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Scaling properties of passive scalars in one dimension
PHYSICA ELSEVIER
Physica A 244 (1997) 190-212
Scaling properties of passive scalars in one dimension Leo P. Kadanoff*, Scott Wunsch, Tong Zhou The James Franck Institute, The University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
Abstract This paper is a set of notes about work in progress. Since the work is directed toward scaling, it seems quite appropriate to report this in a context devoted to Ben Widom's scientific contributions. Model equations for the motion of a passive scalar in one dimension are set up and some scaling properties of their solutions are derived. All these models follow the approach set up by Kraichnan in which the driving velocities have Gaussian correlations with scaling properties in space, but zero-range correlations in time. The simplest and most natural versions of the model fail to satisfy an incompressibility condition. In another version, the model does have an incompressible flow. To permit the incompressibility, there are two pipes - each of which is described by temperature and velocity fields which are functions of a single coordinate, x, and time. Flow from one pipe to the other transfers both mass and heat. This model has two conserved quantities in the limit of zero dissipation. In still another version, fluid elements are interchanged in position so that there are an infinite number of conservation laws in the nodissipation limit. Scaling properties of the resulting two-point functions are derived. P A C S : 47.27.GS; 05.40.+j Keywords: Isotropic turbulence; Random processes; Passive scalars
1. Introduction W e all congratulate Ben on his birthday and wish him many more to come. L P K adds a personal note: Ben W i d o m and I have shared some very exciting times. W e both worked on scaling in critical phenomena. B e n ' s ideas [1,2], which I further developed [3], proved crucial for carrying the field forward. I have enjoyed and profited from following in B e n ' s footsteps. His elegance in both scientific and personal style has been a major influence upon me. * Corresponding author. E-mail: [email protected]. 0378-4371/97/$17.00 Copyright ~) 1997 Elsevier Science B.V. All rights reserved. PH S0378-4371 (97)00239-2
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1.1. The problem
In recent years, ideas of scaling have been very broadly applied. Specifically, there has been a resurgence of applications to turbulence, in which the work of Kolmogorov [4,5], stands as the granddaddy of all scaling analysis, Kolmogorov's two papers prefigure the recent literature [6] which contains two alternative views of the nature of well-developed turbulence. In one view, the simple scaling caught by the 1941 paper of Kolmogorov [4] is the asymptotic truth which holds in the limit of high Reynolds numbers. Then the experimental facts, some of which seems to support a more complicated scaling, are described in terms of nonasymptotic corrections to scaling. In the other view [5], the experiments are better understood and described as a result of a multifractal picture [7] in which cascades produce anomalously large fluctuations in the velocity fields. These two views can both be supported by the experimental evidence. There are theoretical arguments for both. A good summary can be found in Frisch's book [8]. This same issue arises in a simplified problem in which one looks at the motion of a scalar quantity in a specified velocity field with well-defined spatial scaling properties. This passive scalar problem arises when some scalar physical quantity, e.g. temperature, is moved about by a fluid in motion. The word 'passive' means that the flow fluid is not affected by the motion of the scalar. Thus, the behavior of the temperature is defined by a linear problem. The produced distribution of the scalar can be very complicated, even if the underlying flow is rather simple. Recent studies of passive scalars have analyzed a particular case in which the velocity field is (1) (2) (3) (4)
stochastic, with a Gaussian probability distribution, incompressible, correlated only over very short times, having a zero mean, but with a power-law correlation in space over some wide range of scales.
A summary of recent work on this model can be found in a review paper by Shraiman and Siggia [9]. This model, first studied by Kraichnan [10], shows a very interesting behavior in which equal time structure functions formed from different powers of the temperature each show different scaling indices [11]. Furthermore, the scaling indices of the nth order structure function is not linear in n. This non-linear behavior is described by saying that the passive scalars show multiscalin9. The work on the indices is rather controversial in that recent calculations [12-14] show answers which seem to disagree with those of Ref. [11]. If we could see through to the source of that multiscaling, we could finally see who was right. The actual values of the indices do not matter much. The important difference between the two sets of calculations is that the work of Kraichnan and coworkers [11] suggest that the answers are universal, only weakly dependent upon the details of the model, while the other calculations, particularly Chertkov et al. [15] suggest a non-universal answer.
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These calculations might also have an impact upon our understanding of turbulence itself. If we could see how multiscaling arises in the passive scalar problem, we might better understand whether turbulence was likely to show multiscaling. One alternative is that the passive scalar multiscaling might arise from a essentially linear cascade in scales. The behavior of a linear chain of multiplicative and stochastic renormalizations has been studied in the GOY model [16]. The latter model does give multiscaling and some non-universality [6]. Another alternative is that the multiscaling might be a result of the infinite number of conservation laws which apply in the non-dissipative limit of the passive scalar. Each conserved quantity might be connected with its own critical index. Turbulence probably does have a linear cascade in scale; it probably does not have an infinite number of conservation laws. It certainly would be helpful to know how multiscaling might be connected with two, or an infinity of, conservations. There is another, and perhaps deeper, way in which the passive scalar problem might cast light upon the turbulence problem. One issue in turbulence is the way in which information is transferred among the different scales [17]. The ideas of the Kolmogorov's 1941 paper suggested that one consider the information available to an observer looking at inertial range behavior in some region of space and in some interval of distance scales. This information would be the average velocity of the region and its rate of addition of kinetic energy. The details of the dissipative processes and the value of the viscosity would be hidden from such an observer, essentially because there is an information flow from large scales to small. In the renormalization analysis of critical behavior the flow is in precisely the opposite direction: microscopic interactions produce macroscopic consequences. It is interesting to ask about the nature of that flow in a dynamic dissipative system. This paper is the start of an attempt to try to understand issues of information flow and multiscaling by looking at a simplified version of the passive scalar problem. The exact flow equations for a passive scalar are ( ~ t + U(r,t) • V ) T ( r , t ) = D V 2 T ( r , t )
+ F(r, t).
(1.1)
Here D is a diffusion coefficient and F is a random forcing term. To be realistic, we must take the flow to be incompressible. This requires the divergence condition
v.u=o.
(1.2)
Because of the divergence-free condition, the U term is anti-Hermetian. The diffusion term is Hermetian. The U-term can force a complicated pattern on T; the diffusion term tends to smooth and simplify that pattern. Here, we say a word about the structure of the equation used to study the problem. Like the previous workers [12-14] we use an equation called the H o p f equation which describes the time derivative of the equal time correlation function. It has the form (t)t - ~°)( T(1, t)T(2, t ) . . . T(n, t)) = F~(1,2 . . . . . n ) .
(1.3)
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Here the numbers 1,2 .... stand for space indices, and Fn is a term which describes the forcing. The time development operator, Lf depends only on the structure of the velocity correlations and does not depend upon either the structure of the forcing, or the initial conditions for the temperature, or upon the temperature boundary conditions. The forcing term on the right-hand side of the equation depends upon lower-order correlation functions. Thus, the Hopf equation provides a closed set of equations for successively determining the higher-order correlation functions. One can take several different approaches to the random forcing term, F. One method, followed by Kraichnan [10] and later workers is to take the forcing term to be a Gaussian random variable, with correlations of zero range in time, but having the force be completely independent of U. The other possibility, conceptually somewhat neater [13], is to imagine that the temperature field has an average behavior which includes a linear gradient in space, g. The T(r,t) in Eq. (1.1) is the deviation for the temperature from the linear gradient and the forcing term is F -- - g. U. These two possibilities lead to correlation functions which can be invariant under space and time translations. For most purposes, the scaling properties of these correlation functions are quite similar. Hence these two approaches mostly give the same conclusions. Another conceivable approach, less followed up to now, is to take the random forcing to be strictly zero but set some initial data for the temperature. For example, one can pick the temperature at t-----0 to be a delta function in space. Then one can follow the subsequent time development, using the known form of the average time development operator Lf. In this paper we shall follow all three possibilities in different calculations. We can summarize the recent scaling studies by imagining the form of the Hopf Eq. (1.3) in the time-independent situation. The equation then reads S ( T ( 1 ) T ( 2 ) - - • T(n)) = F , ( I , 2 .... n).
(1.4)
The classical analysis of passive scalars appears in the work of Obukhov and Corrsin [18]. Their results can be derived by looking at Eq. (1.4) in a scaling re-. gion in which all distances are of the order of r, and r stands in the interior of an inertial region. In this region, we notice that 5 a is a second-order differential operator which has coefficients which depend upon distance as a power law, r". Hence, the left-. hand side has .~ scale a s r (r/-2). Estimate the scaling of the right-hand side, and then divide by this estimate of L~ a. You will then find the classical form of the scaling of the correlation function. In contrast, the more recent papers [12-14], ask about the secondorder differential operator c~,. It has eigenstates, with eigenvalues zero, which then satisfy the homogeneous equation f ~ 9 = 0 . These eigenmodes scale differently from the Obukhov Corrsin result. Do they change the scaling of the correlation function? To answer that question, one must do the eigenmode analysis of Eq. (1.4). The particular linear combination of modes which enter is set by boundary conditions (See, for example, the discussion in [14].), so one must be careful to understand these boundary. conditions in some detail. Since there are many variables involved, the analysis of this equation for three-dimensional flow is quite difficult. A simpler analysis might be
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expected in lower dimensions or in the limit of infinite dimension [12]. Here we look to the simplest case: one dimension.
1.2. Models Of course the simplest extension of the passive scalar equation, Eq. (1.1), to one dimension is the exact transcription g2
(Ot + u(x,t)Ox)T(x,t) = D ~ x 2 T ( x , t ) + F(x,t)
Compressible Model.
(1.5)
In this equation we can no longer demand incompressibility, since that would make u independent of x. But in view of the incompressibility condition, one can put the derivative either to the right of or to the left of U in Eq. (1.1). If we write the equation with the derivative to the left, we have 02
dtTx(x,t)+3x(U(x,t)Tx(x,t))=Off-jTx(x,t)+Fx(x,t)
Adjoint Model.
(1.6)
We have written this equation with the symbols T~ and Fx used for the basic variable and the driving force, in place of the T and F which we have used up to now. That is because the model defined by Eq. (1.6) is not essentially different from the one defined by Eq. (1.5). To get Eq. (1.6) differentiate Eq. (1.5) with respect to x, and write Tx and Fx for, respectively, the x-derivatives of T and F. Neither of these one-dimensional equations (Eq. (1.5) or (1.6)) has an incompressibility condition. Hence, they both fail to have an anti-Hermetian convective term. It is possible to construct a one-dimensional flow which does have that condition, and consequently that flow displays the effects of anti-Hermeticity. Simply use two parallel one-dimensional pipes. Let there be a velocity field which drives a passive scalar in each of the pipes. The physical idea is to have any fluid which leaves one pipe and enters the other. The mathematical idea is to have the convection term in the equation for the temperature be an antisymmetric differential operator. (Since the operator will be real, it will be anti-Hermetian.) To do this, let the pipes be distinguished by an index, a, which takes on the values ±1. There is a specified velocity of the form uo(x, t) which describes the flow along the pipes. There is another velocity, or rather flux, v(x, t), which describes the rate (per unit length) of interchange of fluid between the two pipes. The equations of motion for the temperatures in the pipes, T~(x,t), are
-~ + ua(x,t) 02
T~(x,t) + ~v(x,t)[T+(x,t) - T_(x,t)]
= DO--~ T~(x, t) - 2(T,(x, t) - T_,(x, t)) + F~(x, t)
(1.7)
with D being a diffusion coefficient for flow along the pipes and 2 being a coefficient which describes a Newtonian rate of leakage of heat from one pipe to another.
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The conservation of fluid is represented by saying that any x-variation of u~(x, t) is produced by the flow of fluid from pipe to pipe. This gives us the condition that
~xU+(x,t)=v(x,t)= -
u_(x,t),
(1.8)
which integrates to the statement that
u~(x, t) = au(x, t),
(1.9)
plus a constant of integration independent of x. Take the constant of integration to be zero so that the velocity in the two pipes point in opposite directions. Notice that u and v are related by ~3
~xU= V .
(1.10)
In this way we have generated three separate first-order PDE models for the flow of a one-dimensional passive scalar. Can these models shed any light upon the issues related to more realistic flows of passive scalars in higher dimensions?
1.3. Conservation laws In the case of real 'passive scalars', as defined by Eqs. (1.1) and (1.2), spatial integrals of all powers of the temperature are conserved by the flow terms. Specifically, if Sp(t) is the space-integral of the pth power of the temperature then the conservation law reads
dSPdt - pD /
d r [ T ( r ) ] p - l ' ~ 2 T(r).
( I. 11 )
So, in the limit of D going to zero, there is some sort of conservation of the Sp's. Here, and in the remainder of this section, we neglect the direct effect of forcing terms upon the conservation laws. The two-pipe model has only two conservation laws which hold in the absence of dissipation (2 = D = 0). The two conserved quantities are the 'energy',
I4: f dx ~
r~(x,t)
(1,12)
6r
and the 'entropy'
s= fax y~ r ~ ( x , t ) 2 .
(l.13)
(7
One immediate consequence of the fluid conservation condition (1.8) is that the u and v terms together leave expression (1.12) unchanged in time. The resulting conservation law is the statement: dH/dt = 0. On the other hand, the second-order invariant, S, is
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diminished by dissipation. To derive the entropy condition, define an inner product ($,, q~) via ($,q~)= f a x
Z
Oa(x)d?a(x).
(1.14)
Or
The velocity term in Eq. (1.7) is the anti-Hermetian operator, U. We define this operator by giving its effect upon a vector ~b which depends upon x and a. Let ~ = U~b. Then ~k has the value c3u ~Or(x)=~ru(x,t) ~x q~a(x)+ 51 ~x(qS+(x) - q~_(x)).
(1.15)
Notice that, with this definition, U is antisymmetric (~, UqS)= - ( ( U $ ) , qS). Naturally, the diagonal element of this antisymmtric operator is zero, (T, U T ) = 0. This last condition then, gives a situation in which the convective terms conserve the entropy. The resulting entropy conservation law is then derived as
ldS l d -(T, T) = - (T, UT) -D(OxT, OxT) - ),(aT, a T ) . 2 dt 2 dt
(1.16)
Since U is anti-symmetric, the first term on the right-hand side vanishes, then the remaining terms in the time derivative are negative semi-definite: Real 'passive scalars', as defined by Eq. (1.1), have an infinity of conservation laws in the zero-dissipation .... ,m,." ilowever, in the two-pipe model integrals of higher powers of the temperature are not conserved by the flow terms. For example, the time derivative of the cube is d 2 ~ ( T , T) = - 3 ( T 2, UT) + dissipation terms = - 3 ( T 2, cru~T) - 3(T 2, ~(Ou)T) + dissipation terms. Here ~ is an operator which acts in the spin space and has the effect = ~b implies ~ ( x ) = [qS+(x) - ~b_ ( x ) ] / 2 .
Then we find d
2
~ ( T , T) =
/
dxv[T+(x) -- T ( x ) ] 2 [ T + ( x )
(1.17)
+ T_ (x)]/2 + dissipation terms, (1.18)
so that the flow process does not conserve the sum of cubes. More complicated versions of the same argument indicate that higher powers are not conserved either. Despite this weakness, we might hope that the model might have some features in common with the standard models of passive scalars. Besides, it would be very interesting to know what happens with just two conservation laws, The simple models described by Eqs. (1.5) and (1.6) have even fewer conserved quantities. The first model (Eq. (1.5)) gives no powers of T being conserved by the
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flow processes. The second (Eq. (1.6)) has the x-integral of T~(x,t) conserved by the flow, but that is apparently the only conservation law. Despite these paucities of conservation laws, we should ask about the properties of these models. To make them solvable, we pick a simple scaling behavior for v, e.g. Gaussian with zero time correlation range, and some x-scaling. Next, we solve for the temperature correlation functions. We can then ask: Do they show scaling? Do they have the large fluctuations in T that are associated with multiscaling? Do we have any hints of a universal behavior?
1. 4~ A rearrangement model We construct this model in the spirit of the work of Kerstein [19]. It is indeed possible to achieve a situation in which there are an infinite number of conservation laws. Imagine a situation in which we have a discrete labeling for the space variable. Thus, we have T(n,t) with n = 1,2 . . . . . perhaps with periodic behavior, with period L, or maybe with the shifted periodic behavior
T(n + L,t) = g L + T ( n , t ) .
(1.19)
We can have a discrete version of a dissipative process
T(n,t + 1)= T(n,t) + D[T(n - l,t) + T(n + l,t) - T(n,t)] .
(1.20)
One could also imagine rearrangement processes, which are the analog of flow processes in which we have
T(n+m,t+l)={T(n+q-m,t) T(n + m, t)
ifO
( 1.21 )
To get scaling, we choose the size of the re-arrangements at different time steps according to a power law. This rearrangement does not preserve any continuity in the temperature, but it does permit sums over n of all powers T(n,t) p to be conserved. Thus, if the conservation laws are crucially important this rearrangement model is an attractive candidate for study.
2. Correlation functions
2. I. Velocity correlations Gaussian correlations are defined by averages of the first and second moments of the Gaussian quantities. We take the average of u(x, t) to be zero. We further take the correlation between two velocities to have a zero range in time:
{u(x, t)u(y, s)) = 2c~(t - s)C(x - y ) .
(2.1)
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In what follows, it will be important that C ( x - y ) is an even function of its argument. In the usual approach, one would like the equal time velocity structure fimction defined by Su(x, y )
=
fds
u ( y , t ) ] [ u ( x , s ) - u ( y , s ) ] ) = 4 [ C ( 0 ) - C ( x - y)]
(2.2)
to grow as the power )1 of Ix - Yl in the inertial range. If S is a simple power, one could have an additive constant in the correlation function, viz., C ( x - y ) = C(O) - C ' i x - YI"
(2.3)
in the inertial range. In the previous work, t / w a s picked to be between zero and two. Since the triangle inequality requires the magnitude of C ( x - y ) to be smaller than C(0), it is then true that both C(0) and C* must be positive. Since we wish the correlation to fall to zero near the integral scale, L, we might as well define L by (2.4)
C(O) = C * L ~ .
The dissipative scale is defined by the condition that eddy diffusion and the ordinary diffusion match at that scale. Thus, l is defined by the condition D=C*
(2.5)
I~ .
Notice the structure of the correlation function in Eq. (2.3), in which the scaling term in the correlation function increases with increasing I values of r = [x - y[. This form is used because, in higher dimensions, this correlation is dominated by swirls of size comparable to r. These swirls have velocities point in different directions on different sides of the swirl. Larger swirls involve higher velocities. Thus, we have a negative term which increases with r. There is also no simple scaling. The big constant term, C(0) describes the summed effect on velocity squared of all sizes of swirls. That term dominates the term in q. In order to get a simple scaling one considers the structure function defined by Eq. (2.2). I f we did not have a time dependence this structure function would be proportional to the average squared fluctuations in the velocity difference [ u ( x ) - u(y)]. The structure function has a simple scaling behavior in the inertial range, l ~ I x - y [ ~ L . From Eqs. (2.2) and (2.3) we see that Su(x - y ) = 4 C * l x
- YI".
(2.6)
2.2. Force correlations
In doing our subsequent calculations, we shall need information about the force correlations. We take the force, F ( x , t ) to be a random Gaussian variable with zero mean and a vanishing correlation time (F(x, t ) F ( y , s)) = 2 6 ( t - s ) g ( x - y ) .
(2.7)
1This growth is almost unknown in the large body of scaling studies built around critical phenomena. In that case, the scaling terms typically decay with distance.
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If we have to make a specific choice of the correlation function, we shall choose it to be constant over the inertial range and be zero outside that range, specifically.
Z(x)
= ~ ;(o (0
if o < Ixl < L, otherwise.
(2.8)
We will use this simple forcing to drive our temperature problem. As we have said, one can alternatively drive the system by an overall temperature gradient, 9. Then the formulation of the problem is much the same as in the externally forced case. Instead of calculating correlations of a temperature variable, the correlations to be calculated are of the deviations of the temperature from its average value, 9x. For many purposes, this gradient-driven problem is that same as the externally forced problem except that the forcing correlation, Z is replaced by (2.9)
Z(X) ~ g 2 C ( x ) .
2.3. Hopf equations In every case, our equation of motion for the temperature is
~t T ÷ U(t)T= WT + F .
(2.10)
Here, U is a (differential) operator linear in the stochastic velocity field, W is a dissipation term, and F is a stochastic forcing. This equation has a formal solution T ( t ) = l- -
exp
oc
d s ( - g ( s ) + W)
.
(2.11)
+
In this equation, the subscript ' + ' is an indication that the integral is a time-ordered product in which operators referring to larger times appear to the left of those describing smaller times. In general, evaluating time-ordered products may be quite a mess, but because of the short-ranged nature of the time correlations, this particular operator is quite simple. We define correlation functions like S2(xa; y/~) = (T~(x)Tu(y))
(2.12)
as well as higher-order correlations. Here we have included the subscripts appropriate for the two-pipes, but we can do a much more general job by using the symbol Sn(1,2 .... ) with 1 and 2 and ... standing for the needed arguments. One of the contributions of Kraichnan [10] is to notice that one can set up simple equations for such correlation functions for the special case in which the velocities have Gaussian behavior and short-ranged correlations in time. The equations are derived by writing a formal solution of the linear equation of motion, Eq. (2.10). We then form the equal time version of the n-point correlation function, and calculate its time derivative, which
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must be in fact zero. The time derivative contains terms of zeroth and first order in the stochastic velocities. This term is n
5~(T(1,t)T(2,t)... T(n,t)) = E ([-U(j,t) + Wj]T(I,t)T(2, t)... T(n,t)). j=l
(2.13) The averages can be calculated quite generally [14], using the time-ordered product of Eq. (2.11) to get the time-development operator for our Hopf equation. This equation takes the form of Eqs. (1.3) and (1.4), in which the time-development operator has the form
~ = E Dij.
(2.14)
i,j
Here, the operators on the right-hand side are produced by correlations of the U's. Specifically, if Ui(t) and Uj(s) act on different variables so that they are in some sense in different subspaces, then they define the quantities on the left-hand side of Eq. (2.14) as
(Ui(t)Uj(s)) = 2 6 ( s - t)Dij
for i C j . .
(2.15)
Correspondingly, for the case in which the operators act on the same variables, we have
Wi÷~1 /ds(Ui(t)Ui(s))=Dii.
(2.16)
The story is completed by specifying the forcing terms. These are evaluated in terms of the same stochastic averages as went into the time-development operator. Specifically, the forcing term Fn of Eq. (1.3) is given by Fn(1,2 . . . . . n)=2)~(1,2)(T(3, t)T(4,t)... T(n,t)) + interchange terms.
(2.17)
This equation works whenever we have a prescribed Gaussian forcing given by F(1, t). If n is odd the forcing and the corresponding correlation function vanishes. A similar form applies when the forcing is given by a linear gradient 9 and the velocity field U. For even n, Eq. (2.17) still applies but the force F ( 1 , t ) is replaced by U(1)gxl so that in Eq. (2.17) Z(1,2) is replaced by g2C(1,2). In this case of internal forcing, the odd-order correlation functions are not zero. Instead they are given by F~(1,2 ..... n) = 2 f as((U(1,t)gx, )U(2,s))(T(2,t)T(3, t)... T(n, t)) + interchange terms.
(2.18)
2.4. A first solution The simplest calculation is for the average of a temperature at a single point. We do not include any force term. Instead we include the initial data that at time zero the
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temperature has the value T * 6 ( x ) . We can determine the subsequent behavior from the equation (c3t - Dll )(T(1, t)) = T * 6 ( x ) 6 ( t ) .
(2.19)
In the compressible model of Eq. (1.5), the diagonal contribution to the time development operator is easily evaluated as Dll = (D + C(0))(~3xf.
(2.20)
Thus, the flow produces an eddy diffusion, De(¢ = D + C(O)
(2.21)
in which the motion of the fluid enhances the diffusion so that the effective diffusion constant is increased by adding C(0). The solution to the heat flow problem stated in Eqs. (2.19) and (2.20) is the familiar form 1
(T(x, t)) = a + bx + - exp(-x2/(4tDe~.))T * v/4ntDeff
(2.22)
in which a and b are constants to be set by the boundary conditions. If we use the boundary condition that T vanishes at spatial infinity then we are required to set these constants to zero. So far, everything is easy. But this warmup exercise is too easy. Next try a force F(x, t) which is a delta function at x = 0 but has been in place forever. Thus, the equation is ( a, - De#(ax)2)(T(x, t)) = F* 6(x) .
(2.23)
We can take a time-independent solution of the form (T(x, t)) = a + bx - F *
[xl/(2Oe~).
(2.24)
Now we can no longer just ignore the 'zero modes', the ones with coefficient a and b. The magnitude of the solution certainly grows as one goes to infinity. One can no longer set the solution equal to zero there. Instead, one must be careful to set two boundary conditions, corresponding to the second-order nature of the problem. One could do this, for example, by saying that T ( x ) vanishes at the two points x = ± B/2, where B is some huge distance. Then we set the undetermined coefficients and find that the solution of Eq. (2.24) reads (T(x, t)) = F * (B/2 - ]x])/(2De~ ).
(2.25)
We can now see the reason that the zero mode must enter. The forcing term adds heat to the system. That heat cannot disappear, it must flow to infinity. Consequently, the solution must extend over a large distance. The adjoint model of Eq. (1.6) has identical equations for its one-point function. Hence, we need not discuss it separately. This model does not really have a conservation law for energy, i.e. the space integral of T~. But the one-point function behaves as if there were such a conservation law.
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2.5. Boundary conditions The use of a boundary condition in which the temperature vanishes at some far-away boundary of the system is perfectly workable but it does have one (mostly aesthetic) disadvantage. It produces correlation functions which are not translationally invariant. We can get the same result and have translational invariance in both space and time if we slightly modify the equation for the time derivative of the temperature by adding to the diffusion operator W = D V 2 a small negative constant so that Eq. (1.1) is replaced by (~
+ U(r,t). V) T(r,t)=(DV 2- e)V(r,t)+ F(r,t).
(2.26)
Here e is small and positive. If it is small enough it will not affect our answer. Once again, use the boundary condition that T vanishes at spatial infinity, both :kc¢. The new term destroys heat and reduces the temperture to zero far away. If the eddy conductivity term vanishes for large x, then in this region, the equation has two solutions, one exponentially growing the other exponentially decaying as x goes to infinity. The right solution can have only an exponentially decaying term. Using g, we introduce a cut off at very large scale to control infrared divergences. Now we can demand that the correlation functions be finite everywhere. This should provide enough boundary conditions to determine the solution. We hope and expect that the interesting parts of the solution, especially the scaling indices in the inertial range, are independent of ~. 2.6. The two-pipe model Precisely, the same analysis carries over to the two-pipe case. The only additional complication is the behavior in the two-component space of the variables which label the pipes. The operators of the Hopf equation depend upon some spin matrices. We specifically use ~ = I + ) ( - I and r = [-)(+[. In this notation, [+) is a projector onto the sum of temperatures from the two pipes and [-) is a projector onto the difference. In the equation of motion, the convection term is U = CtOxU+ 3UOx.
(2.27)
Since the two matrices ~t and fl are transposes of one another, and since the derivative is odd under transposition, U is an anti-Hermetian operator. Calculate the off-diagonal term, D12 in the Hopf equation. The result is a direct product of two matrices one in the spin space of x, the other in the spin space of y, so that D12 = flxflyC(x - y)axay -~- O~xO~yaxayC(x - y ) + flxOtyayC(x - y)ax
+ exflyOxC(x - y)0y.
(2.28)
The diagonal term is slightly tricky because one has to calculate derivatives of correlation functions at zero separation. We use the notation C(0) and C"(0) for the
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correlation function and its second derivative at the origin. The first derivative is zero by the evenness of the correlation function. Using these, we find that the contribution to the diagonal term from velocity correlations is DII = (c¢[30xC(x - y)¢3x + f l ~ x O x C ( x - y ) ) y =x = =
fi + I - > ( - I ( C " ( 0 ) + c(0)
x2)
+ I-><-IC"(o).
(2.29)
In addition, the time-development operator has contributions for the ordinary diffusion coefficient, D, and a leakage coefficient from pipe to pipe, 2. These coefficients are enhanced by the eddy contributions to give (2.30)
Oil = O e f f ~ 2 - [ - ) ( - [ 2 e f t / 2
with De#" = D + C ( 0 ) ,
2eff = 2 -- C " ( 0 ) / 2 .
The flow equations are easily described. Consider situations in which there is an input of heat at x = 0. The energy density, the sum of the temperatures in the two pipes obeys an equation precisely like Eq. (2.23). The flow is once more determined by eddy diffusion and the boundary conditions at infinity are necessary to set the coefficients of the zero modes. Once more, at least one coefficients is nonzero. The other equation is for T_, the difference in temperatures between the two pipes. If we have a flux entering the pipes at x = 0 and the difference between the two fluxes is F_*, then T_ obeys (2.31 )
(-D¢ff(Ox) 2 q- ;~¢ff) ( T _ ( x ) ) = F*_b(x) .
Here the flow between the two pipes produces exponentially increasing (or decreasing) terms. The boundary conditions far away make those terms quite small, leaving us with a solution localized near the heat source. ( T _ ( x ) ) = F *---~--~ exp(-lx[/~) - 2Deff
with
~ = v/De#/2e#.
(2.32)
Thus, the zero modes for the one-point function are directly determined by the boundary conditions.
3. Temperature correlations: Two-point functions 3.1. T w o p o i n t s a n d one pipe
We need the values of the operators D12 to calculate the two-point functions. A brief calculation shows that for the compressible model and the adjoint models we
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have D12 = C ( x l - x2)0102,
compressible model,
D12 = ~71~2C(Xl - x2),
adjoint model.
(3.1)
Specifically, if we have a Gaussian forcing, F, with zero range in time, Eq. (2.14) implies a two-point function which obeys (Dll ÷D22 ÷ 2 D 1 2 ) S 2 ( 1 , 2 ) = Z ( 1 , 2 )
(3.2)
with Z being the time-integral of the force correlation function. We can now put it all together and find an equation for the two-point function of the compressible model as [ A ( x - Y)(Ox
-
~y)2/4 ÷ Deff(~x
÷ Oy)2]S2(x, Y ) =
- Z(X - y ) .
(3.3)
Where the distance-dependent diffusion coefficient is A ( x - y ) = - C ( x - y ) + C(O) ÷ D .
(3.4)
This fourth-order equation requires four boundary conditions. One nice set of boundary conditions is to make S 2 ( x , y ) vanish whenever x or y is equal to ± B / 2 . One can certainly solves the problem with this boundary condition. However, one would expect that, when one is close to the source but far from the walls, the solution would depend almost entirely upon the distance from the source. Instead, we employ the boundary conditions decided in Section 2.5, while taking $2 to be a function of x - y. Thus, we replace Eq. (3.3) and its boundary conditions by the statement A(x)(~x)ZS2(x) = - Z(x)
while
(3.5)
lim S 2 ( x ) = 0.
(3.6)
X----+ OO
Eq. (3.5) was first derived and analyzed by Kraichnan [10]. In t h e inertial range, the space-dependent coefficient takes the form A ( x ) = - C ( x ) ÷ C(O) + D = C *
Ixl
+ D.
(3.7)
Here C(0) cancels the leading term in C ( x ) , leaving a remainder which have a simple scaling property. Let us analyze the behavior in the scaling region, using only the information which is available in that region. If we take the forcing term to be constant, Eq. (3.5) takes the form
c* Ix["(~x)2S2(x)=
-
z0,
(3.8)
which has the simple solution S2(x) = a + bx -
Z°lxl2-"
(1 - q ) ( 2 - r / ) C *
(3.9)
involving two undetermined coefficients a and b. In general, solutions of the homogenous equation can be called 'zero modes' since they are eigenfunctions of some
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linear operator with eigenvalue zero. The last term in Eq. (3.9) in the scaling range is produced as a power-law solution to the homogeneous equation. This kind of term might correspondingly be termed a 'scaling mode'. In the passive scalar case, and other cases too, it is easy to find the scaling modes by a kind of dimensional analysis argument. However, it is hard to generate all the zero terms and set all their coefficients. In particular, the coefficients must be set by a careful analysis of the effects of boundary conditions. Now let us analyze the two-point function in somewhat more detail. For simplicity, we use the model in which the forcing can be replaced by gu, so that Z(X-y) becomes g2C(x - y). We thus find, following Kraichnan, the exact solution
S2(x) =A + Bx - g2
// dy
0
dzC(z)/A(z) .
(3.10)
0
Once again the first two terms are the zero modes. We can eliminate B by noticing that all other terms in the equation are even in x while this one is odd. It cannot balance against anything else and must be zero. The A-term can be eliminated by the trick of considering the structure function instead of the correlation function. We find
dz2C(z)/A(z).
([T(x) - T(0)] 2) = 2Sz(0) - 2 S z ( x ) = 0
(3.11)
0
The zero-modes seem to have disappeared. However, we have not seen the last of them in this situation. Let us calculate the structure function in the scaling region. The leading term in the numerator on the right-hand side is given by taking C(x) to be C(0) which is approximately C'L". Now comes the integral over z. There are two contributions. One comes from taking z in the inertial range; the other comes from the dissipative range. The dissipative range integral gives a contribution to the structure function which is of the order of C(O)xl/D. The inertial range contribution from L ~ x ~ l gives a contribution to the structure function which is ( [ T ( x ) - T(0)] 2) ~
// dy
0
dz2C(z)/A(z) l
,-~(2/(l - tl))g2L" (~x]?~
xl'-") .
(3.12)
The final bracket in Eq. (3.12) is a sum of two terms. Note that the contribution from the dissipative range of z is of the same order as the second term in the parentheses. Therefore, Eq. (3.12) gives a correct estimate of the main behavior in the scaling range. If r/ is smaller than one the first term in the bracket dominates and we find a result in which the scaling mode dominates ([T(x) - r ( 0 ) ] 2) ~
2(g2x2)( 2 (L/x)"__
(3.13) q)
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L.P. Kadanoff et al. I Physica A 244 (1997) 190-212
Note the proportionality of the result to (gx) 2. Given a temperature with gradient g, this first factor is precisely the order of magnitude of variation of the squared temperature if we have somewhat imperfect mixing over the distance x. But our calculation shows that, in this situation, the RMS fluctuation in the temperature is actually enhanced over this 'default' value by the extra scaling factor (L/x)". It is just this kind of enhancement which makes the whole problem interesting, not only for this two temperature correlation but also in the context of higher-order correlations. We call this first term which was the direct result of the forcing a 'direct scaling term'. We can find such a term by doing a simple order-of-magnitude (or dimension counting) scaling analysis. In this kind of analysis, we demand that the left- and right-hand sides of Eq. (3.5) are of the same order of magnitude. Then the estimate of Eq. (3.13) pops fight out. Now look at the opposite case. Assume that q is greater than one. Then, the second term in Eq. (3.12) dominates. In this situation, we have the estimate fiT(x) - T(0)] 2) ,~ (g212)(L/l)~(x/l).
(3.14)
Thus, the main behavior of the structure function is given by the zero mode. The fluctuations are enhanced over the size one might have expected at the dissipative length, l, but they are not necessarily enhanced over the size we might have expected with mixing over a distance x. In this way, we see that the zero mode has returned to dominate the behavior in the situation in which r/ is greater than one. Thus, even this one-dimensional model, has some intricacy in its scaling behavior. I f r/is greater than one, then the structure function will scale as x, so that the scaling is set by a zero-mode. I f r/ is less than one, then the structure function will scale as x 2-~, so that the behavior is set by the scaling mode. This distinction is carried over to the higher-order correlations terms in the recent calculations [11-14] and all these authors conclude that zero modes dominate over the scaling modes. The zero modes are rather intricate, and they cannot be set by any easy dimensional argument. One wonders whether turbulent velocities will show a similarly delicate structure. In the calculation we have just done, we have used the dissipative terms in the equation to generate a mixture of different modes. The scaling mode has index 2 - r/, the zero modes have indices zero and one. Because the dissipative terms act at the characteristic distance l, the each mode in the sum has a contribution of the same order of magnitude when Ix - Yl has a value of this order. Then, as the difference variable increases into the scaling region, the term with the largest scaling index will grow the fastest, so that only the largest-index term will be important for scaling. Later on, we shall look at some higher-order correlation functions. If 2 - q is bigger than one, then the scaling mode will dominate in the short distance behavior. The reamining issue will be the satisfactor of the boundary conditions which apply at large differences, those of order L. Once again there will be a variety of modes which matter. But because the fits are at large distances, in these higher-order correlation functions, it is the term with the smallest value of the scaling index which will dominate the behavior.
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3.2. A d j o i n t model." scaling a n d zero modes
Precisely, this same analysis can be carried out for the adjoint model 1.6. Compared with the compressible model, the adjoint model has the same diagonal terms (Eq. (2.20)); but the interchange in the order of u and differentiation makes the offdiagonal operator different: DI2
=
~x(~yS(x
-
y),
(3.15)
so that the equation for the Tx correlator becomes, in the inertial regime, ~2C*lx - YI" S ( x - y)xx = Zxx(X - y ) .
(3.16)
Carry out the same analysis as before and find that there are once more three terms in the correlation function. In the inertial regime, the scaling mode is proportional to I x - yl 2-" just as before, while the zero modes are now I x - y [ - " and I x - yl 1-". Once again we match terms at large distances and identify the leading order behavior as the one which has the smallest power. So now, the leading behavior always comes from the zero mode and is S ( x - y)xx ~ Ix - yl -~ •
(3.17)
So do zero modes dominate even in the two-point correlator? This would be at variance with the standard wisdom and with part of the result in the case just analyzed. However, recall that the variable in this case is just the x-derivative of the variable in the previous case. Therefore, two derivatives of the solution of Eq. (1.5) should give the solution of Eq. (1.6). A comparison of solutions shows that this rule works just fine. Two integrals of the zero m o d e of Eq. (3.17) produces the scaling m o d e which is the last term in Eq. (3.9). The lesson: by rearranging an equation, we can convert zero-modes into scaling modes and vice versa. Some care is required. 3.3. The two-pipe case
Now let us return to the two-pipes case. We want to calculate two point correlations for this example. The correlation function of two forcings generates the function Z(X, ax; y, a y) There are two terms in this correlation, one independent of the tr's and the other proportional to their product. The same is true of the correlations, we thus write Z(X, tyx; y, t r y ) = Z+(x - y ) q- 6xfy~(-(x -- y ) ,
(3.18)
S2(x, ax; y, ay) = S ~ ( x - y ) + a x a y S f (x - y ) .
(3.19)
For simplicity, we once again use the model in which there is a linear gradient so that the velocity itself can drive the flow. In this situation, X+ is zero while Z- is g2C( X -- y).
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We can now write down the equations for the correlation functions. We find ~32x[C(x - y ) S 2 ( x - y)] = (C(O) + D)a2xS+(x - y ) ,
(3.20)
C(x - y)O2[S+(x - y)] = (C(O) + D)c?2x[S]-(x - y)]
+ (C"(0) - 22)$2(x - y) (3.21)
+ g2C(x - y ) .
Before we calculate the behavior of the two-pipe correlations, we should comment on the different terms in these equation. Notice again how the diffusion naturally built into the system, D is enhanced by the eddy diffusivity term C(0). This eddy diffusion is larger than the natural diffusion by a factor of (L/l)", which we take to be very large. Similarly, the leakage from pipe to pipe, which is naturally given by 2 is enhanced by the addition of the extra term - C " ( 0 ) . This second derivative is negative. Its value is set by the triangle inequality on the autocorrelation function of v. To see this setting define T
(3.22)
V(x) = v/1/2T / dtv(x, t) 0
so that we can write - C"(0) : (V(x)
V(x)) >~ l(V(x) V ( y ) ) [ - - ]C"(x -
Y)I •
(3.23)
According to our definitions, the last quantity reaches its maximum value (at least for the inertial range) at the difference I x - Yl at the dissipative value, l, so that - C t l ( O ) >/C* 1q-2
(3.24)
which can be very large for small values of l. In our previous analysis, the eddy diffusion term C(0) was very large, but it canceled out of the calculation of the correlation function. We might similarly expect - C " ( 0 ) to be large, and we might similarly hope that it cancels. Unfortunately, it will not do so. To establish that this term is large, we should compare it with the eddy diffusion term C*xq( d /dx ) 2 ~ C * x q- 2 .
(3.25)
This term is to be compared with 2 and - C ' ( O ) . The eddy diffusion term in Eq. (3.25) reaches its minimum at the integral scale L. Hence 2 and - C ' ( O ) will be effectively small if they are much less than C * L n-2. We can set 2 to achieve this smallness but, according to the estimate of Eq. (3.24), - C ' ( 0 ) is a factor of (L/l) 2-~ too large! Therefore, this term is dangerous. The dangerous term is one which permits fluid to flow from pipe to pipe. It is so big because we must have a certain amount of flow to produce the eddy diffusion. The danger is that this flow will be so large as to equalize the temperature in the two pipes and damp away all fluctuations.
L. P. Kadanoff et al. I Physica A 244 (1997) 190-212
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Return to Eqs. (3.20) and (3.21). We can solve them in part by substituting the expression for S+ from the first of these into the second. We find C(x)~Zx[C(x)S;(x)]/(C(O) + D) = (C(0) + D)O2[S~(x)] + ( C ' ( 0 ) - 22))$2(x) (3.26)
q- g Z C ( x ) .
There are, in effect, two large constants in this equation, C(0) and - C ' ( 0 ) . The former appears in C(x) since C(x) = C(0) + D - A(x), where A is not large. As before, we hold on to only the lowest order part of the forcing term on the right-hand side and use the smallness of A to simplify the equation to -A(x){~Zs~-(x)] + 82x[A(x)Sf(x)] - (C"(0) - 22)$2(x ) = 92C(0).
(3.27)
The last term on the left-hand side is much larger than the remaining terms on the left so that the solution becomes S2 (x) ~ 9212(L/I) n .
(3.28)
Thus, the fluctuations are small and independent of distance in the inertial range. Then Eq. (3.20) gives us that S+(x) = S f (x) + ax + b
(3.29)
which also seems quite uninteresting. So, for the moment, the two-pipe model looks discouraging. There is a way of rescuing the model. To do this, notice that all our trouble come from the very large value of - C ' ( 0 ) . If that would go away, the solutions would look very similar to the ones which have been analyzed by Kraichnan. But this constant is produced by behavior well within a non-physical region of very short-length correlations. Why do not we just redefine C(x - y) in this unphysical region and proceed blithely on'? We can do it. The trouble is that no real (as distinct from complex) velocity fields can have such correlations. However if u(x, t) is complex, albeit with real correlations, the triangle inequality does not apply and the model can be defined. There may be additional problems. Probability distributions for temperatures need not converge. Computed probabilities might turn out to be negative. However, correlation functions will make sense. Maybe the continued theory might offer some insight into some more real situation. Set - C " ( 0 ) equal to zero in Eq. (3.27). Then that equation will have the solution C(0) S ~ ( x ) = C*(2 + q ( r / - 1)) x2-~/ q- axP+ q- bxp
"
(3.30)
Here the exponents obey
(I P±=
-.)± gT-.~ 2
(3.31)
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To lowest order, Eq. (3.20) implies that the two correlation functions are equal, S+(x) = S f ( x ) . Both indices, p+ have a real part which is always smaller than 2 - ~/, for 2 > q > 0, so the scaling mode in S~(x) will dominate over the zero modes. Then S ] will have additional zero-modes linear and constant in x. The linear mode can dominate if r/ is greater than one. It would be interesting to see what happened to higher-order correlations in this situation.
4. Higher-order correlations 4.1. Third order Using the approach set up by Shraiman and Siggia [13] and following the calculation of [20], calculate an equation of motion for a third-order correlation function. The equation takes the form
5f 3S3(X, Y,Z) = F3(X, Y,Z) .
(4.1)
Since the result is translationally invariant, we can write the third-order correlation function as
S3(X, Y,Z) =s3(Y - X , Z - Y) =s3(x, y) .
(4.2)
We use the abbreviations x and y for the two difference variables in the correlation function. In the calculations performed below both x and y are taken to be positive. The time development operator may be expressed in terms of these difference variables, as
Lt'3 = A(x)(-Ox)(ax - Oy) + A(y)(Oy)(Ox - ay) + a ( x + y)(-Oxdy).
(4.3)
The equation for the forcing term is easily found by an expansion of the time-ordered products in the expression for the forcing term. The result is
ds
u ( X , t ) u ( X , s ) - ~ + u ( Y , t ) u ( Y , s ) - ~ + u(Z, tlu(Z,s)~-~2
kt--z
+2u(X,t)u(~,s)-g2- ~ + 2u(X,t)u(Z,s) +D
-~ t
t--T
+ ~
+ ~
~ + 2u(r,t)u(Z,s)-~-~-~
(T(X)T(YIT(Z))
L.P. Kadanoff et al. l Physica A 244 (1997) 190-212
211
t
t--"C t
+ u(Y,s) ~----~]I ( T ( X ) T ( Y ) ) . l--'f
This expression then gives us an evaluation of the force term which is
F3(x, y) = [C(x + y) - C(x)]S~(y) - [C(x + y) - C(y)]S~(x) - [C(x) - C(y)]S~(x + y ) .
(4.4)
Notice that this forcing term is odd under the interchange of x and y. Hence, it vanishes when these coordinates are equal. The third-order correlation function has the same properties. These can be derived by consider the joint effect o f flipping the sign o f the coordinate and o f interchanging T and - T . This invariance of the flow interchanges x and y and also changes the sign o f the correlation function. This boundary condition can be used in constructing the scaling form of the thirdorder correlation function. Write the correlation function as
s3(x, y) = rSdp(z)
(4.5)
where x - y
(4.6)
x+y and
r=(x + y).
(4.7)
If we match indices on the two sides o f Eq. (4.1) we find s=3 - q.
(4.8)
Then the derivatives translate to: 3x=
rOr + (1 - Z)Oz r
and
t3y=
rot - (1 + z)Oz
(4.9)
F
The equation o f motion for the scaling function is obtained as
[-z~+(s - 1 + (1 - Z)Oz)2Oz + z~_(s - 1 - (1 + z)d~)2Oz + (s - 1 - (1 + z)Oz)(s + (1 -- z)Oz)]c~(z) = const[(1 - z~+)z~" -- (1 - 2_)z~_-n - [z~ -- z n l l ,
(4.10)
where (z:e) is an abbreviation for ( l + z ) / 2 . Eq. (4.10) is to be solved with the symmetry condition that ~b(z) is an odd function of z. We need one additional condition. The
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authors o f this paper are hotly debating the best way to formulate that condition for the scaling function and the zero modes, given the known boundary conditions at infinity. 4.2. The n e x t steps
So the next step is the actual calculation o f the higher-order correlation functions. This work is not done yet. Nor is the scientific work on Ben Widom by any means at its culmination. So we leave both, in progress. Happy birthday, Ben.
Acknowledgements This research was supported in part by the N S F - D M R and the ONR. The work of Scott Wunsch was supported in part by a fellowship from the Fannie and John Hertz Foundation. We have profited very much from conversations on this subject with Boris Shraiman, Alan Kerstein, Itamar Procaccia, Omri Gat, Peter Constantin, Gregory Falkovich, and Saleh Tanveer. Some of this article is based on unpublished materials in a review by Boris Shraiman and Eric Siggia. We thank them for sending us this material prior to publication.
References [1] B. Widom, J. Chem. Phys. 43 (1965) 3892. [2] B. Widom, J. Chem. Phys. 43 (1965) 3898. [3] L.P. Kadanoff, Physics 2 (1966) 263. [4] A.N. Kolmogorov, CR. Acad. Sci. USSR. 30 (1941) 299. [5] A.N. Kolmogorov, J. Fluid Mech. 13 (1962) 82. [6] L. Kadanoff, D. Lohse, J. Wang, R. Benzi, Phys. Fluids 7 (3) (1995) 617-629. This paragraph is a paraphrase of the introductorysection in the referenced paper. Many earlier references are contained in this paper. [7] B.B. Mandelbrot, J. Fluid Mech. 62 (1974) 331. [8] U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995, reviews turbulence and particularly the Kolmogorov approach. [9] Boris I. Shraiman, Eric D. Siggia, Fluctuation and mixing of a passive scalar in a turbulent flow, preprint, 1996. [10] R.H. Kraichnan,Phys. Fluids 11 (1968) 945. [11] R.H. Kraichnan,Phys. Rev. Lett. 72 (1994) 1016; R.H. Kraichnan,V. Yakhut, Z.-S. Chen, Phys. Rev. Lett. 75 (1995) 240. [12] K. Gawedzki, A. Kupiainen, Phys. Rev. Lett. 75 (1995) 3844. [13] B.I. Shraiman, E.D. Siggia, C.R. Acad. Sci. (Paris) (1995). 279-321. [14] M. Chertkov, G. Falkovich, I. Kolokolov, V. Lebedev, Phys. Rev. E 52 (1995) 4924. [15] M. Chertkov, G. Falkovieh, V. Lebedev, Phys. Rev. Lett. 76 (1996) 3707. [16] R. Benzi, L. Biferale, G. Parisi, Physica D 65 (1993) 163. [17] L. Kadanoff, Phys. Today, September 1995, I1. [18] A.M. Obukhov, Izv. Akad. Nauk SSSR. Geogr. Geofiz. 13 (1949) 58; S. Corrsin, J. Appl. Phys. 22 (1951) 469. [19] A.R. Kerstein, Phys. Fluids A - Fluid Dyn. 3 (12) (1991) 2838. [20] Omri Gat, Victor L'vov, Evgenii Podiviloc, Itamar Procaecia, private communication.
Physica A 244 (1997) 190-212
Scaling properties of passive scalars in one dimension Leo P. Kadanoff*, Scott Wunsch, Tong Zhou The James Franck Institute, The University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
Abstract This paper is a set of notes about work in progress. Since the work is directed toward scaling, it seems quite appropriate to report this in a context devoted to Ben Widom's scientific contributions. Model equations for the motion of a passive scalar in one dimension are set up and some scaling properties of their solutions are derived. All these models follow the approach set up by Kraichnan in which the driving velocities have Gaussian correlations with scaling properties in space, but zero-range correlations in time. The simplest and most natural versions of the model fail to satisfy an incompressibility condition. In another version, the model does have an incompressible flow. To permit the incompressibility, there are two pipes - each of which is described by temperature and velocity fields which are functions of a single coordinate, x, and time. Flow from one pipe to the other transfers both mass and heat. This model has two conserved quantities in the limit of zero dissipation. In still another version, fluid elements are interchanged in position so that there are an infinite number of conservation laws in the nodissipation limit. Scaling properties of the resulting two-point functions are derived. P A C S : 47.27.GS; 05.40.+j Keywords: Isotropic turbulence; Random processes; Passive scalars
1. Introduction W e all congratulate Ben on his birthday and wish him many more to come. L P K adds a personal note: Ben W i d o m and I have shared some very exciting times. W e both worked on scaling in critical phenomena. B e n ' s ideas [1,2], which I further developed [3], proved crucial for carrying the field forward. I have enjoyed and profited from following in B e n ' s footsteps. His elegance in both scientific and personal style has been a major influence upon me. * Corresponding author. E-mail: [email protected]. 0378-4371/97/$17.00 Copyright ~) 1997 Elsevier Science B.V. All rights reserved. PH S0378-4371 (97)00239-2
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1.1. The problem
In recent years, ideas of scaling have been very broadly applied. Specifically, there has been a resurgence of applications to turbulence, in which the work of Kolmogorov [4,5], stands as the granddaddy of all scaling analysis, Kolmogorov's two papers prefigure the recent literature [6] which contains two alternative views of the nature of well-developed turbulence. In one view, the simple scaling caught by the 1941 paper of Kolmogorov [4] is the asymptotic truth which holds in the limit of high Reynolds numbers. Then the experimental facts, some of which seems to support a more complicated scaling, are described in terms of nonasymptotic corrections to scaling. In the other view [5], the experiments are better understood and described as a result of a multifractal picture [7] in which cascades produce anomalously large fluctuations in the velocity fields. These two views can both be supported by the experimental evidence. There are theoretical arguments for both. A good summary can be found in Frisch's book [8]. This same issue arises in a simplified problem in which one looks at the motion of a scalar quantity in a specified velocity field with well-defined spatial scaling properties. This passive scalar problem arises when some scalar physical quantity, e.g. temperature, is moved about by a fluid in motion. The word 'passive' means that the flow fluid is not affected by the motion of the scalar. Thus, the behavior of the temperature is defined by a linear problem. The produced distribution of the scalar can be very complicated, even if the underlying flow is rather simple. Recent studies of passive scalars have analyzed a particular case in which the velocity field is (1) (2) (3) (4)
stochastic, with a Gaussian probability distribution, incompressible, correlated only over very short times, having a zero mean, but with a power-law correlation in space over some wide range of scales.
A summary of recent work on this model can be found in a review paper by Shraiman and Siggia [9]. This model, first studied by Kraichnan [10], shows a very interesting behavior in which equal time structure functions formed from different powers of the temperature each show different scaling indices [11]. Furthermore, the scaling indices of the nth order structure function is not linear in n. This non-linear behavior is described by saying that the passive scalars show multiscalin9. The work on the indices is rather controversial in that recent calculations [12-14] show answers which seem to disagree with those of Ref. [11]. If we could see through to the source of that multiscaling, we could finally see who was right. The actual values of the indices do not matter much. The important difference between the two sets of calculations is that the work of Kraichnan and coworkers [11] suggest that the answers are universal, only weakly dependent upon the details of the model, while the other calculations, particularly Chertkov et al. [15] suggest a non-universal answer.
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These calculations might also have an impact upon our understanding of turbulence itself. If we could see how multiscaling arises in the passive scalar problem, we might better understand whether turbulence was likely to show multiscaling. One alternative is that the passive scalar multiscaling might arise from a essentially linear cascade in scales. The behavior of a linear chain of multiplicative and stochastic renormalizations has been studied in the GOY model [16]. The latter model does give multiscaling and some non-universality [6]. Another alternative is that the multiscaling might be a result of the infinite number of conservation laws which apply in the non-dissipative limit of the passive scalar. Each conserved quantity might be connected with its own critical index. Turbulence probably does have a linear cascade in scale; it probably does not have an infinite number of conservation laws. It certainly would be helpful to know how multiscaling might be connected with two, or an infinity of, conservations. There is another, and perhaps deeper, way in which the passive scalar problem might cast light upon the turbulence problem. One issue in turbulence is the way in which information is transferred among the different scales [17]. The ideas of the Kolmogorov's 1941 paper suggested that one consider the information available to an observer looking at inertial range behavior in some region of space and in some interval of distance scales. This information would be the average velocity of the region and its rate of addition of kinetic energy. The details of the dissipative processes and the value of the viscosity would be hidden from such an observer, essentially because there is an information flow from large scales to small. In the renormalization analysis of critical behavior the flow is in precisely the opposite direction: microscopic interactions produce macroscopic consequences. It is interesting to ask about the nature of that flow in a dynamic dissipative system. This paper is the start of an attempt to try to understand issues of information flow and multiscaling by looking at a simplified version of the passive scalar problem. The exact flow equations for a passive scalar are ( ~ t + U(r,t) • V ) T ( r , t ) = D V 2 T ( r , t )
+ F(r, t).
(1.1)
Here D is a diffusion coefficient and F is a random forcing term. To be realistic, we must take the flow to be incompressible. This requires the divergence condition
v.u=o.
(1.2)
Because of the divergence-free condition, the U term is anti-Hermetian. The diffusion term is Hermetian. The U-term can force a complicated pattern on T; the diffusion term tends to smooth and simplify that pattern. Here, we say a word about the structure of the equation used to study the problem. Like the previous workers [12-14] we use an equation called the H o p f equation which describes the time derivative of the equal time correlation function. It has the form (t)t - ~°)( T(1, t)T(2, t ) . . . T(n, t)) = F~(1,2 . . . . . n ) .
(1.3)
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Here the numbers 1,2 .... stand for space indices, and Fn is a term which describes the forcing. The time development operator, Lf depends only on the structure of the velocity correlations and does not depend upon either the structure of the forcing, or the initial conditions for the temperature, or upon the temperature boundary conditions. The forcing term on the right-hand side of the equation depends upon lower-order correlation functions. Thus, the Hopf equation provides a closed set of equations for successively determining the higher-order correlation functions. One can take several different approaches to the random forcing term, F. One method, followed by Kraichnan [10] and later workers is to take the forcing term to be a Gaussian random variable, with correlations of zero range in time, but having the force be completely independent of U. The other possibility, conceptually somewhat neater [13], is to imagine that the temperature field has an average behavior which includes a linear gradient in space, g. The T(r,t) in Eq. (1.1) is the deviation for the temperature from the linear gradient and the forcing term is F -- - g. U. These two possibilities lead to correlation functions which can be invariant under space and time translations. For most purposes, the scaling properties of these correlation functions are quite similar. Hence these two approaches mostly give the same conclusions. Another conceivable approach, less followed up to now, is to take the random forcing to be strictly zero but set some initial data for the temperature. For example, one can pick the temperature at t-----0 to be a delta function in space. Then one can follow the subsequent time development, using the known form of the average time development operator Lf. In this paper we shall follow all three possibilities in different calculations. We can summarize the recent scaling studies by imagining the form of the Hopf Eq. (1.3) in the time-independent situation. The equation then reads S ( T ( 1 ) T ( 2 ) - - • T(n)) = F , ( I , 2 .... n).
(1.4)
The classical analysis of passive scalars appears in the work of Obukhov and Corrsin [18]. Their results can be derived by looking at Eq. (1.4) in a scaling re-. gion in which all distances are of the order of r, and r stands in the interior of an inertial region. In this region, we notice that 5 a is a second-order differential operator which has coefficients which depend upon distance as a power law, r". Hence, the left-. hand side has .~ scale a s r (r/-2). Estimate the scaling of the right-hand side, and then divide by this estimate of L~ a. You will then find the classical form of the scaling of the correlation function. In contrast, the more recent papers [12-14], ask about the secondorder differential operator c~,. It has eigenstates, with eigenvalues zero, which then satisfy the homogeneous equation f ~ 9 = 0 . These eigenmodes scale differently from the Obukhov Corrsin result. Do they change the scaling of the correlation function? To answer that question, one must do the eigenmode analysis of Eq. (1.4). The particular linear combination of modes which enter is set by boundary conditions (See, for example, the discussion in [14].), so one must be careful to understand these boundary. conditions in some detail. Since there are many variables involved, the analysis of this equation for three-dimensional flow is quite difficult. A simpler analysis might be
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expected in lower dimensions or in the limit of infinite dimension [12]. Here we look to the simplest case: one dimension.
1.2. Models Of course the simplest extension of the passive scalar equation, Eq. (1.1), to one dimension is the exact transcription g2
(Ot + u(x,t)Ox)T(x,t) = D ~ x 2 T ( x , t ) + F(x,t)
Compressible Model.
(1.5)
In this equation we can no longer demand incompressibility, since that would make u independent of x. But in view of the incompressibility condition, one can put the derivative either to the right of or to the left of U in Eq. (1.1). If we write the equation with the derivative to the left, we have 02
dtTx(x,t)+3x(U(x,t)Tx(x,t))=Off-jTx(x,t)+Fx(x,t)
Adjoint Model.
(1.6)
We have written this equation with the symbols T~ and Fx used for the basic variable and the driving force, in place of the T and F which we have used up to now. That is because the model defined by Eq. (1.6) is not essentially different from the one defined by Eq. (1.5). To get Eq. (1.6) differentiate Eq. (1.5) with respect to x, and write Tx and Fx for, respectively, the x-derivatives of T and F. Neither of these one-dimensional equations (Eq. (1.5) or (1.6)) has an incompressibility condition. Hence, they both fail to have an anti-Hermetian convective term. It is possible to construct a one-dimensional flow which does have that condition, and consequently that flow displays the effects of anti-Hermeticity. Simply use two parallel one-dimensional pipes. Let there be a velocity field which drives a passive scalar in each of the pipes. The physical idea is to have any fluid which leaves one pipe and enters the other. The mathematical idea is to have the convection term in the equation for the temperature be an antisymmetric differential operator. (Since the operator will be real, it will be anti-Hermetian.) To do this, let the pipes be distinguished by an index, a, which takes on the values ±1. There is a specified velocity of the form uo(x, t) which describes the flow along the pipes. There is another velocity, or rather flux, v(x, t), which describes the rate (per unit length) of interchange of fluid between the two pipes. The equations of motion for the temperatures in the pipes, T~(x,t), are
-~ + ua(x,t) 02
T~(x,t) + ~v(x,t)[T+(x,t) - T_(x,t)]
= DO--~ T~(x, t) - 2(T,(x, t) - T_,(x, t)) + F~(x, t)
(1.7)
with D being a diffusion coefficient for flow along the pipes and 2 being a coefficient which describes a Newtonian rate of leakage of heat from one pipe to another.
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The conservation of fluid is represented by saying that any x-variation of u~(x, t) is produced by the flow of fluid from pipe to pipe. This gives us the condition that
~xU+(x,t)=v(x,t)= -
u_(x,t),
(1.8)
which integrates to the statement that
u~(x, t) = au(x, t),
(1.9)
plus a constant of integration independent of x. Take the constant of integration to be zero so that the velocity in the two pipes point in opposite directions. Notice that u and v are related by ~3
~xU= V .
(1.10)
In this way we have generated three separate first-order PDE models for the flow of a one-dimensional passive scalar. Can these models shed any light upon the issues related to more realistic flows of passive scalars in higher dimensions?
1.3. Conservation laws In the case of real 'passive scalars', as defined by Eqs. (1.1) and (1.2), spatial integrals of all powers of the temperature are conserved by the flow terms. Specifically, if Sp(t) is the space-integral of the pth power of the temperature then the conservation law reads
dSPdt - pD /
d r [ T ( r ) ] p - l ' ~ 2 T(r).
( I. 11 )
So, in the limit of D going to zero, there is some sort of conservation of the Sp's. Here, and in the remainder of this section, we neglect the direct effect of forcing terms upon the conservation laws. The two-pipe model has only two conservation laws which hold in the absence of dissipation (2 = D = 0). The two conserved quantities are the 'energy',
I4: f dx ~
r~(x,t)
(1,12)
6r
and the 'entropy'
s= fax y~ r ~ ( x , t ) 2 .
(l.13)
(7
One immediate consequence of the fluid conservation condition (1.8) is that the u and v terms together leave expression (1.12) unchanged in time. The resulting conservation law is the statement: dH/dt = 0. On the other hand, the second-order invariant, S, is
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diminished by dissipation. To derive the entropy condition, define an inner product ($,, q~) via ($,q~)= f a x
Z
Oa(x)d?a(x).
(1.14)
Or
The velocity term in Eq. (1.7) is the anti-Hermetian operator, U. We define this operator by giving its effect upon a vector ~b which depends upon x and a. Let ~ = U~b. Then ~k has the value c3u ~Or(x)=~ru(x,t) ~x q~a(x)+ 51 ~x(qS+(x) - q~_(x)).
(1.15)
Notice that, with this definition, U is antisymmetric (~, UqS)= - ( ( U $ ) , qS). Naturally, the diagonal element of this antisymmtric operator is zero, (T, U T ) = 0. This last condition then, gives a situation in which the convective terms conserve the entropy. The resulting entropy conservation law is then derived as
ldS l d -(T, T) = - (T, UT) -D(OxT, OxT) - ),(aT, a T ) . 2 dt 2 dt
(1.16)
Since U is anti-symmetric, the first term on the right-hand side vanishes, then the remaining terms in the time derivative are negative semi-definite: Real 'passive scalars', as defined by Eq. (1.1), have an infinity of conservation laws in the zero-dissipation .... ,m,." ilowever, in the two-pipe model integrals of higher powers of the temperature are not conserved by the flow terms. For example, the time derivative of the cube is d 2 ~ ( T , T) = - 3 ( T 2, UT) + dissipation terms = - 3 ( T 2, cru~T) - 3(T 2, ~(Ou)T) + dissipation terms. Here ~ is an operator which acts in the spin space and has the effect = ~b implies ~ ( x ) = [qS+(x) - ~b_ ( x ) ] / 2 .
Then we find d
2
~ ( T , T) =
/
dxv[T+(x) -- T ( x ) ] 2 [ T + ( x )
(1.17)
+ T_ (x)]/2 + dissipation terms, (1.18)
so that the flow process does not conserve the sum of cubes. More complicated versions of the same argument indicate that higher powers are not conserved either. Despite this weakness, we might hope that the model might have some features in common with the standard models of passive scalars. Besides, it would be very interesting to know what happens with just two conservation laws, The simple models described by Eqs. (1.5) and (1.6) have even fewer conserved quantities. The first model (Eq. (1.5)) gives no powers of T being conserved by the
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flow processes. The second (Eq. (1.6)) has the x-integral of T~(x,t) conserved by the flow, but that is apparently the only conservation law. Despite these paucities of conservation laws, we should ask about the properties of these models. To make them solvable, we pick a simple scaling behavior for v, e.g. Gaussian with zero time correlation range, and some x-scaling. Next, we solve for the temperature correlation functions. We can then ask: Do they show scaling? Do they have the large fluctuations in T that are associated with multiscaling? Do we have any hints of a universal behavior?
1. 4~ A rearrangement model We construct this model in the spirit of the work of Kerstein [19]. It is indeed possible to achieve a situation in which there are an infinite number of conservation laws. Imagine a situation in which we have a discrete labeling for the space variable. Thus, we have T(n,t) with n = 1,2 . . . . . perhaps with periodic behavior, with period L, or maybe with the shifted periodic behavior
T(n + L,t) = g L + T ( n , t ) .
(1.19)
We can have a discrete version of a dissipative process
T(n,t + 1)= T(n,t) + D[T(n - l,t) + T(n + l,t) - T(n,t)] .
(1.20)
One could also imagine rearrangement processes, which are the analog of flow processes in which we have
T(n+m,t+l)={T(n+q-m,t) T(n + m, t)
ifO
( 1.21 )
To get scaling, we choose the size of the re-arrangements at different time steps according to a power law. This rearrangement does not preserve any continuity in the temperature, but it does permit sums over n of all powers T(n,t) p to be conserved. Thus, if the conservation laws are crucially important this rearrangement model is an attractive candidate for study.
2. Correlation functions
2. I. Velocity correlations Gaussian correlations are defined by averages of the first and second moments of the Gaussian quantities. We take the average of u(x, t) to be zero. We further take the correlation between two velocities to have a zero range in time:
{u(x, t)u(y, s)) = 2c~(t - s)C(x - y ) .
(2.1)
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In what follows, it will be important that C ( x - y ) is an even function of its argument. In the usual approach, one would like the equal time velocity structure fimction defined by Su(x, y )
=
fds
u ( y , t ) ] [ u ( x , s ) - u ( y , s ) ] ) = 4 [ C ( 0 ) - C ( x - y)]
(2.2)
to grow as the power )1 of Ix - Yl in the inertial range. If S is a simple power, one could have an additive constant in the correlation function, viz., C ( x - y ) = C(O) - C ' i x - YI"
(2.3)
in the inertial range. In the previous work, t / w a s picked to be between zero and two. Since the triangle inequality requires the magnitude of C ( x - y ) to be smaller than C(0), it is then true that both C(0) and C* must be positive. Since we wish the correlation to fall to zero near the integral scale, L, we might as well define L by (2.4)
C(O) = C * L ~ .
The dissipative scale is defined by the condition that eddy diffusion and the ordinary diffusion match at that scale. Thus, l is defined by the condition D=C*
(2.5)
I~ .
Notice the structure of the correlation function in Eq. (2.3), in which the scaling term in the correlation function increases with increasing I values of r = [x - y[. This form is used because, in higher dimensions, this correlation is dominated by swirls of size comparable to r. These swirls have velocities point in different directions on different sides of the swirl. Larger swirls involve higher velocities. Thus, we have a negative term which increases with r. There is also no simple scaling. The big constant term, C(0) describes the summed effect on velocity squared of all sizes of swirls. That term dominates the term in q. In order to get a simple scaling one considers the structure function defined by Eq. (2.2). I f we did not have a time dependence this structure function would be proportional to the average squared fluctuations in the velocity difference [ u ( x ) - u(y)]. The structure function has a simple scaling behavior in the inertial range, l ~ I x - y [ ~ L . From Eqs. (2.2) and (2.3) we see that Su(x - y ) = 4 C * l x
- YI".
(2.6)
2.2. Force correlations
In doing our subsequent calculations, we shall need information about the force correlations. We take the force, F ( x , t ) to be a random Gaussian variable with zero mean and a vanishing correlation time (F(x, t ) F ( y , s)) = 2 6 ( t - s ) g ( x - y ) .
(2.7)
1This growth is almost unknown in the large body of scaling studies built around critical phenomena. In that case, the scaling terms typically decay with distance.
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If we have to make a specific choice of the correlation function, we shall choose it to be constant over the inertial range and be zero outside that range, specifically.
Z(x)
= ~ ;(o (0
if o < Ixl < L, otherwise.
(2.8)
We will use this simple forcing to drive our temperature problem. As we have said, one can alternatively drive the system by an overall temperature gradient, 9. Then the formulation of the problem is much the same as in the externally forced case. Instead of calculating correlations of a temperature variable, the correlations to be calculated are of the deviations of the temperature from its average value, 9x. For many purposes, this gradient-driven problem is that same as the externally forced problem except that the forcing correlation, Z is replaced by (2.9)
Z(X) ~ g 2 C ( x ) .
2.3. Hopf equations In every case, our equation of motion for the temperature is
~t T ÷ U(t)T= WT + F .
(2.10)
Here, U is a (differential) operator linear in the stochastic velocity field, W is a dissipation term, and F is a stochastic forcing. This equation has a formal solution T ( t ) = l- -
exp
oc
d s ( - g ( s ) + W)
.
(2.11)
+
In this equation, the subscript ' + ' is an indication that the integral is a time-ordered product in which operators referring to larger times appear to the left of those describing smaller times. In general, evaluating time-ordered products may be quite a mess, but because of the short-ranged nature of the time correlations, this particular operator is quite simple. We define correlation functions like S2(xa; y/~) = (T~(x)Tu(y))
(2.12)
as well as higher-order correlations. Here we have included the subscripts appropriate for the two-pipes, but we can do a much more general job by using the symbol Sn(1,2 .... ) with 1 and 2 and ... standing for the needed arguments. One of the contributions of Kraichnan [10] is to notice that one can set up simple equations for such correlation functions for the special case in which the velocities have Gaussian behavior and short-ranged correlations in time. The equations are derived by writing a formal solution of the linear equation of motion, Eq. (2.10). We then form the equal time version of the n-point correlation function, and calculate its time derivative, which
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must be in fact zero. The time derivative contains terms of zeroth and first order in the stochastic velocities. This term is n
5~(T(1,t)T(2,t)... T(n,t)) = E ([-U(j,t) + Wj]T(I,t)T(2, t)... T(n,t)). j=l
(2.13) The averages can be calculated quite generally [14], using the time-ordered product of Eq. (2.11) to get the time-development operator for our Hopf equation. This equation takes the form of Eqs. (1.3) and (1.4), in which the time-development operator has the form
~ = E Dij.
(2.14)
i,j
Here, the operators on the right-hand side are produced by correlations of the U's. Specifically, if Ui(t) and Uj(s) act on different variables so that they are in some sense in different subspaces, then they define the quantities on the left-hand side of Eq. (2.14) as
(Ui(t)Uj(s)) = 2 6 ( s - t)Dij
for i C j . .
(2.15)
Correspondingly, for the case in which the operators act on the same variables, we have
Wi÷~1 /ds(Ui(t)Ui(s))=Dii.
(2.16)
The story is completed by specifying the forcing terms. These are evaluated in terms of the same stochastic averages as went into the time-development operator. Specifically, the forcing term Fn of Eq. (1.3) is given by Fn(1,2 . . . . . n)=2)~(1,2)(T(3, t)T(4,t)... T(n,t)) + interchange terms.
(2.17)
This equation works whenever we have a prescribed Gaussian forcing given by F(1, t). If n is odd the forcing and the corresponding correlation function vanishes. A similar form applies when the forcing is given by a linear gradient 9 and the velocity field U. For even n, Eq. (2.17) still applies but the force F ( 1 , t ) is replaced by U(1)gxl so that in Eq. (2.17) Z(1,2) is replaced by g2C(1,2). In this case of internal forcing, the odd-order correlation functions are not zero. Instead they are given by F~(1,2 ..... n) = 2 f as((U(1,t)gx, )U(2,s))(T(2,t)T(3, t)... T(n, t)) + interchange terms.
(2.18)
2.4. A first solution The simplest calculation is for the average of a temperature at a single point. We do not include any force term. Instead we include the initial data that at time zero the
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temperature has the value T * 6 ( x ) . We can determine the subsequent behavior from the equation (c3t - Dll )(T(1, t)) = T * 6 ( x ) 6 ( t ) .
(2.19)
In the compressible model of Eq. (1.5), the diagonal contribution to the time development operator is easily evaluated as Dll = (D + C(0))(~3xf.
(2.20)
Thus, the flow produces an eddy diffusion, De(¢ = D + C(O)
(2.21)
in which the motion of the fluid enhances the diffusion so that the effective diffusion constant is increased by adding C(0). The solution to the heat flow problem stated in Eqs. (2.19) and (2.20) is the familiar form 1
(T(x, t)) = a + bx + - exp(-x2/(4tDe~.))T * v/4ntDeff
(2.22)
in which a and b are constants to be set by the boundary conditions. If we use the boundary condition that T vanishes at spatial infinity then we are required to set these constants to zero. So far, everything is easy. But this warmup exercise is too easy. Next try a force F(x, t) which is a delta function at x = 0 but has been in place forever. Thus, the equation is ( a, - De#(ax)2)(T(x, t)) = F* 6(x) .
(2.23)
We can take a time-independent solution of the form (T(x, t)) = a + bx - F *
[xl/(2Oe~).
(2.24)
Now we can no longer just ignore the 'zero modes', the ones with coefficient a and b. The magnitude of the solution certainly grows as one goes to infinity. One can no longer set the solution equal to zero there. Instead, one must be careful to set two boundary conditions, corresponding to the second-order nature of the problem. One could do this, for example, by saying that T ( x ) vanishes at the two points x = ± B/2, where B is some huge distance. Then we set the undetermined coefficients and find that the solution of Eq. (2.24) reads (T(x, t)) = F * (B/2 - ]x])/(2De~ ).
(2.25)
We can now see the reason that the zero mode must enter. The forcing term adds heat to the system. That heat cannot disappear, it must flow to infinity. Consequently, the solution must extend over a large distance. The adjoint model of Eq. (1.6) has identical equations for its one-point function. Hence, we need not discuss it separately. This model does not really have a conservation law for energy, i.e. the space integral of T~. But the one-point function behaves as if there were such a conservation law.
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2.5. Boundary conditions The use of a boundary condition in which the temperature vanishes at some far-away boundary of the system is perfectly workable but it does have one (mostly aesthetic) disadvantage. It produces correlation functions which are not translationally invariant. We can get the same result and have translational invariance in both space and time if we slightly modify the equation for the time derivative of the temperature by adding to the diffusion operator W = D V 2 a small negative constant so that Eq. (1.1) is replaced by (~
+ U(r,t). V) T(r,t)=(DV 2- e)V(r,t)+ F(r,t).
(2.26)
Here e is small and positive. If it is small enough it will not affect our answer. Once again, use the boundary condition that T vanishes at spatial infinity, both :kc¢. The new term destroys heat and reduces the temperture to zero far away. If the eddy conductivity term vanishes for large x, then in this region, the equation has two solutions, one exponentially growing the other exponentially decaying as x goes to infinity. The right solution can have only an exponentially decaying term. Using g, we introduce a cut off at very large scale to control infrared divergences. Now we can demand that the correlation functions be finite everywhere. This should provide enough boundary conditions to determine the solution. We hope and expect that the interesting parts of the solution, especially the scaling indices in the inertial range, are independent of ~. 2.6. The two-pipe model Precisely, the same analysis carries over to the two-pipe case. The only additional complication is the behavior in the two-component space of the variables which label the pipes. The operators of the Hopf equation depend upon some spin matrices. We specifically use ~ = I + ) ( - I and r = [-)(+[. In this notation, [+) is a projector onto the sum of temperatures from the two pipes and [-) is a projector onto the difference. In the equation of motion, the convection term is U = CtOxU+ 3UOx.
(2.27)
Since the two matrices ~t and fl are transposes of one another, and since the derivative is odd under transposition, U is an anti-Hermetian operator. Calculate the off-diagonal term, D12 in the Hopf equation. The result is a direct product of two matrices one in the spin space of x, the other in the spin space of y, so that D12 = flxflyC(x - y)axay -~- O~xO~yaxayC(x - y ) + flxOtyayC(x - y)ax
+ exflyOxC(x - y)0y.
(2.28)
The diagonal term is slightly tricky because one has to calculate derivatives of correlation functions at zero separation. We use the notation C(0) and C"(0) for the
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correlation function and its second derivative at the origin. The first derivative is zero by the evenness of the correlation function. Using these, we find that the contribution to the diagonal term from velocity correlations is DII = (c¢[30xC(x - y)¢3x + f l ~ x O x C ( x - y ) ) y =x = =
fi + I - > ( - I ( C " ( 0 ) + c(0)
x2)
+ I-><-IC"(o).
(2.29)
In addition, the time-development operator has contributions for the ordinary diffusion coefficient, D, and a leakage coefficient from pipe to pipe, 2. These coefficients are enhanced by the eddy contributions to give (2.30)
Oil = O e f f ~ 2 - [ - ) ( - [ 2 e f t / 2
with De#" = D + C ( 0 ) ,
2eff = 2 -- C " ( 0 ) / 2 .
The flow equations are easily described. Consider situations in which there is an input of heat at x = 0. The energy density, the sum of the temperatures in the two pipes obeys an equation precisely like Eq. (2.23). The flow is once more determined by eddy diffusion and the boundary conditions at infinity are necessary to set the coefficients of the zero modes. Once more, at least one coefficients is nonzero. The other equation is for T_, the difference in temperatures between the two pipes. If we have a flux entering the pipes at x = 0 and the difference between the two fluxes is F_*, then T_ obeys (2.31 )
(-D¢ff(Ox) 2 q- ;~¢ff) ( T _ ( x ) ) = F*_b(x) .
Here the flow between the two pipes produces exponentially increasing (or decreasing) terms. The boundary conditions far away make those terms quite small, leaving us with a solution localized near the heat source. ( T _ ( x ) ) = F *---~--~ exp(-lx[/~) - 2Deff
with
~ = v/De#/2e#.
(2.32)
Thus, the zero modes for the one-point function are directly determined by the boundary conditions.
3. Temperature correlations: Two-point functions 3.1. T w o p o i n t s a n d one pipe
We need the values of the operators D12 to calculate the two-point functions. A brief calculation shows that for the compressible model and the adjoint models we
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have D12 = C ( x l - x2)0102,
compressible model,
D12 = ~71~2C(Xl - x2),
adjoint model.
(3.1)
Specifically, if we have a Gaussian forcing, F, with zero range in time, Eq. (2.14) implies a two-point function which obeys (Dll ÷D22 ÷ 2 D 1 2 ) S 2 ( 1 , 2 ) = Z ( 1 , 2 )
(3.2)
with Z being the time-integral of the force correlation function. We can now put it all together and find an equation for the two-point function of the compressible model as [ A ( x - Y)(Ox
-
~y)2/4 ÷ Deff(~x
÷ Oy)2]S2(x, Y ) =
- Z(X - y ) .
(3.3)
Where the distance-dependent diffusion coefficient is A ( x - y ) = - C ( x - y ) + C(O) ÷ D .
(3.4)
This fourth-order equation requires four boundary conditions. One nice set of boundary conditions is to make S 2 ( x , y ) vanish whenever x or y is equal to ± B / 2 . One can certainly solves the problem with this boundary condition. However, one would expect that, when one is close to the source but far from the walls, the solution would depend almost entirely upon the distance from the source. Instead, we employ the boundary conditions decided in Section 2.5, while taking $2 to be a function of x - y. Thus, we replace Eq. (3.3) and its boundary conditions by the statement A(x)(~x)ZS2(x) = - Z(x)
while
(3.5)
lim S 2 ( x ) = 0.
(3.6)
X----+ OO
Eq. (3.5) was first derived and analyzed by Kraichnan [10]. In t h e inertial range, the space-dependent coefficient takes the form A ( x ) = - C ( x ) ÷ C(O) + D = C *
Ixl
+ D.
(3.7)
Here C(0) cancels the leading term in C ( x ) , leaving a remainder which have a simple scaling property. Let us analyze the behavior in the scaling region, using only the information which is available in that region. If we take the forcing term to be constant, Eq. (3.5) takes the form
c* Ix["(~x)2S2(x)=
-
z0,
(3.8)
which has the simple solution S2(x) = a + bx -
Z°lxl2-"
(1 - q ) ( 2 - r / ) C *
(3.9)
involving two undetermined coefficients a and b. In general, solutions of the homogenous equation can be called 'zero modes' since they are eigenfunctions of some
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linear operator with eigenvalue zero. The last term in Eq. (3.9) in the scaling range is produced as a power-law solution to the homogeneous equation. This kind of term might correspondingly be termed a 'scaling mode'. In the passive scalar case, and other cases too, it is easy to find the scaling modes by a kind of dimensional analysis argument. However, it is hard to generate all the zero terms and set all their coefficients. In particular, the coefficients must be set by a careful analysis of the effects of boundary conditions. Now let us analyze the two-point function in somewhat more detail. For simplicity, we use the model in which the forcing can be replaced by gu, so that Z(X-y) becomes g2C(x - y). We thus find, following Kraichnan, the exact solution
S2(x) =A + Bx - g2
// dy
0
dzC(z)/A(z) .
(3.10)
0
Once again the first two terms are the zero modes. We can eliminate B by noticing that all other terms in the equation are even in x while this one is odd. It cannot balance against anything else and must be zero. The A-term can be eliminated by the trick of considering the structure function instead of the correlation function. We find
dz2C(z)/A(z).
([T(x) - T(0)] 2) = 2Sz(0) - 2 S z ( x ) = 0
(3.11)
0
The zero-modes seem to have disappeared. However, we have not seen the last of them in this situation. Let us calculate the structure function in the scaling region. The leading term in the numerator on the right-hand side is given by taking C(x) to be C(0) which is approximately C'L". Now comes the integral over z. There are two contributions. One comes from taking z in the inertial range; the other comes from the dissipative range. The dissipative range integral gives a contribution to the structure function which is of the order of C(O)xl/D. The inertial range contribution from L ~ x ~ l gives a contribution to the structure function which is ( [ T ( x ) - T(0)] 2) ~
// dy
0
dz2C(z)/A(z) l
,-~(2/(l - tl))g2L" (~x]?~
xl'-") .
(3.12)
The final bracket in Eq. (3.12) is a sum of two terms. Note that the contribution from the dissipative range of z is of the same order as the second term in the parentheses. Therefore, Eq. (3.12) gives a correct estimate of the main behavior in the scaling range. If r/ is smaller than one the first term in the bracket dominates and we find a result in which the scaling mode dominates ([T(x) - r ( 0 ) ] 2) ~
2(g2x2)( 2 (L/x)"__
(3.13) q)
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Note the proportionality of the result to (gx) 2. Given a temperature with gradient g, this first factor is precisely the order of magnitude of variation of the squared temperature if we have somewhat imperfect mixing over the distance x. But our calculation shows that, in this situation, the RMS fluctuation in the temperature is actually enhanced over this 'default' value by the extra scaling factor (L/x)". It is just this kind of enhancement which makes the whole problem interesting, not only for this two temperature correlation but also in the context of higher-order correlations. We call this first term which was the direct result of the forcing a 'direct scaling term'. We can find such a term by doing a simple order-of-magnitude (or dimension counting) scaling analysis. In this kind of analysis, we demand that the left- and right-hand sides of Eq. (3.5) are of the same order of magnitude. Then the estimate of Eq. (3.13) pops fight out. Now look at the opposite case. Assume that q is greater than one. Then, the second term in Eq. (3.12) dominates. In this situation, we have the estimate fiT(x) - T(0)] 2) ,~ (g212)(L/l)~(x/l).
(3.14)
Thus, the main behavior of the structure function is given by the zero mode. The fluctuations are enhanced over the size one might have expected at the dissipative length, l, but they are not necessarily enhanced over the size we might have expected with mixing over a distance x. In this way, we see that the zero mode has returned to dominate the behavior in the situation in which r/ is greater than one. Thus, even this one-dimensional model, has some intricacy in its scaling behavior. I f r/is greater than one, then the structure function will scale as x, so that the scaling is set by a zero-mode. I f r/ is less than one, then the structure function will scale as x 2-~, so that the behavior is set by the scaling mode. This distinction is carried over to the higher-order correlations terms in the recent calculations [11-14] and all these authors conclude that zero modes dominate over the scaling modes. The zero modes are rather intricate, and they cannot be set by any easy dimensional argument. One wonders whether turbulent velocities will show a similarly delicate structure. In the calculation we have just done, we have used the dissipative terms in the equation to generate a mixture of different modes. The scaling mode has index 2 - r/, the zero modes have indices zero and one. Because the dissipative terms act at the characteristic distance l, the each mode in the sum has a contribution of the same order of magnitude when Ix - Yl has a value of this order. Then, as the difference variable increases into the scaling region, the term with the largest scaling index will grow the fastest, so that only the largest-index term will be important for scaling. Later on, we shall look at some higher-order correlation functions. If 2 - q is bigger than one, then the scaling mode will dominate in the short distance behavior. The reamining issue will be the satisfactor of the boundary conditions which apply at large differences, those of order L. Once again there will be a variety of modes which matter. But because the fits are at large distances, in these higher-order correlation functions, it is the term with the smallest value of the scaling index which will dominate the behavior.
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3.2. A d j o i n t model." scaling a n d zero modes
Precisely, this same analysis can be carried out for the adjoint model 1.6. Compared with the compressible model, the adjoint model has the same diagonal terms (Eq. (2.20)); but the interchange in the order of u and differentiation makes the offdiagonal operator different: DI2
=
~x(~yS(x
-
y),
(3.15)
so that the equation for the Tx correlator becomes, in the inertial regime, ~2C*lx - YI" S ( x - y)xx = Zxx(X - y ) .
(3.16)
Carry out the same analysis as before and find that there are once more three terms in the correlation function. In the inertial regime, the scaling mode is proportional to I x - yl 2-" just as before, while the zero modes are now I x - y [ - " and I x - yl 1-". Once again we match terms at large distances and identify the leading order behavior as the one which has the smallest power. So now, the leading behavior always comes from the zero mode and is S ( x - y)xx ~ Ix - yl -~ •
(3.17)
So do zero modes dominate even in the two-point correlator? This would be at variance with the standard wisdom and with part of the result in the case just analyzed. However, recall that the variable in this case is just the x-derivative of the variable in the previous case. Therefore, two derivatives of the solution of Eq. (1.5) should give the solution of Eq. (1.6). A comparison of solutions shows that this rule works just fine. Two integrals of the zero m o d e of Eq. (3.17) produces the scaling m o d e which is the last term in Eq. (3.9). The lesson: by rearranging an equation, we can convert zero-modes into scaling modes and vice versa. Some care is required. 3.3. The two-pipe case
Now let us return to the two-pipes case. We want to calculate two point correlations for this example. The correlation function of two forcings generates the function Z(X, ax; y, a y) There are two terms in this correlation, one independent of the tr's and the other proportional to their product. The same is true of the correlations, we thus write Z(X, tyx; y, t r y ) = Z+(x - y ) q- 6xfy~(-(x -- y ) ,
(3.18)
S2(x, ax; y, ay) = S ~ ( x - y ) + a x a y S f (x - y ) .
(3.19)
For simplicity, we once again use the model in which there is a linear gradient so that the velocity itself can drive the flow. In this situation, X+ is zero while Z- is g2C( X -- y).
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208
We can now write down the equations for the correlation functions. We find ~32x[C(x - y ) S 2 ( x - y)] = (C(O) + D)a2xS+(x - y ) ,
(3.20)
C(x - y)O2[S+(x - y)] = (C(O) + D)c?2x[S]-(x - y)]
+ (C"(0) - 22)$2(x - y) (3.21)
+ g2C(x - y ) .
Before we calculate the behavior of the two-pipe correlations, we should comment on the different terms in these equation. Notice again how the diffusion naturally built into the system, D is enhanced by the eddy diffusivity term C(0). This eddy diffusion is larger than the natural diffusion by a factor of (L/l)", which we take to be very large. Similarly, the leakage from pipe to pipe, which is naturally given by 2 is enhanced by the addition of the extra term - C " ( 0 ) . This second derivative is negative. Its value is set by the triangle inequality on the autocorrelation function of v. To see this setting define T
(3.22)
V(x) = v/1/2T / dtv(x, t) 0
so that we can write - C"(0) : (V(x)
V(x)) >~ l(V(x) V ( y ) ) [ - - ]C"(x -
Y)I •
(3.23)
According to our definitions, the last quantity reaches its maximum value (at least for the inertial range) at the difference I x - Yl at the dissipative value, l, so that - C t l ( O ) >/C* 1q-2
(3.24)
which can be very large for small values of l. In our previous analysis, the eddy diffusion term C(0) was very large, but it canceled out of the calculation of the correlation function. We might similarly expect - C " ( 0 ) to be large, and we might similarly hope that it cancels. Unfortunately, it will not do so. To establish that this term is large, we should compare it with the eddy diffusion term C*xq( d /dx ) 2 ~ C * x q- 2 .
(3.25)
This term is to be compared with 2 and - C ' ( O ) . The eddy diffusion term in Eq. (3.25) reaches its minimum at the integral scale L. Hence 2 and - C ' ( O ) will be effectively small if they are much less than C * L n-2. We can set 2 to achieve this smallness but, according to the estimate of Eq. (3.24), - C ' ( 0 ) is a factor of (L/l) 2-~ too large! Therefore, this term is dangerous. The dangerous term is one which permits fluid to flow from pipe to pipe. It is so big because we must have a certain amount of flow to produce the eddy diffusion. The danger is that this flow will be so large as to equalize the temperature in the two pipes and damp away all fluctuations.
L. P. Kadanoff et al. I Physica A 244 (1997) 190-212
209
Return to Eqs. (3.20) and (3.21). We can solve them in part by substituting the expression for S+ from the first of these into the second. We find C(x)~Zx[C(x)S;(x)]/(C(O) + D) = (C(0) + D)O2[S~(x)] + ( C ' ( 0 ) - 22))$2(x) (3.26)
q- g Z C ( x ) .
There are, in effect, two large constants in this equation, C(0) and - C ' ( 0 ) . The former appears in C(x) since C(x) = C(0) + D - A(x), where A is not large. As before, we hold on to only the lowest order part of the forcing term on the right-hand side and use the smallness of A to simplify the equation to -A(x){~Zs~-(x)] + 82x[A(x)Sf(x)] - (C"(0) - 22)$2(x ) = 92C(0).
(3.27)
The last term on the left-hand side is much larger than the remaining terms on the left so that the solution becomes S2 (x) ~ 9212(L/I) n .
(3.28)
Thus, the fluctuations are small and independent of distance in the inertial range. Then Eq. (3.20) gives us that S+(x) = S f (x) + ax + b
(3.29)
which also seems quite uninteresting. So, for the moment, the two-pipe model looks discouraging. There is a way of rescuing the model. To do this, notice that all our trouble come from the very large value of - C ' ( 0 ) . If that would go away, the solutions would look very similar to the ones which have been analyzed by Kraichnan. But this constant is produced by behavior well within a non-physical region of very short-length correlations. Why do not we just redefine C(x - y) in this unphysical region and proceed blithely on'? We can do it. The trouble is that no real (as distinct from complex) velocity fields can have such correlations. However if u(x, t) is complex, albeit with real correlations, the triangle inequality does not apply and the model can be defined. There may be additional problems. Probability distributions for temperatures need not converge. Computed probabilities might turn out to be negative. However, correlation functions will make sense. Maybe the continued theory might offer some insight into some more real situation. Set - C " ( 0 ) equal to zero in Eq. (3.27). Then that equation will have the solution C(0) S ~ ( x ) = C*(2 + q ( r / - 1)) x2-~/ q- axP+ q- bxp
"
(3.30)
Here the exponents obey
(I P±=
-.)± gT-.~ 2
(3.31)
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210
To lowest order, Eq. (3.20) implies that the two correlation functions are equal, S+(x) = S f ( x ) . Both indices, p+ have a real part which is always smaller than 2 - ~/, for 2 > q > 0, so the scaling mode in S~(x) will dominate over the zero modes. Then S ] will have additional zero-modes linear and constant in x. The linear mode can dominate if r/ is greater than one. It would be interesting to see what happened to higher-order correlations in this situation.
4. Higher-order correlations 4.1. Third order Using the approach set up by Shraiman and Siggia [13] and following the calculation of [20], calculate an equation of motion for a third-order correlation function. The equation takes the form
5f 3S3(X, Y,Z) = F3(X, Y,Z) .
(4.1)
Since the result is translationally invariant, we can write the third-order correlation function as
S3(X, Y,Z) =s3(Y - X , Z - Y) =s3(x, y) .
(4.2)
We use the abbreviations x and y for the two difference variables in the correlation function. In the calculations performed below both x and y are taken to be positive. The time development operator may be expressed in terms of these difference variables, as
Lt'3 = A(x)(-Ox)(ax - Oy) + A(y)(Oy)(Ox - ay) + a ( x + y)(-Oxdy).
(4.3)
The equation for the forcing term is easily found by an expansion of the time-ordered products in the expression for the forcing term. The result is
ds
u ( X , t ) u ( X , s ) - ~ + u ( Y , t ) u ( Y , s ) - ~ + u(Z, tlu(Z,s)~-~2
kt--z
+2u(X,t)u(~,s)-g2- ~ + 2u(X,t)u(Z,s) +D
-~ t
t--T
+ ~
+ ~
~ + 2u(r,t)u(Z,s)-~-~-~
(T(X)T(YIT(Z))
L.P. Kadanoff et al. l Physica A 244 (1997) 190-212
211
t
t--"C t
+ u(Y,s) ~----~]I ( T ( X ) T ( Y ) ) . l--'f
This expression then gives us an evaluation of the force term which is
F3(x, y) = [C(x + y) - C(x)]S~(y) - [C(x + y) - C(y)]S~(x) - [C(x) - C(y)]S~(x + y ) .
(4.4)
Notice that this forcing term is odd under the interchange of x and y. Hence, it vanishes when these coordinates are equal. The third-order correlation function has the same properties. These can be derived by consider the joint effect o f flipping the sign o f the coordinate and o f interchanging T and - T . This invariance of the flow interchanges x and y and also changes the sign o f the correlation function. This boundary condition can be used in constructing the scaling form of the thirdorder correlation function. Write the correlation function as
s3(x, y) = rSdp(z)
(4.5)
where x - y
(4.6)
x+y and
r=(x + y).
(4.7)
If we match indices on the two sides o f Eq. (4.1) we find s=3 - q.
(4.8)
Then the derivatives translate to: 3x=
rOr + (1 - Z)Oz r
and
t3y=
rot - (1 + z)Oz
(4.9)
F
The equation o f motion for the scaling function is obtained as
[-z~+(s - 1 + (1 - Z)Oz)2Oz + z~_(s - 1 - (1 + z)d~)2Oz + (s - 1 - (1 + z)Oz)(s + (1 -- z)Oz)]c~(z) = const[(1 - z~+)z~" -- (1 - 2_)z~_-n - [z~ -- z n l l ,
(4.10)
where (z:e) is an abbreviation for ( l + z ) / 2 . Eq. (4.10) is to be solved with the symmetry condition that ~b(z) is an odd function of z. We need one additional condition. The
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authors o f this paper are hotly debating the best way to formulate that condition for the scaling function and the zero modes, given the known boundary conditions at infinity. 4.2. The n e x t steps
So the next step is the actual calculation o f the higher-order correlation functions. This work is not done yet. Nor is the scientific work on Ben Widom by any means at its culmination. So we leave both, in progress. Happy birthday, Ben.
Acknowledgements This research was supported in part by the N S F - D M R and the ONR. The work of Scott Wunsch was supported in part by a fellowship from the Fannie and John Hertz Foundation. We have profited very much from conversations on this subject with Boris Shraiman, Alan Kerstein, Itamar Procaccia, Omri Gat, Peter Constantin, Gregory Falkovich, and Saleh Tanveer. Some of this article is based on unpublished materials in a review by Boris Shraiman and Eric Siggia. We thank them for sending us this material prior to publication.
References [1] B. Widom, J. Chem. Phys. 43 (1965) 3892. [2] B. Widom, J. Chem. Phys. 43 (1965) 3898. [3] L.P. Kadanoff, Physics 2 (1966) 263. [4] A.N. Kolmogorov, CR. Acad. Sci. USSR. 30 (1941) 299. [5] A.N. Kolmogorov, J. Fluid Mech. 13 (1962) 82. [6] L. Kadanoff, D. Lohse, J. Wang, R. Benzi, Phys. Fluids 7 (3) (1995) 617-629. This paragraph is a paraphrase of the introductorysection in the referenced paper. Many earlier references are contained in this paper. [7] B.B. Mandelbrot, J. Fluid Mech. 62 (1974) 331. [8] U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995, reviews turbulence and particularly the Kolmogorov approach. [9] Boris I. Shraiman, Eric D. Siggia, Fluctuation and mixing of a passive scalar in a turbulent flow, preprint, 1996. [10] R.H. Kraichnan,Phys. Fluids 11 (1968) 945. [11] R.H. Kraichnan,Phys. Rev. Lett. 72 (1994) 1016; R.H. Kraichnan,V. Yakhut, Z.-S. Chen, Phys. Rev. Lett. 75 (1995) 240. [12] K. Gawedzki, A. Kupiainen, Phys. Rev. Lett. 75 (1995) 3844. [13] B.I. Shraiman, E.D. Siggia, C.R. Acad. Sci. (Paris) (1995). 279-321. [14] M. Chertkov, G. Falkovich, I. Kolokolov, V. Lebedev, Phys. Rev. E 52 (1995) 4924. [15] M. Chertkov, G. Falkovieh, V. Lebedev, Phys. Rev. Lett. 76 (1996) 3707. [16] R. Benzi, L. Biferale, G. Parisi, Physica D 65 (1993) 163. [17] L. Kadanoff, Phys. Today, September 1995, I1. [18] A.M. Obukhov, Izv. Akad. Nauk SSSR. Geogr. Geofiz. 13 (1949) 58; S. Corrsin, J. Appl. Phys. 22 (1951) 469. [19] A.R. Kerstein, Phys. Fluids A - Fluid Dyn. 3 (12) (1991) 2838. [20] Omri Gat, Victor L'vov, Evgenii Podiviloc, Itamar Procaecia, private communication.