Evolution of trajectory correlations in steady random flows
Series Logo Volume 00, 19xx
,
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 random
incompressible ow with nonzero mean and small uctuations. We show that the Fourier transform of the Richardson function of a passive scalar advected by the ow satis es, under certain conditions, a radiative transport equation. We also study the stretching of curves advected by the ow and show that their length grows algebraically in time, and not exponentially as it does for time dependent, zero mean random ows.
Contents
1. Introduction 2. Long time di usive behavior 3. Flow correlations 4. The Richardson function and its evolution 5. The expectation of the Richardson function 6. Higher moment equations 7. The Wigner distribution and connections with radiative transport theory 8. Di usion{transport duality 9. General scaling of the uctuations 10. Evolution of the Jacobian matrix 11. Application to two-dimensional ows 12. Deformation of the length 13. Summary and conclusions 14. Appendix A. A limit theorem for turbulent di usion 15. Appendix B. Oscillatory functions and the Wigner distribution Acknowledgments References
1 2 4 5 8 9 10 12 14 15 18 19 20 20 21 22 22
1. Introduction
We consider motion in a steady, random, incompressible ow with constant drift u. The random part of the ow v(x), with r v = 0, is a smooth spatially homogeneous random eld with zero mean and rapidly decaying correlations. The trajectory X" (t; x) advected by the ow satis es the scaled equation " " p X d X (1.1) dt = u + "v " " X (0; x) = x; 1991 Mathematics Subject Classi cation. Primary 60F05, 76F05, 76R50; Secondary 58F25. 1
c 0000 American Mathematical Society 0160-7634/00 $1.00 + $.25 per page
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 the random uctuations and their correlation length. We are interested in the behavior of the rescaled two-point di erence function 1 fX" (t; x) , X" (t; x , "y)g " (1.2) Y (t; y ; x) = " in the limit of " ! 0. The individual trajectories X" (t) are close to the trajectories of the deterministic ow X = x + ut for nite times t, when " is small, and thus random perturbations do not play an essential role in their behavior. The rescaled trajectory di erences Y" that start nearby are, however, a ected and that is what we study in this paper. We note that this is di erent from the analysis of the long time correlations of 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 (of order ",1 ) behavior of single trajectories, or groups of trajectories that are not close initially. In Section 3 we show 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 for the 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 a radiative transport equation for the Wigner distribution associated with the trajectory di erence function " Y . We study the duality between the limit two-point di usion process and its analog in Fourier space in Section 8. A scaling more general than 1.1 is treated in Section 9. In Section 10 we show that the Jacobian matrix of the ow (1.1) also converges to a di usion process in the limit " ! 0, and we analyze in detail its law in two-dimensional potential ows in Section 11. In Section 12 we apply these results to study the stretching of curves transformed by (1.1). We show that their length grows algebraically in this case and not 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 by G.I.Taylor [25, 32]. He considered the trajectories of the system (2.1) dX(t) = v(t; X(t)) + dW(t) X(0) = 0; where v(t; x) is a mean zero random velocity eld with rapidly decaying correlations, and W(t) is the standard d-dimensional Brownian motion. He argued that in the long time limit the trajectories behave like those of d-dimensional ZBrownian motion with covariance matrix 1 (2.2) E fvi (t; X(t))vj (0; 0) + vj (t; X(t))vi (0; 0)g dt + 2ij : 0
The rst term in expression (2.2), which corresponds to the turbulent enhancement of the di usion coecient, 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( "t2 ; X" ) + dW(t) (2.3) " X (0) = 0 and analyze the behavior of the trajectories as " ! 0. When the molecular di usivity is positive and the 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 by the homogenization formula (2.2) with X(t) the solution of (2.1) corresponding to " = 1 in (2.3). When the 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 time decorrelation of v in [17]. The weak uctuation scaling (described in detail in Appendix A) was intensively studied (with = 0 and > 0 [16, 18, 21]). A typical scaling in this case, with = 0, is dX" = 1 v( t ; X" (t) ); (2.4) dt " "2 " " X (0) = x; with 0 < 1, so that now the random velocity eld is varying on spatial scales large compared to ", but possibly small on the overall scale if 6= 0. Then, as before, the process X" (t) converges weakly as " ! 0 to a di usion process. Its generator is given by d 2f X Lx f (x) = 21 aij (x) @x@ @x : i j i;j =1 This operator is self-adjoint because the velocity eld v(x) is incompressible. The corresponding generator for a compressible ow has an extra drift term [18, 16, 21]. The di usion matrix is given by Z
1
aij (x) = E fvi (0; x)vj (t; x) + vj (0; x)vi (t; x)g : 0 Expression (2.5) is called the Kubo formula. Of particular interest here is the system dX" = 1 v(X" (t) + t u); (2.6) dt" " "2 X (0) = x; where v(x) is a space homogeneous, mean zero divergence free random eld. Then X" (t; x) converges to Brownian motion with the covariance matrix given by the Kubo formula Z 1 (2.7) cij (0) = E fvi (tu)vj (0)g dt; (2.5)
,1
This corresponds to the Taylor prediction (2.2) with the path X = ut being frozen and independent of the random 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" ; X ") converges weakly to a di usion process (X; X0 ) with generator L~ f (x; x0 ) = Lx f + Lx f + L0 f; where the cross-term L0 is 2f L0 f = 12 cij (x; x0 ) @x@ @x 0: 0
0
i
Here
cij (x , x0 ) =
Thus the di erence
d Z X
1
j
E fvi (0)vj (x0 , x + tu)g dt:
i;j =1 ,1 function Y" = X" , X " converges
weakly to a di usion process Y with generator 2f (2.8) LY f = (cij (0) , cij (y)) @y@ @y : i j 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. 0
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 rescaling the dynamics of many quantities of interest can be reduced to a form similar to (2.4). For example, the rescaled di erence function (1.2) for the trajectories of (1.1) converges to a di usion process with the same generator (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 the rescaled di erence variable " " X (t; x) , X (t; x , "y ) " ; Y (t; y ; x) = " which is the scaled separation of the particles at time t. Then X" (t; x), Y" (t; y; x) satisfy dX" = u + p"v X" ; X" (0) = x (3.1) dt " dY" = p1 v X" , v X" , Y" ; Y" (0) = y: dt " " "
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 " X (t; x) , ut , x " (3.2) Z (t; x) = : " Then the system (3.1) becomes " d Z 1 x + ut " " (3.3) dt = p" v " + Z ;Z (0)= 0 dY" = p1 v x + ut + Z" , v x + ut + Z" , Y" ; Y" (0) = y: dt " " "
This system has the general form (14.1) to which the limit theorem of Appendix A applies. The full statement of this theorem and the necessary assumptions on the random velocity eld in the general case are recalled in Appendix A. We shall assume in particular that v(y) is a mean zero, divergence-free, space homogeneous and strongly mixing velocity eld bounded in C 3 (Rd ). Then the limit theorem implies the following. Theorem 3.1. The processes Z" , and
Y
whose joint generator is
Y
" converge weakly to the correlated di usion processes
Z 1 d 1 2f 2f X Lf = 12 Rij (ut)dt @z@ @z + (2Rij (ut) , Rij (ut + y) , Rij (ut , y)) dt @y@ @y i j i j ,1 i;j =1 ,1 Z 1 2f + (2Rij (ut) , 2Rij (ut , y)) dt @z@ @y (3.4) : i j ,1 Here the covariance tensor Rij is de ned by (3.5) E fvi(y)vj (y + h)g = Rij (h); The individual generators for Z and Y are: d Z1 2f X Rij (ut)dt @z@ @z (3.6) ; LZ f (z ) = 21 i j i;j =1 ,1 Z
Z
and
EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS
and (3.7) respectively, where
LY f (y) =
d Z X
1
5
2f ; Rsij (ut) , Rsij (y + ut) dt @y@ @y i j i;j =1 ,1
Rsij = Rij +2 Rji : Thus the uctuations Z" (t) converge to Brownian motion with covariance matrix given by the Kubo formula (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 uctuating because 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 coecient in (3.7) vanishes for y = cu, so that if two particles start at two nearby positions on the same deterministic trajectory then their separation is not changed by the ow in the limit. The generator LY is asymptotically close to LZ for all y that have large component in the direction perpendicular to the mean ow u. The reason for this is that when the two starting points are separated by a large distance in the direction normal to u, their trajectories are almost independent, and the rescaled di erence trajectory behaves like the uctuations of each individual trajectory. The other end of the asymptotics, the small y behavior of LY , is related to the Jacobian 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 @ + v(x) r = 0 x @t (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 equation in y, in the long time limit. This problem was studied extensively by physicists (see [19] for an extensive review and references), especially for -correlated in time velocity elds, but to the best of our knowledge the only mathematical results are those in [18] and [24, 28]. Molchanov and Piterbarg [24] considered in particular a convection-di usion equation of the form @T " + 1 v( t ; x) r T " = T " (4.1) x @t" " "2 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. If the 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 = 1 @ rd,1 (2 + F (r)) @Q (4.2) @t rd,1 @r @r Q(0; r) = Q0(r): Here Z 1 F (r ) = (RL (t; 0) , RL(t; r)) dt; ,1
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): @R r L L Rij = RL(t; r) + d , 1 @r ij , r(xdi,xj1) @R @r : Equation (4.2) is the generalization of (2.8) for 6= 0. We note that the additional term due to non-zero molecular di usivity is additive. This is di erent from the homogenization formulas of the type (2.2) which come 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 @ " + ,u + p"v x r " = 0 (4.3) @t " "(0; x) = "0(x); where the initial density "0(x) is "-oscillatory (see Appendix B for the precise de nition of this notion). It may 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 of oscillatory form are the localized initial densities (4.5) "0 (x) = "d=1 2 0( x" ); where 0(x) is an L1 (Rd ) \ L2 (Rd ) function, and the inhomogeneous wave family (4.6) "(x) = A(x)eiS (x)=" where A(x) and S (x) are smooth functions, and S (x) is real valued. The latter family describes the distribution of tracers which have the form of high frequency waves propagating in the direction rS (x) with amplitude A(x). A particular case of (4.6) is the high frequency plane wave (4.7) "(x) = Aeixp=": The rescaled Richardson function of the family "(x) is de ned by (4.8) W^ " (t; x; y) = Q(t; x , "y; "y) = " (t; x , "y) " (t; x); 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)
Z
A = f ( ; ) : d sup jf ( ; )j < 1 ; x y
y
x y
x introduced in [22]. The Richardson function of the family (4.4) is (4.9) W^ (x; y) = R0(y); with probability one, by the ergodic theorem, and is independent of x. Here R0(x) is the covariance function of the random process 0(x). The Richardson functions of the localized deterministic family (4.5) is Z (4.10) W^ (x; y) = (x) Q^ 0 (z; y)dz;
where (4.11) Q^ 0 (x; y) = 0(x , y) 0 (x): The limit Richardson function of the WKB family (4.6) is (4.12) W^ (x; y) = jA(x)j2 e,irS (x)y ;
EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS
7
which reduces to (4.13) W^ (x; y) = jAj2e,ipy 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 the form 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 " 1 t @ " (4.14) @t + " v x + u "2 rx = 0 " (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 x , ut " " W (t; x; y) = Q t; " ; y : When the initial data for (4.3) is random, space homogeneous of the form (4.4), both the scaled and unscaled Richardson functions are independent of x in the limit " ! 0, hence they coincide in this limit, and the analysis 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 Section 9. 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 A0 as " ! 0, and W^ 0 (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^ + u r W^ = L W^ (4.15) Y x @t W^ (0; x; y) = W^ 0 (x; y); where the operator LY is given by (3.7). The initial data W^ 0 (x; y) in (4.15) are given by (4.9) in the random case, 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 the modi 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 W^ 0(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 diculty in generalizing this result arises in the second step of the proof, that is, replacing the initial data W^ 0" by the limit W^ 0. The limit theorem of Appendix A does not provide any information on the uniformity of the convergence to the limit process with respect to the starting point in all of Rn. In particular, 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 W^ 0 (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) = "1d Q^ 0( x" ; y), and equation (4.3) implies that W^ " satis es the initial value problem @ W^ " + u r W^ " + p"v x r W^ " + p1 nv x , v x , y o r W^ " = 0 (5.1) x x y @t " " " " " with (5.2) W^ " (0; x; y) = "1d Q^ 0 ( x" ; y); where Q^ 0 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 W^ " (t; x; y) = W^ 0(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 W^ 0 (x; y), which is given by (4.10). We claim that (5.3) E fW^ "(t; x; y) , W^ "(t; x; y)g ! 0 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 Z Z E dxdy(W^ "(t; x; y) , W^ `" (t; x; y))f (x; y) 0
0
ZZ
dxdy(W^ 0"(x; y) , W^ 0 (x; y))E ff (x + ut + "Z" (t; x); Y" (t; y; x))g Z ZZ x 1 ^ = dxdy( "d Q0( " ; y) , (x) dpQ^ 0(p; y))E ff (x + ut + "Z" (t; x); Y"(t; y; x))g ZZ = dxdyQ^ 0(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 (5.5) E ff (ut + "Z" (t; 0); Y"(t; y; 0))g ! f(0; y); strongly in the uniform norm on compact sets in y. Here the function f(x; y) satis es the di usion equation with a drift @ f = u r f + L f x Y @t 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 dZ~ " = p1 v ut + Z~ " ; Z~ " (0) = x dt " " e ~ dY ps = p1 v ut + Z~ " , v ut + Z~ " , Y~ " ; dt " " " (5.4)
=
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))g has the same limit as in (5.5). Then by the dominated convergence theorem (5.5) vanishes in the limit " ! 0 since Q^ 0(x; y) 2 L1 (Rd Rd ). Thus (5.3) holds, and we may replace the initial data W^ 0" by W^ 0 in (5.1), and n o n o ^ " (t; x; y) = lim E W^ " (t; x; y) : W^ (t; x; y) = "lim E W !0 "!0 0
The limit theorem applies to the weak form of the resulting system, that is, for any test function f (x; y) we have ZZ ZZ n o E W^ " (t; x; y) f (x; y)dxdy = W^ 0(x; y)E ff (x + ut + "Z" (t; x); Y" (t; x; y))g dxdy 0
=
ZZ
dxdyQ^ 0(x; y)E ff (ut + "Z" (t; 0); Y"(t; 0; y))g dxdy !
ZZ
dxdyQ^ 0(x; y)f(ut; y)dxdy
by the dominated convergence theorem. Thus the function W^ (t; x; y) is the weak solution of @ W^ + u r W^ = L W^ (5.6) x Y @t W^ (0; x; y) = W^ 0 (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 limit theorem 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 Zj and Yj satisfy the following system: dZ"1 = p1 v x1 + ut + Z" ; Z" (0) = 0 (6.1) 1 1 dt " " " dY1 = p1 v x1 + ut + Z" , v x1 + ut + Z" , Y" ; Y" (0) = y 1 1 1 1 1 dt " " " dZ"2 = p1 v x2 + ut + Z" ; Z" (0) = 0 2 2 dt " " " dY2 = p1 v x2 + ut + Z" , v x2 + ut + Z" , Y" ; Y" (0) = y : 2 2 2 2 2 dt " " "
The joint process (Z"1 ; Y1"; Z"2; Y2" ) 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. The cross term L12 involves terms of the type 2 2 cij (x1 ; x2; y1 ; y2) @i j ; dij (x1 ; x2; y1 ; y2) @i j ; @z1 @y2 @y1 @y2 and other similar ones. The coecients cij , dij , : : : are non-zero only for points x1, x2 which lie on the same deterministic 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 independent and 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 " " Z1 = Z2 . Then the system (6.1) reduces to dZ" = p1 v x + ut + Z" ; Z" (0) = 0 (6.2) dt " " " dY1 = p1 v x + ut + Z" , v x + ut + Z" , Y" ; Y" (0) = y 1 1 1 dt " " " dY2" = p1 v x + ut + Z" , v x + ut + Z" , Y" ; Y" (0) = y : 2 2 2 dt " " " The joint generator for the limit processes Y1 , Y2 has the form: @2f L(2) f (y1 ; y2) = 12 a ij (y1 ; y2) @y @y ; i j where the Greek indices label the points and the Latin indices label the coordinates. The coecients a ij (y1 ; y2) are given by
a ij (y1 ; y2) =
1
Z
,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) n o ^ "(t; x; y1 )W^ "(t; x; y2) G^ (t; x; y1; y2) = "lim E W !0 " " " " = "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 initial value problem: @ G^ = L(2) G; ^ (6.3) @t 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 the rescaled Richardson function Z " W (t; x; k) = (2dy)d eiky "(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] (see Appendix B). The limit Wigner distribution may be interpreted as the limit energy (or particle) density of an "-oscillatory family " (x), travelling in the direction k at position x. In particular, we have (7.2)
lim j "(x)j2 = "!0
Z
W (t; x; k)dk
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 the limit values as " ! 0 of various others quantities of interest, for example, the correlation functions of the derivatives of ". The limit of the correlation matrix of the gradient " @ " (x) ( x , "y )" W~ ij" (t; x; y) = " @ @xi @xj is the Hessian in y of W^ (t; x; k): 2W ^ (7.3) : W~ ij" (t; x; y) ! @y@ @y i j The limits of the higher order derivatives can be similarly expressed via W (t; x; k). Thus, in the high frequency limit the Wigner distribution gives a complete description both of the eld " (t; x) itself and its derivatives. Relations of the type (7.2) and (7.3) are the main reason for the recent studies of the Wigner distribution for waves in random media [30]. Let us brie y recall how radiative transport equations for waves arise [30]. Let w" (t; x) 2 C N be the solution of a symmetric hyperbolic system (this scaling corresponds to = 1 in (4.3)) n p o w" + Dj @ w" = 0 (7.4) A(x) + "V x" @@t @xj " " w (0; x) = w0 (x): Here the matrix A(x) is positive de nite and the matrices Dj are symmetric. The initial data w0" (x) is assumed to be "-oscillatory and deterministic, and V (y) is a matrix valued, space homogeneous random process. Then the N N limit Wigner matrix W (t; x; k) has a special form (7.5)
W (t; x; k) =
X
;i;j
Wij (t; x; k)b ;ib ;j :
Here the vectors b ;j 2 C N form a basis for the eigenspace of the dispersion matrix L(x; k) = A,1 (x)kj Dj of the system (7.4), corresponding to the eigenvalue ! . The size of the square matrices W is equal to the degeneracy of the eigenvalue. They satisfy a system of radiative transport equations. This system is decoupled when the eigenvalues are simple and ! (k) 6= ! (p) for all , k and p. Then the radiative transport equation for the scalar W has the form @W + r ! r W , r ! r W (7.6) k x x k @tZ = dk0 (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 spectrum of V (y). Equation (7.6) has a long history. It was proposed phenomenologically by Rayleigh in the beginning of this century and then studied extensively by physicists [10]. Various derivations of this equation were given in 1960's (see [30] for references) when w" is a solution of the wave equation or Maxwell's equations. A general case was treated in [30] but the results were derived only formally. The only case when the radiative transport equation was obtained rigorously, to the best of our knowledge, is for the Schrodinger equation with 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 following theorem.
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 converges to W (t; x; k) weakly in A0 , where W (t; x; k) satis es the radiative transport equation @W + u r W = Z dk0 k k R^ s (k0 , k) ((k0 , k) u)(W (k0 ) , W (k)) (7.7) x @t (2)d,1 i j ij
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 time and for random velocity elds that are not necessarily Gaussian. This tells us that in more general cases the radiative 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 to plane 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 Markovian jump process [5]. Consider the backward characteristics of (7.6) _ = ,rk !(X; K); (8.1) X _ = rx !(X; K); K starting at x(0) = x and K(0) = k. Let a particle move along a trajectory of (8.1) for a random time 1 and then switch wave vector randomly from K to K0 . After that it follows trajectories of (8.1) for a random time 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) =
Z
dk0(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 of the previous jump. The probability density that the wave number jumps from direction k into direction k0 is given by (x; k; k0) : p(x; k; k0) = ( x; k) The resulting process is well de ned if the total scattering cross-section is bounded from above, so that only a nite number of jumps occur during any given time interval, with probability one. The Kolmogorov equation 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) Z 0 (8.2) Qf = (2d)kd,1 ki kj R^ sij (k0 , k) ((k0 , k) u)(f (k0 ) , f (k)) is of the same form as the integral operator in (7.7). It is the generator of a jump process for which the particle is moving along the deterministic trajectory x(t) = x(0) + ut, while the wave vector is the jump process described above with the di erential scattering cross-section (k; k0 ) = (21)d,1 kikj R^ sij (k0 , k) ((k0 , k) u) and the total scattering cross-section Z (8.3) (k) = dk (21)d,1 kikj R^ sij (k0 , k) ((k0 , k) u): Thus the projection of the wave vector on the mean velocity direction u is not changed after the jump. The total scattering cross-section (8.3) is an unbounded function of k, and so the standard argument [5] that K(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)eiky = Q(k)eiky and hence n o n o (8.4) EY eikY(t;y) = EK eiK(t;k)y : 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 function f (t; x; y) we may view f(t; y) = E ff (Y(t; y))g either as a solution of @ f + u r f = L f Y x @t f (0; x; y) = f (x; y) or as the Fourier transform of the solution of @ f^ + u r f^ = Qf^ x @t ^ f^(0; x; k) = f ((2x;,)dk) :
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 implies that solutions of (8.5) tend to a delta function in the directions orthogonal to the mean ow u, and so the in the long time limit Y(t) tends to be parallel to the mean velocity u. In terms of the ow, this means that particles 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 oscillating on a scale ner than ", but their strength is also scaled appropriately. The trajectories X" (t; x) satisfy the scaled equation dX" = u + " v X" ; (9.1) dt " " X (0; x) = x: The trajectory uctuation, de ned as before by " X (t; x) , ut , x " Z (t; 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 on scale " of order one. The case = 1 corresponds to (1.1), while = 2 corresponds to the homogenization scaling (2.3), which we do not consider here. In that case = 0 and uctuations of the velocity eld are no longer weak, which leads to entirely di erent results. The uctuations Z" , de ned by (9.2), and Y" , de ned by " " X (t; x) , X (t; x , "y ) " ; Y (t; y ; x) = " satisfy a scaled system similar to (3.3) dZ" = p1 v x + ut + Z" ; Z" (0) = 0 (9.3) dt " ,1 " " dY" = p1 v x + ut + Z" , v x + ut + Z" , Y" ; Y" (0) = y: (9.4) dt " " ,1 " " ,1 " ,1 " The limit theorem of Appendix A applies also to this system [16] as does the limit theorem for the process Z" of Section 3 in this scaling. The limit theorem for the process Y" has to modi ed because the two terms in (9.4) become decorrelated in the limit " ! 0, as opposed to (3.3). That means that the y-dependent cross term in (3.7) vanishes in the scaling of (9.1). The two particles behave independently of each other and there is no interaction. This happens because the initial separation "y is large compared to the scale of the randomness and the two particles sample nearly uncorrelated parts of the random medium. The results of 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 that are comparable to the wavelength [30]. Inclusions that are smaller than the wavelength but with higher contrast than in the scaling (4.3) will cause dissipation of energy on smaller scales. The contrast still has to be small, which corresponds to > 0 in (9.1). Let "(t; x) be the solution of the rescaled version of (4.3): @ " + u + "1, =2 v( x ) r " = 0: (9.5) x @t " The limit equation for W (t; x; k) in this scaling has the form @W + u r W = , Z dk0 k k R^ s (k0 , k) ((k0 , k) u)W (k); (9.6) x @t (2)d,1 i j ij and ZZ ZZ Z Z " 2 " 2 W (t; x; k)dxdk < W0 (x; k)dxdk = "lim !0 j 0(x)j dx = "lim !0 j (t; x)j dx:
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 is the subject of recent study, both numerical [7] and theoretical [6, 8, 19, 20, 31]. One of the main quantities of interest in such a ow is the Lyapunov exponent responsible for the exponential growth in time of the norm of the Jacobian matrix. Its positivity was established in [8] for velocity elds that are a nite-dimensional Ornstein-Uhlenbeck process. The Jacobian and the evolution of curves in isotropic stochastic ows with zero 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 not grow exponentially in time. Let J " (t; x) be the Jacobian matrix of the transformation x ! X" (t; x): " i (t; x) (10.1) Jik" (t; x) = @X@x k This map is volume preserving because the ow is incompressible, and thus, det J " = 1. We are interested in 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 process with generator
(10.2) starting at J = I .
Z 1 2 d X 1 @ Rim(ut) dtJ J @2f ; LJ f = , 2 jk ln @yj @yl @Jik @Jmn i;k;m;n=1 0
We derive (10.2) at the end of this section. First we note that the generator (10.2) for the limit di usion process J (t) may be interpreted naturally in terms of the generator (3.7) for the limit process Y(t). We recall that " " X (t; x) , X (t; x , "y ) " J " (t; x)y Y (t; y ; x) = " for small y. Thus the processes Y and S = J y should behave in a similar way for small y. In particular we should have that (10.3) E ff (Y(t; y))g E ff (S(t; y))g for small y. To show that (10.3) holds we expand the generator (3.7) of Y for small y and obtain Z 1 2 LY f = 21 (Rsim(ut) , Rsim (ut + y)) dt @y@ @yf i m ,1 Z 1 2 = 12 Rim (ut) , 21 Rim (ut + y) , 12 Rim (ut , y) dt @y@ @yf i m ,1 Z 1 2 2f 2f 1 @ R ( ut) @ @ im ,2 (10.4) @yj @yl yj yl dt @yi @ym = cimjl yj yl @yi @ym ; 0 where Z 1 2 @ Rim (ut) dt: 1 (10.5) cimjl = , 2 @y @y 0
j l
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 the process S(t; y; !) = J (t; A; !)y, starting at s = Ay is given by (10.4). Theorem 10.1 implies that given any function f (s), the expectation f(t; y; A) = E ff (S(t))g = E ff (J (t)y)g satis es the di usion equation @ 2 f ; @ f = c A A imjl jk ln @t @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 = c s s @ 2 g @t imjl j l @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 recall that the large y asymptotics of LY gives rise to the generator LZ for the limit process Z(t). Thus the limit di erence function Y(t) contains information about both limit processes Z(t) and J (t). In physical terms the behavior of the Jacobian matrix can be recovered from the large k behavior of the Wigner distribution which corresponds to the small y behavior of the Richardson function. The behavior of the uctuations Z(t) may be recovered from small k limit of the Wigner distribution or, equivalently, large y behavior of the Richardson function. We note further that the generator (10.2) is formally of the form @2f : LJ f =< cJ; J > @J@J (10.6) Such di usion processes have typically positive non-zero Lyapunov exponents, that is, the limit ln jjJ (t)jj = Tlim !1 T 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 strong degeneracy 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. The evolution equation for J " is obtained by di erentiating (1.1) with respect to xk : " " dJ 1 @v X i ik " p (10.7) = dt " @xj " Jjk 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 suitable for the limit theorem of Appendix A " 1 x + ut d Z " " (10.8) dt = p" v " + Z ; Z (0) = 0 dJ " = p1 rv x + ut + Z" J "; J " (0) = I: dt " "
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 (10.9) L = LZ + LJ + LJZ + LZJ : The operator LZ is given by (3.6) and the operator LJ is Z 1 d X @vi x + z J @ @vm x + ut + z J @f (z; J ) dt (10.10) LJ f = 21 E @x jk @J ln @J j " ik @xl " mn i;k;m;n=1 0
Z 1 d X 1 @vi x + z @vm x + ut + z dtJ @ J @f (z; J ) : E @x = 2 jk @J ln @J @xl " j " ik mn i;k;m;n=1 0 To simplify (10.10) we note that if we let @vi (x) @vm (x + y) ; Fijml (y) = E @x @xl j then F^ijml (q) (p + q) = ip iq E fv^ (p)^v (q)g = qj ql R^ im (q) (p + q); j l i m (2)d (2)d and thus 2 Fijml (y) = , @ @yRim@y(y) : j l Therefore the generator LJ has the form Z 1 2 d X @ Rim (ut) dtJ @ J @f (10.11) LJ f = , 12 jk @J ln @J @yj @yl ik mn i;k;m;n=1 0
Z 1 2 d X @ Rim (ut) dtJ J @2f ; = , 12 jk ln @yj @yl @Jik @Jmn i;k;m;n=1 0 the last equality being due to incompressibility. Note that LJ is independent of z. Next we compute LZJ : d Z 1 x X @vi x + ut + z J @f dt LZJ f = 12 (10.12) E vk " + z @z@ @x nj @J k n " nj k;i;j =1 0
d Z1 @v x X 1 @2f : i = 2 E vk x" + z @x + ut + z dtJ nj @zk @Jij n " k;i;j =1 0 The coecients of LZJ are also independent of z. The rst order derivative term vanishes after taking the expectation because of the incompressibility condition. The operator LJZ is d Z 1 @v x @f X @ x 1 i (10.13) E @x " + z Jnj @J vk " + ut + z @z dt LJZ f = 2 n ij k i;j;k=1 0
d Z1 X 1 @vi x + z v x + ut + z dtJ @ 2 f ; = 2 E @x k " nj @z @J n " k ij i;j;k=1 0
it does not have rst order derivative terms as well, and has coecients independent of z. This shows that J " 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 ow in R2 has the form @ @ (11.1) v (x) = @x ; , @x ; 2
1
where (x) is the stream function. We assume that the random eld (x) is space homogeneous, zero mean and isotropic with covariance function 2 R (x) = E f (y) (x + y)g = F r2 ; where r2 = x21 + x22 . The Jacobian matrix of two-dimensional time dependent ows with zero mean u was studied in detail in [31], in the di usion approximation. The results there are entirely di erent but may be formally recovered from our calculations as we explain below. The covariance matrix Rij for the ow (11.1) is 22 x1 x2 00 , 1 0 , x 0 (11.2) R = 0 ,1 F + x x ,x2 F : 1 2
1
The tensor cimjl (10.5) in the generator (10.11) has now the form , c1111 = , ju1 j I2 + I3 + u^21u^22 I4 , c = , 1 3^u u^ I + u^ u^3 I 1112
j j
1 23
u
1 24
c1122 = , ju1 j 3I2 + 6^u22I3 + u^42I4 , c1211 = ju1 j 3^u1 u^2I3 + u^31u^2I4 , c = 1 I + I + u^2u^2I
(11.3)
1212
j j
,
2
u
3
1 24
c1222 = ju1 j 3^u1 u^2I3 + u^1u^32I4 , c2211 = , ju1 j 3I2 + 6^u21I3 + u^41I4 , c = , 1 3^u u^ I + u^3u^ I 2212
,
j j
1 23
u
1 24
c2222 = , ju1 j I2 + I3 + u^21u^22 I4 : p Here u = (u1; u2) is the mean ow, juj = u21 + u22, u^j = juujj , and the constants I2 =
,
2 2 2 Z 1 Z 1 F 00 t2 dt; I3 = t2 F 000 t2 dt; I4 = t4 F (iv) t2 dt 0 0 0
Z
1
are related by (11.4) I2 = ,I3 ; I4 = 3I2 : The other entries are determined by the symmetries cimjl = cmijl = cimlj . The generator (11.5) with cimjl as 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 our calculation by setting formally I3 = I4 = 0 in (11.3). Then the generator (10.2) given by Theorem 10.1 reduces 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 = , 3juI2j and the generator LJ is
2 2 2 2 3I2 @ @ @ @ @ I 2 2 2 LJ = 2juj 3J21 @J 2 + 6J21J22 @J @J + 3J22 @J 2 = 2juj J21 @J + J22 @J : 11 12 11 12 11 12 The matrix J (0) = I , and thus the only entry which is changed during the evolution in the limit is J12 . The generator LJ has the simple form @2f : (11.5) LJ f = 23jIu2j @J 2 12 The Lyapunov exponent of the matrix valued process J (t) is manifestly zero. The reason that only J12 is changed by the dynamics can be seen from the generator LY . Its coecients depend only on y2 and are independent 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 has length of order one, and is parameterized by x = x(s), 0 s 1, its length is given by Z 1 l = ddsx ds: 0 The length of the curve X"(t; s) = X" (t; x(s)) is Z 1 Z 1 " X" ds = J (t; x(s)) dx (s) ds: l" (t) = dds ds 0 0 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 = cu for some c, which means that the points lie on the same deterministic trajectory. This is similar to the joint behavior of trajectories of (1.1) starting at points lying on di erent deterministic trajectories as described in Sectionn 6. The same is true for any number of xed initial points. Thus l" (t) ! l(t), so that E fln (t)g = (E fl(t)g) , 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 ddsx = (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 coecient that depends only on J2 . In the time dependent case [31] there is no such degeneracy, the process (J1 ; J2) is not restricted to a line, and its norm grows exponentially in time. We have J1(0) = cos , and l(t) = E fl(t)g = lim E fl" (t)g = "!0
Z
0
1
E
q
J12() + J22 ()
ds = E
q
J12() + 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 that the limit length l (t) = g(t; cos ), where the function g(t; x) satis es the initial value problem: @g = 3I2 sin2 @ 2 g (12.1) @t juj @x2 p g(0; x) = sin2 + x2 : p This means that unless = 0 the length l(t) C () t, 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 growth is 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 of the length.
13. Summary and conclusions
We have shown that weak, time independent, incompressible uctuations of a uniform ow produce certain non-trivial e ects on time scales of order one. These are important in advection of passive scalars when the initial density varies on scales comparable to that of the inhomogeneities of the ow. This regime is similar to the one for which the radiative transport theory [30] holds, and we show that the Wigner distribution of the passive scalar satis es the radiative transport equation (7.7). The limit Richardson function, or the two point correlation function, satis es the degenerate di usion equation (4.15). This result does 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 its direction. As we explain in Section 8, using the duality between the limit di usion process Y(t) and the corresponding jump process in Fourier space, the two point di erence vector tends to be aligned with the mean 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 show that its limit is a di usion process. We show that because of the degeneracy caused by the mean ow the corresponding Lyapunov exponent vanishes. This implies in particular that the length of curves moving with the ow is growing only algebraically, which is quite di erent from strong, time dependent, mean zero ows studied in [8, 31]. Physically this is because of the sweeping e ect of the mean ow and the fact that we are 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 results regarding the two point motion remain essentially the same allowing for an additive term in the di usion equation 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 by a limit theorem, which was proven by Kesten and Papanicolaou [ ] for = 1, and later by Komorowski [ ] for 0 < 1. They considered equations of the form 18, 21
16
" 1 t Q"(t) d Q (14.1) dt = " G "2 ; " ; " ; " 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 C 3(Rd ) as a function of 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 Gst ("; M ) = fG(u; q; "; )js u t; jqj M g; and (t; M ) = sup sup jP (AB ) , P (A)P (B )j: s s0;0
,
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 random
incompressible ow with nonzero mean and small uctuations. We show that the Fourier transform of the Richardson function of a passive scalar advected by the ow satis es, under certain conditions, a radiative transport equation. We also study the stretching of curves advected by the ow and show that their length grows algebraically in time, and not exponentially as it does for time dependent, zero mean random ows.
Contents
1. Introduction 2. Long time di usive behavior 3. Flow correlations 4. The Richardson function and its evolution 5. The expectation of the Richardson function 6. Higher moment equations 7. The Wigner distribution and connections with radiative transport theory 8. Di usion{transport duality 9. General scaling of the uctuations 10. Evolution of the Jacobian matrix 11. Application to two-dimensional ows 12. Deformation of the length 13. Summary and conclusions 14. Appendix A. A limit theorem for turbulent di usion 15. Appendix B. Oscillatory functions and the Wigner distribution Acknowledgments References
1 2 4 5 8 9 10 12 14 15 18 19 20 20 21 22 22
1. Introduction
We consider motion in a steady, random, incompressible ow with constant drift u. The random part of the ow v(x), with r v = 0, is a smooth spatially homogeneous random eld with zero mean and rapidly decaying correlations. The trajectory X" (t; x) advected by the ow satis es the scaled equation " " p X d X (1.1) dt = u + "v " " X (0; x) = x; 1991 Mathematics Subject Classi cation. Primary 60F05, 76F05, 76R50; Secondary 58F25. 1
c 0000 American Mathematical Society 0160-7634/00 $1.00 + $.25 per page
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 the random uctuations and their correlation length. We are interested in the behavior of the rescaled two-point di erence function 1 fX" (t; x) , X" (t; x , "y)g " (1.2) Y (t; y ; x) = " in the limit of " ! 0. The individual trajectories X" (t) are close to the trajectories of the deterministic ow X = x + ut for nite times t, when " is small, and thus random perturbations do not play an essential role in their behavior. The rescaled trajectory di erences Y" that start nearby are, however, a ected and that is what we study in this paper. We note that this is di erent from the analysis of the long time correlations of 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 (of order ",1 ) behavior of single trajectories, or groups of trajectories that are not close initially. In Section 3 we show 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 for the 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 a radiative transport equation for the Wigner distribution associated with the trajectory di erence function " Y . We study the duality between the limit two-point di usion process and its analog in Fourier space in Section 8. A scaling more general than 1.1 is treated in Section 9. In Section 10 we show that the Jacobian matrix of the ow (1.1) also converges to a di usion process in the limit " ! 0, and we analyze in detail its law in two-dimensional potential ows in Section 11. In Section 12 we apply these results to study the stretching of curves transformed by (1.1). We show that their length grows algebraically in this case and not 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 by G.I.Taylor [25, 32]. He considered the trajectories of the system (2.1) dX(t) = v(t; X(t)) + dW(t) X(0) = 0; where v(t; x) is a mean zero random velocity eld with rapidly decaying correlations, and W(t) is the standard d-dimensional Brownian motion. He argued that in the long time limit the trajectories behave like those of d-dimensional ZBrownian motion with covariance matrix 1 (2.2) E fvi (t; X(t))vj (0; 0) + vj (t; X(t))vi (0; 0)g dt + 2ij : 0
The rst term in expression (2.2), which corresponds to the turbulent enhancement of the di usion coecient, 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( "t2 ; X" ) + dW(t) (2.3) " X (0) = 0 and analyze the behavior of the trajectories as " ! 0. When the molecular di usivity is positive and the 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 by the homogenization formula (2.2) with X(t) the solution of (2.1) corresponding to " = 1 in (2.3). When the 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 time decorrelation of v in [17]. The weak uctuation scaling (described in detail in Appendix A) was intensively studied (with = 0 and > 0 [16, 18, 21]). A typical scaling in this case, with = 0, is dX" = 1 v( t ; X" (t) ); (2.4) dt " "2 " " X (0) = x; with 0 < 1, so that now the random velocity eld is varying on spatial scales large compared to ", but possibly small on the overall scale if 6= 0. Then, as before, the process X" (t) converges weakly as " ! 0 to a di usion process. Its generator is given by d 2f X Lx f (x) = 21 aij (x) @x@ @x : i j i;j =1 This operator is self-adjoint because the velocity eld v(x) is incompressible. The corresponding generator for a compressible ow has an extra drift term [18, 16, 21]. The di usion matrix is given by Z
1
aij (x) = E fvi (0; x)vj (t; x) + vj (0; x)vi (t; x)g : 0 Expression (2.5) is called the Kubo formula. Of particular interest here is the system dX" = 1 v(X" (t) + t u); (2.6) dt" " "2 X (0) = x; where v(x) is a space homogeneous, mean zero divergence free random eld. Then X" (t; x) converges to Brownian motion with the covariance matrix given by the Kubo formula Z 1 (2.7) cij (0) = E fvi (tu)vj (0)g dt; (2.5)
,1
This corresponds to the Taylor prediction (2.2) with the path X = ut being frozen and independent of the random 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" ; X ") converges weakly to a di usion process (X; X0 ) with generator L~ f (x; x0 ) = Lx f + Lx f + L0 f; where the cross-term L0 is 2f L0 f = 12 cij (x; x0 ) @x@ @x 0: 0
0
i
Here
cij (x , x0 ) =
Thus the di erence
d Z X
1
j
E fvi (0)vj (x0 , x + tu)g dt:
i;j =1 ,1 function Y" = X" , X " converges
weakly to a di usion process Y with generator 2f (2.8) LY f = (cij (0) , cij (y)) @y@ @y : i j 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. 0
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 rescaling the dynamics of many quantities of interest can be reduced to a form similar to (2.4). For example, the rescaled di erence function (1.2) for the trajectories of (1.1) converges to a di usion process with the same generator (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 the rescaled di erence variable " " X (t; x) , X (t; x , "y ) " ; Y (t; y ; x) = " which is the scaled separation of the particles at time t. Then X" (t; x), Y" (t; y; x) satisfy dX" = u + p"v X" ; X" (0) = x (3.1) dt " dY" = p1 v X" , v X" , Y" ; Y" (0) = y: dt " " "
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 " X (t; x) , ut , x " (3.2) Z (t; x) = : " Then the system (3.1) becomes " d Z 1 x + ut " " (3.3) dt = p" v " + Z ;Z (0)= 0 dY" = p1 v x + ut + Z" , v x + ut + Z" , Y" ; Y" (0) = y: dt " " "
This system has the general form (14.1) to which the limit theorem of Appendix A applies. The full statement of this theorem and the necessary assumptions on the random velocity eld in the general case are recalled in Appendix A. We shall assume in particular that v(y) is a mean zero, divergence-free, space homogeneous and strongly mixing velocity eld bounded in C 3 (Rd ). Then the limit theorem implies the following. Theorem 3.1. The processes Z" , and
Y
whose joint generator is
Y
" converge weakly to the correlated di usion processes
Z 1 d 1 2f 2f X Lf = 12 Rij (ut)dt @z@ @z + (2Rij (ut) , Rij (ut + y) , Rij (ut , y)) dt @y@ @y i j i j ,1 i;j =1 ,1 Z 1 2f + (2Rij (ut) , 2Rij (ut , y)) dt @z@ @y (3.4) : i j ,1 Here the covariance tensor Rij is de ned by (3.5) E fvi(y)vj (y + h)g = Rij (h); The individual generators for Z and Y are: d Z1 2f X Rij (ut)dt @z@ @z (3.6) ; LZ f (z ) = 21 i j i;j =1 ,1 Z
Z
and
EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS
and (3.7) respectively, where
LY f (y) =
d Z X
1
5
2f ; Rsij (ut) , Rsij (y + ut) dt @y@ @y i j i;j =1 ,1
Rsij = Rij +2 Rji : Thus the uctuations Z" (t) converge to Brownian motion with covariance matrix given by the Kubo formula (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 uctuating because 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 coecient in (3.7) vanishes for y = cu, so that if two particles start at two nearby positions on the same deterministic trajectory then their separation is not changed by the ow in the limit. The generator LY is asymptotically close to LZ for all y that have large component in the direction perpendicular to the mean ow u. The reason for this is that when the two starting points are separated by a large distance in the direction normal to u, their trajectories are almost independent, and the rescaled di erence trajectory behaves like the uctuations of each individual trajectory. The other end of the asymptotics, the small y behavior of LY , is related to the Jacobian 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 @ + v(x) r = 0 x @t (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 equation in y, in the long time limit. This problem was studied extensively by physicists (see [19] for an extensive review and references), especially for -correlated in time velocity elds, but to the best of our knowledge the only mathematical results are those in [18] and [24, 28]. Molchanov and Piterbarg [24] considered in particular a convection-di usion equation of the form @T " + 1 v( t ; x) r T " = T " (4.1) x @t" " "2 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. If the 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 = 1 @ rd,1 (2 + F (r)) @Q (4.2) @t rd,1 @r @r Q(0; r) = Q0(r): Here Z 1 F (r ) = (RL (t; 0) , RL(t; r)) dt; ,1
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): @R r L L Rij = RL(t; r) + d , 1 @r ij , r(xdi,xj1) @R @r : Equation (4.2) is the generalization of (2.8) for 6= 0. We note that the additional term due to non-zero molecular di usivity is additive. This is di erent from the homogenization formulas of the type (2.2) which come 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 @ " + ,u + p"v x r " = 0 (4.3) @t " "(0; x) = "0(x); where the initial density "0(x) is "-oscillatory (see Appendix B for the precise de nition of this notion). It may 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 of oscillatory form are the localized initial densities (4.5) "0 (x) = "d=1 2 0( x" ); where 0(x) is an L1 (Rd ) \ L2 (Rd ) function, and the inhomogeneous wave family (4.6) "(x) = A(x)eiS (x)=" where A(x) and S (x) are smooth functions, and S (x) is real valued. The latter family describes the distribution of tracers which have the form of high frequency waves propagating in the direction rS (x) with amplitude A(x). A particular case of (4.6) is the high frequency plane wave (4.7) "(x) = Aeixp=": The rescaled Richardson function of the family "(x) is de ned by (4.8) W^ " (t; x; y) = Q(t; x , "y; "y) = " (t; x , "y) " (t; x); 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)
Z
A = f ( ; ) : d sup jf ( ; )j < 1 ; x y
y
x y
x introduced in [22]. The Richardson function of the family (4.4) is (4.9) W^ (x; y) = R0(y); with probability one, by the ergodic theorem, and is independent of x. Here R0(x) is the covariance function of the random process 0(x). The Richardson functions of the localized deterministic family (4.5) is Z (4.10) W^ (x; y) = (x) Q^ 0 (z; y)dz;
where (4.11) Q^ 0 (x; y) = 0(x , y) 0 (x): The limit Richardson function of the WKB family (4.6) is (4.12) W^ (x; y) = jA(x)j2 e,irS (x)y ;
EVOLUTION OF TRAJECTORY CORRELATIONS IN STEADY RANDOM FLOWS
7
which reduces to (4.13) W^ (x; y) = jAj2e,ipy 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 the form 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 " 1 t @ " (4.14) @t + " v x + u "2 rx = 0 " (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 x , ut " " W (t; x; y) = Q t; " ; y : When the initial data for (4.3) is random, space homogeneous of the form (4.4), both the scaled and unscaled Richardson functions are independent of x in the limit " ! 0, hence they coincide in this limit, and the analysis 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 Section 9. 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 A0 as " ! 0, and W^ 0 (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^ + u r W^ = L W^ (4.15) Y x @t W^ (0; x; y) = W^ 0 (x; y); where the operator LY is given by (3.7). The initial data W^ 0 (x; y) in (4.15) are given by (4.9) in the random case, 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 the modi 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 W^ 0(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 diculty in generalizing this result arises in the second step of the proof, that is, replacing the initial data W^ 0" by the limit W^ 0. The limit theorem of Appendix A does not provide any information on the uniformity of the convergence to the limit process with respect to the starting point in all of Rn. In particular, 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 W^ 0 (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) = "1d Q^ 0( x" ; y), and equation (4.3) implies that W^ " satis es the initial value problem @ W^ " + u r W^ " + p"v x r W^ " + p1 nv x , v x , y o r W^ " = 0 (5.1) x x y @t " " " " " with (5.2) W^ " (0; x; y) = "1d Q^ 0 ( x" ; y); where Q^ 0 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 W^ " (t; x; y) = W^ 0(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 W^ 0 (x; y), which is given by (4.10). We claim that (5.3) E fW^ "(t; x; y) , W^ "(t; x; y)g ! 0 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 Z Z E dxdy(W^ "(t; x; y) , W^ `" (t; x; y))f (x; y) 0
0
ZZ
dxdy(W^ 0"(x; y) , W^ 0 (x; y))E ff (x + ut + "Z" (t; x); Y" (t; y; x))g Z ZZ x 1 ^ = dxdy( "d Q0( " ; y) , (x) dpQ^ 0(p; y))E ff (x + ut + "Z" (t; x); Y"(t; y; x))g ZZ = dxdyQ^ 0(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 (5.5) E ff (ut + "Z" (t; 0); Y"(t; y; 0))g ! f(0; y); strongly in the uniform norm on compact sets in y. Here the function f(x; y) satis es the di usion equation with a drift @ f = u r f + L f x Y @t 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 dZ~ " = p1 v ut + Z~ " ; Z~ " (0) = x dt " " e ~ dY ps = p1 v ut + Z~ " , v ut + Z~ " , Y~ " ; dt " " " (5.4)
=
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))g has the same limit as in (5.5). Then by the dominated convergence theorem (5.5) vanishes in the limit " ! 0 since Q^ 0(x; y) 2 L1 (Rd Rd ). Thus (5.3) holds, and we may replace the initial data W^ 0" by W^ 0 in (5.1), and n o n o ^ " (t; x; y) = lim E W^ " (t; x; y) : W^ (t; x; y) = "lim E W !0 "!0 0
The limit theorem applies to the weak form of the resulting system, that is, for any test function f (x; y) we have ZZ ZZ n o E W^ " (t; x; y) f (x; y)dxdy = W^ 0(x; y)E ff (x + ut + "Z" (t; x); Y" (t; x; y))g dxdy 0
=
ZZ
dxdyQ^ 0(x; y)E ff (ut + "Z" (t; 0); Y"(t; 0; y))g dxdy !
ZZ
dxdyQ^ 0(x; y)f(ut; y)dxdy
by the dominated convergence theorem. Thus the function W^ (t; x; y) is the weak solution of @ W^ + u r W^ = L W^ (5.6) x Y @t W^ (0; x; y) = W^ 0 (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 limit theorem 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 Zj and Yj satisfy the following system: dZ"1 = p1 v x1 + ut + Z" ; Z" (0) = 0 (6.1) 1 1 dt " " " dY1 = p1 v x1 + ut + Z" , v x1 + ut + Z" , Y" ; Y" (0) = y 1 1 1 1 1 dt " " " dZ"2 = p1 v x2 + ut + Z" ; Z" (0) = 0 2 2 dt " " " dY2 = p1 v x2 + ut + Z" , v x2 + ut + Z" , Y" ; Y" (0) = y : 2 2 2 2 2 dt " " "
The joint process (Z"1 ; Y1"; Z"2; Y2" ) 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. The cross term L12 involves terms of the type 2 2 cij (x1 ; x2; y1 ; y2) @i j ; dij (x1 ; x2; y1 ; y2) @i j ; @z1 @y2 @y1 @y2 and other similar ones. The coecients cij , dij , : : : are non-zero only for points x1, x2 which lie on the same deterministic 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 independent and 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 " " Z1 = Z2 . Then the system (6.1) reduces to dZ" = p1 v x + ut + Z" ; Z" (0) = 0 (6.2) dt " " " dY1 = p1 v x + ut + Z" , v x + ut + Z" , Y" ; Y" (0) = y 1 1 1 dt " " " dY2" = p1 v x + ut + Z" , v x + ut + Z" , Y" ; Y" (0) = y : 2 2 2 dt " " " The joint generator for the limit processes Y1 , Y2 has the form: @2f L(2) f (y1 ; y2) = 12 a ij (y1 ; y2) @y @y ; i j where the Greek indices label the points and the Latin indices label the coordinates. The coecients a ij (y1 ; y2) are given by
a ij (y1 ; y2) =
1
Z
,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) n o ^ "(t; x; y1 )W^ "(t; x; y2) G^ (t; x; y1; y2) = "lim E W !0 " " " " = "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 initial value problem: @ G^ = L(2) G; ^ (6.3) @t 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 the rescaled Richardson function Z " W (t; x; k) = (2dy)d eiky "(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] (see Appendix B). The limit Wigner distribution may be interpreted as the limit energy (or particle) density of an "-oscillatory family " (x), travelling in the direction k at position x. In particular, we have (7.2)
lim j "(x)j2 = "!0
Z
W (t; x; k)dk
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 the limit values as " ! 0 of various others quantities of interest, for example, the correlation functions of the derivatives of ". The limit of the correlation matrix of the gradient " @ " (x) ( x , "y )" W~ ij" (t; x; y) = " @ @xi @xj is the Hessian in y of W^ (t; x; k): 2W ^ (7.3) : W~ ij" (t; x; y) ! @y@ @y i j The limits of the higher order derivatives can be similarly expressed via W (t; x; k). Thus, in the high frequency limit the Wigner distribution gives a complete description both of the eld " (t; x) itself and its derivatives. Relations of the type (7.2) and (7.3) are the main reason for the recent studies of the Wigner distribution for waves in random media [30]. Let us brie y recall how radiative transport equations for waves arise [30]. Let w" (t; x) 2 C N be the solution of a symmetric hyperbolic system (this scaling corresponds to = 1 in (4.3)) n p o w" + Dj @ w" = 0 (7.4) A(x) + "V x" @@t @xj " " w (0; x) = w0 (x): Here the matrix A(x) is positive de nite and the matrices Dj are symmetric. The initial data w0" (x) is assumed to be "-oscillatory and deterministic, and V (y) is a matrix valued, space homogeneous random process. Then the N N limit Wigner matrix W (t; x; k) has a special form (7.5)
W (t; x; k) =
X
;i;j
Wij (t; x; k)b ;ib ;j :
Here the vectors b ;j 2 C N form a basis for the eigenspace of the dispersion matrix L(x; k) = A,1 (x)kj Dj of the system (7.4), corresponding to the eigenvalue ! . The size of the square matrices W is equal to the degeneracy of the eigenvalue. They satisfy a system of radiative transport equations. This system is decoupled when the eigenvalues are simple and ! (k) 6= ! (p) for all , k and p. Then the radiative transport equation for the scalar W has the form @W + r ! r W , r ! r W (7.6) k x x k @tZ = dk0 (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 spectrum of V (y). Equation (7.6) has a long history. It was proposed phenomenologically by Rayleigh in the beginning of this century and then studied extensively by physicists [10]. Various derivations of this equation were given in 1960's (see [30] for references) when w" is a solution of the wave equation or Maxwell's equations. A general case was treated in [30] but the results were derived only formally. The only case when the radiative transport equation was obtained rigorously, to the best of our knowledge, is for the Schrodinger equation with 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 following theorem.
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 converges to W (t; x; k) weakly in A0 , where W (t; x; k) satis es the radiative transport equation @W + u r W = Z dk0 k k R^ s (k0 , k) ((k0 , k) u)(W (k0 ) , W (k)) (7.7) x @t (2)d,1 i j ij
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 time and for random velocity elds that are not necessarily Gaussian. This tells us that in more general cases the radiative 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 to plane 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 Markovian jump process [5]. Consider the backward characteristics of (7.6) _ = ,rk !(X; K); (8.1) X _ = rx !(X; K); K starting at x(0) = x and K(0) = k. Let a particle move along a trajectory of (8.1) for a random time 1 and then switch wave vector randomly from K to K0 . After that it follows trajectories of (8.1) for a random time 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) =
Z
dk0(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 of the previous jump. The probability density that the wave number jumps from direction k into direction k0 is given by (x; k; k0) : p(x; k; k0) = ( x; k) The resulting process is well de ned if the total scattering cross-section is bounded from above, so that only a nite number of jumps occur during any given time interval, with probability one. The Kolmogorov equation 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) Z 0 (8.2) Qf = (2d)kd,1 ki kj R^ sij (k0 , k) ((k0 , k) u)(f (k0 ) , f (k)) is of the same form as the integral operator in (7.7). It is the generator of a jump process for which the particle is moving along the deterministic trajectory x(t) = x(0) + ut, while the wave vector is the jump process described above with the di erential scattering cross-section (k; k0 ) = (21)d,1 kikj R^ sij (k0 , k) ((k0 , k) u) and the total scattering cross-section Z (8.3) (k) = dk (21)d,1 kikj R^ sij (k0 , k) ((k0 , k) u): Thus the projection of the wave vector on the mean velocity direction u is not changed after the jump. The total scattering cross-section (8.3) is an unbounded function of k, and so the standard argument [5] that K(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)eiky = Q(k)eiky and hence n o n o (8.4) EY eikY(t;y) = EK eiK(t;k)y : 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 function f (t; x; y) we may view f(t; y) = E ff (Y(t; y))g either as a solution of @ f + u r f = L f Y x @t f (0; x; y) = f (x; y) or as the Fourier transform of the solution of @ f^ + u r f^ = Qf^ x @t ^ f^(0; x; k) = f ((2x;,)dk) :
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 implies that solutions of (8.5) tend to a delta function in the directions orthogonal to the mean ow u, and so the in the long time limit Y(t) tends to be parallel to the mean velocity u. In terms of the ow, this means that particles 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 oscillating on a scale ner than ", but their strength is also scaled appropriately. The trajectories X" (t; x) satisfy the scaled equation dX" = u + " v X" ; (9.1) dt " " X (0; x) = x: The trajectory uctuation, de ned as before by " X (t; x) , ut , x " Z (t; 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 on scale " of order one. The case = 1 corresponds to (1.1), while = 2 corresponds to the homogenization scaling (2.3), which we do not consider here. In that case = 0 and uctuations of the velocity eld are no longer weak, which leads to entirely di erent results. The uctuations Z" , de ned by (9.2), and Y" , de ned by " " X (t; x) , X (t; x , "y ) " ; Y (t; y ; x) = " satisfy a scaled system similar to (3.3) dZ" = p1 v x + ut + Z" ; Z" (0) = 0 (9.3) dt " ,1 " " dY" = p1 v x + ut + Z" , v x + ut + Z" , Y" ; Y" (0) = y: (9.4) dt " " ,1 " " ,1 " ,1 " The limit theorem of Appendix A applies also to this system [16] as does the limit theorem for the process Z" of Section 3 in this scaling. The limit theorem for the process Y" has to modi ed because the two terms in (9.4) become decorrelated in the limit " ! 0, as opposed to (3.3). That means that the y-dependent cross term in (3.7) vanishes in the scaling of (9.1). The two particles behave independently of each other and there is no interaction. This happens because the initial separation "y is large compared to the scale of the randomness and the two particles sample nearly uncorrelated parts of the random medium. The results of 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 that are comparable to the wavelength [30]. Inclusions that are smaller than the wavelength but with higher contrast than in the scaling (4.3) will cause dissipation of energy on smaller scales. The contrast still has to be small, which corresponds to > 0 in (9.1). Let "(t; x) be the solution of the rescaled version of (4.3): @ " + u + "1, =2 v( x ) r " = 0: (9.5) x @t " The limit equation for W (t; x; k) in this scaling has the form @W + u r W = , Z dk0 k k R^ s (k0 , k) ((k0 , k) u)W (k); (9.6) x @t (2)d,1 i j ij and ZZ ZZ Z Z " 2 " 2 W (t; x; k)dxdk < W0 (x; k)dxdk = "lim !0 j 0(x)j dx = "lim !0 j (t; x)j dx:
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 is the subject of recent study, both numerical [7] and theoretical [6, 8, 19, 20, 31]. One of the main quantities of interest in such a ow is the Lyapunov exponent responsible for the exponential growth in time of the norm of the Jacobian matrix. Its positivity was established in [8] for velocity elds that are a nite-dimensional Ornstein-Uhlenbeck process. The Jacobian and the evolution of curves in isotropic stochastic ows with zero 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 not grow exponentially in time. Let J " (t; x) be the Jacobian matrix of the transformation x ! X" (t; x): " i (t; x) (10.1) Jik" (t; x) = @X@x k This map is volume preserving because the ow is incompressible, and thus, det J " = 1. We are interested in 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 process with generator
(10.2) starting at J = I .
Z 1 2 d X 1 @ Rim(ut) dtJ J @2f ; LJ f = , 2 jk ln @yj @yl @Jik @Jmn i;k;m;n=1 0
We derive (10.2) at the end of this section. First we note that the generator (10.2) for the limit di usion process J (t) may be interpreted naturally in terms of the generator (3.7) for the limit process Y(t). We recall that " " X (t; x) , X (t; x , "y ) " J " (t; x)y Y (t; y ; x) = " for small y. Thus the processes Y and S = J y should behave in a similar way for small y. In particular we should have that (10.3) E ff (Y(t; y))g E ff (S(t; y))g for small y. To show that (10.3) holds we expand the generator (3.7) of Y for small y and obtain Z 1 2 LY f = 21 (Rsim(ut) , Rsim (ut + y)) dt @y@ @yf i m ,1 Z 1 2 = 12 Rim (ut) , 21 Rim (ut + y) , 12 Rim (ut , y) dt @y@ @yf i m ,1 Z 1 2 2f 2f 1 @ R ( ut) @ @ im ,2 (10.4) @yj @yl yj yl dt @yi @ym = cimjl yj yl @yi @ym ; 0 where Z 1 2 @ Rim (ut) dt: 1 (10.5) cimjl = , 2 @y @y 0
j l
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 the process S(t; y; !) = J (t; A; !)y, starting at s = Ay is given by (10.4). Theorem 10.1 implies that given any function f (s), the expectation f(t; y; A) = E ff (S(t))g = E ff (J (t)y)g satis es the di usion equation @ 2 f ; @ f = c A A imjl jk ln @t @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 = c s s @ 2 g @t imjl j l @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 recall that the large y asymptotics of LY gives rise to the generator LZ for the limit process Z(t). Thus the limit di erence function Y(t) contains information about both limit processes Z(t) and J (t). In physical terms the behavior of the Jacobian matrix can be recovered from the large k behavior of the Wigner distribution which corresponds to the small y behavior of the Richardson function. The behavior of the uctuations Z(t) may be recovered from small k limit of the Wigner distribution or, equivalently, large y behavior of the Richardson function. We note further that the generator (10.2) is formally of the form @2f : LJ f =< cJ; J > @J@J (10.6) Such di usion processes have typically positive non-zero Lyapunov exponents, that is, the limit ln jjJ (t)jj = Tlim !1 T 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 strong degeneracy 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. The evolution equation for J " is obtained by di erentiating (1.1) with respect to xk : " " dJ 1 @v X i ik " p (10.7) = dt " @xj " Jjk 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 suitable for the limit theorem of Appendix A " 1 x + ut d Z " " (10.8) dt = p" v " + Z ; Z (0) = 0 dJ " = p1 rv x + ut + Z" J "; J " (0) = I: dt " "
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 (10.9) L = LZ + LJ + LJZ + LZJ : The operator LZ is given by (3.6) and the operator LJ is Z 1 d X @vi x + z J @ @vm x + ut + z J @f (z; J ) dt (10.10) LJ f = 21 E @x jk @J ln @J j " ik @xl " mn i;k;m;n=1 0
Z 1 d X 1 @vi x + z @vm x + ut + z dtJ @ J @f (z; J ) : E @x = 2 jk @J ln @J @xl " j " ik mn i;k;m;n=1 0 To simplify (10.10) we note that if we let @vi (x) @vm (x + y) ; Fijml (y) = E @x @xl j then F^ijml (q) (p + q) = ip iq E fv^ (p)^v (q)g = qj ql R^ im (q) (p + q); j l i m (2)d (2)d and thus 2 Fijml (y) = , @ @yRim@y(y) : j l Therefore the generator LJ has the form Z 1 2 d X @ Rim (ut) dtJ @ J @f (10.11) LJ f = , 12 jk @J ln @J @yj @yl ik mn i;k;m;n=1 0
Z 1 2 d X @ Rim (ut) dtJ J @2f ; = , 12 jk ln @yj @yl @Jik @Jmn i;k;m;n=1 0 the last equality being due to incompressibility. Note that LJ is independent of z. Next we compute LZJ : d Z 1 x X @vi x + ut + z J @f dt LZJ f = 12 (10.12) E vk " + z @z@ @x nj @J k n " nj k;i;j =1 0
d Z1 @v x X 1 @2f : i = 2 E vk x" + z @x + ut + z dtJ nj @zk @Jij n " k;i;j =1 0 The coecients of LZJ are also independent of z. The rst order derivative term vanishes after taking the expectation because of the incompressibility condition. The operator LJZ is d Z 1 @v x @f X @ x 1 i (10.13) E @x " + z Jnj @J vk " + ut + z @z dt LJZ f = 2 n ij k i;j;k=1 0
d Z1 X 1 @vi x + z v x + ut + z dtJ @ 2 f ; = 2 E @x k " nj @z @J n " k ij i;j;k=1 0
it does not have rst order derivative terms as well, and has coecients independent of z. This shows that J " 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 ow in R2 has the form @ @ (11.1) v (x) = @x ; , @x ; 2
1
where (x) is the stream function. We assume that the random eld (x) is space homogeneous, zero mean and isotropic with covariance function 2 R (x) = E f (y) (x + y)g = F r2 ; where r2 = x21 + x22 . The Jacobian matrix of two-dimensional time dependent ows with zero mean u was studied in detail in [31], in the di usion approximation. The results there are entirely di erent but may be formally recovered from our calculations as we explain below. The covariance matrix Rij for the ow (11.1) is 22 x1 x2 00 , 1 0 , x 0 (11.2) R = 0 ,1 F + x x ,x2 F : 1 2
1
The tensor cimjl (10.5) in the generator (10.11) has now the form , c1111 = , ju1 j I2 + I3 + u^21u^22 I4 , c = , 1 3^u u^ I + u^ u^3 I 1112
j j
1 23
u
1 24
c1122 = , ju1 j 3I2 + 6^u22I3 + u^42I4 , c1211 = ju1 j 3^u1 u^2I3 + u^31u^2I4 , c = 1 I + I + u^2u^2I
(11.3)
1212
j j
,
2
u
3
1 24
c1222 = ju1 j 3^u1 u^2I3 + u^1u^32I4 , c2211 = , ju1 j 3I2 + 6^u21I3 + u^41I4 , c = , 1 3^u u^ I + u^3u^ I 2212
,
j j
1 23
u
1 24
c2222 = , ju1 j I2 + I3 + u^21u^22 I4 : p Here u = (u1; u2) is the mean ow, juj = u21 + u22, u^j = juujj , and the constants I2 =
,
2 2 2 Z 1 Z 1 F 00 t2 dt; I3 = t2 F 000 t2 dt; I4 = t4 F (iv) t2 dt 0 0 0
Z
1
are related by (11.4) I2 = ,I3 ; I4 = 3I2 : The other entries are determined by the symmetries cimjl = cmijl = cimlj . The generator (11.5) with cimjl as 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 our calculation by setting formally I3 = I4 = 0 in (11.3). Then the generator (10.2) given by Theorem 10.1 reduces 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 = , 3juI2j and the generator LJ is
2 2 2 2 3I2 @ @ @ @ @ I 2 2 2 LJ = 2juj 3J21 @J 2 + 6J21J22 @J @J + 3J22 @J 2 = 2juj J21 @J + J22 @J : 11 12 11 12 11 12 The matrix J (0) = I , and thus the only entry which is changed during the evolution in the limit is J12 . The generator LJ has the simple form @2f : (11.5) LJ f = 23jIu2j @J 2 12 The Lyapunov exponent of the matrix valued process J (t) is manifestly zero. The reason that only J12 is changed by the dynamics can be seen from the generator LY . Its coecients depend only on y2 and are independent 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 has length of order one, and is parameterized by x = x(s), 0 s 1, its length is given by Z 1 l = ddsx ds: 0 The length of the curve X"(t; s) = X" (t; x(s)) is Z 1 Z 1 " X" ds = J (t; x(s)) dx (s) ds: l" (t) = dds ds 0 0 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 = cu for some c, which means that the points lie on the same deterministic trajectory. This is similar to the joint behavior of trajectories of (1.1) starting at points lying on di erent deterministic trajectories as described in Sectionn 6. The same is true for any number of xed initial points. Thus l" (t) ! l(t), so that E fln (t)g = (E fl(t)g) , 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 ddsx = (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 coecient that depends only on J2 . In the time dependent case [31] there is no such degeneracy, the process (J1 ; J2) is not restricted to a line, and its norm grows exponentially in time. We have J1(0) = cos , and l(t) = E fl(t)g = lim E fl" (t)g = "!0
Z
0
1
E
q
J12() + J22 ()
ds = E
q
J12() + 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 that the limit length l (t) = g(t; cos ), where the function g(t; x) satis es the initial value problem: @g = 3I2 sin2 @ 2 g (12.1) @t juj @x2 p g(0; x) = sin2 + x2 : p This means that unless = 0 the length l(t) C () t, 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 growth is 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 of the length.
13. Summary and conclusions
We have shown that weak, time independent, incompressible uctuations of a uniform ow produce certain non-trivial e ects on time scales of order one. These are important in advection of passive scalars when the initial density varies on scales comparable to that of the inhomogeneities of the ow. This regime is similar to the one for which the radiative transport theory [30] holds, and we show that the Wigner distribution of the passive scalar satis es the radiative transport equation (7.7). The limit Richardson function, or the two point correlation function, satis es the degenerate di usion equation (4.15). This result does 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 its direction. As we explain in Section 8, using the duality between the limit di usion process Y(t) and the corresponding jump process in Fourier space, the two point di erence vector tends to be aligned with the mean 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 show that its limit is a di usion process. We show that because of the degeneracy caused by the mean ow the corresponding Lyapunov exponent vanishes. This implies in particular that the length of curves moving with the ow is growing only algebraically, which is quite di erent from strong, time dependent, mean zero ows studied in [8, 31]. Physically this is because of the sweeping e ect of the mean ow and the fact that we are 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 results regarding the two point motion remain essentially the same allowing for an additive term in the di usion equation 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 by a limit theorem, which was proven by Kesten and Papanicolaou [ ] for = 1, and later by Komorowski [ ] for 0 < 1. They considered equations of the form 18, 21
16
" 1 t Q"(t) d Q (14.1) dt = " G "2 ; " ; " ; " 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 C 3(Rd ) as a function of 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 Gst ("; M ) = fG(u; q; "; )js u t; jqj M g; and (t; M ) = sup sup jP (AB ) , P (A)P (B )j: s s0;0
