A closure method for random advection of a passive scalar

James P. Gleeson�

Applied Mathematics, Caltech, Pasadena, CA 91125

Email: gleeson@math.la.asu.edu,

telephone: (480) 965 2352,

fax: (480) 965 0461.

Revised January 27, 2000.

Abstract

A novel functional method is applied to calculate the statistics of a passive scalar in an isotropic

turbulent velocity �eld. The method yields asymptotic series expansions for small velocity correlation

time, from which approximate closure equations are derived. The closure method admits a diagram

expansion, and is implemented as a Mathematica program. Pad�e summation of the asymptotic series

yields accurate values for the e�ective di�usivity and gives formulas expressing the Lagrangian corre-

lation of the velocity in terms of the Eulerian correlation. The approximations compare very favorably

with numerical simulations of advection by a Gaussian velocity �eld.

�Current address: Department of Mathematics, Arizona State University, Tempe AZ 85287-1804.

1

1 Introduction

When a pollutant tracer or small amount of heat is transported (or advected) by a turbulent uid, it does

not signi�cantly a�ect the ow by its presence. Such a quantity is therefore called a passive scalar. Since

the advecting uid is in turbulent motion, it can only be described by its statistical characteristics, notably

the mean value of the uid velocity and the energy spectrum of the ow. It is to be expected that the

passive scalar will also require such a statistical representation. An outstanding problem in uid mechanics

is to describe the statistics of the scalar, given the velocity statistics.

In particular, there is considerable interest1{3 in the calculation of the mean scalar concentration, the

dispersion rate of tracers and the e�ective (or \turbulent") di�usivity. The e�ective di�usivity has been

examined by various workers using di�erent closure schemes, e.g., Roberts'4 application of Kraichnan's5

direct interaction approximation and the self-consistent theory of Phythian and Curtis.3 In this work, we

introduce a functional method which enables us to derive successively improving closure approximations.

We generate a diagram expansion similar to that employed by Dean et al.2 for time-independent Gaus-

sian velocity �elds, and demonstrate the interpretation of the diagram series as an asymptotic series in

the velocity correlation time, denoted by ��. Thus, �� is the characteristic time over which the velocity

remains appreciably correlated. The vanishing �� limit leads to a white-noise (delta-correlated) advecting

velocity, which was introduced by Kraichnan6 and has attracted considerable recent attention, particularly

in regard to the anomalous scaling of the scalar moments.7{10 Silant'ev11 notes the possibility of using an

iteration scheme to generate an expansion in �� for the long-term e�ective di�usivity, but highlights its

poor convergence in the physically relevant case ��u=l = O(1), i.e., when the velocity correlation time is on

the order of the eddy circulation time. (Here l is the integral scale of the turbulent velocity �eld and u is its

r.m.s. value). The importance of this case is highlighted by the available data on the velocity correlation

function in turbulent ows|see, for example, the experimental results of Comte-Bellot and Corrsin12 and

2

the numerical calculations of Sanada and Shanmugasundaram13 and McComb et al.14 We implement the

diagram rules as an algorithm, using the symbolic manipulation capabilities of Mathematica, and calculate

up to �ve terms in the �� power series for the long-term e�ective di�usivity. Silant'ev argues that such a

series is useful only for very short correlation times, but we show the corresponding Pad�e approximations

converge rapidly and give accurate results, even when �� is on the order of the eddy circulation time.

Predictions of tracer dispersion and pollutant spread are most easily expressed in terms of the La-

grangian velocity correlation, i.e., the correlation measured when following a uid particle (see Taylor,15

Csanady1). On the other hand, experimental data is almost always measured at �xed points, giving the

so-called Eulerian correlation functions (or spectra). A classical turbulence problem is the determination

of a connection between the Lagrangian and Eulerian correlation functions|see, for example, Sa�man's16

application of Corrsin's conjecture17 to obtain a quasi-normal type closure. This and related methods are

reviewed in the context of one-dimensional �elds by Davis.18 A simple generalization of the method out-

lined above allows us to write Pad�e-type approximations which give explicit expressions for the Lagrangian

velocity correlation in terms of the Eulerian correlation functions. We perform numerical simulations

of Gaussian velocity �elds and �nd the approximations for the e�ective di�usivity and the Lagrangian

correlation to be accurate for correlation times which are on the order of the eddy circulation time.

Consider the advection of a passive scalar �(x; t) (representing, say, pollutant concentration) by a

random, three-dimensional, incompressible velocity �eld u(x; t):

@

@t� + u � r� � �0r

2� = 0 (1)

where �0 is the molecular di�usivity. The velocity can be calculated by solving the Navier-Stokes equations

(at least in principle, if not often in practice), but for simplicity here we will suppose it to be a Gaussian

(i.e., multivariate normal) random function with known correlation hu�(x; t)u�(x0; t0)i. We will further

assume the velocity statistics to be homogeneous and isotropic.

3

If we take the initial condition for the passive scalar equation (1) to be

�(x; 0) = Æ(x); (2)

then � dx is the probability, for one realization (i.e., an experiment giving u(x; t)) of the turbulence, that

a marked particle which was at the origin at time t = 0 will be in the volume element dx at time t. The

average probability density h�(x; t)i is found by averaging over the velocity statistics. The dispersion or

mean-square displacement of the marked particles at a time t is a measurement of their distance from the

source at the origin and can be calculated as

D(t) =1

3

Zx�x� h�(x; t)i dx; (3)

with summation over repeated indices and integration over all of space. The quantity

�(t) =1

2

dD(t)

dt(4)

is often called the e�ective di�usivity or eddy di�usivity. These names arise from the use of this quantity

in di�usion equation approximations for turbulent advection.1

Averaging equations (1) and (2) yields the following equations for the mean concentration �(x; t) =

h�(x; t)i:

@

@t�� �0r

2� = �r � hu�i :

Evaluation of the term hu�i is complicated by the fact that � is a non-trivial functional of the random

velocity u. In the next section we outline a method for expanding hu�i as an in�nite series which may be

truncated to give integrodi�erential approximating equations for �(x; t). Having obtained equations for �

in this manner, we may calculate series approximations to the dispersion D(t) and the e�ective di�usivity

�(t) according to equations (3) and (4).

4

2 The Functional Derivative Closure

Exposition of our method is eased by transforming the governing equations to wavenumber space, i.e.,

writing them in terms of Fourier-transformed variables such as

�(k; t) =1

(2�)3

Z�(x; t)e�ik�xdx:

(Henceforth we consider only such Fourier-transformed variables and so can use the same symbol as in

physical space without ambiguity). The transform of equations (1) and (2) yields the following pair of

equations for �(k; t):

@

@t� + �0k

2� = �i

Z(k � p) � u(p; t)�(k � p; t) dp (5)

�(k; 0) = 1; (6)

which may be averaged as above to �nd equations for �(k; t) = h�(k; t)i:

@

@t�+ �0k

2� = �i

Z(k � p) � hu(p; t)�(k � p; t)i dp

�(k; 0) = 1: (7)

Again the diÆculty is the evaluation of the stochastically nonlinear term hu�i.

We introduce the functional derivative closure method (FDC hereafter, for further details see Ref. 19).

The goal is the representation of hu�i as an in�nite series, truncations of which give closed equations for

�. The FDC series is generated by repeated application of a theorem due to E. A. Novikov:20

Theorem 1 (Novikov, 1965) For Gaussian random functions u�(p; t) with zero mean, the cross-correlation

hu�(p; t)F [u]i may be computed as

hu�(p; t)F [u]i =

Z 1

0

dt1

Zdp hu�(p; t)u�(q; t1)i

�ÆF [u]

Æu�(q; t1)

�:

Here F stands for any functional of u.

5

In particular, taking F [u] = �(k� p; t), we have

hu�(p; t)�(k � p; t)i =

Z 1

0

dt1Q��(p; t� t1)

��(k� p; t)

Æu�(�p; t1)

�; (8)

where Q�� is de�ned by the relation for homogeneous stationary turbulence:

hu�(k; t)u�(p; t1)i = Æ(k+ p)Q��(k; t� t1):

For incompressible velocity with isotropic and stationary statistics, Q�� may be written21

Q��(k; �) =E(k)R(�; k)

4�k2

��� �

k�k�k2

�;

where E(k) is the usual energy spectrum and R(�; k) is the time correlation function of the velocity (with

R(0; k) = 1).

An aside on functional di�erentiation is in order here. The standard de�nition (e.g., Ref. 22) obeys the

rules of normal di�erentiation, i.e.,

Æ

Æf(x)fF1[f ] + F2[f ]g =

ÆF1[f ]

Æf(x)+ÆF2[f ]

Æf(x)

Æ

Æf(x)fF1[f ]F2[f ]g = F1[f ]

ÆF2[f ]

Æf(x)+ÆF1[f ]

Æf(x)F2[f ];

and

Æf(y)

Æf(x)= Æ(x� y):

In order to apply Novikov's theorem (8), a formal solution of (5) for �(k � p; t) is written as:

�(k � p; t) = e��0jk�pj2t � i

Z t

0

dt2 e��0jk�pj

2(t�t2)

Zdq (k � p� q) � u(q; t2)�(k� p� q; t2): (9)

This is simply the integral equation corresponding to the di�erential equation (5) with initial condition

(6). Taking the functional derivative and averaging yields��(k� p; t)

Æu�(�p; t1)

�= �ie��0jk�pj

2(t�t1)k� h�(k; t1)i

� i

Z t

0

dt2 e��0jk�pj

2(t�t2)

Zdq (k � p� q) �

�u(q; t2)

�(k � p� q; t2)

Æu�(�p; t1)

�: (10)

6

The second term on the right-hand side contains an integrand of form hu�F [u]i, and so it may be expanded

by applying Novikov's theorem again. If, instead, we ignore the second term and consider the �rst term on

the right-hand side of (10) as a �rst approximation to hÆ�=Æu�i, then Novikov's theorem (8) gives a �rst

approximation to (7):

@

@t�+ �0k

2� = �

Z t

0

dt1

Zdp (k� � p�)Q��(p; t� t1)k�e

��0jk�pj2(t�t1)�(k; t1); (11)

and we call this the FDC1 approximation to (7). Note this is a closed equation for �, the solution of which

gives an approximation to the exact average probability density. The correction to this approximation is

found by applying Novikov's theorem to the second term on the right-hand side of (10). This correction

involves a second order functional derivative, which is evaluated from the formal solution (9) and another

application of Novikov's theorem. Successive applications of Novikov's theorem to higher functional deriva-

tives of (9) enable us to approximate the right-hand side of (7) as a series of integrals over �, each weighted

by multiple factors of Q�� . The FDCn approximation is de�ned to be the truncation of this series which

contains the integrals over m factors of Q�� , for all m � n. Thus, the FDC2 approximation to (7) contains

7

integrals in which Q appears once and twice:

@

@t�(k; t) + �0k

2�(k; t) =

�

Z t

0

dt1

Zdp [(k� p) �Q(p; t� t1) � k] e

��0jk�pj2(t�t1)�(k; t1)

+

Z t

0

dt2

Z t2

0

dt1

Z t1

0

dt3

ZZdp dq

�[(k� p) �Q(p; t� t1) � (k� q)]

� [(k� p� q) �Q(q; t2 � t3) � k]

� e��0[jk�pj2(t�t2)+jk�p�qj

2(t2�t1)+jk�qj2(t1�t3)]

��(k; t3)

+

Z t

0

dt2

Z t2

0

dt3

Z t3

0

dt1

ZZdp dq

�[(k� p) �Q(p; t� t1) � k]

� [(k� p� q) �Q(q; t2 � t3) � (k� p)]

� e��0[jk�pj2(t�t2)+jk�p�qj

2(t2�t3)+jk�pj2(t3�t1)]

��(k; t1):

(12)

We have adopted the notation k �Q � p � k�Q��p�.

In the next section we introduce the diagram expansion, which allows us to concisely represent the

complicated approximation equations like (12) in terms of geometrical diagrams. This representation also

facilitates the implementation of the approximation scheme in Mathematica (see section 5).

3 Diagram expansions

Clearly the FDC approximation equations quickly become very complicated to write down. However, the

equations may be reproduced from a diagram expansion by applying a few simple diagram rules. Diagram

expansions are an accepted bookkeeping device for perturbation series.2, 23, 24 First, de�ne the diagrams

of order n to be n-polygons with dotted lines joining pairs of vertices. Then the FDC diagrams of order

n are those diagrams of order n which are connected, i.e., which cannot be split into two separate parts

by cutting one solid line. For example, the FDC diagrams of order 1 and 2 (which represent the FDC2

8

approximation (12) above) are shown in Fig. 1. The FDC equation is recovered from the diagrams by

applying the following diagram rules.

Consider the diagram

k-p

q p

k-p-q

k-q

t

t1 t2

t3

(13)

which represents the �rst FDC2 term

Z 1

0

dt2

Z 1

0

dt1

Z 1

0

dt3Htt2Ht2t1Ht1t3

ZZdp dq [(k� p) �Q(p; t� t1) � (k� q)]

� [(k� p� q) �Q(q; t2 � t3) � k] e��0[jk�pj2(t�t2)+jk�p�qj2(t2�t1)+jk�qj2(t1�t3)]�(k; t3): (14)

Here Hts represents the Heaviside function

Hts =

8>><>>:

1 if t > s,

0 otherwise

Observe that (14) may be produced from (13) by applying the following rules:

1. Vertex labels are the time integration variables, except for the �rst vertex (which is always labeled

t).

2. The time integral limits are determined by associating a factor of Htitj with the solid line joining

vertices labeled ti and tj .

3. The wavevector integration variables are the wavevectors labeling the internal dotted lines; these

integrals are over all wavevector space.

9

4. The vector sum of wavevectors at each vertex is zero, except for the �rst vertex (labeled t) which has

sum +k, and the �nal (circled) vertex which has sum �k.

To compose the integrand, we multiply the factors resulting from each of the following rules:

5. For each internal dotted line, consider the start and end vertices. For example, in (13) for the

internal dotted line labeled p, the start vertex is labeled t and the end vertex is labeled t1. Both

the start and the end vertex have solid lines emanating from them; suppose the wavevector labels

on these lines are a and b respectively. Then the factor we seek is �a � Q(p; t � t1) � b where p

is the dotted line label and t and t1 are the start and end vertex labels. (If the end vertex is the

circled vertex, then let b = k). In the example (13), a = k � p and b = k� q, so that the factor is

�(k�p) �Q(p; t� t1) � (k�q). By applying this rule again to the second dotted line, we �nd another

factor of �(k� p� q) �Q(q; t2 � t3) � k.

6. For each solid line joining ti to tj say, and labeled by a, multiply by a factor of exp���0jaj

2(ti � tj)�.

In the example (13) this gives us three factors of exp���0jk� pj2(t� t2)

�, exp

���0jk� p� qj2(t2 � t1)

�and exp

���0jk� qj2(t1 � t3)

�.

7. Finally, the circled vertex carries a factor of �(k; ti), where ti is the label of the circled vertex.

These rules form an algorithm for �nding the FDC equations and so may be implemented using a symbolic

manipulation program like Mathematica.

Recalling the de�nition (3) of the dispersion D(t), we �nd

D(t) = �(2�)31

3

@2

@k�@k��(k; t)

����k=0

: (15)

This relation allows us to �nd expressions for the dispersion (or more readily the e�ective di�usivity _D=2)

from the FDC evolution equations for �(k; t), for example (12). Taking the evolution equation for �(k; t),

10

applying the operator �(2�)3 13@2

@k�@k�, then setting k to zero (and noting �(0; t) = (2�)�3 since � is a

probability density) yields the FDC2 equation for the e�ective di�usivity �(t) = 12dDdt :

�(t) = �0 + �1(t) + �2(t); (16)

with

�1(t) =2

3

Z t

0

dt1

Zdp

4�p2E(p)R(t� t1; p) e

��0p2(t�t1);

and

�2(t) = �1

3

Z t

0

dt2

Z t2

0

dt1

Z t1

0

dt3

ZZdp dq

(4�pq)2

�pq�(�2 � 1)E(p)E(q)R(t � t1; p)R(t2 � t3; q)

� e��0[p2(t�t2)+(p2+q2+2pq�)(t2�t1)+q

2(t1�t3)]

+ 2p2(1� �2)E(p)E(q)R(t � t3; p)R(t2 � t1; q)e��0[p2(t�t2)+(p2+q2+2pq�)(t2�t1)+p

2(t1�t3)]�:

We denote by � the cosine of the angle between p and q, i.e., � = p � q=pq. Here �n(t) represents the new

term appearing in the FDCn approximation and (16) is the truncation at order two of the series

�(t) = �0 +

1Xn=1

�n(t): (17)

In Ref. 19 it is demonstrated that

�n(t) = O��2n�1�

�as �� ! 0;

where �� denotes the velocity correlation time, nondimensionalized by, say, the eddy circulation time. In

other words, the series (17) is an asymptotic series for small velocity correlation time. In section 5 we

consider truncations of this series|for example (16)|each of which yields a quadrature expression for the

e�ective di�usivity �(t), depending only on the energy spectrum E(k) and the time correlation function

R(�; k). The diagram rules for � are easily generalized to give a diagram-based algorithm for the FDC

series for �(t).

11

4 Interference of turbulent and molecular di�usion

In the absence of molecular di�usivity (�0 = 0) the dispersive e�ect denoted by D�0=0(t) is referred to as

pure turbulent di�usion. In general �0 > 0 and the molecular di�usivity interferes non-trivially with the

turbulent di�usivity. Sa�man25 has considered this question and by considering solutions of the passive

scalar equation on short time and length scales he demonstrates that

D(t) = D�0=0(t)�1

9�0t

3!2 +O(t4); (18)

for small times t. Here !2 is the mean-square vorticity. We consider the FDC expansion for the dispersion

in a similar form to (17):

D(t) =1Xn=1

Dn(t);

and for small t obtain

D1(t) = D�0=01 (t)�

2

9�0t

3

Z 1

0

dp p2E(p) +

+1

18t4��20

Z 1

0

dp p4E(p)� 2�0

Z 1

0

dp p2E(p)@R

@t

����t=0

�+O(t5) (19)

D2(t) = D�0=02 (t) +O(t5): (20)

Since (see, for example Batchelor21) Z 1

0

dk k2E(k) =1

2!2

and Z 1

0

dk k4E(k) =1

2[r�!]2;

12

we may rewrite (19) and (20) as

D1(t) = D�0=01 (t)�

1

9�0t

3!2 +

1

18t4��20

1

2[r�!]2 � 2�0

Z 1

0

dp p2E(p)@R

@t

����t=0

�+O(t5); (21)

D2(t) = D�0=02 (t) +O(t5): (22)

Evidently to O(t3) Sa�man's conclusion of destructive interference of the molecular di�usivity with the

turbulent di�usivity is con�rmed by this small-time expansion of the FDC series.

5 The long-term e�ective di�usivity

We now set the molecular di�usivity �0 to zero, in order to examine the mixing e�ects of the turbulent

velocity �eld. The long-term e�ective di�usivity is de�ned to be

�(1) � limt!1

�(t)

when the limit exists. As a �rst example, consider a velocity �eld with energy spectrum

E(k) =3

2u2Æ(k � k0) (23)

and time correlation function

R(t; k) = e�jtj=�� :

We nondimensionalize using a reference length k�10 and a reference time u�1k�1

0 (henceforth we use �� to

represent the nondimensional correlation time), and implement the diagram rules in Mathematica. The

delta-function spectrum reduces all wavevector integrals to angular integrals, while the time integrals may

be done separately and straightforwardly to yield the appropriate power of ��. Finally the angular integrals

are also performed on Mathematica. All integration is done analytically so the results for each diagram

13

are exact. The following is the expression for the nondimensional long-term di�usivity, correct to order �9�

(i.e. retaining terms up to, and including, �5(t) in (17)):

�(1) = ��

�1�

1

2�2� +

11

24�4� �

4061

7200�6� +

8775029

10080000�8� + : : :

�; (24)

Note the alternating signs of the coeÆcients, and the fact that all coeÆcients are O(1). This series does

not converge quickly (if at all), nor should we expect it to|the FDC method is justi�ed as a perturbation

method for small correlation time,19 so we must accept (24) as a possibly divergent asymptotic series.

5.1 Pad�e approximation

We are thus motivated to examine methods for summing perturbation series. One well-known method is

Pad�e approximation.26 Brie y, given a power series f(z) =P1

n=0 anzn, the Pad�e approximant PN

M (z) is a

rational function, i.e., a ratio of two polynomials, with numerator of degree N and denominator of degree

M , whose Taylor series agrees with f(z) for the �rst N +M +1 terms. To examine the Pad�e approximants

to (24) we let z = �2� , and consider the Pad�e approximants for the term in square brackets in (24):

P 01 (z) =

1

1 + 12z

P 11 (z) =

1 + 512z

1 + 1112z

P 12 (z) =

1 + 16611500z

1 + 24111500z +

259750z

2

P 22 (z) =

1 + 85146374351200z +

1848204152214400z

2

1 + 106902374351200 z +

5869186352214400z

2: (25)

From Fig. 2 we can see that the Pad�e approximants appear to be converging and doing so much more

rapidly than the basic series (24).

The convergence theory of Pad�e approximants is chie y based upon Stieltjes series, i.e., series of the

14

form1Xn=0

an(�z)n;

where the coeÆcients an are the moments of a real nonegative function �(s):

an =

Z 1

0

sn�(s)ds; �(s) � 0 (0 � s <1):

For these series it can be shown26, 27 that when z is �xed, PNN (z) decreases monotonically, PN

N+1(z) increases

monotonically, and PNN (z)! PN

N+1(z) as N increases. Although we cannot prove that our series (24) is a

Stieltjes series, the available Pad�e approximants do indeed have these properties. Thus we conjecture that

��P12 (�

2� ) provides a lower bound for the nondimensional e�ective di�usivity and that ��P

22 (�

2� ) provides

an upper bound, i.e.,

��P12 (�

2� ) � �(1) � ��P

22 (�

2� ): (26)

Since P 12 (�

2� ) and P 2

2 (�2� ) di�er by only about 1% even at �� = 1, we conclude that (26) gives an accurate

approximation for the long term e�ective di�usivity for (dimensional) correlation times as large as (uk0)�1.

5.2 More general spectra

Even when the energy spectrum does not have the simple delta-function form considered in (23) above,

the FDC integrals can always be reduced to multiple integrals over wavenumbers and time by doing all

the angular integrals exactly (see appendix). Moreover, for simple forms of the time correlation R, the

time integrals may also be done exactly, leaving just wavenumber integrals over the energy spectrum E(k).

Using the Mathematica program based on the diagram expansion, we have calculated the �rst few terms in

the FDC series for the long-term e�ective di�usivity for a variety of spectral shapes and time correlation

functions. Speci�cally, we list in Tables I and II results of the form (24) for the following energy spectra

15

and time correlation functions:

Ea(k) =3

2u2Æ(k � k0)

Eb(k) =4u2k4

�1

2 k50exp(�k2=k20)

Ec(k) =

8>><>>:

1

1�(1+�)�2

3

u2k2

3

0 k� 5

3 for k0 < k < (1 + �)k0,

0 otherwise

Ra(t; k) = exp(�!kjtj)

Rb(t; k) = exp(�!2kt

2)

Rc(t; k) = exp(�!kjtj) cos(2!kt)

where the \inverse correlation time" !k equals one of 1=��; k=�� or k2=��. Spectrum Eb is often used to

approximate the �nal stages of decaying turbulence and spectrum Ec models an inertial range. Each

spectrum is normalized so that Z 1

0

E(k)dk =3

2u2:

The time correlation function Ra makes the time integrals very simple; however, it is not di�erentiable at

t = 0 and so we include Rb as a more realistic model. Note that both the above time correlation functions

are positive for all t; we also consider the e�ect of negative loops in the time correlation function when we

use Rc.

Pad�e approximants like (25) for certain cases from Table I are plotted in Figs. 2 to 7. For each case

we calculate the integral length

l =3�

4

R10

k�1E(k)dkR10

E(k)dk

and plot the available approximants for values of the dimensionless correlation time �� running from zero

up to the (dimensionless) eddy circulation time lk0. In general there is not much di�erence between the

16

approximations resulting from correlations Ra and Rb (see Figs. 3 and 4), indicating that the shape of

the correlation function near t = 0 is not critical to the value of �(1). Note also that the convergence of

the approximations is improved when passing from !k = 1=�� to !k = k=�� to !k = k2=�� (compare Figs.

5 and 6). In all cases with time correlation Ra and Rb we �nd behavior of the Pad�e approximants which

we term Stieltjes-like, i.e., the power series coeÆcients alternate in sign, PNN decreases monotonically and

PNN+1 increases monotonically as N increases, with no poles of the approximants being on the positive real

axis. This leads us to conjecture that Pad�e approximants provide successively improving upper and lower

bounds on the long-term e�ective di�usivity (as they are known to do for Stieltjes series), when the time

correlation function is always positive. However, for time correlation Rc the sign pattern of the power

series coeÆcients violates the Stieltjes rule, and indeed we �nd a pole of the P 01 approximant for energy

spectrum Ea at �� = 1:96. Nevertheless the higher order Pad�e approximants still converge rapidly (see Fig.

7), so we can still �nd close approximations for the value of the long-term e�ective di�usivity, although

without the neat bounding behavior of the Stieltjes-like series. The accuracy of these approximations is

demonstrated by comparison with numerical calculations of the e�ective di�usivity in section 8.

6 Generalized Pad�e approximation for �(t)

We wish to generalize the ideas of the preceding section to the calculation of �(t), the e�ective di�usivity

at �nite time. Each diagram contribution can be calculated as before and now generates a function of t.

We alter the time variable to ~t = t=�� and change each integration variable to ~ti = ti=�� to pull a factor of

�2n�1� outside each FDCn integral. The generalization of (24) then has the form

�(t) = ���1(t=��) + �3��2(t=��) + �5��3(t=��) + : : : (27)

17

with �n(1) being the FDCn contribution to the long-term e�ective di�usivity as calculated previously. For

example, for the energy spectrum Ea(k) we know from (24) that �1(1) = 1; �2(1) = � 12 and �3(1) = 11

24 .

Now we generate the generalized Pad�e approximants to (27) by treating each �i(t=��) as if it were a constant

coeÆcient in a �� power series. Thus we approximate �(t)=�� by

P 01 (�

2� ; t) =

�1(t=��)

1� �2(t=��)�1(t=��)

�2�;

P 11 (�

2� ; t) =

�1(t=��)��1(t=��)�3(t=��)��

2

2(t=��)

�2(t=��)�2�

1� �3(t=��)�2(t=��)

�2�; (28)

and so on. The quadrature expressions for �1(~t), �2(~t) and �3(~t) are listed in the appendix.

In Fig. 8 we �x �� = 1 and plot the approximations ��P01 (�

2� ; t) and ��P

11 (�

2� ; t) to �(t) as functions of

the time t. The generalized Pad�e approximants are found to give good approximations for all t and for ��

up to order one, i.e., for dimensional correlation times on the order of the eddy circulation time.

7 The Lagrangian correlation

Having introduced the generalized Pad�e approximation to �(t) for �� of order one, we go a step further

and consider how this method can approximate the Lagrangian correlation for isotropic turbulence

L(t) =1

3hu�(x0; 0)u�(r(t); t)i ;

where r(t) is the position vector of the uid particle which was at x0 at time 0, i.e.,

d

dtr(t) = u(r(t); t)

r(0) = x0: (29)

The Lagrangian correlation is a very useful quantity in studies of turbulent di�usion and particle

dispersion,1 but for experimental ows the Eulerian correlation (which is measured at two points �xed

18

in space, instead of following a uid particle as for L(t)) is much easier to measure. It is therefore of

considerable interest to investigate whether a connection can be made between the Lagrangian correlation

and the known Eulerian correlation.

It follows from the de�nition of the Lagrangian correlation that

L(t) =d

dt�(t) =

1

2

d2

dt2D(t): (30)

As we have already detailed an accurate approximation scheme for �(t), it is straightforward to apply it

to L(t). We take the series of FDCn integrals (27) for �(t):

�(t) = ���1(t=��) + �3��2(t=��) + : : :

and di�erentiate with respect to t to get a series for L(t):

L(t) =d

dt�(t) = �01(t=��) + �2��

02(t=��) + : : : (31)

As each of the �i(~t) functions is produced from repeated time integrals, the derivative �0i(~t) reduces to an

integral of one dimension less than �i(~t). For example, the FDC2 term �2(~t) is de�ned by (see appendix):

�2(~t) = �

Z 1

0

dk1

Z 1

0

dk2

Z ~t

0

d~t1

Z ~t1

0

d~t2

Z ~t2

0

d~t3 E(k1)E(k2)4k219eR(~t� ~t3; k1) eR(~t1 � ~t2; k2);

where eR(~t; k) = R(��~t; k). This can be put in a more convenient form by employing the change of variables

s1 = ~t� ~t1; s2 = ~t1 � ~t2; s3 = ~t2 � ~t3 to obtain:

�2(~t) = �

Z 1

0

dk1

Z 1

0

dk2

Z ~t

0

ds1

Z ~t�s1

0

ds2

Z ~t�s1�s2

0

ds3E(k1)E(k2)4k219eR(s1 + s2 + s3; k1) eR(s2; k2)

and then the di�erentiation is easy to perform:

�02(~t) = �

Z 1

0

dk1

Z 1

0

dk2

Z ~t

0

ds1

Z ~t�s1

0

ds2 E(k1)E(k2)4k219eR(~t; k1) eR(s2; k2)

= �

Z 1

0

dk1

Z 1

0

dk2

Z ~t

0

d~t1

Z ~t1

0

d~t2 E(k1)E(k2)4k219eR(~t; k1) eR(~t1 � ~t2; k2):

19

From (31) we generate the generalized Pad�e approximants for L(t) as we did above for �(t). In the

cases we have calculated these converge remarkably quickly, so that P 01 (�

2� ; t) and P 1

1 (�2� ; t) give very good

approximations even at �� = 1. It is therefore worth explicitly recording the approximations these give for

L(t) in terms of the Eulerian correlation functions:

L(t) = L1(t=��) + �2�L2(t=��) + : : : (32)

with

L1

�~t�=

2

3

Z 1

0

dk1E(k1) eR(~t; k1)L2

�~t�= �

Z 1

0

dk1

Z 1

0

dk2

Z ~t

0

d~t1

Z ~t1

0

d~t2 E(k1)E(k2)4k219eR(~t; k1) eR(~t1 � ~t2; k2);

and the Pad�e approximants to L(t) are then

P 01 (�

2� ; t) =

L1(t=��)

1� L2(t=��)L1(t=��)

�2�

P 11 (�

2� ; t) =

L1(t=��)�L1(t=��)L3(t=��)�L

2

2(t=��)

L2(t=��)�2�

1� L3(t=��)L2(t=��)

�2�(33)

In the next section we generate a random velocity �eld with a given Eulerian spectrum and advect

particles according to (29). These numerical simulations show that the FDC-Pad�e approximations to the

Lagrangian correlation are indeed accurate for �� of order 1, i.e., for dimensional correlation times on the

order of the eddy circulation time (see Fig. 11).

8 Numerical simulation of advection by a random velocity �eld

We create a random velocity �eld with prescribed statistics and follow uid particles as they advect,

recording the statistical quantities for comparison with the theory of the previous sections. We consider

20

Gaussian velocity �elds with prescribed energy spectra and time correlation functions. The velocity �eld

is generated using a method based on that used by Kraichnan.28 In each realization, we set

u(x; t) = ANXn=1

fzn cos [kn � x+ !nt] + yn sin [kn � x+ !nt]g :

To ensure incompressibility, we have

zn = kn � an and yn = kn � bn;

with an and bn chosen from independent Gaussian distributions. The frequencies !n are chosen from a

random distribution to produce the desired time correlation function R(t; k). For example, a Gaussian dis-

tribution with standard deviation 1=�� results in the time correlation function Rb, with inverse correlation

time !k = 1=��. The vectors kn are chosen from a distribution shaped so as to produce the desired energy

spectrum E(k): for Ea(k) the kn are isotropically distributed on a sphere of radius k0, whereas for Eb(k)

each component of kn is selected from independent Gaussian distributions of standard deviation k0. The

amplitude A is chosen so that

hu�(x; t)u�(x; t)i = 2

Z 1

0

E(k)dk

= 3u2;

so for Ea, A = (3=2N)1=2

uk�10 and for Eb, A = (1=N)

1=2uk�1

0 . The number of modes N is taken to be

100. In Fig. 9 we plot the average over 2000 realizations of u�(x; t)u�(x+ r; t) as a function of r = jrj for

the spectrum Ea(k)|this is compared to the exact correlation function which is

hu�(x; t)u�(x+ r; t)i = 2

Z 1

0

Ea(k)sin(kr)

krdk

= 3u2sin(k0r)

k0r:

Having obtained a satisfactory velocity �eld, we proceed to follow uid particles as they are advected

21

by:

d

dtr(t) = u(r(t); t)

r(0) = 0: (34)

In each realization, (34) is solved by using a fourth-order predictor-corrector scheme due to Hamming,29

with starting values formed by iteration of Newton's interpolation formula.30 A time step of 0:2 was

found to be satisfactory, and each uid particle was advected for up to 75 steps (i.e., for a dimensional

time t = 15u�1k�10 ). The numerical approximations for the e�ective di�usivity �(t) and the Lagrangian

correlation L(t) are then calculated from

�(t) =1

3Nr

NrXi=1

r(i)(t) � u(i)(r(i)(t); t);

L(t) =1

3Nr

NrXi=1

u(i)(0; 0) � u(i)(r(i)(t); t): (35)

9 Numerical results

In Figs. 10 and 11 the statistical quantities (35) are compared to the corresponding FDC-Pad�e approxi-

mations which are calculated as described in sections 6 and 7. The number of realizations, Nr, is recorded

in the legend. The 95% con�dence intervals are marked as error bars. Figure 10 uses energy spectrum Ea

and time correlation Rb with !k = 1=�� = (3=2)1

2 . In Fig. 11 we use the spectrum Eb and time correlation

Rb with !k = k=21

2 . Thus the space-time correlation function is

E(k)R(t; k)

4�k2=

u2k2

�3

2 k50exp(�k2=k20) exp(�

1

2u2k2t2);

which was also used by Sa�man.16 Sa�man employs Corrsin's17 conjecture to obtain a quasi-normal type

closure and further assumes that the mean scalar \cloud" has a Gaussian pro�le. This results in a nonlinear

22

di�erential equation for the dispersion which must be solved numerically. The FDC-Pad�e approximations

are very close to the results of Sa�man's approximation|they are almost indistinguishable in Figs. 10 and

11. We stress again that the FDC-Pad�e approximations are explicit quadrature expressions, in contrast to

Sa�man's di�erential equation which requires numerical solution.

It is clear that the FDC-Pad�e approximations are extremely good, even for values of the correlation

time �� of order one. In each case the second approximation P 11 gives a small but de�nite improvement

over the �rst approximation P 01 . Even the lowest order Pad�e approximants give reasonable results which

means the lowest order FDC diagrams are all that need to be calculated in order to closely approximate

the e�ective di�usivity and Lagrangian correlation using the methods of sections 6 and 7.

10 Conclusion

We have demonstrated a systematic method for expanding the stochastically nonlinear term which arises in

problems of advection by a Gaussian velocity �eld. Our major results are the approximating equations for

the average probability density (12), the e�ective di�usivity (16), and the Lagrangian correlation (32). The

approximations are valid for small velocity correlation time, and are extended to physically relevant times

by use of Pad�e approximants. The extension of these results to non-Gaussian velocity �elds is currently

under investigation.

11 Acknowledgements

Discussions with Professor P. G. Sa�man and Professor D. I. Pullin are gratefully acknowledged. This re-

search was partially supported by the Fulbright Commission and a National University of Ireland Traveling

Studentship in Mathematical Physics.

23

A Appendix

When the molecular di�usivity is zero, the diagram integrals de�ned in section 3 may be reduced to multiple

integrals over wavenumbers and time by doing all the angular integrals exactly. We list here the results for

the e�ective di�usivity up to order three, as de�ned in equation (16).

�1(~t) =2

3

Z 1

0

dk1

Z ~t

0

d~t1E(k1) eR(~t� ~t1; k1); (36)

�2(~t) = �

Z 1

0

dk1

Z 1

0

dk2

Z ~t

0

d~t1

Z ~t1

0

d~t2

Z ~t2

0

d~t3E(k1)E(k2)4k219eR(~t� ~t3; k1) eR(~t1 � ~t2; k2); (37)

�3(~t) =

Z 1

0

dk1

Z 1

0

dk2

Z 1

0

dk3

Z ~t

0

d~t1

Z ~t1

0

d~t2

Z ~t2

0

d~t3

Z ~t3

0

d~t4

Z ~t4

0

d~t5E(k1)E(k2)E(k3)

�

��4k21k

23

135eR(~t� ~t3; k1) eR(~t1 � ~t4; k2) eR(~t2 � ~t5; k3)

�4k21k

22

135eR(~t� ~t3; k1) eR(~t1 � ~t5; k2) eR(~t2 � ~t4; k3)

�4k21k

23

135eR(~t� ~t4; k1) eR(~t1 � ~t3; k2) eR(~t2 � ~t5; k3)

�8k21k

22

135eR(~t� ~t4; k1) eR(~t1 � ~t5; k2) eR(~t2 � ~t3; k3)

+8k4127

eR(~t� ~t5; k1) eR(~t1 � ~t2; k2) eR(~t3 � ~t4; k3)

+8k4127

eR(~t� ~t5; k1) eR(~t1 � ~t3; k2) eR(~t2 � ~t4; k3)

+8k4127

eR(~t� ~t5; k1) eR(~t1 � ~t4; k2) eR(~t2 � ~t3; k3)

+8k21k

22

27eR(~t� ~t5; k1) eR(~t1 � ~t4; k2) eR(~t2 � ~t3; k3)

�: (38)

Di�erentiation of these formulas with respect to time (see (31)) gives the approximation (32) for the

Lagrangian correlation L(t).

24

References

[1] G. T. Csanady, Turbulent di�usion in the environment, (D. Reidel Publishing Company, 1973).

[2] D. S. Dean, I. T. Drummond and R. R. Horgan, \Perturbation schemes for ow in random media,"

J. Phys. A 27, 5135 (1994).

[3] R. Phythian and W. D. Curtis, \The e�ective long-term di�usivity for a passive scalar in a Gaussian

model uid ow," J. Fluid Mech. 89, 241 (1978).

[4] P. H. Roberts, \Analytical theory of turbulent di�usion," J. Fluid Mech. 11, 257 (1961).

[5] R. H. Kraichnan, \The structure of isotropic turbulence at very high Reynolds numbers," J. Fluid

Mech. 5, 497 (1959).

[6] R. H. Kraichnan, \Small-scale structure of a scalar �eld convected by turbulence," Phys. Fluids 11,

945 (1968).

[7] T. Elperin, N. Kleeorin and I. Rogachevskii, \Turbulent thermal di�usion of small inertial particles,"

Phys. Rev. Lett. 76, 224 (1996).

[8] D. Bernard, K. Gawedzki and A. Kupiainen, \Anomalous scaling in the N-point functions of a passive

scalar," Phys. Rev. E 54, 2564 (1996).

[9] M. Chertkov and G. Falkovich, \Anomalous scaling exponents of a white-advected passive scalar,"

Phys. Rev. Lett. 76, 2706 (1996).

[10] T. Elperin, N. Kleeorin and I. Rogachevskii, \Anomalous scalings for uctuations of inertial particles

concentration and large-scale dynamics," Phys. Rev. E 58, 3113 (1998).

25

[11] N. A. Silant'ev, \Comparison of methods for calculating turbulent di�usion coeÆcients," Sov. Phys.

JETP 84, 479 (1997).

[12] G. Comte-Bellot and S. Corrsin, \Space-time correlation measurements in isotropic turbulence," J.

Fluid Mech. 48, 273 (1971).

[13] T. Sanada and V. Shanmugasundaram, \Random sweeping e�ect in isotropic numerical turbulence,"

Phys. Fluids A 4, 1245 (1992).

[14] W. D. McComb, V. Shanmugasundaram and P. Hutchinson, \Velocity-derivative skewness and two-

time velocity correlations of isotropic turbulence as predicted by the LET theory," J. Fluid Mech.

208, 91 (1989).

[15] G. I. Taylor, \Di�usion by continuous movements," Proc. Lond. Math. Soc. A 20, 196 (1921).

[16] P. G. Sa�man, \An approximate calculation of the Lagrangian auto-correlation coeÆcient for station-

ary homogeneous turbulence," Appl. Sci. Res. A 11, 245 (1961).

[17] S. Corrsin, \Progress report on some turbulent di�usion problems," Adv. Geophys. 6, 161 (1959).

[18] R. E. Davis, \On relating Eulerian and Lagrangian velocity statistics: single particles in homogeneous

ows," J. Fluid Mech. 114, 1 (1982).

[19] J. P. Gleeson, Random advection of a passive scalar, Ph.D. thesis, (California Institute of Technology,

1999).

[20] E.A. Novikov, \Functionals and the random-force method in turbulence," Sov. Phys. JETP 20, 1290

(1965).

[21] G. K. Batchelor, The theory of homogeneous turbulence, p.49 (Cambridge University Press, 1953).

26

[22] J. Zinn-Justin, Quantum �eld theory and critical phenomena, 3rd ed. p.5 (Oxford University Press,

1996).

[23] R. H. Kraichnan, \Dynamics of nonlinear stochastic systems," J. Math. Phys. 2, 124 (1961).

[24] S. A. Orszag, \Lectures on the statistical theory of turbulence" in Fluid Dynamics, Les Houches 1973,

eds. R. Balian and J. L. Peule, (Gordon and Breach, 1973).

[25] P. G. Sa�man, \On the e�ect of the molecular di�usivity in turbulent di�usion," J. Fluid Mech. 8,

273 (1960).

[26] C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and engineers,

(McGraw-Hill, 1978).

[27] G. A. Baker, Essentials of Pad�e approximants, (Academic Press, 1975).

[28] R. H. Kraichnan, \Di�usion by a random velocity �eld," Phys. Fluids 13, 22 (1970).

[29] R. W. Hamming, \Stable predictor corrector methods for ordinary di�erential equations," J. Assoc.

Comp. Math. 6, 37 (1959).

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Mathematical methods for digital computers, eds. A. Ralston and H. S. Wilf, (John Wiley & Sons,

1960).

27

List of Tables

I CoeÆcients of the FDC series for the long-term e�ective di�usivity. . . . . . . . . . . . . . . 29

II Pad�e approximations to the FDC series in Table I. . . . . . . . . . . . . . . . . . . . . . . . 30

28

Energy Time !k Coe�. Coe�. Coe�. Coe�. Figure

spectrum correlation of �� of �3� of �5� of �7�

Ea Ra 1=�� 1.000 -0.500 0.459 -0.564 2

Ea Rb 1=�� 0.886 -0.184 0.050

Ea Rc 1=�� 0.200 0.052 -0.016 0.0047

Eb Ra 1=�� 1.000 -1.250 3.906 -19.13

Eb Ra k=�� 0.753 -0.352 0.323 -0.413 3

Eb Ra k2=�� 0.667 -0.222 0.140 -0.116

Eb Rb 1=�� 0.886 -0.459 0.429

Eb Rb k=�� 0.667 -0.128 0.034 4

Eb Rb k2=�� 0.591 -0.087 0.019

Eb Rc k=�� 0.113 0.034 -0.007 0.0013

Ec ; � = 1 Ra 1=�� 1.000 -1.027 2.221 -6.925

Ec ; � = 10 Ra 1=�� 1.000 -7.35 318.1 -23711

Ec ; � = 1 Ra k=�� 0.741 -0.363 0.333 -0.416

Ec ; � = 10 Ra k=�� 0.492 -0.211 0.189 -0.254 5

Ec ; � = 1 Ra k2=�� 0.569 -0.162 0.085 -0.060 6

Ec ; � = 10 Ra k2=�� 0.313 -0.049 0.015 -0.006

Table I: CoeÆcients of the FDC series for the long-term e�ective di�usivity.

29

Energy Time !k P 01 (�

2� ) P 1

1 (�2� ) P 1

2 (�2� ) Figure

spectrum correlation at �� = 1 at �� = 1 at �� = 1

Ea Ra 1=�� 0.667 0.739 0.714 2

Ea Rb 1=�� 0.734 0.742

Ea Rc 1=�� 0.270 0.240 0.240

Eb Ra 1=�� 0.444 0.697 0.541

Eb Ra k=�� 0.513 0.569 0.549 3

Eb Ra k2=�� 0.500 0.530 0.522

Eb Rb 1=�� 0.584 0.649

Eb Rb k=�� 0.559 0.565 4

Eb Rb k2=�� 0.515 0.520

Eb Rc k=�� 0.160 0.140 0.140

Ec ; � = 1 Ra 1=�� 0.493 0.675 0.578

Ec ; � = 10 Ra 1=�� 0.120 0.834 0.208

Ec ; � = 1 Ra k=�� 0.497 0.551 0.532

Ec ; � = 10 Ra k=�� 0.345 0.381 0.367 5

Ec ; � = 1 Ra k2=�� 0.443 0.463 0.458 6

Ec ; � = 10 Ra k2=�� 0.271 0.275 0.274

Table II: Pad�e approximations to the FDC series in Table I.

30

List of Figures

1 The FDC diagrams of order 1 and 2 (which represent the FDC2 approximation (12)). . . . 32

2 The FDC5 approximation to �(1) and its Pad�e approximants for spectrum Ea, time corre-

lation Ra and !k = 1=��. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Pad�e approximants for �(1) for spectrum Eb, time correlation Ra and !k = k=��. . . . . . 34

4 Pad�e approximants for �(1) for spectrum Eb, time correlation Rb and !k = k=��. . . . . . 35

5 Pad�e approximants for �(1) for spectrum Ec (� = 10), time correlation Ra and !k = k=��. 36

6 Pad�e approximants for �(1) for spectrum Ec (� = 10), time correlation Ra and !k = k2=��. 37

7 Pad�e approximants for �(1) for spectrum Ea, time correlation Rc and !k = 1=��. P01 has

a pole at �� = 1:96, but P 11 and P 1

2 have no poles on the positive real axis. . . . . . . . . . . 38

8 Generalized Pad�e approximants for �(t) for spectrum Ea, time correlation Ra and !k = 1=��,

with �� = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9 Comparison of exact (solid line) and numerical (symbols) correlation functions for spectrum

Ea. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10 Generalized Pad�e and the Sa�man (1961) approximations (lines) and numerical values (dots)

for �(t) for spectrum Ea, time correlation Rb and !k = 1=��. Here �� is �xed, �� = (2=3)1

2 . 41

11 Generalized Pad�e and the Sa�man (1961) approximations (lines) and numerical values (dots)

for L(t) for spectrum Eb, time correlation Rb and !k = k=��. Here �� is �xed, �� = 21

2 . . . . 42

31

+ +k-p

q p

k-p-q

k-q k-p

k-p-q

k-pp

q

k-p

p

t t

t1 t3

t1t3

t2 t2

t1 t@@tb�(k; t) + �0k

2b�(k; t) =

Figure 1: The FDC diagrams of order 1 and 2 (which represent the FDC2 approximation (12)).

32

0 0.5 1 1.5 2τ∗

0

0.2

0.4

0.6

0.8

1

1.2

Long

-ter

mef

f.di

ff.

P10

P21

P22

P11

5-term series

Figure 2: The FDC5 approximation to �(1) and its Pad�e approximants for spectrum Ea, time correlation

Ra and !k = 1=��.

33

0 0.5 1 1.5τ∗

0

0.2

0.4

0.6

0.8

1

Long

-ter

mef

f.di

ff.

P01

P21

P11

Figure 3: Pad�e approximants for �(1) for spectrum Eb, time correlation Ra and !k = k=��.

34

0 0.5 1 1.5τ∗

0

0.2

0.4

0.6

0.8

1

Long

-ter

mef

f.di

ff. P0

1

P11

Figure 4: Pad�e approximants for �(1) for spectrum Eb, time correlation Rb and !k = k=��.

35

0 0.5 1τ∗

0

0.1

0.2

0.3

0.4

0.5

Long

-ter

mef

f.di

ff.

P01

P11

P12

Figure 5: Pad�e approximants for �(1) for spectrum Ec (� = 10), time correlation Ra and !k = k=��.

36

0 0.5 1τ∗

0

0.1

0.2

0.3

0.4

0.5

Long

-ter

mef

f.di

ff.

P01

P11 P1

2

Figure 6: Pad�e approximants for �(1) for spectrum Ec (� = 10), time correlation Ra and !k = k2=��.

37

0 0.5 1 1.5 2τ∗

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Long

-ter

mef

f.di

ff.

P01

P11

P12

Figure 7: Pad�e approximants for �(1) for spectrum Ea, time correlation Rc and !k = 1=��. P01 has a pole

at �� = 1:96, but P 11 and P 1

2 have no poles on the positive real axis.

38

0 2 4 6 8 10t

0

0.25

0.5

0.75

1

Eff

.Diff

usiv

ity

P11

P12

Figure 8: Generalized Pad�e approximants for �(t) for spectrum Ea, time correlation Ra and !k = 1=��,

with �� = 1.

39

r

Cor

rela

tion

0 2 4 6 8 10-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Figure 9: Comparison of exact (solid line) and numerical (symbols) correlation functions for spectrum Ea.

40

t

Eff

.Diff

usiv

ity

0 1 2 30

0.2

0.4

0.6

0.8

First approx. P01

Second approx. P11

Numerical, Nr=10000Saffman approx.

Figure 10: Generalized Pad�e and the Sa�man (1961) approximations (lines) and numerical values (dots)

for �(t) for spectrum Ea, time correlation Rb and !k = 1=��. Here �� is �xed, �� = (2=3)1

2 .

41

t

L(t)

0 2 4

0

0.2

0.4

0.6

0.8

1

First approx. P10

Second approx. P11

Numerical, Nr=2000Saffman approx.

Figure 11: Generalized Pad�e and the Sa�man (1961) approximations (lines) and numerical values (dots)

for L(t) for spectrum Eb, time correlation Rb and !k = k=��. Here �� is �xed, �� = 21

2 .

42