(x;k) = A�1(x)kjDj

of the system (7.4), corresponding to the eigenvalue !�. The size of the square matrices W� is equal tothe degeneracy of the eigenvalue. They satisfy a system of radiative transport equations. This system isdecoupled when the eigenvalues are simple and !�(k) 6= !�(p) for all �, k and p. Then the radiativetransport equation for the scalar W� has the form

@W�

@t+rk!� � rxW� �rx!� � rkW�(7.6)

=

Zdk0�(x;k;k0)�(!�(k) � !�(k0))(W�(k0)�W�(k)):

The function �(x;k;k0) is the di�erential scattering cross-section and is determined by the power spectrumof V (y). Equation (7.6) has a long history. It was proposed phenomenologically by Rayleigh in the beginningof this century and then studied extensively by physicists [10]. Various derivations of this equation weregiven in 1960's (see [30] for references) when w" is a solution of the wave equation or Maxwell's equations. Ageneral case was treated in [30] but the results were derived only formally. The only case when the radiativetransport equation was obtained rigorously, to the best of our knowledge, is for the Schr�odinger equationwith V (y) Gaussian and only for small t [11, 15].

For the Wigner distribution of the density of a passive scalar in a random ow we have the followingtheorem.

12 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

Theorem 7.1. Let �"(t;x) be the solution of (4.3), with the initial data of the form (4.4), (4.5) or (4.7).Let W0(x;k) be the expectation of the limit Wigner distribution of �"0(x). Then E fW "(t;x;k)g convergesto W (t;x;k) weakly in A0, where W (t;x;k) satis�es the radiative transport equation

@W

@t+ u � rxW =

Zdk0

(2�)d�1kikjR

sij(k

0 � k)�((k0 � k) � u)(W (k0)�W (k))(7.7)

with the initial condition

W (0;x;k) = W0(x;k):

This theorem follows immediately from Theorem 4.1 by applying the inverse Fourier transform to (5.6).Equation (7.7) has the usual form of a radiative transport equation (7.6). The dispersion law of (4.3) is

!(k) = u � k:The scattering operator on the right side of (7.7) is symmetric since (k; R(k � k0)k) = (k0; R(k0 � k)k0)because of the incompressibility of the random �eld v(y). The transport equation is valid globally in timeand for random velocity �elds that are not necessarily Gaussian. This tells us that in more general cases theradiative transport equation should also be valid globally in time and for non-Gaussian random uctuations.It should also be valid for general inhomogeneous high frequency waves of the form (4.6), the restriction toplane wave initial data is technical as explained in the remarks after Theorem 4.1.

8. Diffusion{transport duality

The radiative transport equations (7.6) for waves have a nice interpretation in terms of a certain Mar-kovian jump process [5]. Consider the backward characteristics of (7.6)

_X = �rk!(X;K);(8.1)

_K = rx!(X;K);

starting at x(0) = x and K(0) = k. Let a particle move along a trajectory of (8.1) for a random time �1and then switch wave vector randomly fromK to K0. After that it follows trajectories of (8.1) for a randomtime �2, at which moment it switches it direction again. The process is continued in an obvious manner.The probability distribution of the j-th jump time �j is

P f�j > tg = exp

��Z t

0�(x(s;xj�1;kj�1);k(s;xj�1;kj�1))ds

�Here �(x;k) is the total scattering cross-section:

�(x;k) =

Zdk0�(x;k0;k)�(!�(k) � !�(k0));

and x(s;xj�1;kj�1), k(s;xj�1;kj�1) is the trajectory of (8.1) starting at the position and wave number ofthe previous jump. The probability density that the wave number jumps from direction k into direction k0

is given by

p(x;k;k0) =�(x;k;k0)

�(x;k):

The resulting process is well de�ned if the total scattering cross-section is bounded from above, so thatonly a �nite number of jumps occur during any given time interval, with probability one. The Kolmogorovequation for the resulting process is (7.6), that is, given any function f(x;k), the function �f (t;x;k) =E ff(X(t; x;k);K(t;x;k))g is the solution of (7.6) with the initial data �f (0;x;k) = f(x;k).

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 13

The integral operator on the right side of (7.7)

Qf =

Zdk0

(2�)d�1kikjR

sij(k

0 � k)�((k0 � k) � u)(f(k0) � f(k))(8.2)

is of the same form as the integral operator in (7.7). It is the generator of a jump process for which theparticle is moving along the deterministic trajectory x(t) = x(0) + ut, while the wave vector is the jumpprocess described above with the di�erential scattering cross-section

�(k;k0) =1

(2�)d�1kikjR

sij(k

0 � k)�((k0 � k) �u)

and the total scattering cross-section

�(k) =

Zdk

1

(2�)d�1kikjR

sij(k

0 � k)�((k0 � k) �u):(8.3)

Thus the projection of the wave vector on the mean velocity direction u is not changed after the jump. Thetotal scattering cross-section (8.3) is an unbounded function of k, and so the standard argument [5] thatK(t;k) starting at k does not go to in�nity in a �nite time does not apply. We note that the process K(t;k)is dual to the di�usion process Y(t;x;y) with generator LY given by (3.7). In fact we have that

LY (y)eik�y = Q(k)eik�y

and hence

EY

neik�Y(t;y)

o= EK

neiK(t;k)�y

o:(8.4)

This means that the process K(t;k) is well de�ned as long as the di�usion process Y(t;y) is well de�ned,and so it exists for all time. The duality (8.4) has an interesting qualitative implication. Given any functionf(t;x;y) we may view

�f(t;y) = E ff(Y(t;y))geither as a solution of

@ �f

@t+ u � rx �f = LY �f

f(0;x;y) = f(x;y)

or as the Fourier transform of the solution of

@f

@t+ u � rxf = Qf

f(0;x;k) =f (x;�k)(2�)d

:

The limit process K(t;k) has a unique invariant measure, which is Lebesgue measure on the plane k � u =const. This means that in the long time limit solutions of (8.5) are nearly functions of k � u only, that is,their support �lls out the whole plane, and the function is close to a small constant on this set. This impliesthat solutions of (8.5) tend to a delta function in the directions orthogonal to the mean ow u, and so thein the long time limitY(t) tends to be parallel to the mean velocity u. In terms of the ow, this means thatparticles starting nearby tend to be aligned with the ow no matter what their relative position is initially.

14 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

9. General scaling of the fluctuations

The results described above may be generalized to the case when random uctuations are oscillatingon a scale �ner than ", but their strength is also scaled appropriately. The trajectories X"(t;x) satisfy thescaled equation

dX"

dt= u+ "�v

�X"

"�

�;(9.1)

X"(0;x) = x:

The trajectory uctuation, de�ned as before by

Z"(t;x) =X"(t;x)� ut� x

"(9.2)

has a non-trivial limit if � = 1 � �

2, and 1 � � < 2. The strength of the uctuations � = 1 � �=2, which

increases as the scale of the uctuations decreases, is chosen so as to make the e�ect of the uctuations onscale " of order one. The case � = 1 corresponds to (1.1), while � = 2 corresponds to the homogenizationscaling (2.3), which we do not consider here. In that case � = 0 and uctuations of the velocity �eld are nolonger weak, which leads to entirely di�erent results.

The uctuations Z", de�ned by (9.2), and Y", de�ned by

Y"(t;y;x) =X"(t;x)�X"(t;x� "y)

";

satisfy a scaled system similar to (3.3)

dZ"

dt=

1p"�v

�x+ ut

"�+

Z"

"��1

�; Z"(0) = 0(9.3)

dY"

dt=

1p"�

�v

�x+ ut

"�+

Z"

"��1

�� v

�x+ ut

"�+

Z"

"��1� Y"

"��1

��; Y"(0) = y:(9.4)

The limit theorem of Appendix A applies also to this system [16] as does the limit theorem for theprocess Z" of Section 3 in this scaling. The limit theorem for the process Y" has to modi�ed because the twoterms in (9.4) become decorrelated in the limit "! 0, as opposed to (3.3). That means that the y-dependentcross term in (3.7) vanishes in the scaling of (9.1). The two particles behave independently of each otherand there is no interaction. This happens because the initial separation "y is large compared to the scale ofthe randomness and the two particles sample nearly uncorrelated parts of the random medium. The resultsof Theorems 4.1 and 7.1 for the limit Richardson and Wigner functions are modi�ed because of this. Thus,the regime of validity of the radiative transport theory in is restricted to the case of inhomogeneities thatare comparable to the wavelength [30]. Inclusions that are smaller than the wavelength but with highercontrast than in the scaling (4.3) will cause dissipation of energy on smaller scales. The contrast still has tobe small, which corresponds to � > 0 in (9.1).

Let �"(t;x) be the solution of the rescaled version of (4.3):

@�"

@t+�u+ "1��=2v(

x

"�)�� rx�" = 0:(9.5)

The limit equation for W (t;x;k) in this scaling has the form

@W

@t+ u � rxW = �

Zdk0

(2�)d�1kikjR

sij(k

0 � k)�((k0 � k) � u)W (k);(9.6)

and ZZW (t;x;k)dxdk <

ZZW0(x;k)dxdk = lim

"!0

Zj�"0(x)j2dx = lim

"!0

Zj�"(t;x)j2dx:

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 15

Thus, Proposition 1 of Appendix B implies that if the family �"0(x) is "-oscillatory, then the solution �"(t;x)does not remain "-oscillatory for all times t > 0, and randomness generates oscillations on its own scale when� > 1.

10. Evolution of the Jacobian matrix

The behavior of the Jacobian matrix of the map induced by a mean zero time dependent random ow isthe subject of recent study, both numerical [7] and theoretical [6, 8, 19, 20, 31]. One of the main quantitiesof interest in such a ow is the Lyapunov exponent responsible for the exponential growth in time of the normof the Jacobian matrix. Its positivity was established in [8] for velocity �elds that are a �nite-dimensionalOrnstein-Uhlenbeck process. The Jacobian and the evolution of curves in isotropic stochastic ows withzero drift is studied in [4]. The trajectories of (1.1) behave very di�erently from mean zero time dependent ows. In particular, as we shall see in Section 8, the length of the curves advected by this ow does notgrow exponentially in time.

Let J"(t;x) be the Jacobian matrix of the transformation x! X"(t;x):

J"ik(t;x) =@X"

i (t;x)

@xk(10.1)

This map is volume preserving because the ow is incompressible, and thus, det J" = 1. We are interestedin the limit of J"(t;x) as "! 0. The main result of this section is the following theorem.

Theorem 10.1. The Jacobian J"(t;x) of the map x! X"(t;x) converges weakly to the di�usion processwith generator

LJf = �1

2

dXi;k;m;n=1

Z 1

0

@2Rim(ut)

@yj@yldtJjkJln

@2f

@Jik@Jmn;(10.2)

starting at J = I.

We derive (10.2) at the end of this section. First we note that the generator (10.2) for the limit di�usionprocess J(t) may be interpreted naturally in terms of the generator (3.7) for the limit process Y(t). Werecall that

Y"(t;y;x) =X"(t;x)�X"(t;x� "y)

"� J"(t;x)y

for small y. Thus the processes Y and S = Jy should behave in a similar way for small y. In particular weshould have that

E ff(Y(t;y))g � E ff(S(t;y))g(10.3)

for small y. To show that (10.3) holds we expand the generator (3.7) of Y for small y and obtain

LY f =1

2

Z 1

�1

(Rsim(ut)� Rsim(ut+ y)) dt@2f

@yi@ym

=1

2

Z 1

�1

�Rim(ut)� 1

2Rim(ut+ y) � 1

2Rim(ut� y)

�dt

@2f

@yi@ym

� �1

2

Z 1

0

@2Rim(ut)

@yj@ylyjyldt

@2f

@yi@ym= cimjlyjyl

@2f

@yi@ym;(10.4)

where

cimjl = �1

2

Z 1

0

@2Rim(ut)

@yj@yldt:(10.5)

16 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

Let J(t; A; !) be the di�usion with generator LJ starting at J(0) = A. We claim that the generator for theprocess S(t;y; !) = J(t; A; !)y, starting at s = Ay is given by (10.4). Theorem 10.1 implies that given anyfunction f(s), the expectation

�f (t;y; A) = E ff(S(t))g = E ff(J(t)y)gsatis�es the di�usion equation

@ �f

@t= cimjlAjkAln

@2 �f

@Aik@Amn;

�f (0; y; A) = f(Ay):

Then a simple calculation shows that

�f (t; A;y) = �g(t; Ay);

where the function �g(t; s) satis�es

@ �f

@t= cimjlsjsl

@2�g

@si@sm�g(0; s) = f(s):

Thus, the generator for S coincides with the small y asymptotics (10.4) of the generator LY . We recallthat the large y asymptotics of LY gives rise to the generator LZ for the limit process Z(t). Thus the limitdi�erence function Y(t) contains information about both limit processes Z(t) and J(t). In physical termsthe behavior of the Jacobian matrix can be recovered from the large k behavior of the Wigner distributionwhich corresponds to the small y behavior of the Richardson function. The behavior of the uctuationsZ(t) may be recovered from small k limit of the Wigner distribution or, equivalently, large y behavior of theRichardson function.

We note further that the generator (10.2) is formally of the form

LJf =< cJ; J >@2f

@J@J:(10.6)

Such di�usion processes have typically positive non-zero Lyapunov exponents, that is, the limit

� = limT!1

ln jjJ(t)jjT

exists with probability one, and � > 0. The computation of � for some time dependent ows was done in[4, 8, 31, 19, 20]. We shall show in Section 11 that in our case � = 0. The reason for this is the strongdegeneracy of (1.1) in the direction of the mean ow u.

We derive now formula (10.2) for the generator of the limit process J(t) and prove Theorem 10.1. Theevolution equation for J" is obtained by di�erentiating (1.1) with respect to xk:

dJ"ikdt

=1p"

@vi@xj

�X"

"

�J"jk(10.7)

J(0;x) = I:

We rewrite this equation using the uctuations Z"(t; x), de�ned by (3.2), so as to put it in a form suitablefor the limit theorem of Appendix A

dZ"

dt=

1p"v

�x+ ut

"+ Z"

�; Z"(0) = 0(10.8)

dJ"

dt=

1p"rv

�x + ut

"+ Z"

�J"; J"(0) = I:

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 17

From this theorem, the joint generator for the limit di�usion processes Z and J has the form

L = LZ + LJ + LJZ + LZJ :(10.9)

The operator LZ is given by (3.6) and the operator LJ is

LJf =1

2

dXi;k;m;n=1

Z 1

0

E

�@vi@xj

�x"+ z

�Jjk

@

@Jik

�@vm@xl

�x"+ ut+ z

�Jln

@f(z; J)

@Jmn

��dt(10.10)

=1

2

dXi;k;m;n=1

Z 1

0

E

�@vi@xj

�x"+ z

� @vm@xl

�x"+ ut+ z

��dtJjk

@

@Jik

�Jln

@f(z; J)

@Jmn

�:

To simplify (10.10) we note that if we let

Fijml(y) = E

�@vi@xj

(x)@vm@xl

(x+ y)

�;

then

Fijml(q)

(2�)d�(p+ q) = ipjiqlE fvi(p)vm(q)g = qjqlRim(q)

(2�)d�(p+ q);

and thus

Fijml(y) = �@2Rim(y)

@yj@yl:

Therefore the generator LJ has the form

LJf = �1

2

dXi;k;m;n=1

Z 1

0

@2Rim(ut)

@yj@yldtJjk

@

@Jik

�Jln

@f

@Jmn

�(10.11)

= �1

2

dXi;k;m;n=1

Z 1

0

@2Rim(ut)

@yj@yldtJjkJln

@2f

@Jik@Jmn;

the last equality being due to incompressibility. Note that LJ is independent of z. Next we compute LZJ :

LZJf =1

2

dXk;i;j=1

Z 1

0

E

�vk

�x"+ z

� @

@zk

�@vi@xn

�x"+ ut+ z

�Jnj

@f

@Jnj

��dt(10.12)

=1

2

dXk;i;j=1

Z 1

0

E

�vk

�x"+ z

� @vi@xn

�x"+ ut+ z

��dtJnj

@2f

@zk@Jij:

The coe�cients of LZJ are also independent of z. The �rst order derivative term vanishes after taking theexpectation because of the incompressibility condition. The operator LJZ is

LJZf =1

2

dXi;j;k=1

Z 1

0

E

�@vi@xn

�x"+ z

�Jnj

@

@Jij

�vk

�x"+ ut+ z

� @f

@zk

��dt(10.13)

=1

2

dXi;j;k=1

Z 1

0E

�@vi@xn

�x"+ z

�vk

�x"+ ut+ z

��dtJnj

@2f

@zk@Jij;

it does not have �rst order derivative terms as well, and has coe�cients independent of z. This shows thatJ" converges by itself to a di�usion process J with generator (10.11), and thus Theorem 10.1 holds.

18 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

11. Application to two-dimensional flows

We apply the results of Section 10 to two dimensional ows. Any two dimensional incompressible owin R2 has the form

v(x) =

�@

@x2;� @

@x1

�;(11.1)

where (x) is the stream function. We assume that the random �eld (x) is space homogeneous, zero meanand isotropic with covariance function

R (x) = E f (y) (x + y)g = F

�r2

2

�;

where r2 = x21 + x22. The Jacobian matrix of two-dimensional time dependent ows with zero mean u wasstudied in detail in [31], in the di�usion approximation. The results there are entirely di�erent but may beformally recovered from our calculations as we explain below. The covariance matrix Rij for the ow (11.1)is

R =

��1 00 �1

�F 0 +

��x22 x1x2x1x2 �x21

�F 00:(11.2)

The tensor cimjl (10.5) in the generator (10.11) has now the form

c1111 = � 1

juj�I2 + I3 + u21u

22I4�

c1112 = � 1

juj�3u1u2I3 + u1u

32I4�

c1122 = � 1

juj�3I2 + 6u22I3 + u42I4

�c1211 =

1

juj�3u1u2I3 + u31u2I4

�c1212 =

1

juj�I2 + I3 + u21u

22I4�

(11.3)

c1222 =1

juj�3u1u2I3 + u1u

32I4�

c2211 = � 1

juj�3I2 + 6u21I3 + u41I4

�c2212 = � 1

juj�3u1u2I3 + u31u2I4

�c2222 = � 1

juj�I2 + I3 + u21u

22I4�:

Here u = (u1; u2) is the mean ow, juj =pu21 + u22, uj =

ujjuj , and the constants

I2 =

Z 1

0

F 00�t2

2

�dt; I3 =

Z 1

0

t2F 000�t2

2

�dt; I4 =

Z 1

0

t4F (iv)

�t2

2

�dt

are related by

I2 = �I3; I4 = 3I2:(11.4)

The other entries are determined by the symmetries cimjl = cmijl = cimlj . The generator (11.5) with cimjlas above is very di�erent form the generator the limit of the Jacobian J" for mean zero, time dependent ows [19, 20, 31]. We may not set even formally u = 0 in (11.3) because the resulting expression diverges

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 19

as it always happens with the Kubo formula. However, the results of [19, 20, 31] can be recovered in ourcalculation by setting formally I3 = I4 = 0 in (11.3). Then the generator (10.2) given by Theorem 10.1reduces formally to the one obtained in [19, 20, 31].

Let us choose a coordinate system so that u2 = 0, and u1 = juj. Then the only non-zero entry is

c1122 = �3I2juj and the generator LJ is

LJ =I22juj

�3J221

@2

@J211+ 6J21J22

@2

@J11@J12+ 3J222

@2

@J212

�=

3I22juj

�J21

@

@J11+ J22

@

@J12

�2:

The matrix J(0) = I, and thus the only entry which is changed during the evolution in the limit is J12. Thegenerator LJ has the simple form

LJf =3I22juj

@2f

@J212:(11.5)

The Lyapunov exponent of the matrix valued process J(t) is manifestly zero. The reason that only J12 ischanged by the dynamics can be seen from the generator LY . Its coe�cients depend only on y2 and areindependent of the y-coordinate along the mean velocity u. This is re ected in (11.5).

12. Deformation of the length

Theorem 10.1 allows us to study stretching of curves by the ow (1.1) in R2. If the initial curve haslength of order one, and is parameterized by x = x(s), 0 � s � 1, its length is given by

l =

Z 1

0

����dxds����ds:

The length of the curve X"(t; s) = X"(t;x(s)) is

l"(t) =

Z 1

0

����dX"

ds

���� ds = Z 1

0

����J"(t;x(s))dxds (s)���� ds:

The limit theorem of Appendix A implies that for any two distinct points x and y, the processes J"(t;x)and J"(t;y) are not only identically distributed, but also independent in the limit "! 0, unless x� y = cufor some c, which means that the points lie on the same deterministic trajectory. This is similar to the jointbehavior of trajectories of (1.1) starting at points lying on di�erent deterministic trajectories as describedin Section 6. The same is true for any number of �xed initial points. Thus l"(t)! l(t), so that E fln(t)g =(Efl(t)g)n, and the length l"(t) becomes deterministic in the limit.

Let us consider stretching of an interval of length one which is initially at angle � with respect to the

mean ow u = (juj; 0), so that dxds

= (cos �; sin �), under the potential ow (11.1). We introduce the variables

J1(�) = J11 cos � + J12 sin � = cos � + J12 sin �

J2(�) = J21 cos � + J22 sin � = sin �:

The behavior of the joint process (J1(�); J2(�)) in our case is trivial and degenerate. The point (J1; J2)performs an ordinary Brownian motion along the horizontal lines J2 = const with di�usion coe�cient thatdepends only on J2. In the time dependent case [31] there is no such degeneracy, the process (J1; J2) is notrestricted to a line, and its norm grows exponentially in time.

We have J1(0) = cos �, and

�l(t) = Efl(t)g = lim"!0

Efl"(t)g =Z 1

0

E

�qJ21 (�) + J22 (�)

�ds = E

�qJ21 (�) + sin2 �

�;

20 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

because the law for the limit process J is independent of the starting point. Then, using (11.5) we see thatthe limit length �l(t) = g(t; cos �), where the function g(t; x) satis�es the initial value problem:

@g

@t=

3I2juj sin

2 �@2g

@x2(12.1)

g(0; x) =psin2 � + x2:

This means that unless � = 0 the length �l(t) � C(�)pt, and does not grow exponentially. The length does

not change at all when � = 0, that is, when the interval is parallel to the mean ow. The algebraic growthis similar to the algebraic growth of the length in a shear ow [6]. The ow (1.1) is dominated by the mean ow, and randomness is not strong enough to generate the kind of mixing needed for exponential growth ofthe length.

13. Summary and conclusions

We have shown that weak, time independent, incompressible uctuations of a uniform ow producecertain non-trivial e�ects on time scales of order one. These are important in advection of passive scalarswhen the initial density varies on scales comparable to that of the inhomogeneities of the ow. This regimeis similar to the one for which the radiative transport theory [30] holds, and we show that the Wignerdistribution of the passive scalar satis�es the radiative transport equation (7.7). The limit Richardsonfunction, or the two point correlation function, satis�es the degenerate di�usion equation (4.15). This resultdoes not require that the velocity �eld or the initial tracer distribution be Gaussian.

The non-zero mean ow has a strong e�ect on the two-point motion introducing a degeneracy in itsdirection. As we explain in Section 8, using the duality between the limit di�usion process Y(t) and thecorresponding jump process in Fourier space, the two point di�erence vector tends to be aligned with themean ow in the long time limit.

We also study the evolution of the Jacobian of the ow map x ! X"(t) in the limit " ! 0 and showthat its limit is a di�usion process. We show that because of the degeneracy caused by the mean ow thecorresponding Lyapunov exponent vanishes. This implies in particular that the length of curves moving withthe ow is growing only algebraically, which is quite di�erent from strong, time dependent, mean zero owsstudied in [8, 31]. Physically this is because of the sweeping e�ect of the mean ow and the fact that weare not looking at the long time limit but rather at �nite time e�ects.

Our results can be generalized to the case of non-zero but small molecular di�usivity. The resultsregarding the two point motion remain essentially the same allowing for an additive term in the di�usionequation for the Richardson function, and an absorption term in the radiative transport equation.

14. Appendix A. A limit theorem for turbulent diffusion

The limit as "! 0 of the trajectories of the dynamical systems with the scaling as in (2.4) is described bya limit theorem, which was proven by Kesten and Papanicolaou [18, 21] for = 1, and later by Komorowski[16] for 0 � < 1. They considered equations of the form

dQ"

dt=

1

"G

�t

"2;Q"(t)

" ; "

�;(14.1)

Q"(0) = q;

where the function G satis�es the following conditions.

(A1) The function G(t;q; "; !) is jointly measurable in all its arguments and a.s. in C3(Rd) as a functionof q for each t, ".

EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS 21

(A2) The process fG(t; �; "; !)g is stationary in t for each �xed ".(A3) Let

Gts(";M ) = �fG(u;q; "; �)js � u � t; jqj �Mg;and

�(t;M ) = sups�0;0<"�1

supA2Gs

0(";M);B2G1

s+t(";M)

jP (AB)� P (A)P (B)j:

Then Z 1

0

j�(t;M )j1=pdt <1

for p = 6 + 2d.(A4) EfG(t;q; ")g = 0:(A5) For each M <1 there exists a constant C(M ) independent of t, k, and " such that

E

(sup

jq�kj�M

jD�G(t;q; ")jmax(8;d))� C; 0 � j�j � 3;(14.2)

and these integrals converge uniformly in k. All the derivatives in (14.2) are with respect to q.(A6) The following limits exist uniformly on compact sets and are bounded functions of q:

Aij(q) = lim"!0

Z 1

0

E fGi(0;q; ")Gj(t;q; ")gdt(14.3)

cij(q) = lim"!0

Z 1

0

E

�Gi(0;q; ")

@

@qiGj(t;q; ")

�dt:

In addition the matrix aij = Aij +Aji is twice continuously di�erentiable.(A7) The velocity �eld is incompressible: divG" = 0.

By a solution of (14.1) we mean a continuous function Q"(t) which satis�es

Q"(t; !) = q +1p"

Z t

0

G��";Q"(�; !); "; !

�d�:(14.4)

Theorem 14.1. [18, 16] Let G be a random �eld that satis�es conditions (A1-A7). Then with proba-bility one (14.4) has a unique solution for all t � 0. Let R" be the measure on C([0;1);Rd) induced by thissolution. For each f 2 C2(Rd) put

Lf(q) =1

2

dXi;j=1

aij(q)@2f(q)

@qi@qj(14.5)

Then R" converges weakly to the probability measure R on C([0;1);Rd), which corresponds to the di�usionwith in�nitesimal generator L and starting at the point q, that is, R fQ(0) = qg = 1.

The most general statement of this theorem, including more general time dependence and compressiblevelocity �elds, may be found in [16].

15. Appendix B. Oscillatory functions and the Wigner distribution

We review brie y some de�nitions and basic facts about oscillatory families of functions and the Wignerdistribution [14]. Let f"(x) be a bounded family of functions in L2loc(R

d).

22 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

Definition 1. The family f"(x) is "-oscillatory if for any smooth test function of compact support �(x)

lim sup"!0

Zjkj�R="

���d�f"(k)���2 dk! 0 as R! +1:(15.1)

A su�cient condition for (15.1) to hold is

9k > 0; such that "k��������@kf"@xk

��������L2loc

� C:

Definition 2. The family f"(x) is said to be compact at in�nity if

lim sup"!0

Zjxj�R="

jf"(x)j2 dx! 0 as R! +1:(15.2)

A su�cient condition for (15.2) to hold is that there exists a set K such that suppf"(x) � K.

Definition 3. The Wigner distribution is de�ned by

W "(x;k) =

Zdy

(2�)deik�yf"(x� "y

2)f"�(x+

"y

2):

It has a weak limit point W (x;k) in the space S0 of Schwartz distributions provided that f"(x) isbounded in L2loc . We shall assume that such a limit point is unique. The limit Wigner distribution may beinterpreted as a phase space energy density. For instance, the limit Wigner distribution of the WKB familyf"(x) = A(x)eiS(x)=" is

W (x;k) = jA(x)j2 �(k�rS(x)):The limit Wigner distribution is mostly relevant for "-oscillatory and compact at in�nity families f"(x).

Proposition 1. [14] Let f" be a bounded family in L2loc with limit Wigner measure W (x;k). Then

(i) The limit Wigner distribution W (x;k) 2 S0 is non-negative (a measure).(ii) If two families f"(x) and g"(x) coincide on an open ball B � Rd then Wf (x;k) = Wg(x;k) for x 2 B

and all k 2 Rd.(ii) For any smooth function of compact support �(x)ZZ

j�(x)j2W (dx; dk) � lim sup"!0

ZRd

j�(x)f"(x)j2dx(15.3)

with equality holding if and only if f" is "-oscillatory. In this case lim sup can be replaced by lim onthe right side of (15.3).

Acknowledgments

This work was partially sponsored by a grant from the NSF, DMS-9622854, and by the Air Force O�ce ofScienti�c Research, Air Force Materials Command, USAF, under grant number F49620-95-1-0315. The USGovernment is authorized to reproduce and distribute reprints for governmental purposes notwithstandingany copyright notation thereon. The views and conclusions contained herein are those of the authors andshould not be interpreted as necessarily representing the o�cial policies or endorsements, either expressedor implied, of the Air Force O�ce of Scienti�c Research or the US Government.

We are grateful to Joseph B. Keller for numerous discussions on the subject.

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24 ALBERT FANNJIANG, LEONID RYZHIK, AND GEORGE PAPANICOLAOU

Department of Mathematics, University of California at Davis, Davis CA 95616, fannjian@math.ucdavis.edu

Department of Mathematics, Stanford University, Stanford CA, 94305, ryzhik@math.stanford.edu, papan-

ico@math.stanford.edu