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(continued on back flap)

I \\\Ill 1\lllli~iiiiiliiilli~iijl\l\ Ill\ Ill\ 1203854351

'''"r1r1 i 11n iTffiiii 7070 125222 49

I I 5 b

{)

r

Wave Propagation in a Turbulent Medium

WAVE PROPAGATION IN A TURBULENT MEDIUM

by V. I. Tatarski INSTITUTE OF ATMOSPHERIC PHYSICS

ACADEMY OF SCIENCES OF THE USSR

TRANSLATED FROM THE RUSSIAN

by R. A. Silverman

DOVER PUBLICATIONS, INC. NEW YORK

Copyright © 1961 by R. A. Silverman. All rights reserved under Pan American and In­

ternational Copyright Conventions.

Published in Canada by General Publishing Com­pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.

Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London we 2.

This Dover edition, first published in 1967, is an unabridged republication of the English translation originally published in 1961 by the McGraw-Hill Book Company, Inc.

Library of Congress Catalog Card Number: 67-26484

Manufactured in the United States of America Dover Publications, Inc.

180 Varick Street New York, N. Y. 10014

TRANSLATOR'S PREFACE

This book is a translation of' V.I.. Tatarski '.s "TeopYUi q>;nyKTya~HOHHlllX

flBneHHH ITpH PacnpocTpaHeHHH Bona B Typ6yneHTHOH ATMocwepe"

literally "The Theory of Fluctuation Phenomena in Propagation of' Waves in a Tur-

bulent Atmosphere". It is hoped that Tatarski 1 s book) together with the transla-

tion of L.A. Chernov's "Wave Propagation in a Random Medium" (McGraw-llill Book Co.)

1960) will furnish a comprehensive and authoritative survey of the present state

of research in the field of wave propagation in turbulent mediaJ with special

emphasis on important Russian contributions.

For typographical convenience} the numerous f'ootnotes appearing in the Rus-

sian original have been collated in the ~ ~ Remarks section at the end of

the book; I have taken the liberty of' adding some remarks of' my ownJ all identi-

fied by the symbol T in parentheses. The Russian original has also been supple­

mented in two other ways: 1) Dr. R.H. Kraichnan has written an Appendix qualifying

the material in Chapter 5; 2) In the References sectionJ I have cited some readily

available English and German translations of Russian papers. (The origin of' one

reference} No. 61J was not clear to me.)

The time has come to thank the team. of J' acqueline Ellis and Maureen Kelly

for their expert performance in preparing the masters for both this book and the

Chernov translation. I also take this occasion to thank my wife for her pains-

taking proofreading of both books.

v

AUTHOR 1 S PREFACE

In contemporary radiophysics, atmospheric optics and acoustics, one often studies the pro-

pagation of electromagnetic and acoustic waves in the atmosphere; in doing so,it is increasing-

ly often necessary to take into account the turbulent state of the atmosphere, a state which

produces fluctuations in the refractive index of the air. In some cases the turbulence mani-

fests itself as atmospheric "noise", causing fluctuations in the parameters of waves propaga-

ting through the atmosphere; in other cases the atmospheric turbulence behaves like a source of

inhomogeneities which produce scattering. This latter phenomenon has attracted the attention

of numerous investigators, since it is connected with the long distance propagation of V.H.F.

and U.H.F. radio waves by scattering in the ionosphere and in the troposphere. Thus the prob-

lem of "waves and turbulencen is at present one of the important problems of radiophysics,

atmospheric optics and acoustics.

In the last decade, a large number of papers pertaining to this problem have been published.

These papers are reviewed in the special monograph by D.M. Vysokovski, entitled nsome Topics in

the Long Range Tropospheric Propagation of U.H.F. Radio Waves" (Izdat. Ak.ad. Nauk SSSR, Moscow,

1958); this monograph is chiefly concerned with papers by foreign authors. Recently there has

also appeared a monograph by L.A. Chernov entitled "Wave Propagation in a Medium with Random

* Inhomogeneitiesn (Izdat. Akad. Nauk SSSR, Moscow, 1958). However, the present monograph differs

from those cited in that the author has tried to make more complete and consistent use of the

results of turbulence theory.

In recent years the study of turbulence (in particular, atmospheric turbulence) has ad-

vanced considerably. In this regard, a large role has been played by the work of Soviet sci­

entists ~.g. 8,9,11-15,17,21,22,30]. However, the results of turbulence theory are often not

used in solving problems related to wave propagation in a turbulent atmosphere. In a consider-

able number of radiophysics and astronomy papers devoted to radio scattering, the twinkling and

* Translated by R.A. Silverman as nwave Propagation in a Random Medium", McGraw-Hill Book Co.,

New York, 1960. (T)

vii

quivering of stellar images in telescopes, etc., only crude models, which do not correspond to

reality, are used to describe the atmospheric inhomogeneities. Naturally, the results obtained

in such papers can only be a very rough and purely qualitative description of the properties of

wave propagation in the atmosphere.

In the present monograph we try to give a general treatment of the theory of scattering of

electromagnetic and acoustic waves and of the theory of parameter fluctuations of short waves

propagating in a turbulent atmosphere. We take as our starting point the Kolmogorov theory of

locally isotropic turbulence, which gives a sufficiently good description of the turbulent

atmosphere. In Part I we give a brief exposition of some topics from the theory of random

fields and turbulence theory which are necessary to understand what follows. There we give

special attention to the representation of random fields by using generalized spectral expan­

sions. Spectral representations are very appropriate both for formally solving many problems

in the theory of wave propagation in a turbulent medium and for interpreting these problems

physically.

Part II is devoted to the scattering of electromagnetic waves (Chapter 4) and acoustic

waves (Chapter 5) by turbulent atmospheric inhomogeneities. The radio scattering theories of

Booker and Gordon, Villars and Weisskopf, and Silverman are studied from a general point of

view, as being different special cases which follow from a general formula. In Part III we

consider amplitude and phase fluctuations of short waves propagating in a turbulent atmosphere,

first fluctuations of a plane wave (Chapters 6 and 7), then amplitude and phase fluctuations

of a plane wave in a medium with a smoothly varying "intensity" of turbulence ( Chapter 8) , and

finally fluctuations of a spherical wave (Chapter 9). In Part IV we present some results of

experimental studies of atmospheric turbulence (Chapter 10) and the results of experiments on

the propagation of sound and light in the layer of tbe atmosphere near the earth. The results

of observations of twinkling and quivering of stellar images in telescopes and the interpreta­

tion of these results are given in Chapter 13. In presenting experimental material we give

the corresponding theoretical considerations.

Some problems which have much in common with the foregoing have not been considered• in

this book. Foremost among these is the question of radio scattering by the turbulent iono­

sphere, despite the fact that the mechanism for this effect has very much in common with that

for radio scattering in the troposphere. We did not think it possible to go into the specific

viii

details which would have to be considered in studying this phenomenon. We have also not

included in this monograph the interesting problem of radiation of sound by a turbulent flow,

considered in the papers of Lighthill.

I wish to express my deep gratitude to A.M. Obukhov and A.M. Yaglom for the help they

gave me while I was writing this book.

ix

l

Translator's preface

Author's preface

INTRODUCTORY REMARKS

CONTENTS

Part I

SOME TOPICS FROM TEE THEORY OF RANDOM FIELDS

AND TURBULENCE THEORY

Chapter 1. METHODS FOR STATISTICAL DESCRIPTION OF CONTINUOUS RANDOM FIELDS •

1.1 Stationary random functions

1.2 Random functions with stationary increments

1.3 Homogeneous and isotropic random fields

1.4 Locally homogeneous and isotropic random fields •

Chapter 2. THE MICROSTRUCTURE OF TURBULENT FLOW •

Introductory Remarks

iii

v

1

3

3

8

15

19

27

27

2.1 Onset and development of turbulence • 27

2.2 Structure functions of the velocity field in developed turbulent flow • 29

2.3 Spectrum of the velocity field in turbulent flow 34

Chapter 3· MICROSTRUCTURE OF THE CONCENTRATION OF A CONSERVATIVE

PASSIVE ADDITIVE IN A TURBULENT FLOW • 4o

3.1 Turbulent mixing of conservative passive additives • 4o

3.2 Structure functions and spectral functions of the field of a conservative

passive additive in a turbulent flow 44

xi

3.3 Locally isotropic turbulence with smoothly varying mean characteristics •

3.4 Microstructure of the refractive index in a turbulent flow •

Part II

SCATTERING OF ELECTROMAGNETIC AND ACOUSTIC WAVES

IN TEE TURBULENT ATMOSPHERE

Chapter 4. SCATTERING OF ELECTROMAGNETIC WAVES IN THE TURBULENT ATMOSPHERE •

Introductory Remarks

4.1

4.2

4.3

4.4

4.5

Solution of Maxwell's equations

The mean intensity of scattering •

Scattering by inhomogeneous turbulence •

Analysis of various scattering theories

Evaluation of the size of refractive index fluctuations from data on the

scattering of radio waves in the troposphere •

Chapter 5· TEE SCATTERING OF SOUND WAVES IN A LOCALLY ISOTROPIC TURBULENT FLOW

Part III

PARAMETER FLUCTUATIONS OF ELECTROMAGNETIC AND ACOUSTIC WAVES

PROPAGATING IN A TURBULENT ATMOSPHERE

INTRODUCTORY REMARKS

Chapter 6. SOLUTION OF THE PROBLEM OF AMPLITUDE AND PHASE FLUCTUATIONS OF A PLANE

MONOCHROMATIC WAVE BY USING TBE EQUATIONS OF GEOMETRICAL OPTICS •

6.1 Derivation and solution of the equations of geometrical optics •

51

55

59

59

59

64

69

70

77

81

91

93

93

6.2 The structure function and the spectrum of the phase fluctuations of the wave 97

6.3 Solution of the equations of geometrical optics by using spectral expansions 102

6.4 Amplitude and phase fluctuations of a wave propagating in a locally isotropic

turbulent flow • 110

xii

6.5 A consequence of the law of conservation of energy •

6.6 Amplitude and phase fluctuations of sound waves •

6.7 Limits of applicability of geometrical optics

Chapter 7. CALCULATION OF AMPLITUDE AND PHASE FLUCTUATIONS OF A PLANE MONOCHROMATIC

WAVE FROM TEE WAVE EQUATION USING TBE METHODS OF "SMALL" AND "SMOOTHn

PERTURBATIONS

7.1 Solution of the wave equation by the method of small perturbations

7.2 The equations of the method of smooth perturbations

7. 3 Solution of the equations of the method of "smooth" perturbations by using

spectral expansions

Qualitative analysis of the solutions

Amplitude and phase fluctuations of a wave propagating in a locally isotropic

turbulent medium

Relation between amplitude and phase fluctuations and wave scattering

Chapter 8. PARAMETER FLUCTUATIONS OF A WAVE PROPAGATING IN A TURBULENT MEDIUM WITH

SMOOTHLY VARYING CHARACTERISTICS

Chapter 9. AMPLITUDE FLUCTUATIONS OF A SPHERICAL WAVE •

Part IV

EXPERIMENTAL DATA ON PARAMETER FLUCTUATIONS OF LIGHT

AND SOUND WAVES PROPAGATING IN TEE ATMOSPHERE

Chapter 10. EMPIRICAL DATA ON FLUCTUATIONS OF TEMPERATURE AND WIND VELOCITY IN TEE

LAYER OF TEE ATMOSPHERE NEAR TEE EARTH AND IN THE LOWER TROPOSPHERE

Chapter 11.. EXPERIMENTAL DATA ON THE AMPLITUDE AND PHASE FLUCTUATIONS OF SOUND WAVES

PROPAGATING IN TEE LAYER OF THE ATMOSPHERE NEAR THE EARTH

xiii

113

115

120

122

122

124

128

137

150

156

164

173

189

198

Chapter 12. EXPERIMENTAL INVESTIGATION OF THE SCINTILLATION OF TERRESTRIAL LIGHT

SOURCES

Introductory Remarks

206

206

12.1 The probability distribution ~ction of the fluctuations of light intensity 208

12.2 Dependence of the amount of scintillation on the distance and on the meteor-

ological conditions 210

12.3 The correlation function of the fluctuations of light intensity in the plane

perpendicular to the ray 212

12.4 Frequency spectra of the fluctuations of the logarithm of the light intensity

(theory) 215

12.5 Frequency spectrum of fluctuations of light intensity (experimental results) 219

Chapter 13. TWINKLING AND QUIVERING OF STELLAR IMAGES IN TELESCOPES 224

APPENDIX

NOTES AND REMARKS 260

REFERENCES 280

xiv

Wave Propagation in a Turbulent Medium

Part I

SOME TOPICS FROM THE THEORY OF RANDOM FIELDS

AND TURBULENCE THEORY

Introductory Remarks

The index of refraction of the atmosphere for electromagnetic waves is a function of the

temperature and humidity of the air. Similarly) the velocity of sound in the atmosphere is a

function of the temperature) wind velocity and humidity. Therefore) in studying microfluctua-

tions of the refractive index of electromagnetic and acoustic waves in the atmosphere)we must

first of all explain the basic laws governing the structure of meteorological fields like the

temperature) humidity and wind velocity fields.

For us the most important fact about the atmosphere is that it is usually in a state of

turbulent motion. The values of the wind velocity at every point of space undergo irregular

fluctuations; similarly) the values of the wind velocity taken at different spatial points at

the same instant of time also differ from one another in a random fashion. What has been said

0.1 sec ~

Fig. l Simultaneous record of temperature and wind velocity.

l

applies as well to all other meteorological quantities} in particular to temperature and humid-

ity. In Fig. 1 we give as an example a sample of the record of the instantaneous values of the

wind velocity and the temperature at one pointJ obtained by using a low inertia measuring

device. We see that both of these quantities undergo irregular oscillations} which differ in

amplitude and frequency and are superimposed in a random manner. It is natural that statis-

tical methods are used to describe the laws characterizing the structure of such fluctuating

quantities.

Chapter 1

METHODS FOR STATISTICAL DESCRIPTION OF CO}~INUOUS RANDOM FIELDS [a]

1.1 Stationary random functions

The curves shown in Fig. 1 serve as examples of realizations of random functions. The

value of any such function f(t) at a fixed instant of time is a random variable} i.e. can

assume a set of different valuesJ where there exists a definite probability F(tJf1 ) that

f(t) < f1

. But to completely specify the random function f(t) it is not enough to know only

the probability F(tJf1

); one must also know all possible multidimensional probability distri­

butions} i.e. all the probabilities

(1.1)

sible Nand t 1Jt2J ... JtNJ f 1}f2J•••JfN. HoweverJ in the applications it is usually difficult

to determine all the functions (1.1). Therefore in practiceJ instead of the distribution

function (1.1L one ordinarily uses much more "meager" (but muc".h simpler) characteristics of

the random field. Of these statistical characteristics of the random function f(t)} which

are widely used in practiceJ the most important and simplest is the mean value f(t) [b].

The next simplest and very important characteristic of the function is its correlation func-

(1.2)

It is clear that the relation B(t1Jt2 ) = B(t

2Jt

1) holds for real functions f. The correla­

tion function vanishes when the quantities f(t1

) - r(t1

) and f(t2

) - f(t2

) are statistically

independent} i.e.} when the fluctuations of the quantity f(t) at the times t1

and t 2 are not

related to each other. In this case the mean value of the product in the right hand side of

2 3

(l.2) factors into the product of the quantities

each of which equals zero. Thus) the correlation function Bf(tl)t2 ) characterizes the mutual

relation between the fluctuations of the quantity f(t) at different instants of time. In

analogy to the correlation function Bf(tl)t2 )) one can also construct more complicated charac­

teristics of the random field f(t)) for example) the quantities

However) we shall use only the mean value f(t) and the correlation function Bf(tl)t2 ).

The mean value of a function can be a constant or can change with time (for example) as

the wind gradually increases) the mean value of the wind velocity u(;Jt) at any point~

increases). Similarly) the correlation function Bf(tl)t2 ) can either depend only on the "dis­

tance" between the times tl and t 2 (in which case the statistical relation between the fluc­

tuations of the quantity f at different instants of time does not change in the course of

time) or else it can depend also on the positions of these points on the time axis. A random

function f(t) is called stationary [d] if its mean value f(t) does not depend on the time and

if its correlation function Bf(tl)t2 ) depends only on the difference tl - t2

) i.e. if

It is easy to show that Bf(T) satisfies the condition !Bf(T)I ~ Bf(O). We shall always

assume below that the mean value of a stationary random function f(t) is zero [e].

(l.3)

For stationary random functions f(t) there exist expansions similar to the expansions of

non-random functions in Fourier integrals) namely a stationary random function can be repre-

sented in the form of a stochastic (random) Fourier-Stieltjes integral with random complex

amplitudes dcp( w) [ l J :

f(t)

00

J imt ( ) e dcp m . (l.4)

4

T Using the expansion (l.4) we can obtain an expansion of the correlation function Bf(tl - t 2 )

of the stationary random function f(t) in the form of a Fourier integral. In fact) substitu-

ting the expansion (l.4) in the left hand side of (l.2)) we obtain

Since in the stationary case the correlation function must depend only on the difference

t - t) the quantity dcp(w )dcp*(m2 ) must have the following form [f]: l 2 l

where) obviously) W(m) > 0. From this it follows that [_g]

i.e. J

00

the functions B (T) and W(m) are Fourier transforms of each other. f

(l.5)

(l. 6)

Thus) the Fourier

transform of a correlation function B (T) must be nonnegative; if it is negative at even one f

point) this means that the function Bf(T) cannot be the correlation function of any stationary

random function f(t). Khinchin [6] showed that the converse assertion is also true: if the

Fourier transform of the function B (T) is nonnegative) then there exists a stationary random f

function f(t) with Bf(T) as its correlation function. This fact) which we shall use below)

makes it easy to construct examples of correlation functions. When the specified conditions

are met) the non-random function W(m) is called the spectral density of the stationary random

function f( t).

We now explain the physical meaning of the spectral density. For example) let f(t)

represent a current flowing through a unit resistance. Then [f(t)]2

is the instantaneous

power dissipated in this resistance) and the mean value of this power is [f(t)]2

= Bf(o).

Using Eq. (l.6)) we obtain

5

J W(ill)dm.

Thus} in this case W(ill) represents the spectral density of the power} so that in the litera-

ture of radiophysics this function is often called the noise power spectrum. In the case

where f(t) is the magnitude of the velocity vector of a fluid} W(ill) represents the spectral

density of the energy of a unit mass of fluid} so that in the literature of turbulence theory

this function is often called the spectral density of the energy distribution.

We now give some examples of correlation functions and their spectral densities.

a) The correlation function

B(-r) (1. 7)

is often used in the applications. The co1·responding spectral density is easily found to be

W(ill) 1 2:rr

00

J -iffi-r 2 ( I I I ) e a exp - -r -r0

d-r

2 a -r 0

2 2 • :rr(l + (J) -r )

0

(1.8)

Here W(ill) >OJ so that the function a2

exp(-l-r/-r I) can actually be the correlation function of 0

a stationary random process.

b) The correlation function

B(-r) (1.9)

corresponds to the spectral density

W(ill) (1.10)

6

c) To the spectral density

2 1 ar(v +2) -r

1 W(ill) 0 0 1 > v >-2}

J-;i r( v) 2 2 v +-(l + (J) 'f )

2 0

(1.11)

corresponds the correlation function

2 v

~(o) a2) J B(-r) a (2_) K (2...) } 'f > 0}

2v-l r(v) 'f v -r 0 0

( 1.12)

where K (x) is the Bessel function of the second kind of imaginary argument. This correlation v

function is also used in some applications.

In Fig. 2 we show how the correlation functions (1.7)J (1.9) and (1.12) depend on -r/-r0

J

and in Fig. 3 we show the corresponding spectral densities.

Fig. 2 The correlation functions

22/3 l/3 l) r(l/3) (:) Kl/3(-r/-ro);

0

2) 3)

7

w

Fig. 3 The spectral densities of the correlation functions of Fig. 2.

The time ~0 required for an appreciable decrease in the correlation function} for

exampleJthe time in which B('r) falls to 0.5 or Q.l of the value B(OL is called the correla-

tion time. As can be seen from the examples considered} the quantity T0

is related to the

"width" m of the spectrum (i.e. to the frequency at which the spectral density W(m) falls off 0

appreciably) by the relation m ~ ~ l. 0 0

l.2 Random functions with stationary increments

Actual random processes can very often be described with sufficient accuracy by using

stationary random functions. An example of such a process is the fluctuating voltage appear-

8

ing across a resistance in a state of thermodynamic equilibrium with the surrounding medium.

However} the opposite case can also occur} where the random processes cannot be regarded as

stationary. As an example of such a process in radiophysics we cite the phase fluctuations

of a vacuum tube oscillator [7]. Such examples also occur very often in meteorology. For

example} as already noted} as the strength of the wind gradually increase~ the mean value of

the velocity at any point increases} so that in this case the wind velocity is not a station-

ary random function. The mean values of other meteorological variables of the atmosphere}

e.g.} temperature} pressure and humidity} also undergo comparatively slow and smooth changes.

In analyzing these variables the same difficulty continually arises} i.e.} which changes of

the function f(t) are to be regarded as changes of the mean value and which are to be regarded

as slow fluctuations? Such characteristics of a random function as the mean square fluctua-

tionJ the correlation time} the shape of the correlation function B(~) and of the spectral

density W(m)J very often depend to a considerable extent on the answer to this question.

To avoid this difficulty and to describe random functions which are more general than

stationary random functions} in turbulence theory one uses instead of correlation functions

(l.2) the so-called structure functions} first introduced in the papers of Kolmogorov [8}9].

The basic idea behind this method consists of the following. In the case where f(t) repre-

sents a non-stationary random function} i.e.} where f(t) changes in the course of time} we can

consider instead of f(t) the difference F (t) = f(t + ~) - f(t). For values ofT which are ~

not too large} slow changes in the function f(t) do not affect the value of this difference}

and it can be a stationary random function of time} at least approximately. In the case where

F (t) is a stationary random function} the function f(t) is called a random function with ~

stationary first increments} or simply a random function with stationary increments [h].

If we use the algebraic identity

l[ 2 2 2 2] (a - b)( c - d) = - (a - d) + (b - c) - (a - c) - (b - d) J 2-

then we can represent the correlation function of the increments in the following form:

9

Thus BF(t1,t2 ) is expressed as a linear combination of the functions

Df( t., t.) l J

[f( t. ) - f( t . ) J 2 • l J

( 1.13)

The function (1.13) of the arguments t. and t., where t. and t. take the values t 1+ T, t 1, t2

, l J l J

t2+ T, is called the structure function of the random process. In order for BF(t

1,t

2) to

depend only on t1

- t 2 , it is sufficient that Df(t1,t2 ) depend only on this difference, i.e.,

that the relation Df(t1,t2 ) = Df(t

1- t 2 ) holds.

The structure function Df(T) = [f(t + T) - f(t)]2

is th~ basic characteristic of a random

process with stationary increments. Roughly speaking, the value of Df(T) characterizes the

intensity of those fluctuations of f(t) with periods which are smaller than or comparable with

T. Of course, the function Df(T) can also be constructed for ordinary stationary functions,

which are a special case of functions with stationary increments. If f(t) is a stationary ran-

dom function with mean value 0, then

l}'(t + T)]2 + [!(t)]

2- 2f(t + T)f(t).

It follows from the stationarity of f(t) that

Thus, for a stationary process

(1.14)

In the case where Bf(oo) = 0 (and in practice this condition is almost always met), we have

Df(oo) = 2Bf(o). This relation allows us to express the correlation function Bf(T) in terms of

the structure function Df(T), i.e.

(1.14')

Thus, in the case of stationary random processes, the structure functions Df(T) can be used

along with the correlation functions, and in some cases their use is even more appropriate [i].

10

l

As we have already seen, the expansion

00

J cos(ayr)W(m)dm (1.6)

is valid for the correlation function of a stationary random process. From this we can obtain

a similar expansion for the corresponding structure function. Indeed, substituting (1.6) in

(1.14), we obtain

2 J (1 - cos mT)W(m)dm. ( 1.15)

It turns out that the same expansion is also valid for the structure function of the general

random function with stationary increments, the only difference being that the spectral den-

sity W(m) can now have a singularity at the origin (in this regard see below).

Just as a stationary random function can be represented as a stochastic Fourier-Stieltjes

integral

f(t) J eimt dcp(m)' (1.4)

a random function with stationary increments can be represented in the form

f(t) f ( 0) + J ( 1 - e imt) dcp ( m) , (1.16)

where f(O) is a random variable, and the amplitudes dcp(m) obey the condition

(1.17)

11

Substituting the expansion (1.16) in the right hand side of (1.13) and using the relation

(1.17)) we obtain

JJ

(1.15 I)

Thus) the expansion (1.15) for the structure function of a stationary random process is also

valid for a random process with stationary increments. Since the spectral density W(m) which

figures in (1.15) signifies the average spectral density of the power (or energy) of the fluc­

tuations) it is natural to assume that the spectral density W(m) figuring in (1.15') for pro-

cesses with stationary increments has the same physical meaning. We note that for convergence

of the integral (1.6) it is necessary that the integral

J W(m)dill

exist) i.e.J that the power of the fluctuations be finite. On the other hand the integral

(1.15) also converges when W(m) has a singularity at zero of the form m-a (a< 3)) i.e.J when

the low freg_uency components of the fluctuation spectrum have infinite "energy" [j].

We now consider some examples.

a) We construct the structure function of the stationary random process considered in

2 ~ 21-v v -r ~ 2a 1 - nr:::~ ( ..::_) K (-) · r, v 1 -r v -r 0 0

12

--For -r << -r

0 we can use the first two terms of the series expansion of the function K)x) [k].

After some simple calculations we obtain

2v i.e. Df(-r) "' -r

the constant 2a2

•

For -r"' -r0

the growth of the function Df(-r) slows down) and it approaches

The spectral density corresponding to Df(-r) is the same as in the example

on page 7 •

b) Consider the spectral density W(w) = AjwJ-(p+l)J (A > 0) 0 < p < 2).

this function in Eg_. (1.15) and carrying out the integration [t]J we obtain

2kc -rP.

sin i r(l + p)

Thus to the structure function

corresponds the spectral function

With v

W (w)

p/2 and

2 a -rv=

'[ 0

r(l + p) sin np 21 ~-(p+l) 2n 2c w •

22v - 1 r(l + v) 2 r(l - v) c

Substituting

the structure function considered in the preceding example coincides with the structure func­

tion c2

-rp for -r << -r . The spectra of these structure functions coincide in the region 0

w-r0

>> 1. Fig. 4 shows the structure functions of examples a) and b)) and Fig. 5 shows their

spectra.

13

...

~----------------L-----------------~T/To 2

Fig. 4 The structure functions:

l) l 2/3

--y]3 'r 'r

0

~-22/3 'r l/3

Kl/3(-r-r 8 2) canst. "RmJ (-;r) 0 0

W(w)/c 2

0.10

0.08

0.06

0.04

0.02

4 5 WTo

Fig. 5 The spectral densities of' the

structure functions of' Fig. 4.

l.3 Homogeneous and isotropic random fields

We turn now to random functions of' three variables (random fields). The concept of' a ran-

dom field is completely analogous to the concept of' a random process. Examples of' random

fields are the wind velocity field in the turbulent atmosphere (a vector random field compri-

sing three random velocity components), and the (scalar) fields of' temperature, humidity and

dielectric constant. For a random field f'(;) we can also define the mean value f'(;) and the

correlation function

(l.l8)

In the case of' random fields the concept of' stationarity generalizes to the concept of' homo-

geneity. A random field is called homogeneous if' its mean value is constant and if' its corre­

lation function does not change when the pair of' points ;l and r2 are both displaced by the

same amount in the same direction, i.e. if'

Choosing ; 0

ous field, i.e., the correlation function of' a homogeneous random field depends only on rl- r2,

so that Bf'(;l,;2 )

depends only on r

Bf'(;l_ ;2

). A homogeneous random field is called isotropic if' Bf'(r)

j;j, i.e. only on the distance between the observation points. Of' course,

a homogeneous field may also not be isotropic; for example, the field with correlation f'unc-

tion of' the form

is homogeneous, but not isotropic.

If' in a homogeneous and isotropic field we single out any straight line and consider the

values of' the field only along this line, then as a result we obtain a random function of' one

variable x, to which we can apply all the results pertaining to stationary random functions.

l5

In particular) we can expand the correlation function as a Fourier integral.

J cos (Kx) V(K)d.K. (1.19)

-co

However it is more natural to use three-dimensional expansions. A homogeneous random field

can be represented in the form of a three-dimensional stochastic Fourier-Stieltjes integral:

co

Iff (1. 20)

-co

Here the amplitudes d~(K) satisfy the relation [m]

(1.21)

where ~(K) ~ 0. Substituting this expansion in the formula

(assuming that f(;) =~and taking into account the relation (1.21), we obtain

( 1. 22)

-co

B(;2

- ;1

)J so that the formula can also be written in the form

co

j f j cos (/(. ;) ~(/()d/(. ( 1. 23)

-co

16

function ~(K) can be expressed in terms of B(r):

(1. 24)

ThusJ the functions Bf(;) and ~(K) are Fourier transforms of each other.

If the random field f(;) is isotropic) the function Bf(;) depends only on 1;1. Then in

the integral (1.24) we can introduce spherical coordinates and carry out the angular integra-

tions. As a result we obtain the expression

1 -2-2n K

J rBf(r)sin (Kr)drJ

0

(1. 25)

wher~K = 1/(J · ThusJ in an isotropic random field the spectral density ~(K) is a function

of only one variable) the magnitude of the vector K· This allows us to simplify the expres­

sion (1.23) in the case of an isotropic field. Introducing spherical coordinates in the space

of the vector K and carrying out the angular integrations) we obtain the relation

Bf(r) = ~n J K~(K) sin (Kr)dK. (1. 26)

0

It should be noted that the three-dimensional spectral density ~(K) of an isotropic random

field is related to the one-dimensional spectral density V(K) by the simple relation

l 2nK

which can be obtained by substituting ( 1.19) in the right hand side of ( 1. 25) [ n J . We now give some examples of spatial correlation functions and their spectra:

a)

17

(1. 27)

(1.28)

Using the results of example a) on page 6 and the fact that the expansion (1.19) is completely

analogous to the expansion (1.6) for a stationary random process, we can write down immediately

the one-dimensional spectral density V(K):

V(K)

2 a r 0 2 2 .

1((1 + K r ) 0

We use Eq. (1.27) to determine ~(K):

2 2 2 2 . 1( (1 + K r )

0

b) Similarly, for the correlation function

B(r)

we obtain

V(K) 2 [ 2 2J a r K r

__ o exp- T 2.j;.

and

c) Finally, for the correlation function

B(r)

we have

V(K)

2 1 a r(v + 2)

J;{r(v)

r 0

+! 2 2 v 2 (1 + K r ) 0

18

(1. 29)

(1. 30)

(1. 31)

(1. 32)

(1. 33)

(1.)4)

and

r ( v + ~)

1( ,;;. r ( v) 3 •

2 2 v +­(1 + K r )

2 0

1.4 Locally homogeneous and isotropic random fields

(1. 35)

It should be noted that it is a very rough approximation to regard actual meteorological

fields as homogeneous and isotropic random fields. Atmospheric turbulence always contains

large scale components which usually destroy the homogeneity and isotropy of the fields of the

meteorological variablesj moreover, these components cause the meteorological fields to be

non-stationary. Thus, there is a close relation between the non-stationarity, inhomogeneity

and anisotropy of the meteorological field of an atmospheric variablej basically they are due

to the same causes. Therefore, in analyzing the spatial structure of meteorological fields

(and some others) it is again appropriate to apply the method of structure functions. In

fact, the difference between the values of the field f(;) at two points ;l and ;2

is chiefly

affected only by inhomogeneities of the field f with dimensions which do not exceed the dis-

tance 1~1 - ~2 1. If this distance is not too large, the largest inhomogeneities have no

effect on f(~1 ) - f(~2 ) and therefore the structure function

(1. 36)

can depend only on ;l - ~2 . At the same time, the value of the correlation function Bf(;1,;

2)

-+ --+ is affected by inhomogeneities of all scales, so t~at for the same values r1

and r2

the

function Bf(;1,;2 ) can depend on each of the arguments separately and not just on the differ­

ence ; 1-;2 . Thus, we arrive at the concept of local homogeneity [8]. The random field f(;)

is called locally homogeneous in the region G if the distribution functions of the random vari-

(--+ --+ -+ --+

able f r 1 ) - f(r2) are invariant with respect to shifts of the pair of points r1,r

2, as longas

these points are located in the region G. Thus, the mean value f(;1

) - f(;2

) and the struc-

--+ ture function (1.36) of a locally homogemeous random field depend only on r

1- r

2:

19

(1. 37)

A locally homogeneous random field is called locally isotropic in the region G if the distri­

bution functions of the quantity f(;1

) - f(;2) are invariant with respect to rotations and

mirror reflections of the vector ;1

- ; 2, as long as the points ;l and ; 2 are located in G [8].

The structure function of a locally isotropic random field depends only on 1;1- ; 21:

(1. 38)

A locally homogeneous random field f(;) can be represented in a form similar to (1.16):

00

f(o) + f f f (l - ei"K·r)dcp(/(). ( 1. 39)

Here f(O) is a random variable, and the random amplitudes ~(~) satisfy the relation

(1.40)

where~(/() ~ 0 is the spectral density of the random field f. Substituting the expansion

(1.39) in Eq. (1.37) and using the relation (1.40), we obtain

2 f f f (l - cos l(.r)~(/()dl<. (1.41)

-oo

In the case where the field f is locally isotropic, Df(r)

case

00

(1.42)

0

20

It should be noted that the integral (1.42) also converges in the case where ~(K) has a singu­

-a. larity at zero of the type K (a.< 5), which corresponds to the case where the large-scale

components of the turbulence have infinite energy {j] .

We can also examine a locally isotropic field along any line in space. The corresponding

expansion of the field f has a form similar to the expansion of a random function with sta-

tionary increments. In this case the expansion of the structure function is similar to Eq.

(1.15):

00

2 J (l - cos Kr) V(K)dK,

The functions !(K) and V(K) are connected by the same relation as in the case of an isotropic

random field:

l dV - 2rtK dK (1.44)

In addition to the expansions (1.39) and (1.42) of a locally isotropic random field and

its structure function as three-dimensional Fourier integrals, we shall also use two-dimen-

sional expansions in the plane x = canst:

00

f(x,o,o) + J J {1 - exp[i(K2y + K3

z)J} d<jr(K2,K3,x).

-oo

Here f(x,O,O) is a random function and <jr(K2

,K3,x) obeys the relation

( 1.46)

21

Consider the difference of the values of f(x,y,z) at two points of the plane x = canst. Using

the expansion (l.45) we obtain

f(x,y,z) - f(x,y 1 ,z 1 )

-co

We calculate the correlation fUnction of two such differences taken in the planes x and x 1 :

[!(x,y,z)- f(x,y 1 ,z 1 [J[f(x 1 ,y,z)- f(x 1 ,y 1 ,z 1 }]

[!(x,y, z) - f(x,y 1, z 1 [J [!*(x 1 ,y, z) - f*(x 1 ,y 1 , z 1 )]

J J J J {exp[_i(K2y' + K3z• )] - exp[i(Kc7 + K3z)]} {exp[-i(K;,Y' + K3z')] --co

Using Eq. (l.46), we obtain

l}'(x,y,z)- f(x,y 1 ,z 1 )][f(x 1 ,y,z)- f(x 1 ,y 1 ,z 1 )]

2 JJ {l- cos[K2(y- y') + K3(z- z')J} X -co

( l.47)

Clearly, correlation between the difference f(x,y,z) - f(x,y 1 ,z 1 ) and f(x 1 ,y,z) - f(x 1 ,y 1 ,z 1 )

is produced only by those inhomogeneities with scales exceeding the distance lx - x 1 I between

the planes, i.e. t ~ lx - x' I. Since the wave number K N 2n/t corresponds to the scale t ,

22

correlation between these differences is caused only by that part of the spectrum for which

the wave numbers obey the condition K I x - x 1 I $ l. Consequently, the function F( K2

, K3' I x-x 1 I), which is the spectral density of the quantity

[f(x,y,z)- f(x,y',z 1 )][f(x 1 ,y,z)- f(x 1 ,y 1 ,z 1 )],

fallS off rapidly for Klx - X1 I > l. Using the algebraic identity

l[ 2 2 2 2] (a - b)(c - d) = 2 (a- d) + (b - c) - (a- c) - (b - d) ,

we can express the left hand side of Eq. (l.47) in terms of the structure function of the

field f, with the result that the formula takes the following form:

Df(x- X1

, y- y 1, z- z') - Df(x- x', o,o)

Setting x = x 1, y- y 1 TJ , z - z 1 ~ , we obtain

00

2 J J [l- cos(K2TJ + K30]F(K2,K3'0)dK2dK3' (L49) -co

i.e., the function F(K2,K3,o) is the two-dimensional spectral density of the quantity

Df(O,TJ,~). In the case of local isotropy in the plane x =canst, F(K2

,K3

, lxl) depends only

on K = JK~ + K~ and then [o]

( l. 50) 0

23

Here p2 = ~ 2 + ~ 2 and F(K2,K3,o) = F(\/K~ + K~, 0). In the case where the field f(;) is

homogeneous and isotropic in the plane x = canst, its correlation function in this plane can

be expressed in terms of F(K,O) by using the formula

(1.51)

0

The function F(K,x) of a locally isotropic random field can be expressed in terms of its

three-dimensional spectral density ~(K). Substituting.the expansion (1.41) in the left hand

side of (1.48) and using the evenness of the function I(K1,K2

,K3

) in K1, we obtain the rela-

tion

( 1. 52)

Inverting this Fourier integral, we find

(1.53)

We now consider some examples of structure functions.

a) The structure function of a homogeneous and isotropic random field can be expressed

in terms of its correlation function by using the formula

Setting (see example c on page 18)

24

here, we obtain

The spectral density corresponding to (1.55) is

For r << ro

i.e.

r(v + ~)

n Jri.r(v)

2 3 a r 0

+~ 2 2 v 2 (l + K r )

0

( 2 r{l - v) ( r ) Df r)"' 2a r(l + v) 2r

0

2v

2v "'r

b) Consider the structure function

c2rP (o < p < 2).

(1. 55)

( 1. 35)

(1. 56)

The one-dimensional spectral density corresponding to this function (see example b on page 13)

is

V(K) r(p + 1) sin ~ 2 -(p + 1) 2n 2 c K •

(1.57)

We use the relation (1.44) to find the three-dimensional spectral density I(K):

1 dV r(p + 2) s;n np 2 -(p + 3) - 2nK dK =

41\2 ..._ 2 c K •

(1. 58)

25

We also calculate the two-dimensional spectral density F(KJx) corresponding to the structure

function c2rP. Substituting the expression (1.58) in the right hand side of Eq. (1.52) and

carrying out the integration) we obtain [P]

1 + p

2 R - 1 (Kx) 2K p (Kx) 22

1 + 2 c sin rep r(1 + ~) 2 2 p + 2 rc K

( 1.59)

Since for z >> l} K (z) N ~z e-zJ for Kx >> 1 the function (1.59) rapidly approaches zeroJ v

which corresponds to the abovementioned property of the ·function F(KJx).

For

v R 2

2 a

2V r

0

22v-l r(l + v) 2 r(1 - v) c

the structure function of the preceding example coincides with the structure function c 2~ for

r << r. The spectra of these functions agree for Kr >> 1 (see Fig. 3). 0 0

26

Chapter 2

THE MICROSTRUCTURE OF TURBULENT FLOW

Introductory Remarks

In what follows we shall repeatedly need basic information about the statistical proper-

tieS of developed turbulent flow. The statistical theory of turbulence) which was initiated

in the papers of Friedmann and Keller) has undergone great development in the last two decades.

A very important advance was achieved in the year 1941} when Kolmogorov and Obukhov established

the laws which characterize the basic properties of t~e microstructure of turbulent flow at

very large Reynolds numbers; some years later certain foreign scientists (OnsagerJ von Weiz-

sacker} Heisenberg) arrived at the same results. In this chapter we present only those results

of the Kolmogorov theory which are most important for our purposes) and refer to the original

sources [8 J 9 J 11-16] for more detailed information.

2.1 Onset and development of turbulence

Consider an initially laminar flow of a viscous fluid. This flow can be characterized by

the values of the kinematic viscosity vJ the characteristic velocity scale v and the charac-

teristic length L. The quantity L characterizes the dimensions of the flow as a whole} and

arises from the boundary conditions of the fluid dynamics problem. The laminar flow of the

fluid is stable only in the case where the Reynolds number Re = vL/v does not exceed a certain

critical value Re cr As the number Re is increased (e.g. by increasing the velocity of the

flow) the motion becomes unstable. This stability criterion can be explained by the following

simple considerations.

Suppose that for some reason or other a velocity fluctuation vt occurs in a region of

size t of the basic laminar flow. The characteristic period T = t/vl which corresponds to

this fluctuation specifies the order of magnitude of the time required for the occurrence of

the fluctuation. The energy (per unit mass) of the given fluctuation is vt2

. Thus} when the

velocity fluctuation under consideration occurs) the amount of energy per unit time which goes

over from the initial flow to the fluctuational motion is equal in order of magnitude to

27

vt2/~ N vl 3/t. On the other hand) the local velocity gradients of our fluctuation are given.

by the ratio vt/tJ and therefore the energy dissipated as heat per unit mass of the fluid per

unit time is of the order of magnitude E = vvt2/t

2. If the velocity fluctuation which arises

is to exist) it is clearly necessary that the inequality vt 3/t >vvl2/t

2 hold) i.e.

tv' t

v

Since all these calculations are accurate only to within undetermined numerical factors) it

would be more correct to write the relation we have just obtained in the form Ret> Recr'

Here Ret denotes the 11 inner11 Reynolds number corresponding to fluctuations of size tJ and

Re is some fixed number which cannot be determined precisely. These considerations show cr

tha~ generally speaking) large perturbations) corresponding to large values of the number

RetJ are most easily excited. But if the condition vL/v > Re cr

whole) then the laminar motion is stable.

is not met for the flow as a

Let us assume that as the number Re is gradually increased the laminar motion loses sta-

bility and there occur velocity fluctuations vt with geometric dimensions t. If the initial

number Re = vL/v was only a little larger than RecrJ then the fluctuations which arise have

small velocities and Ret = vtt/v < RecrJ i.e.J the velocity fluctuations which occur are

stable. As Re = vL/v is increased further) the velocities of the fluctuations which occur

increase and their inner Reynolds number Ret may exceed the critical value. This means that

the 'qfirst order'' velocity fluctuations which arise lose stability themselves and can transfer

energy to new "second order11 fluctuations. As the number Re is increased further the 11 second

order11 fluctuations become unstable J and so on.

Let the geometrical dimensions of the smallest fluctuations which occur be t0

J and let

their velocities be v0

• For all the velocity fluctuations with sizes t>t0

the inner number

Re. is large (exceeds Re ). It follows from this that their direct energy dissipation is -v cr

small compared to the energy which they receive from larger perturbationsj thus) these flue-

tuations transfer almost all the energy they receive to smaller perturbations. Consequently)

the quantity vl 3/tJ which represents the energy per unit mass received per unit time by eddies

of the n'th order from eddies of the (n-l)'th order and transferred by them to eddies of the

(n+l)'th order) is constant for perturbations of almost all sizes (with the exception

28

of the very smallest). In the smallest velocity perturbations with sizes t0

J this energy is

converted into heat. The rate of dissipation of energy into heat is determined by the local

2/ 2 velocity gradients in these smallest perturbation~) i.e. is of order E~ vv0

t0

• ThusJ for

velocity fluctuations of all scales) except the very smallest) we have vt 3/t ~ E and

( 2.1)

i.e.J the size of the fluctuational energy belonging to perturbations with sizes of the order

tis proportional to tl/3. Moreover} for all scales the size of these fluctuations depends

only on one parameter} the energy dissipation rate E·

We now calculate the dimension t0

of the smallest inhomogeneities. For them the rela­

tions v ~ (Et )l/3 and vv2/t 2 .... E hold. Solving this system of equations} we find that

0 0 0 0

t 0

v .... 0

4 \/VE . ( 2. 2)

The quantity t0

can also be expressed in terms of the dimensions of the largest eddies LJ

which are comparable with the dimension of the flow as a whole. Since v~/L .... EJ then substi­

tuting this expression for E in Eq. (2.2) we obtain

t 0

L

ThusJ the larger the Reynolds number of the flow as a whole} the smaller the size of the

velocity inhomogeneities which can arise.

( 2.3)

The considerations given above are essentially only of a qualitative nature} but they can

be used as the basis for constructing a more rigorous theory.

2.2 Structure functions of the velocity field

in developed turbulent flow

The largest eddies which arise as a result of the instability of the basic flow are of

course not isotropic) since they are influenced by the special geometric properties of the

29

flow. However these special properties no longer influence the eddies of sufficiently high

order, and therefore there are good grounds for considering the latter to be isotropic. Since

eddies with dimensions much larger than lrl- r21 do not influence the two point velocity dif­

ference ~(~l) - ~(~2), then for values of lrl- r21 which are not very large, this difference

will depend only on isotropic eddies. Thus, we arrive at the scheme of a locally isotropic

random field. Since the field v(r) is a vector field, it is characterized by a set of nine

structure functions (instead of by one structu~e function) composed of the different components

of the vector ~:

( 2.4)

Here i,k = 1,2,3, the vi are the components with respect to the x,y,z axes of the velocity

vector at the point rl, and the vi are the components of the velocity at the point ;i = ;l+ ;.

It follows from the local isotropy of the velocity field that Dik(;) has the form (see

e.g. [14,15,16])

where oik = l for i = k, oik ~ 0 for i t k, and the ni are the components of the unit vector

directed along;, D rr

(v- v 1)2

, where v is the projection of the velocity at the point r r r ~ ~ ~ ~ ~

r 1 along the direction of r, and v~ is the same quantity at the point rl = r1

+ r;

Dtt = (vt - vt)2

, where vt is the projection of the velocity at the point ;l along some direc-

tion perpendicular to the vector;, and vt is the same quantity at the point ;i· D is called rr

the longitudinal structure function and Dtt the transverse structure function of the velocity

field. In the case where the velocities v are small compared to the velocity of sound, the

~

motion of the fluid can be regarded as incompressible. In this case div v = 0. It follows

from this relation that the tensor Dik(;) satisfies the condition

where we sum from l to 3 with respect to repeated indices. Substituting from Eq. (2.5), we

30

find that

( 2.6)

Thus, the tensor Dik is determined by the single function Drr(r) (or Dtt(r)). For values of r

Which are not very large, the form of the function D (r) can be established by using the rr

qualitative considerations developed above.

Let r be large compared to the inner scale t of the turbulence (i.e. compared to the 0

size of the smallest eddies) and small compared with the outer scale 1 of the turbulence (i.e.

compared to the size of the largest, anisotropic eddies). Then the velocity difference at the

points ;l and ;l = ;l +; is mainly due to eddies with dimensions comparable to r. As

explained above, the only parameter which characterizes such eddies is the energy dissipation

rate E. Thus, we can assert that D (r) is a function only of rand E, i.e. D (r) = F(r,E). rr rr

But the only combination of the quanti ties r and E with the dimensions of velocity squared is

the quantity (Er)2

/ 3, and it is impossible to construct dimensionless combinations of these

quantities. Thus, we arrive at the conclusion that D is proportional to (Er) 2/3, i.e. rr

D (r) rr ( t << r << 1), 0

where C is a dimensionless constant of order unity [a].

( 2. 7)

We can arrive at the aame conclusion by starting with Eq. (2.1). In fact,

D (r) rr

2 is mainly due to eddies with sizes r, i.e. D (r) ~ v • rr r But by (2.1) v ~ (Er) 1/ 3, whence we

r

again arrive at (2.7). Eq. (2.7) was first obtained by Kolmogorov and Obukhov [8,11] and

bears the name of the "two-thirds law". Using the relation ( 2.6) we can also obtain the quan-

(t << r << 1). 0

31

( 2.8)

We now consider the value of the structure functions for r << t . In this case the 0

changes of velocity occur smoothly) since now the relative motions are laminar. The velocity

difference v (; ) - v(; + ;) can therefore be expanded as a series of powers of the quantity r l l

r. Retaining only the first non-vanishing term of the expansion) we obtain

where ~ is a constant vector. From this we find that

D (r) rr 2 ar

for r << t 0

) where a is some constant. The quantity a can be related to the quanti ties v and

E by a more careful argument (see e.g. [8 J llf] ) . It turns out that

D (r) l E

15 v r rr

It follows from (2.6) that

Dtt(r) 2 E

15 v r

Thus

D (r) rr

2 (r << t ) .

0

in this case

2 (r << t ) .

0

(for r >> t ) J 0

(for r << t L 0

(for r >> t L 0

(for r << t ) . 0

( 2.9)

(2.10)

( 2.11)

In Eqs. (2.11) it is assumed that r << LJ where Lis the outer scale of the turbulence. Fur-

ther hypotheses are needed to determine the form of the functions Drr and Dtt for intermediate

values of the argument r (see [1~ and [13]) ·

32

When r is increased) the condition r << L is violated. Then the large eddies) which can­

not be regarded as isotropic and homogeneous) begin to influence the value of vr(;1 )- vr(;1+;).

In this case) the structure functions Drr and Dtt depend on the coordinates of both observa­

tion points) and no universal law can be given which describes the structure functions for

large values of r. We can only state that the growth of the structure functions D (r) and rr

D ( r) slows down for r >> L [b J . Fig. 6 shows the general shape of the function D ( r) • For tt rr

small values of r,the curve can be replaced by a parabola with great accuracy) then the part

of the curve corresponding to the "two-thirds law" begins) and finally) in the region of the

outer scale of the turbulence) the curve starts to "saturate". The dashed parts of the curve

show the asymptotic behavior of the function D (r) for r << t B.J.J.d r >> t . To make more rr o o

precise the definition of t0

J the inner scale of the turbulence) we shall call the inner scale

of the turbulence that value of r for which the functions Er2/l5v and CE 2/ 3r 2/3 intersect) i.e.

t 0

Drr(r)

Fig. 6

Lo

General shape of the structure function D (r). rr

(L is the outer scale and t the inner scale of the turbulence.) 0 0

33

( 2.12)

In many applications of turbulence theory an important role is played by the isotropic

eddies described by the structure functions (2.11). In these case~ it is often expedient to

regard the values of r in Eqs. (2.11) as not being bounded above by the value L. This is

almost always the case in the problems considered in this book. However) if we cannot neglect

the "saturatioh1

of the structure function) it is necessary to use interpolation formulas which

approximately describe the behavior of the structure function for large values of r. For small

values of r these formulas must reduce to the same values of D as given in the expression rr

(2.11). One of the functions satisfying the stipulated requirements is the function

( 2.13)

Proposed b V n K ~ 1 Here -v1 2 1' s the mean 1 · t Y o arman. square ve oc1 y fluctuation and L is the outer

0 scale of the turbulence. As shown in example a) on page 12J for r << L the function (2.13) is

0

approximately equal to

v 12 2/3 -;--2/3 r J

L 0

i.e. coincides for r << L with the 11 two-thirds law11 • 0

satisfying the same requirements.

(2.14)

We can also write down other functions

2.3 Spectrum of the velocity field in turbulent flow

We now study the spectrum of turbulence. As shown in Chapter lJ the structure function

Df(;) of a locally isotropic scalar field can be represented in the form

2 J J J [1 - cos(/(.r)]!Ci<)d!C.

'larly the structure tensor D (;) can be represented in the form Siilll ) ik

( 2.15)

where ~ik(K) is the spectral tensor of the velocity field. The form of this tensor can be

determined from the incompressibility equation and the local isotropy condition) i.e.J the ten­

sor ~ik(K) can be expressed in terms of the vector; and the unit tensor oik [16j as

( 2.16)

-7

where G(K) and E(K) are scalar functions of a single argument) the magnitude of the vector K·

It follows from the incompressibility equation that

0

(see page 30) . Taking the divergence of Eq. (2.15)) we obtain

Iff -7 -7 ih -7 -7

sin(K•r)K. ~·k(K)dK = 0. l l

Thus the condition Ki Iik(;) = 0 must be met. Substituting Eq. (2.16) into this condition)

2 I 2 we get GK Kk + EKk = OJ whence G = - E K . Consequently we have

(2.17)

Thus) Eq. (2.15) takes the form

(2.18)

35

To explain the physical meaning of the function E(K) let us assume temporarily that the

velocity field is isotropic and that the correlation tensor Bik(;) exists as well as the struc~

ture tensor Dik(;). Then we have

Iff -oo

We contract this expression with respect to the indices i and k. Taking into account that

oii = 3 and KiKi = K2J we obtain

Setting r = 0 in this equation} we get

l ~ 2v J J J E(K)d;.

ThusJ the quantity E(K) is the spectral density in three-dimensional wave vector space of the

distribution of the energy of the velocity fluctuations.

We now find the form of the function E(K) corresponding to the "two-thirds law". To do

this we contract the expression (2.18) with respect to the indices i and k

00

( 2.19)

It follows from (2.5) that

36

D (r) rr

we obtain

00

4 J J J [l - cos("K.r)]E(K)di<.

-oo

( 6) d ( l 58) in example b) on page 25 J we find that Comparing this expression with Eqs. 1.5 an ·

E(K)

where

A=

A 2/3 -ll/3 E K J

(8) . :rc liT 3 Sln 3

24:rc2

c = o.o61c.

( 2. 20)

th one-dimensional spectral expansion of the quantity Many papers on turbulence theory use e -+

D with respect to the magnitude of the vector K) i.e. rr

00

D (r) rr

(2.21)

· of the structure tensor (2.18). Using instead of the three-dimensional spectral expanslon

Eqs. (1.56) and (1.57) of example b) on page 25) we obtain

where

tion D rr

( 2. 22)

r(~) sin 3 _:::__ __ .:::.c. 2:rc

( ) and ( 2.22) correspond to the struction func-The spectral energy densities 2.20

2/3 2/3 th ~ t that the structure functions D (r) and CE r and do not reflect e ~ac rr

37

Dtt(r) have a parabolic character for r << t0

) because of the smoothing action of the viscous

forces. As we have already seen above) the action of these forces is apparent from the fact

that eddies with sizes of the order of t0

are stable. Thus) in a turbulent flow there are no

inhomogeneities with sizes much smaller than t0

. This means that the spectral density E(K)

rapidly dies off to zero when K > 2~/t • The character of this cutoff is related to the form _,y 0

of the structure function D (r) in the transition region from r << t to r >> t0

) and at pre-rr o

sent has not yet been ascertained exactly. The spectral density (2.20) goes to infinity at

K = 0. This is related to the fact that the structure function D = CE 2/3r2/3 goes to infin­rr

ity at r = oo, In all actual cases) of course) Drr(oo) < ~ and the growth of the spectral den-

sity E(K) for K < K . slows down (the quantity K . ~ 2~/L corresponds to the outer scale of rm_n Inln 0

the turbulence) . The form of the function E(K) for K < K . obviously has to depend on the rm_n

concrete conditions which determine the formation of the largest eddies) and therefore it can-

not be universal [c].

As an example of a spectral density E(K) which is finite for K = 0) we can give the

spectrum of the von Karman function (2.13). Instead of the rather formidable expression for

E(K) we give here the relatively simple formula for E1

(K):

I'(5/6)

3 j; r(l/3)

L v' 2 0

(2.23)

For K = 0) E1 is finite and for KL0

>> l) E(K) N K-5/ 3) i.e. coincides with the spectral den­

sity E1 (K) corresponding to the "two-thirds law".

Numerous experimental investigations of the form of structure functions in the atmosphere

(mainly in the layer of the atmosphere near the earth)) where the Reynolds numbers are very

large [a]) give good confirmation of the "two-thirds law" for distances of the order of centi­

meters and meters [17]. These experiments also allow the constant C figuring in (2.11) to be

rather accurately determined. Some as yet rather sparse measurements of the structure func-

tion in the lower troposphere also agree with this law for distances of the order of tens of

meters [18] . One can arrive at a similar conclusion by analyzing the spectrum of airplane

buffeting [i~ .

We defer a more detailed presentation of the results of experimental investigation of

38

tructure of the wind in the atmosphere until Part IV. We turn now to the microstructure the s

of the concentration of a passive additive in a turbulent flow.

39

Chapter 3

MICROSTRUCTURE OF THE CONCENTRATION OF A CONSERVATIVE

PASSIVE ADDITIVE IN A TURBULENT FLOW

3.1 Turbulent mixing of conservative passive additives

As already noted above) the microstructure of the refractive index of the atmosphere is

determined by the structure of the temperature) humidity and wind velocity fields. The tem­

perature) humidity and some other characteristics of the atmosphere can very often be regarded

approximately (to a high degree of accuracy) as conservative passive additives.

If a volume of air is characterized by a concentration -a of additive J then by saying that

the additive is conservative we mean that the quantity -a does not change when the volume ele-

ment is shifted about in space. By the dd1't' b · · a 1 ve e1ng pass1 ve is meant that the quantity -a

does not affect the dynamical reg1'me of the turbulence [aJ. Th e problem of the microstructure

of the concentration of a conservative passive additive was first considered by Obukhov and

Yaglom [21, 2~ for the case of the temperature field.

We shall start from the equation of molecular diffusion

dt) dt + div(- D grad -a) ( 3.1)

which must be obeyed by the concentration -a of a passive conservativ~ additive. Here D is the

molecular diffusion coefficient of the additive and

is the total time derivative taken along the moving parcel of air. We shall assume that the

air can be regarded as incompressible) i.e. that div ~ = o. In this case

~.grad -a div(~-a) J

4o

and Eq. (3.1) takes the form

~ + div(;-a - D grad -a)

or

o. ( 3.2)

AS usualJ we separate the value of the quantity -a into the mean value ":§ and the departure -a'

from the mean value) i.e. -a=":§+ -a•. Similarly) we set vi

we obtain

a-a a (-- - a-a ~t + ~ v. -a+ v!t)' - D ) oc; oxi 1 1 ~

o.

v. + v!. Averaging Eq. (3.2)J 1 1

( 3.3)

The expression in parentheses represents the mean density of flow of the quantity -a. The quan-

tity ~ = - D grad ":§ represents the mean flow of -a caused by molecular diffusion. The quan-

~ -:::;--tity qa = v -a is connec~ed with the transport of -a by the mean velocity of the flow; it is

usually constant and dro~s out of the equation. FinallyJ the quantity ~ = -;r -a' represents

the density of turbulent flow of -a. It is natural to assume that qT is proportional to the

gradient of the mean concentration [b]J i.e.

- K grad :§. ( 3.4)

The quantity K is called the coefficient of turbulent diffusion and usually exceeds the coef-

ficient of molecular diffusion by several orders of magnitude. Of courseJ the value of K

depends on the intensity of the turbulence.

It should be remarked that there is an essential difference between the mechanisms of

molecular and turbulent diffusion) which the following example clarifies. Let the mean value

~ depend only on one coordinate zJ say. Assume for definiteness that ":§ increases as z in-

creases. Consider the values of -a at two different levels z1

and z2 . Because of the turbu­

lent mixingJ parcels of air from the level z2

will arrive at the level z1 J and conversely

parcels of air from the level z1

will arrive at the level z2 . ThusJ at each of these levels

41

parcels of air characterized by the value~ (z1 ) appear next to parcels of air characterized

by the value~ (z2 ). As a result, the mean concentrations taken over each level change in

such a way that the difference between them decreases, while the "variegation" of the values

of ~ at each level is greatly increased. Thus, as a result of turbulent mixing the inhomo-

geneity of the spatial distribution of ~ is increased and large "local" gradients of ~ are

created. Only after large local gradients of ~ have appeared does the process of molecular

diffusion play a significant role by smoothing out the spatial distribution of ~.

The inhomogeneity of the spatial distribution of ~ can be characterized quantitatively

by the following measure of inhomogeneity in the volume V:

G = ~ J ~ dV ~ - ~. ( 3-5) v

Clearly G = 0 only when ~~ = 0 in the whole volume v. Subtracting Eq. (3.2) from Eq. (3.3),

we obtain an equation which determines d~ 1 /dt:

o. ( 3.6)

Multiplying this equation by ~~ and using the obvious relations [cJ

d d~l d (l ~ 12) d (1 ~12) ~I £( V • ~ I ) ~~v. dx:"" = v. dx:"" 2 v. ) i l l l 'd"X,"" 2 X. l l l l

d - d~ d (D d~ 1 ) = d (~ 1 D d~ 1 ) D(d~ I)

2 ~I d'x."'(v!~) v!~~ dX,"" '

~I - ) X. l l ~ ~ ~ ~ 'd"X,"" l l l

we obtain

o.

42

gl·ng this equation and using the fact that A'.rera

we have

~~ d (v!~~) ~l

o,

d ~ ( ---2 ~ dt "2 + div ~ :;;~ 1 - D ~ 1 grad ~~)

2 + D (grad ~~) 0.

+ ~ 1-; • grad ~ +

( 3. 7)

(3 7) th l V The l'ntegral of the divergence can be transformed we now integrate • over e vo ume .

into a surface integral, which is small compared to the volume integrals. Neglecting the sur-

face integral, we find

dG + dt

-=;-­Substituting into (3.8) the value of the quantity V 1 ~

~ = J [K(grad ;1)2

- D(grad ~· )~dV = O.

v

( 3.8)

~ from Eq. ( 3.4), we obtain

( 3.9)

Thus, the amount G of inhomogeneity increases due to the presence of turbulent mixing ("turbu­

lent flow") of the substance in question and decreases due to the presence of the process of

molecular diffusion. In the stationary case, dG/dt = 0 and both processes must balance each

other!

J K(grad ~) 2 dV = J D(grad ~ 1 )2

dV. ( 3.10)

v v

If the inhomogeneity measure is stationary not only for the volume as a whole but for its

separate parts as wellJ then Eq. (3.10) can be written in the form

- 2 2 K(grad ~) = D(grad ~') . ( 3.11)

The quantity

- 2 N = D( grad ~' )

represents the amount of inhomogeneity which disappears per unit time due to molecular diffu­

sion; this quantity is analogous to the energy dissipation rate E. The quantity K(grad ~) 2

represents the amount of inhomogeneity which appears per unit time due to the turbulence) and

is similar to v3/~ the rate of production of the energy of the fluctuations. We can carry

out an even more detailed analogy between the velocity fluctuations in a turbulent flow and

the concentration fluctuations of a conservative passive additive ~.

3.2 Structure functions and spectral functions

of the field of a conservative passive additive in a turbulent flow

Concentration inhomogeneities ~1 with geometrical dimensions t appear as a result of the

action of velocity field perturbations with dimensions t and characteristic velocities vt.

The amount of inhomogeneity appearing per unit time is clearly equal to vt~1 2/t (t/vt is the

time of formation of the velocity fluctuation vt and ~1 2 is the measure of inhomogeneity).

According to the second term on the right of Eq. (3.9)J the rate of levelling out of the inho-

inhomogeneity ~t which appears is not dissipated by the action of molecular diffusion) but

rather has a stable existence and can subsequently subdivide into smaller eddies. This pro­

cess of subdivision proceeds until inhomogeneities appear for which vt ~1 2/t1 ~ D~l2/t~ or

l l 1 t 1vt N D. These inhomogeneities are dissipated by the process of molecular diffusion. It

l follows from Eq. (3.10) that GJ the amount of inhomogeneity appearing per unit time due to

the largest eddiesJ is equal to NJ the rate at which the inhomogeneity is levelled out in the

smallest eddies. ThusJ the amount of inhomogeneity transferred per unit time from the largest

to the smallest eddies is constant and is equal to the rate N at which the inhomogeneity is

dissipated. Consequently) for inhomogeneities with tvt >>D (i.e. for all inhomogeneities

44

N t2/3

---~· E

( 3.12)

The size of the smallest inhomogeneities in the distribution of ~is defined by the relation

. )1/3 N DJ whence) s1nce vt N (Et1 J we obtain

tlVt 1 1

( 3.13)

The quantity tv t/DJ which determines the "stability" of inhomogeneities ~t with dimen­

sions t J is analogous to the Reynolds number which determines th: ~--stability of velocity per­

turbations. Since the values of D are near vJ the numbers tvt/v and tvt/D are always of the

Sflffie order. (For air v = 0.15 cm2/sP:c the coefficient of temperature conductivi.ty

Dt = 0.19 cm2

/secJ the diffusion coefficient for atmospheric water vapor is De= 0.20 cm2

/secJ

the diffusion coefficie~t for atmospheric co2 is DCO = 0.14 cm2

/secJ etc}) ThusJ the range

of sizes within which the relations vt N (Et)1 / 3 and2

~t2 - Nt2/ 3E-l/3 holdJare always the same.

The size of the smallest eddies and the size t of the smallest inhomogeneities in the distri­

bution of~ have the same order of magnitude) i.e. (v3 E-1 ) 3/ 4 N (D3 E-1 )3/ 4 . Because of

thisJ we shall henceforth make no distinction between these quantities.

A more rigorous theory can be constructed on the basis of the above qualitative consider-

ations. The largest inhomogeneities in the distribution of ~ originate from the largest

eddies and are not isotropic.

can be considered isotropic.

However the smallest inhomogeneities in the distribution of ~

Since the difference of the values of ~ at two points ;l and ; 2

is determined mainly by inhomogeneities of sizes 1;1 - ; 2 1J then in the case where

lrl - r2

1 << L0

the quantity ~(;1 ) - ~(;2 ) can be considered statistically isotropic. ThusJ

~ ~I • ~(r) is a locally isotropic random field. For the range of values t0

<< lr1 - r2 << L0Jthe

structure function

( 3.14)

depends on r I ~ ~ I N and the quantity E characterizing the turbulence) i.e. rl - r2 J

F(NJ EJ r). Dimensional considerations lead to the ~ormula [21]

2 N 2/3 a "J13 r

E

( t << r << L ) ) 0 0 ( 3.15)

where a is a numerical constant. The expression (3.15)) which corresponds to Eq. (3.12) and

is derived on the basis o~ qualitative considerations) was ~irst obtained in the paper o~

Obukhov [ 21] and is called the "two-thirds law" ~or the concentration o~ a conservative passive

additive. For r << t0Jthe di~~erence ~(;1 + ;) - ~(;1 ) is a smooth ~ction o~ r and can be

expanded in a seTies beginning with the ~irst power o~.r. 2 Consequently) D~(r) N r ~or

r << t 0

• More detailed considerations lead to the ~ormula [21]

(r << t ). 0

( 3.16)

We now make more precise the de~inition o~ t0

~or inhomogeneities in the distribution of

~. We shall assume that the quantity t0

is de~ined as the point o~ intersection o~ the asymp­

totic expansions (3.15) and (3.16)) i.e.

Solving this equation) we obtain

t 0

4 J 27a:D3

•

Thus) the structure ~ction D~(r) can be represented in the ~orm

~( c2 2/3

~or r >> t0

J ~

r

D~(r) 2

c2 t2/3 (~) ~or r << t0

J ~ 0 t

0

46

(3.17)

( 3.18)

and t is de~ined by Eq. (3.17). 0

( 3.19)

According to the general ~ormu1a (1.39)) a locally isotropic random ~ield can be repre-

sented in the ~orm o~ a stochastic integral

00 -+ -+

(1 - eiK·r)d~(~)J ~(o) + J J J ( 3· 20)

where the random amplitudes d~(K) satis~y the relation

(3.21)

The ~ction ~~(K) is the spectral density o~ the structure ~unction D~(;)J i.e.

00

2 j j j (1 - cos K·r)L(K) d)( • ( 3· 22)

-oo

The expression (3.22) can also be written in the ~orm

00

Brtf ( 3· 23)

0

The ~unction ~~(K) is the spectral density in the three-dimensional space o~ wave numbers

K1

J K2

J K3

o~ the distribution o~ the amount o~ inhomogeneity in a unit volume. The ~orm o~

this ~ction corresponding to the "two-thirds law" ~or the concentration ~ J was given in

example b) on page 25) i • e.

and

( 3· 24)

However) we note that for r << t0

the structure function D~(r) behaves quadratically) corre­

sponding to a rapid decrease of !~(K) forK ;2; 1/t 0

(see· page 38). At present) the exact cut­

off law of !~(K) forK "'l/t0

has not yet been ascertained exactly. In some calculations we

shall use a function !~(K) which vanishes for K > K • m Of course) such a definition of !~(K)

is not rigorously founded) but is recommended by its simplicity; it simply means that a cer-

tain interpolation formula has been chosen for D~(r). Thus we shall assume that

for K > K • m

(3.25)

The quantity Km can be related to the quantity t0

defined in Eq. (3.17). Substituting the

function (3.25) in the expression (3.23), we obtain

K m

J 0

( 1 sin Kr) -11/3 2d ---- K K K. Kr

For Kn[>> 1 the integral (3.26) is practically the same as the function (3.15).

and

2 2 ( 1 _ sin Kr) "' K r

Kr ~

48

( 3· 26)

ForK r << 1 m

Comparing this expression with the formula D~(r)

K and t : m o

K t m o

( -3/4 4 0.033n) = 5· 8.

c2t 2/3(~) 2 we obtain the relation between ~ 0 t )

0

2 2- -1/3 The expression C~ = a N E can be transformed into a form which permits this quantity

to be calculated in terms of data on the mean value profiles of the wind velocity and concen-

tration ~. To do this we have to use the relation

-2 2 -K(grad ~) = D(grad ~·) = N)

which expresses the quantity N in terms of characteristics of the mean profile of ~. A simi-

lar relation holds for the quantity

as well [a]) i.e.

Here u. is the average value of the wind velocity. 1

(3.11) into the formula for c~) we obtain

2 K 2

[

2 ]1/3

a ~ 2 (grad :a) •

( 3· 28)

Substituting the expressions (3.28) and

( 3.29 I)

All the quantities figuring in the right hand side of (3.29 1) can be obtained from observa-

tions of the average atmospheric profiles of the wind) the temperature and the quantity ~.

Methods of determining K from observations of wind and temperature profiles in the atmosphere

are described in [23-26J. It should be noted that a large amount of observational material

is required to check the formulas recommended for calculation of K in the papers cited, but

they do give the correct order of magnitude.

Eq. (3.29') can be used to calculate the range of sizes within which the "two-thirds

is valid. The mean square difference which the fluctuations produce in the difference of the

values of~ at two points grows like t2/3 (tis the distance between the observation points).

At the same time there exists between these two observation points a systematic difference in

the values of~, equal to Jgrad ~Jt, with square Jgradt~j 2t2 • It is clear that for suffi­

ciently small values of t, the fluctuational difference in the values of ~ is much larger than

the systematic difference (since x2

/ 3 is always >>x2

for x << 1). Thus, over small distances

the presence of a systematic difference in ~ does not affect the size of the fluctuations and

does not affect their homogeneity and isotropy. However, as the distance t is increased, the

systematic difference in the values of ~ becomes larger than the fluctuational difference and

for such scales the field of fluctuations of ~ cannot be considered locally isotropic. Let

us designate by L the dimension for which the relation

( - 2 2

grad ~) L

holds. Substituting Eq. (3.29') in this relation, we obtain

(3.30)

We shall call the quantity L0

= La-3/ 2, which differs from 1 only by a numerical factor, the

outer scale of the turbulence. In the free troposphere L0

ranges from tens to hundreds of

meters in order of magnitude [_95] . In practice u = u( z), i.e.

C<Ju:V

2

= (du)2

dx. dz J

132 J

and then

( 3.31)

50

the quantity L0

, we can write Eq. (3.29') in the following convenient form

4/3 - 2 L (grad ~) . 0

( 3.29 ")

Experimental investigations of concentration fluctuations of a conservative passive additive

.-6 have been carried out mainly in the layer of the atmosphere near the earth (temperature

fluctuations) and in the lower troposphere (fluctuations of the air's refractive index for

centimeter radio waves). Measurements of the temperature fluctuations in the layer of the

atmosphere near the earth have confirmed the "two-thirds law" as well as the dependence (3.29)

of the quantity C~ on the mean conditions, and have permitted determination of the numerical

· · (3 29') [ J Measurements of the spectrum of the air's refractive constant a figurlng ln . e .

index fluctuations in the free troposphere also agree very well with the "two-thirds law",

according to which the one-dimensional spectral density of the fluctuations is proportional

to K-5/3 G15, 96] ·

3.3 Locally isotropic turbulence with smoothly varying mean characteristics

So far we have considered the case of locally homogeneous and isotropic turbulence, where

the structure function

satisfies the condition

inside some region G with dimensions of the order of magnitude of the outer scale of turbu­

lence 10

• In the case where t0

<< 1;1

- ; 21 << L0

, Df(r) = C~r2/3 . If we consider another

region G' which also has dimensions of the order L0

and which is separated from G by a dis­

tance of order 1 , then the field f will also be locally homogeneous and isotropic in G' · 0

The structure function Df(;1,r

2) in the region G' will also be expressed by a "two-thirds

2 law", but in general with another value of the constant Cf. Thus, we can assume that the

51

quantity C~ is a smooth function of the coordinates, which changes appreciably only in

tances of order 10

• It is natural that C~ should depend on the position of the center of

-7 1 (-7 -7 R = 2 r1

+ r 2) of the two observation points. Thus, we arrive at the formula

In the general case

The function D~0 )(r), which describes the local properties of the field f, is

the regions G, G') ••. in which the turbulence differs only by its "intensity", characterized

2 the size of Cf. Just as we previously defined the spectral expansion

2 Il J {l - cos[K·(i\ - r2)~ !fP)aK ( 3. 33)

-oo

for points ;1,;

2 belonging to the region G, we can also define a similar expansion in each of

2 the regions G,G', •.. by considering Cf to be constant in each such region. Thus, we obtain

the spectral expansion

where [ f]

( 3-35)

52

( 3- 36)

ThuS, the relative spectral distribution of the fluctuations of the quantity f is identical in

the regions G,G', ••. and is described by the function !~o)(K); only the overall intensity

fluctuations, expressed by the quantity [g]

varieS from region to region. As in Chapter 1 we can introduce the two-dimensional spectral

density

-7 -7

-7 -7 il\ -7 r + r' ( 4 connected with the functions Df(r,r') and ~f(K ~2--) by relations similar to 1. 8), (l.52) '

and ( l. 53) , i. e •

(3.37)

2(;1 + ;~ (In Eqs. (3.34) - (3.37) we neglect the difference between Cf 2 ~ at two nearby

points which are separated from each other by the distance 1;1 - ; 21 << 10

, since this func­

tion changes appreciably only when its argument changes by a quantity of order 1 • ) 0

We now consider briefly the spectral expansion of the random field f(;) itself. For

simplicity we consider an example where there exists a correlation function of the fluctua-

tions of f, of the form

53

(Similar questions were considered by Silverman [92].) Let f(;) be represented by a sto

tic Fourier-Stieltjes integral [h]

Iff ~ .....

iK ·r (~) e dcp K

-oo

where it is assumed that f = 0. Consider the expression

00

JJJJJJ -oo

~ ..... ~ ..... 1(~ ~ and introduce the coordinates p = r- r' and R = 2 r + r').

~ ..... --?- 1(-> -+ -+ ( K - K 1 )·R + 2 K + K

1 ) • P

and

00

JJJJJJ -oo

By assumption

whence it follows that dcp(K)dcp*(K') must have the form

~ ~

dcp(K)dcp*(K') = M(K- /(t)!~o)(K ~ K') dK ~

dK' J

Then we have

( 3.4o)

( 3.41)

where obviously M(O) ~ 0 and !~o)(K) ~ 0. Substituting this expression in (3.40) and carry­

ing out the change of variables, we obtain

( 3.42)

-oo

2 -+ ~ 2 ~ ~ ( ) In the case where crf = const, M(K- K1) = crf o(K- K1

) and Eq. (3.41) reduces to 3.21. As

follows from ( 3.41), in the case of variable cr~, correlation appears between the neighboring

..... ~, 1..... ~,I -l [ J spectral components dcp(K) and dcp*(K ), i.e. if K - K $ L0

, then ~

dcp("K)dcr*("K•) ~ o.

(It follows from (3.42) that the function M(K) differs appreciably from zero in an interval

of order 1/L, since cr:(R) changes appreciably in an interval of order L .) 0 ~ 0

3.4 Microstructure of the refractive index in a turbulent flow

The refractive index n for radio waves in the centimeter range is a function of the

absolute temperature T, the pressure p (in millibars) and the water vapor pressure e (in

millibars), i.e.

n _ 1 = 10-6 x 79 ( P + 48ooe) • T T

( 3 .. 44)

It is not hard to see that the quantities T and e figuring in this formula are not, strictly

speaking,conservative additives. In fact, it is well known that when small parcels of air

are displaced vertically their pressure undergoes a continuous equalization with the pressure

of the surrounding air at the given height. The changes in pressure produce changes in tem-

perature satisfying the equation of the adiabat

55

dT Y - 1 dn T=-r-~'

where Y = c /c is Poisson's constant. The quantity dp is related to the change in p v

the barometric formula dp = -pgdz, where P is the density of' the air and g is the acce..~._c:r·»~'-'

tion due to gravity. Thus we have

dTT = -~ Eg dz = - y - 1 g dz ' Y p -y- RT

and

dT Y - 1 g _ ~ = dz = - -y- R = c - y a·

p

The quantity Ya = 0.98°/100 m is called the adiabatic temperature gradient (a rising parcel

of' air cools of'f' 0.98° f'or every 100m of' elevation). Integrating Eq. (3.45), we obtain

T + raz = const. Consequently, when parcels of' air are displaced vertically, the quantity

called in meteorology the potential temperature, preserves its value and may be regarded as

a conservative additive.

The water vapor pressure e which figures in Eq. (3.44) is also not a conservative quan­

tity, since it depends on the pressure. It can be expressed in terms of' the so-called spec

humidity q, which represents the concentration of' water vapor in the air (i.e. the ratio of'

mass of' water vapor to the mass of' moist air in a unit volume), by using the approximate

formula

e = 1.62 pq. (3.4

The quantity q is a conservative additive. (It is assumed that while the moist air is being

displaced there is no condensation of' water vapor.) Replacing the quantities T and e in

Eq. (3.44) by H - Y z and 1.62 pq, we obtain the formula a

(n - 1) X 106 = N = 79p (1 7800q ) H - Y z + H - y z

a a ( 3.48)

expresses the refractive index in terms of' the conservative passive additives H and q.

quantity N depends on z through p(z), H(z), q(z) as well as on z directly, i.e.

N = N(z, p(z), H(z), q(z)).

Suppose a parcel of' air f'rom the level z1 , characterized by the value

Nl = N(z1

, p(z1

), H(z1

), q(z1

)), appears at the level z2 as a result of' the action of' turbu­

lent mixing. Since the quantities H(z) and q(z) do not change when the parcel is displaced,

while the quantities z and p(z) take on the new values z2 and p(z2) at the level z2, the same

parcel will be characterized at the level z2 by the value N]_ = N(z2, p(z2), H(z1 ), q(z1 )).

The value of' N]_ dif'f'ers f'rom the 11local11 value of' N at the level z2 by the quantity

Thus, the fluctuations of' N which appear are not proportional to the 11f'ull

11 gradient n, but

rather to the quantity

(3.49)

57

The expression (3.49) has to be used to find the size of the fluctuations of the refractive

index n.

The structure function of the refractive index of the atmosphere can be represented by

Eqs. (3.18), i.e.

for t 0

<< r << L,

for r << t0

,

where the quantity en is defined by the expressions (3.29) and (3.49), i.e.

(3-51)

For the spectrum of the refractive index fluctuations we write the formula

0.033C2 K-ll/3 (K < K < K ) J n o m

where K0

"" l/L0

and L0

is the outer scale of the turbulence. The formulas given can be used

to calculate the size of the refractive index fluctuations.

Part II

SCATTERING OF ELECTROMAGNETIC AND ACOUSTIC WAVES

IN THE TURBULENT ATMOSPHERE

Chapter 4

SCATI'ERING OF ELECTROMAGNETIC WAVES IN TEE TURBULENT ATMOSPHERE

Introductory remarks

The problem of scattering of electromagnetic waves in the turbulent atmosphere has

attracted considerable attention,since this phenomenon is related to long-range atmospheric

propagation of short waves beyond the limits of the "radio horizon". We cannot linger here on

all the numerous problems associated with the use of radio scattering for the purpose of long

range communication. Instead, we consider only theoretical aspects and try to indicate the

physical content of the problem.

The problem which we consider in this chapter can be formulated as follows. A plane

monochromatic electromagnetic wave is incident on a volume V of a turbulent medium; because

of turbulent mixing within the volume V) there appear irregular refractive index fluctuations,

which scatter the incident electromagnetic wave. It is required to find the mean density of

the energy scattered in a given direction. To solve the problem, we shall assume that the

refractive index field within the volume V is a random function of the coordinates and does

not depend on time. The time changes in n which actually occur will simply be regarded as

changes in the different realizations of the random field n(;). Thus, we do not consider the

problem of frequency fluctuations and change of the frequency spectrum of the scattered

field [a] .

4.1 Solution of Maxwell's equations

We shall assume that the conductivity of the medium is zero and that the magnetic perme-

ability is unity. Furthermore, we shall assume that the electromagnetic field under consider­

-iwt ation has a time dependence given by the factor e • In this case Maxwell's equations take

59

the following form:

curl E = iklL

curl H = - ikEEt

div EE = 0.

(4.1)

Here k = w/c is the wave numb f th 1 t ti er o e e ec romagne c waveJ E is the dielectric constant (as

already stated above J we regard E as time independent) J and E and H are the amplitudes of the

electric and magnetic fields) so that the fields themselves are equal to Ee-iwt and He-iwt J

respectively. Taking the curl of the first of the equations (4.1) and using the second equa­

tion) we have [b]

( 4.2)

Since

div EE = E divE+ E•grad E 0)

we have

divE= - E•grad log E.

Using this equality and setting E 2 n ) we obtain

~ 2 ~ ~ 6E + k n E + 2 grad(E·grad log n) o. ( 4. 3)

We assume that the fluctuations of the refractive index n are small) i.e. that In - nl << 1.

Let n1 denote the deviation of n from its mean value) so that n = n + n1

• Since n is near

unity) we shall henceforth assume that n = 1 (if necessary we can change k to kn in all the

results).

Substituting n = 1 + n1 in Eq. (4.3)) we obtain

~ 2:! ~

6E + k E =- 2 grad(E•grad log(l + n1 )) (4.4)

60

To solve Eq. (4.4)J we can apply the method of small perturbations) whereby a solution is

sought in the form of a series

--')- ~ ~

E E0

+ E1 + E2 + ••• )

where the k 1th term of the series has the order of smallness n~. Substituting this series in

(4.4) and equating to zero each group of terms of the same order of smallness) we obtain

OJ (4.5)

(4.6)

In deriving Eq. (4.6)J the quantity log(l + n1 ) has to be expanded in a series of powers of

2 n1

J i.e. log(l + n1

) N n1

- (n1/2) + •... The quantity E

0 represents the amplitude of the

electric vector of the incident wave. Assuming that the incident wave is plane [c] J we set

E0

~ A0exp(ik·;). The quantity E

1 represents the amplitude of the electric vector of the

~ ~ ~ ~

scattered wave. (The terms of the series E = E0

+ E1 + E2 + ••• which come after E1 are

neglected because of their smallness [ctJ.)

As is well known) the solution of the equation

(a)

corresponding to outgoing waves is of the form

(b)

~

where r' is a variable vector ranging over the scattering volume v. We choose the origin of

coordinates inside the scattering volume. If the observation point ~ is at a great distance ~

from the scattering volume V as compared to the dimensions of VJ then for all r 1 the quantity

61

j; - ;r I is almost constant and close to r = 1;1. In this case) the quantity I; - ;r I can be

expanded in a series of powers of r'/r) i.e.

where i = ;;r is a unit vector directed from the origin of coordinates (chosen within the

scattering volume) to the observation point. If the inequality

k [ 2 ~ ~ 21 2r r' - ( m' r' ) - < < 1)

holds for all values of r') i.e. if the dimensions L of the scattering volume satisfy the

condition lr >> L 2) then

exp

Moreover) in the denominator of Eq. (b) we can replace I; - ;, j by r.

1 eikr 1hr-r- J

v

Thus) the formula

is valid in the Fraunhofer zone. We use Eq. (c) to solve Eq. (4.6)) obtaining

1 eikr J + 21( --r-

V

J v

~~ ~~

ik·r'~ ~ -ikm·r' grad( e A0

• grad n1 (r') )e dV'

62

(c)

second of the integrals figuring in (4.7) can be transformed by using Gauss' theorem:

J u grad cp dV'

v J cp grad u dV 1 ,

v

The surface integral vanishes) since the surface of integration can be moved beyond the

limits of the volume v. Since

Consequently

~ ~

-ikm·r grad e

J v

J v

ikeikr ~ + ~m J

v

-+ -+ ~ -ikm·r ikm e

J v

(4.8)

where

Cl = J nl (;') ei(k-~). ;r dV')

v

J ~ ~ ~

~ (~ i(k-km)·r' A0

• grad n1 r') e dV'.

v

Both terms in the right hand side of (4.8) represent spherical waves whose amplitudes and

phases depend on the refractive index fluctuations inside the volume v (through the random

variables C1 and c2). The second term is a longitudinal alternating electric field. Trans­

forming the expression for c2 by using Gauss' theorem) we can show that the second term in

(4.8) cancels the longitudinal component of the field contained in the first term, so that

the scattered field is purely transverse [~. Indeed, in calculating the flow of scattered

energy we can simply ignore the second term in (4.8).

4.2 The mean intensity of scattering

To calculate the density of flow of the scattered energy S = (c/8~) Re (E1

x lit) , i.e.

the average value of the density of energy flow during the period of one oscillation [f], we

need the QUantity H1 J which can be found by using the first of the eQuations (4.1) 1

(4.9)

We have neglected the rapidly decreasing term eikr/r2• Substituting (4.8) and (4.9) into the

formula for S, we obtain

64

~

Tbe density of energy flow in the direction m is eQual to

s m

(4.10)

(4.11)

wnere x is the angle between the vectors A0

and i. Substituting the expression for c1 J we

find

s m JJ vv

The g_uanti ty Sm is random. Its mean value is eQual to

s m

k4A2 . 2x C Sln 0 JJ

vv ( 4.12)

Thus) Sm is expressed in terms of the spatial correlation function Bn(-;1,;2) of the refractive

index fluctuations. We assume temporarily that the field of refractive index fluctuations is

homogeneous; later we shall extend our results to the case of locally homogeneous and iso­

tropic fields. Then Bn(;1J;

2) = Bn(-;

1 - ; 2) and the expression in the integrand of (4.12)

depends only on the distance ;1

- ; 2 • Introducing the change of variables ;l - ; 2 = P,

;1

+ ;2

= 2; in (4.12L we carry out the integration with respect to;) which gives as a

result the volume V. Then the expression for S takes the form m

(4.13)

We now use the expansion (1.22) of the correlation function as a Fourier integral:

co co

B (P) n Iff Iff -i"K· "P ih (-+) -+ e ~ K dK n (4.14)

-co -co

Substituting this expression in the integral

I

we obtain

I (4.15)

Let us examine the inner integral

When the region of integration is infinite, the inner integral equals o(r) = o("K - i + ~)

ih -+ -+ and therefore I= ~n(k- km). In the case of a finite volume of integration the function

has a sharp maximum in a region near the point 'A 0 and outside this region oscillates and

66

off rapidly [g] , while

co

J f f F(t)dt = J o(P)dVP 1.

-co V

Moreover, since F(O) = V/(8n3)) the function F(t) is appreciably different from zero in a

region of wave vector space which has a volume of order 8n3/v; of course, in each concrete I

case the shape of this volume and the behavior in it of the function F(t) depend on the

dimensions and shape of the spatial volume V. Thus

co

I = J J J In (K)F(K - k + ki)dK ~

where T represents the region of wave vector space with volume T = 8n3/V near the point ~ -+ -+ K = k - km. Therefore

I (4.16)

where I("K) is the mean value of the function !(K) obtained by averaging it over the region

of wave vector space of volume an 3;v surrounding the point KJ of course, this mean value

should not be confused with the statistical average. Substituting the expression (4.16) in

Eq. (4.13), we obtain

"' ih -+ -+ ~ (k - km). n (4.17)

~f the function !n(K) does not change much in ranging over the volume T = 8n3/V, then

!n(i~ - ~) "' !n(k - ~) and [h]

4 2 2 ck V A

0 sin X ,.., _,. _,.

s = ---2~-- ~ (k - km). m 4r n ( 4.18)

Using the expression (4.17), we can find the formula for the effective scattering cross sec­

tion of the volume V. Denoting by a the effective cross section for scattering into the

solid angle dQ in the direction with unit vector i, we obtain

- 2 8 r dQ m

da = --2- = c A

0

--s;r-

4 2 i _,. _,. 2nk V sin X ~n (k - km)dn • ( 4.19)

It follows from this formula that scattering at the angle e(cos e = k i/k) is determined

only by a narrow portion of the turbulence spectrum near the point K = k - ~. Thus, only a

small group of spectral components of the turbulence participate in the scattering at a given

angle e; these components form a spatial diffraction grating of fixed spacing t(e) which is

determined by the relation

t( e) 2n

lk- ~I 2n A

e ---e' 2k sin 2 2sin 2

( 4. 20)

i.e., satisfies the well known Bragg condition. ~~ directions of the vectors k - ~' k and

~ are related by the "mirror reflection" condition (the 11 nodal planes" of the spatial dif­

fraction grating are perpendicular to the vector k- ~). If the dimensions of the volume v

are of order R, i.e. V = R3, then besides the spectral components of the turbulence corres­

ponding to the spacing t(e) = A/(2sin ~), a part of the scattering at the angle e will be

contributed by spectral components corresponding to nearby periods in the interval

\2 ) 2k sin!!.. + ~ 2 - R 2 • e A ' s~n 2 ± H

68

the point A/(2sin ~) [~. In every concrete case it is easy to evaluate the size of

ll d · th • (e) r.J . There+>ore, we shall interval, and its size is usually sma compare ~ -v LJ .L

nenceforth make no distinction between functions !:u(~) and in(K).

E~· (4.19) was obtained under the assumption that the field of refractive index fluctua-

tions iS homogeneous in the volume v. However, this result can also be extended to the case

h +>' ld In fact, since the effective cross section for scattering at a of locally omogeneous .L~e s.

ll d d 1 on one "spectral component" of the refractive index inhomoge-given angle rea y epen s on y

neities, the remaining "spectral components" can be changed as one pleases; it is only impor-

tant that the quantity ~n (k - :k.i) retain its value for given k and i. Conse~uently, we can

alSO consider functions ln(K) which have a singularity at the origin, i.e. which correspond

to locally homogeneous random fields. Of course, in doing so we assume that we use Eq. (4.19)

onlY for values of k - ~ for which lk - ~~ >> 2n/L0

(L0

is the outer scale of the turbulence;

see page 33), or [k]

A --. -e << Lo. 2s~n 2

( 4. 21)

4.3 Scattering by inhomogeneous turbulence

We now consider the case where the turbulence is not homogeneous inside the scattering

volume and its mean characteristics change smoothly [ t].. The expression

4 2 2 ck A sin X

0 JJ vv

_,. _,. (_,. _,.. ) i(k-km) • r -r l 2

e (4.12)

obtained above for the density of flow of the scattered energy, does not depend on the assump­

tion that the field of the refractive index fluctuations is homogeneous and can therefore also

be applied to this case.. We use the formula

Then Eq. (4.12) takes the fo~

Using Eqs. (3.43) and (4.16)) we can express the inner integral in this formula by

j~ 0 ) (k - J<i), whe:rce, as above, ~ denotes averaging over a volume a,3 /V in wave number space.

Then we have

s m

Ck4

A2 s1·n2X

""'( ) 0 ~ 0 (k - ~)

4r2 n

or) introducing !n(K)R)

4 2 2

J v

s =: m

ck A sin X 0 J t(k-~)R)dVR.

4r2 v

( 4. 24)

( 4.25)

Eqs. (4.24) and (4.25) are the generalization of the expression (4.17) for the case of inhomo-

geneous turbulence.

4.4 Analysis of various scattering theories

1 .. In one of the first papers devoted to the problem of the scattering of radio waves

by atmospheric inhomogeneities (Booker and Gordon [27]) J it was assumed that the correlation

70

refractive index inhomogeneities has the form

(4.26)

th · se ious J·ustl· fication for using this particular correlation function) course) ere lS no r

t f d · 1 1 t• ns As established in it is used only because it is convenien or o1ng ca cu a 10 .

on page 18 (Eq. (l.30))Jthe spectral density !(K) corresponding to the function

the form

Substituting here 2k sin~ instead of K) we obtain

do( e) 2 :n:

4-k Vn~r~ _____ ...:::;......:,_.__--:::'sin~ dn

(, 22. 2e\2

~ + 4k r 0 Sln 2)

(4.27)

(4.28)

for da(e). Eq. (4.28) is the basic result of the paper of Booker and Gordon) and has served

for a long time as the starting point for numerous experimental investigations. In these

investigations) the reaults of measurements of refractive index inhomogeneities were analysed

~ d f. · in the correlation function with the aim of determining the parameters n1 an r 0 1gur1ng

(4.26)j values of the order of 60 meters were usually obtained for the quantity r 0 • For the

usual value of k and e) the quantity 2kr0sin ~ is much larger than unity. In this case

~ V nl sin~

da(e) = 8:n: r- . 4 e dn ) 0 Sln 2

(4.29)

i.e. J in the Booker-Gordon theory da(e) does not depend on the frequency and is determined

~ j wh1·ch characterizes the refractive index inhomogeneities of by the single parameter n1 r 0J

the atmosphere.

71

If we start from the basic formulas obtained in this chapter, it is not hard to show the

fundamental defect of this series of papers. As we have already emphasized, the quantity

da(e) is proportional to In(k- kiL i.e.,it is proportional to the 11 intensity11 of the inhomo~

gcneities with sizes satisfying the Bragg condition. From this point of view, the most

natural way to determine the function In(K) would be to measure it directly for the values of

bulent flow. Substituting Eq. (4.31) in (4.19), we obtain

da( e) 2r(v + 2)

2

J; r( v) (,1 + 4k2 2 . 2 8\ v \: r 0 s1n 2)

3 drt •

+-2

(4.32)

K which may be of interest in the applications; these values of K usually correspond to For 2kr0

sin ~ >> l, we obtain from this that

inhomogeneities with sizes ranging from some tens of centimeters to some tens of meters. How-

ever, in the Booker-Gordon theory and in the papers based on it, the value of the quantity

-;r: (K) ~n corresponding to comparatively small inhomogeneity sizes is determined from the outer

scale r0

of the inhomogeneities and from the characteristic n~ of the refractive index fluc­

tuations (which is also due to the most intense, large-scale inhomogeneities) by using the

essentially arbitrary formula (4.27). We should also note that it is hopeless to evaluate

2 small-scale fluctuations of the refractive index by using the quantities n1 and r

0 charac-

terizing large scale inhomogeneities, for the additional reason that the inhomogeneities of

the largest scale are always inhomogeneous and anisotropic, so that their relation to the

small scale inhomogeneities cannot be universal and must change as the general meteorological

conditions change.

2. In some more recent papers [28], the expression

B (r) n

has been used as a correlation function. The spectral function

r( v + ~) :n:\[;(r(v) 2 2 + 2

(l+Kr)v 2 0

(4.30)

(4.31)

corresponding to (4.30) was considered on page 19. As already shown, for Kr0

>> 1, the

function I (K) coincides with the spectral density corresponding to the structure function n

D (r) = c2r 2v. Thus, for v = l/3, the function (4.31) coincides in the region Kr >> l with n o

the spectral density l (K) N K-ll/3, which expresses the theoretical size distribution of n

inhomogeneities in the concentration of a conservative passive additive in a developed tur-

72

da(e) r( v + ~) (4.33)

22( v+l) J;c r( v)

The quantity da(e) depends on the frequency and on the parameter n~ / r~v which characterizes

the refractive index fluctuations. Of course, Eq. (4.33) for v = l/3 is much more justified

than the expression (4.29), because the spectral density of refractive index fluctuations

(4.31) used to derive it corresponds to the refractive index spectrum in a turbulent flow.

Row important this fact is for the problem being considered can be seen from the following

example. Since the correlation functions (4.26) and (4.30) are represented by outwardly very

2 similar curves (see Fig. 2), the parameters n1

and r0

determined from them will have values

which are very close together. At the same time, the ratio of the quantity da(e) calculated

from Eq. (4.33) to the value of da(e) calculated from Eq. (4.29) is equal to

(kr 0

e l - 2v sin 2)

to within a constant factor. For v l/3 this quantity is equal to

Since we usually have kr0

sin ~ >> l, Eqs. (4.29) and (4.33) will give greatly different

values for da(e). Eq. (4.33), as well as Eq. (4.29), expresses the spectral component

e 2 ! (2k sin-) which interests us in terms of the quantities r0

and n1 depending on the large n 2 .

scale inhomogeneities, and therefore it cannot be reliable enough, because the relation

73

between the small scale and the large scale inhomogeneities just cannot be universal.

call that for large rJ or correspondingly for small KJ formulas of the type (4.30) or

describe the structure of the random field only to a very crude approximation.

3. Villars and Weisskopf [29] have made an interesting attempt to explain the

scattering of electromagnetic waves by a turbulent flow. Their theory also begins by assum-

ing that the deviation of the refractive index of the air from unity is proportional to the

quantity p/T. HoweverJ Villars and Weisskopf neglect temperature fluctuations caused by tur-

bulent mixing of the atmosphere and assume that the refractive index fluctuations of the

sphere are caused by pressure fluctuations [m]. Pressure fluctuations p in a turbulent

are caused by velocity fluctuations v' and are related to them by the formula

2 p "' Pv' J

where p is the density of the fluid. To explain the meaning of this relationJ we recall

that the Bernoulli equation p + Pv2/2 = const is satisfied for stationary flow of a fluid.

For non-stationary flowJ an expression similar to (4.34) can be obtained from the equations

of motion (see [30Jl3]). ThusJ the pressure field in a turbulent field is random. Its

structure fUnction can be expressed in terms of the structure function of the velocity field

by using a relation similar to ( 4. 34) J namely [30 J 13] :

2[ ]2 p D (r) J rr

where Drr is the longitudinal structure function of the velocity field. Since for

L >> r >> t0

J the structure function of the velocity field has the form D (r) = C(Er)2

/ 3 o rr

(see page 32)J then

D (r) p

( t << r << L )J 0 0

where E is the energy dissipation rate.

( 4. 36)

It follows from the relation n - l = const ~that n' N ~~ (since it is assumed in

paper that the temperature is constant). ThusJ according to Villars and WeisskopfJ the

fUnction of the refractive index must have the form

D (r) n (4.37)

in the example on page 25J the spectral density

~n(K) (4.38)

to the structure fUnction (4.37). Substituting this expression into Eq. (4.19)J

da( e) 2 4/3 -l/3 . e -l3/3 const P E Vk (s1n 2) sin2X dn. (4.39)

and Weisskopf's basic assumption that refractive index fluctuations in a turbulent

are caused by pressure fluctuations [n] does not withstand serious criticism. However)

interesting in that it applies turbulence theory considerations to the problem

of radio waves. This feature is expressed by the fact that Eq. (4.39) contains

EJ which actually characterizes inhomogeneities of the sizes which cause scat-

4. We now turn to the model which attributes the refractive index fluctuations to

mixing (see Chapter 3). Assuming that the potential temperature and specific humi-

conservative and passiveJ we obtained in Chapter 3 the following expression for the

function of the refractive index of the air:

D (r) n ( o << r << L ) J

0 0 (4.4o)

D (r) n

(r << t ) . 0

(4.41)

quantity Cn depends on EJ the energy dissipation rate in the turbulent flowJ and on the

rate of levelling out of the amount of refractive index inhomogeneity produced by the

75

processes of molecular diffusion of water vapor and by the temperature conductivity. In

. and the distribution of n are stable, to calculate the the case where the turbulent reglme

2 we can use the formulas (3.51) which express cn2

in terms of quantities character­quantity en

izing the average profiles of the wind, temperature and humidity, i.e.

-6 M = _ 79 X 10 P

T2

{ 15,500~ ~+ T ) QT 7800 dD

dz + 7a - 15,500q dz 1 + T

(4.42)

The spectral density corresponding to the structure function (4.40) is equal to (cf. (3.52))

~ (K) n (4.44) ( ]:_ << K << z ) .

Lo o

Substituting this expression in Eq. (4.19), we obtain

do( e)

l << ( 2k sin ~) << Z · Lo o

An expression equivalent to Eq. (4.45) was obtained by Silverman [o,31] · Eq. (4.45) differs

from all the previously considered expressions for do(e) in the first place by the fact that

to derive it we used the expression (4.44) for the spectral density of refractive index

it . correspondl'ng to the law established in turbulence theory. inhomogene leS In the second

place, the spectral density

-+ -+ ';!;' ( • e) ~n (k - km) = ~n 2k Sln 2 ,

pertaining to the small scale inhomogeneities is now no longer described by quanti ties per­

taining to the large scale inhomogeneities, but rather directly by the quantity C~, which

characterizes the intensity of the small scale refractive index inhomogeneities.

4.5 Evaluation of the size of refractive index fluctuations

from data on the scattering of radio waves in the troposphere

We now consider some concrete examples which illustrate the application of the theory

just presented. We compare the experimentally observed values of electromagnetic fields scat­

tered by turbulence with the values of the same quantities inferred from Eq. (4.45). It is

convenient to carry out this comparison for the ratio of the flux density P of the scattered p

energy to the flux density P0

which would be received at the same distance from the transmit-

ter to the receiver if they were located in free space. Let the distance between the trans-

mitter and the receiver be equal to D. If the transmitter power is E0

and the gain of its

antenna is G, then at the distance D it produces an energy flux density P = GE /4~D2 . The 0 0

energy flux density at the distance D/2, where the scattering volume is located, is

P1

= 4E0G/4~D2 . The amount of power scattered into the solid angle dQ is P

1do =(4E0G/4~D2)d~

At the distance D/2 from the scattering volume, this power is distributed over an area

(D/2) 2dn, and therefore the flux density of the scattered energy is P = (16E G/4~D4)(do/dn).

p 0

Thus we have

(4.46)

To estimate the size of the scattering volume V, which figures in Eq. (4.19) for do,

we use the approximate formula [P]

(4.47)

where r is the effective angular width of the gain pattern of the antenna. (It is assumed

that the receiving and transmitting antennas are identical and have identical gain patterns

in the vertical and horizontal planes.) Using Eq. (4.47), we find

77

~ = 4nk 4 sin~ r 3 ~ (2k sin ~) E

0 n e

for P /P . In the case where the receiving and transmitting antennas are directed at the p 0

horizon, the quantity D/e is constant and is equal to the effective radius R of the earth.

Therefore we have

(4.48)

Substituting the expression

into Eq. (4.48) and noting that sin~ N 1 for e << 1, we obtain

(4.49)

We use Eq. (4.29) to estimate the values of e which are necessary to explain the ob­n

served values of P /P . In Fig. 7 we show a graph of the seasonal trend of the monthly aver­p 0

ages of the quantity P /P , which we have taken from the paper [34 J . The path length was p 0 -

Pp -30r-----------------------------------~

-70r-----------------------------------~

-sor-----------------------------------~

J A S 0 N D J F M A M J J A S 0 N D J F M A M 1953 1954 1955

Fig. 7 Seasonal trend of mean received signal levelS produced by

tropospheric scattering. Pp is in decibels relative to 1 milliwatt.

78

300 km (188 miles), the transmitter frequency was 3600 Mcps (A= 8.17 em), the width of the

gain patterns of the receiving and transmitting antennas was 0.012 radians, and the scattering

angle was o.o48 radians. In this experiment, r < e, so that we can use Eq. (4.47) and its con­

sequence (4.49). Substituting the indicated values of A, !'and e into Eq. (4.49),we find that

the values of e needed to explain the values of 10 log (P /P ) ranging from -67 to -93 deci-n p o -8 4 -9 -1/3 bels experimentally observed must lie in the range 8 X 10 to • 5 X 10 em • The height

of the center of the scattering volume above the earth's surface was 1.5 km, so that the

values of en obtained pertain to this level. It should be noted that the size of the inhomo­

geneities responsible for the scattering is in this case t(e) N A/e = 1.7 m. This size cer­

tainly satisfies the condition t << t( e) << L , i.e. lies in the range where the 11 two-thirds 0 0

law11 can be applied.

In order to judge whether these values are realistic, we now consider the temperature

fluctuation characteristic eT which figures in the "two-thirds law" for the temperature field:

If we assume that the refractive index fluctuations are caused only by temperature fluctua-

tions, then from (3.44) we can obtain the following relation between the quantities en and eT:

e n

-6 79 X 10 p e

T2 T'

where p is the atmospheric pressure in millibars and T is the absolute temperature. If we

substitute here p = 850 mb and T = 273° K (the approximate values of these quanti ties at the

height 1.5 km),we find that the value eT = 0.09 deg cm-l/3 corresponds to the value

-8 -1/3 -1/3 en = 8 X 10 em and that the value eT = 0.005 deg em corresponds to the value

e = 4.5 x 10-9 cm-l/3• An analysis of the results of measurements carried out by Bullington n

()5] at a frequency of 3700 Mcps leads to approximately the same values of en and eT' i.e.

eT N 0.002 to o.oo6 deg cm-l/3.

Direct measurements of the quantity eT were first made by Krecbmer [36] • The author of

this book has also made numerous measurements in the layer of air near the earth and on a

tethered balloon [37]. In the layer of air near the earth, the value of eT varies from zero

79

to 0.2 deg cm-1

/ 3 depending on the meteorological conditions) with the largest values obtained

during the noon hours. In the lowertroposphere(up to a height of 500 m) the size of CT

decreases compared with its value at the earth's surface) and is of the order of 0.03 deg

cm-l/3 and less [q]. ThusJ it is apparent that the observed values of CT are sufficient to

explain the magnitude of the scattered signals. It should also be noted that the values of

Cn obtained by analyzing the phenomena of twinkling and quivering of stellar images in

telescopes have the same order of magnitude as those obtained above (see Chapter 13).

So

Chapter 5

THE SCATTERING OF SOUND WAVES

IN A LOCALLY ISOTROPIC TURBULENT FLOW [~

The scattering of sound waves in a turbulent flow resembles in many ways the phenomenon

of scattering of electromagnetic waves. The velocity of propagation of sound waves depends

both on the wind velocity and on the temperature. Since in a turbulent flow both of these

quantities undergo irregular fluctuations) the velocity of sound is a random function of coor-

dinates and timeJ a fact which leads to the scattering of sound waves. In this chapter we

shall regard the quantity T itself as a conservative additive rather than H = T + razJ since

in what follows it is assumed that the results of calculations of acoustic scattering are

used only in the layer of the atmosphere near the earth) where the difference between H and

Tis unimportant. The scattering of sound waves in a turbulent flow was first considered by

Obukhov [38] in the year 1941; subsequently papers by other authors [33J 32J 35[] have been

devoted to the same problem.

The basic equation for sound propagation in a moving medium can be written in the form

(5.1)

where P is the potential of the sound waveJ the u. are the components of the velocity of l

motion of the medium) and c is the velocity of sound. A derivation of this equation with

some simplifying assumptions is given by .Andreyev and Rusakov [4oJ. Naturally) in using

Eq. (5.1) we do not take into account the rotational component of the acoustic field. How-

everJ under atmospheric conditions) the size of this component is small compared to the poten-

tial component. -+ -+

We assume that the mean flow velocity is equal to zero and that u = u' represents the

instantaneous value of the fluctuational velocity in the turbulent flow. Since the velocity

fluctuations are small compared to the velocity of sound (under the conditions in the earth's

atmosphere)) we shall retain only the lowest power of the quantity u'/c (u' = I~' I) in the

81

equations. Squaring the operator d dt + , we obtain

1 d;; 1 (Jlrr 2 ~ dP 6P - c2 dt2 = c2 dt·grad p + c2 u' ·grad dt (5.2)

with an accuracy up to terms of order u'lc. The first term in the right hand side is of

order no larger than (llc2)n ~~. grad P (where n is the largest frequency of the fluctuations

I 2 ~ of flow velocity); the second term is of order (1 c )w u·grad P, where w is the angular

frequency of the sound. In the case n << w (and this condition is practically always met for

all frequencies in the acoustic spectrum) we can neglect th~ first term in the right hand

side of (5.2), obtaining as a result the equation

(5-3)

The velocity of sound c, which figures in Eq. (5.3), is a function of temperature.

For example, c N ,jT for an ideal gas. If we denote the mean temperature in the flow by T

and the temperature fluctuation by T', then we have

- (, T') c(T) N c(T) ~ + 2T •

In the atmosphere the quantity TVT is of the same order as ulc. Substituting the expression

(5.4) in Eq. (5.3) we obtain

with an accuracy up to terms of order T 'I T. ( F:enceforth instead of T and c(T) we shall

write T and c, understanding these quantities to be the corresponding mean values.) Assuming

that the time dependence of Pis given by the factor e-iwt, i.e. that P =IT e-iwt, we obtain

the equation

2 1i 2 T' 6IT + k IT=- 2ik c·grad IT+ k 21 IT (5.5)

82

the amplitude potential IT, where k = wlc is the wave number of the sound wave [b].

We shall look for a solution of Eq. (5.5) in the form of a series IT= IT0

+ IT1 + IT2 + .•. ,

where ITk has the order of smallness of u'lc or T'IT raised to the power of k. Then we have

6 IT + k~ 0 0

o, ( 5.6)

2 u' IT + k2 ~IT . 6 ITl + k IT1 = - 2ik c · grad 0

T 0

(5.7)

represents the amplitude of the acoustic wave potential incident on the scattering volume no y. Assuming that the incident wave is plane, we obtain

IT 0

~~

ik•r A e

0 (5.8)

where k is the wave vector of the incident wave. Substituting this expression in Eq. (5.7),

we obtain

2k -~+-2 eli·.~ T~ c 2T ( 5.9)

where~ is a unit vector in the direction of k (k= k~). Eq. (5.9) has the form of Eq. (a) on

2 page 61. Consequently, its solution at large distances from the scattering volume (1r >> L ,

where L3 = V) is

1 eikr J - 4;( -r-

v (5.10)

where i is a unit vector directed from the center of the scattering volume to the receiving

point. Thus, IT1(;) represents a spherical wave IT

1 = Q eikrlr with random complex amplitude

(5.11)

As is well known [l4]) the average value of the flux density vector of the scattered

energy (taken over the period of one oscillation) is equal to

where P is the density of the gas. Calculating the gradient of rrl) we obtain

grad lll = grad Q e~r = Q E e~r _ e:~) ;;_ N ikQ e~kr ;;_,

where it is assumed that kr >>l. Therefore we have

wpk ~ = ~ QQ* m

2r

The quantity SJ which depends (via Q) on the velocity and temperature fluctuations of the

flow within the scattering volume) is random. Its mean value equals [~

6 2 ~ wpk --- ~ ~ pck Ao 8=2 QQ* m=m~

2r 8rc r X

(5.l3)

We assume that the random fields~'(;) and T'(;) are homogeneous and isotropic; below it will

be possible to extend our results to the case of locally isotropic fields as well. In this

case it follows from the incompressibility condition for the turbulent flow of the fluid

(valid for u << c) that [d]

o.

84

(5.l4)

is the correlation tensor of the velocity field) and

is the correlation function of the field of temperature fluctuations. Since the integrands

in the right hand side of (5.l4) depend only on ;l - ; 2 ) we can carry out one of the integra­

tions in doing the double integrals over the volume) obtaining as a result

[

-+ -+ """*, J:. J B ("""*') ik(n-m)• r dV' +

2 ni~ ik r e c v

J BT(;') eik("i:-;;_).~' dV' l v

(5.l5)

We now use the representation of correlation functions in the form of Fourier integrals. As

established in Part I) we have

()()

Iff -oo

-+ -+ iK·r e ~ - KiKk)

ik 2 K

Here E(K) is the spectral density of the energy of the turbulence in wave number space) and

~(K) is the spectral density of the temperature fluctuations (more exactly) the spectral den­

sity of the amount of inhomogeneity in the temperature field). Substituting the expressions

(5.16) and (5.17) into the integrals in the right hand side of Eq. (5.15)) we obtain

J v

-+ -+ -iK·r' e

r: K.Kk ~ \ik - :2 E(K) ) (5.18)

(5.19)

where the double overbar over a fUnction like F(K) denotes the average of this fUnction over

the region in wave number space of volume ~3/v surrounding the point;, The derivation of

these formulas is analogous to the derivation of Eq. (4.16). Substituting the expressions

(5.18) and (5.19) into Eq. (5.15)) we obtain

86

+ 4~2 lp(k(;; - ~) ) J . (5. 20)

In the case where the volume Vis so large that averaging over the region 8~3/V of wave num-

ber space does not substantially change the averaged fUnctions) Eq. (5.20) can be simplified

considerably. In this case we have

cos e) where e is the angle between the direction of the vector k and the vector;

going from the center of the scattering volume to the observation point) i.e. the scattering

angle. Therefore

and

2 e cos 2

Since in the case of isotropic turbulence E(K)

and

e E(2k sin 2)

(!) ... ... ih e ~(k(n- m)) = ~(2k sin 2).

(5.21)

E(K) and ~(K) = ~(K)) it follows that

Thus we have

rcpck6A Sr [

--2 ° 12 E(2k sin f?..2 ) cos

2 f?.. + ..l:..... if\ (2k e) J 2 2 ~T sin -2 •

r c 4T ( 5. 22)

Eq. (5.22) can be used to find the effective s t' f cro s sec lOll or the scattering of sound

in the direction e. The acoustic power scattered into the solid angle dQ is equal to Sr2dQ.

The energy flux density in the incident wave IT 0

is equal to

and its absolute value is S 0

1 2 2 2 cpk A0

• Consequently, we have

dcr(e) 2rck4v [ 1

2 E(2k sin~) cos2

§?_ + ..1:.._ "iii (2k sin @.2)] dQ.

c 2 4T2 ~ ( 5-23)

Eq. (5.23) is completely analogous to Eq. (4.19), which defines the effective cross section

for the scattering of electromagnetic waves. (The expression inside square brackets in Eq.

(5.23) signifies the spectral density of the refractive index fluctuations.)

It follows from the expression (5.23) that the effective cross section for scattering at

the angle e depends only on spectral components of the turbulence with wave numbers 2k sin~'

corresponding to "sinusoidal space diffraction gratings" with period

t( e) 2rc 2k . e Slll 2 2 sin ~

2

(5. 24)

satisfying the Bragg condition. This fact allows us to extend Eq. (5.23), obtained by assum­

ing that the velocity and temperature fluctuations are homogeneous d · t · an lSo roplc, to the case

of locally isotropic fields. In fact if t(e) << L0

, then the values of the functions

E(2k sin~) and ~(2k sin~) are determined only by the isotropic inhomogeneities (eddies)

and the anisotropy of the large scale inhomogeneities has no influence whatsoever on these

88

In the case t << t(e) << L , i.e. 0 0

).. << 2 sin ~ << ~ 1 2 t 0 0

the quanti ties E( 2k sin ~) and ~( 2k sin %) are determined by the 11 two-thirds laws" for the

velocity and temperature fields:

D rr C

2 2/3 T r .

(5.26)

Here C~ = CE2/ 3, C~ = a 2 NE-l/~ E is the energy dissipation rate of the turbulence, and N is

the rate of levelling out of the temperature inhomogeneities. In this case, we have

(5.27)

(5.28)

(see (2.20) and (3.24)). Substituting these expressions into Eq. (5.23), we obtain

dcr(e ) I [ c2 c~ J -11/3

0. 030 kl 3 v cv2 2 e + o 13 ( i e ) d" cos 2 . T2 s n 2 ~· · (5.29)

In the layer of the atmosphere near the earth, the quantities Cv/c and CT/T have the same

order of magnitude, so that the temperature and wind fluctuations make approximately the same

contribution to the scattering of sound in the atmosphere [e]. An experimental investigation

of the scattering of sound in the atmosphere was carried out by Kallistratova ~7] . Her

results agree satisfactorily with Eq. (5.29).

Using the general formula (5.23) we can also easily find the quantity da(e) in cases

where the spectral densities of the velocity and temperature fluctuations have a form dif-

ferent from (5.27) and (5.28). Such expressions are given in the papers [39,41]. For

example, in the case where the correlation functions of the fluctuations of wind velocity and

temperature have the exponential form

89

B (r) rr

the expression for d~(e) takes the form [f]

d~(e) v

2nt [

2 0 v3 2 v 2 k2t2 T ~ 8 sin e 2 2 2 e + ~ 3c l + 4k t sin 2 T

(5-30)

dst •

(5-31)

It is interesting to note that in this case d~(O) depends only on the temperature fluctuations,

i.e. the wind inhomogeneities do not scatter at zero angle [g].

90

Part III

PARAMETER FLUCTUATIONS OF ELECTROMAGNETIC AND ACOUSTIC WAVES

PROPAGATING IN A TURBULENT ATMOSPHERE

Introductory Remarks

The influence of atmospheric turbulence on the propagation of electromagnetic and acous-

tic waves involves more than scattering of the waves. As the waves propagate through the

medium, there occur fluctuations of amplitude, phase, frequency and other wave parameters.

These effects are of great importance in a host of problems in atmospheric optics, acoustics

and radio meteorology. (For example, one might mention the fluctuations of frequency and angle

of arrival of electromagnetic and acoustic waves, the twinkling and quivering of stellar

images in telescopes, radio star scintillation, etc.) On the other hand, the study of para-

meter fluctuations of electromagnetic and acoustic waves can give valuable information about

the structure of atmospheric turbulence (see Part IV). The most recent papers devoted to

amplitude and phase fluctuations of electromagnetic waves are concerned with the phenomena of

twinkling and quivering of stellar images in telescopes. In recent years, interest in

this problem has increased greatly, and there already exist a large number of experimental and

theoretical papers dealing with these matters.

The problem of parameter fluctuations of waves propagating in the turbulent atmosphere

can be formulated as follows (with a view to obtaining a theoretical solution): Along the

wave propagation path from the source to the observation point there occur refractive index

fluctuations produced by the turbulence. The wave source may be situated either outside the

region where the fluctuations occur or inside it. In the first case, we can replace the

actual source of waves by an equivalent source located on the boundary of the region. (For

example, a star located at a great distance from the earth can be replaced by a plane wave

located at the boundary of the refracting atmosphere.) In the second case, we can generally

disregard the part of space lying behind the wave source, since its influence on the propaga-

ting waves is negligibly small. Thus, in both cases we can assume that the source of waves

91

lies on the boundary of the region occupied by the refractive index fluctuations. We shall

always assume that the observation point lies inside the region. (If the observation point

lies outside the region) the value of the field at the observation point can be determined

from the values of the field at the boundary of the region occupied by the inhomogeneities.)

Just as in the problem of wave scattering by refractive index inhomogeneities) we shall

assume that the field of refractive index inhomogeneities is quasi-stationary, so that we

shall not be concerned with frequency fluctuations and the frequency spectrum of the ampli-

tude and phase fluctuations of the wave. (However) the problem of the frequency spectrum of

the amplitude and phase fluctuations of the wave can be approached by starting with the spa-

tial spectrum of the fluctuations (in this regard) see Chapter l2)). The actual time changes

of the refractive index field can be regarded as changes of the realizations of the random

field. We shall consider that the field of refractive index fluctuations is a locally iso-

tropic random field. Our problem will be to determine the statistical properties of the wave

field at a distance L from the source of radiation (or from the boundary of the region occu-

pied by the refractive index fluctuations).

In Part III we consider some methods for solving the problem just stated. We begin by

presenting the simplest method) which is based on the equations of geometrical optics [42)43)

92

Chapter 6

SOLUTION OF THE PROBLEM OF AMPLITUDE AND PHASE FLUCTUATIONS

OF A PLANE MONOCHROMATIC WAVE

BY USING THE EQUATIONS OF GEOMETRICAL OPTICS

6.l Derivation and solution of the equations of geometrical optics

we consider first the problem of amplitude and phase fluctuations of short electromag­

netic waves. As was shown in Part II) the process of scattering of electromagnetic waves in

an inhomogeneous medium can be described by the equation (cf. (4.3))

o. (6.l)

We assume that the geometrical dimensions of all the inhomogeneities in the spatial distribu­

tion of the refractive index are much greater than the wavelength A (i.e.) that A<< to) where

t is the inner scale of the turbulence). In this case we can neglect the last term of Eq. 0

(6.l) [a]. Thus) the propagation of short waves (A << t0

) in an inhomogeneous medium is des-

cribed by the equation

--)- 2 2--)---)-6 E + k n (r)E = 0. (6.2)

The vector equation (6.2) reduces to three scalar equations) having the form

iS where u can denote any of the field components. We set u = Ae ) where A is the amplitude

and s is the phase of the wave. Substituting this expression in Eq. (6.3) and setting the

real and imaginary parts of this equation equal to zero) after first representing Eq. (6.3)

in the form

93

.6 u 2 2(~) 2 2 2 ~ -u- + k n r = .6 log u + (v log u) + k n (r) 0

~or convenience, we obtain a system o~ two equations equivalent to (6.3)

(6.4)

.6 S + 2 V log A•V S 0.

To simpli~y Eq. (6.4) fUrther, we note that V S is o~ order k, e.g. (in a plane wave

S = k·; and V S = k). There~ore the two last terms in Eq. (6.4) are 0~ order k2 = 4~2/A2. Moreover, the wave amplitude A can change appreciably only in distances 0~ the order 0~ the

dimensions o~ the inhomogeneities in the medium. There~ore

.6 log A+ (v log A) 2 = 6 A A

is o~ order no greater than 1/t 2• s · h 0 lnce we ave assumed that A << t

0, the ~irst two terms 0~

Eq. (6.4) are small compared to the last two terms and can be neglected. Th E (6 4) us, q. •

takes the ~orm

(6.6)

We shall now use the system o~ equations (6.5) and (6.6), i.e. the equations 0~ geometrical

optics, to solve the problem o~ amplitude d h ~l t an P ase ~ uc uations o~ a plane wave propagating

in a locally isotropic turbulent ~low.

Let the re~ractive index n(;) be a random ~ction o~ the coordinates, with a mean value

equal to 1

<< 1; this condition is accurately met in all real cases. The small-

ness o~ the re~ractive index ~luctuations allows us to use perturbation theory to solve Eqs.

( 6. 5) and ( 6. 6) . We set S = S + S1

and log A = log A- + X, where X= log A/ A is the 11 level" 0 0 0

of the amplitude ~luctuations on a logarithmic scale. Then Eqs. (6.5) and (6.6) take the ~arm

(6.8)

6. S0

+ .6 S1 + 2 V log A0

• V S0

+ 2 V log A0

• V s 1 + 2 V X ·V S0

+ 2 V X• V s 1 = O.

(6.9)

Equating groups o~ terms o~ the zeroth order o~ smallness, we obtain

(6.10)

.6 S 0

+ 2 V log A0

• V S 0

== 0. ( 6.11)

Subtracting these equations ~rom (6.8) and (6.9), we ~ind

( 6.12)

( 6.13)

In the case where IV sll << lv sol= k, i.e. AIV sll << 2~, we can neglect the term (v 81)2

22~ in Eq. (6.12). Moreover, in the right hand side o~ (6.12) we can omit the term k n1(r) o~

the second order o~ smallness. Thus, the linearized equation

(6.14)

is valid ~or AIV s1

1 << 2~, i.e. when the phase changes by a small amount over the distance o~

a wavelength A (note that the smallness requirement is not imposed on the value o~ s1 itsel~).

I~ the same condition AI~ s1 1 << 2~ is met, we can omit the last term in (6.13), which in this

95

case is small compared to the third term of this equation; the result is

( 6 -15)

We now consider the amplitude and phase fluctuations of a plane wave) choosing its direc~

tion of propagation as the x-axis. Then 8 0

= kx and A = const. 0

In this case) Eqs. (6.14)

and (6.15) take the form

(6.16)

A 8 2k ()X w l + dx = o. (6.17)

Let the source of the plane wave be located at the plane x = 0 (this plane can also be

regarded as the boundary of the region occupied by the refractive index fluctuations)) and

let the observation point have the coordinates (L)y)z)) i.e. be located at a distance L from

the source of the wave. Integrating Eq. (6.16) and (6.17)) we obtain

L

81(L)y)z) = k J n1(x)y)z)dx)

0

The quantity

is small compared to the integral figuring in Eq. (6.19).

(6.18)

Therefore we have approximately

L X

Jdx J (6.20)

0 0

Eqs. (6.18) and (6.20) express the amplitude and phase fluctuations at the point (L)y)z) in

terms of the refractive index fluctuations along the propagation path.

6.2 The structure fUnction and the spectrum

of the phase fluctuations of the wave

Averaging Eqs. (6.18) and (6.20) and taking into account that nl = 0) we obtain X= Sl= o.

These equations allow us to express the structure (correlation) functions of the phase and

amplitude fluctuations in terms of the structure fUnction of the refractive index. For exam-

ple) taking the difference of the values of 81

at two points on the plane x = L) we obtain

L

81(L)y1)z) - s1(L)y2)z2) = k J [n1 (x,y1,z1 ) - n1(x,y2)z2)]dx.

0

(6.21)

We square this equation and write the square of the integral in the form of a double integral.

Then performing the average, we find

L

f dx2 [nl (xl)Yl' zl)-nl (xl,y2' z2)] X [pl (x2,yl' zl)-nl (x2,y2' z2)] • 0 0

(6.22)

97

Using the algebraic identity

lr, 2 2 2 21 (a- b)(c - d) = 2Ja- d) + (b - c) - (a- c) - (b - d) J

we express the integrand in terms of the structure fUnction of the refractive index, i.e.

( 6.23)

Since we assume that the refractive index field is a locally isotropic random field, we have

(6.24)

where the indices ~, ••• ,~, ••• ,v can take the values 1 and 2. Thus, the expression (6.23) is

equal to

~ Dn ~(xl - x2)2 + (yl - y2)2 + (zl - z2)~ +

+ ~ Dn ~(xl - x2)2 + (yl - y2)2 + (zl - z2)~

Substituting (6.25) into Eq. (6.22), we obtain

- D n

(6.25)

( 6.25 I)

where P = /(y1- y2)2+ (z1- z2)2 is the distance between the observation points in the plane

x = 1. It is easy to convince oneself that the equality

L

J dxl 0

L

J dx2 f( x1 - x2) = 2 0

L

J (L - x)f(x)dx

0

holds for any even fUnction f( x) • Applying this relation to ( 6. 25 1 ) , we obtain

L

n8(P) = 21<:2 j (L - xl[ Dn 9x

2 + p~- Dn(x)Jdx.

0

(6.26)

(6.27)

The relation (6.27) can still be simplified a bit further. To do so, we should consider the

fact that for x >> P, the expression D ( J.l;?-) - D (x) is very small. For example, if n n D (x) = Cx~ (~ < 2), then

n

99

for x >> p. The chief contribution to the integral (6.27) occurs for x ~ p, If P << LJ

then on the segment x $ pJ L - x N L and

L

Ds(P) ... 2k~ J ~n (/x2 + P2) - Dn(x8dx· 0

Since the integrand is very small for x >LJ the upper limit of integration can be replaced

by oo J and then we obtain the formula

Ds(P) 2k~ j ~n ~x2 + P2) - Dn(x8d.x, ( 6.28)

0

which is valid for p << L.

Eq. (6.28) expresses the structure function of the phase fluctuations of the wave in the

plane x = L in terms of the structure function D (x) of the refractive index. From Eq. (6.28) n

we can obtain a relation between the spectra of the phase fluctuations and of the refractive

index. .Af3 was shown in Chapter 1, a structure function given in some plane can be represented

by the integral (cf. (1.49))

00

( 6.29)

where p2 = '112 + ~ 2 • Here F

8(K2JK

3,o) represents the two-dimensional spectral density of the

structure function n8

(p). In Chapter 1 we also derived the formula ( cf. (1.48))

00

Dn (Jx2 + p

2) - Dn(x) = 2 J J [1- cos(K2 'fl + K3~)]Fn(K2,K3'x)dK2dK3' ( 6.30)

-oo

100

refractive index fluctuations by the relation (cf. (1.53))

00

2n 1n ( Kl' K2, K3

) f F n ( K2, K3' x) cos ( K1x) dx. (6.31)

- 00

It follows from (6.31) that

00 J Fn(K2,K3'x)dx = n ~n(O,K2,K3 ) • (6.32)

0

We substitute the expansion (6.30) in Eq. (6.28) and change the order of integration with

00

[l - cos(K2'fl + K30]dK2dK3 f Fn(K2,Kyx)dx ..

0

Bearing in mind the relation ( 6. 32) , we find

2 f f [1 - cos(K2T} + K30]2nk~ ~n(O,K2,K 3 )dK2dK3 .. -oo

(6.33)

(6 ) i 1 s th t the two-dimensional Comparing the expansion (6.33) and Eq. .29 , we conv nee ourse ve a

spectral density F (K ,K ,o) of the phase fluctuations is equal to s 2 3

101

Since

in a locally isotropic turbulent flow) then

and

Writing K = v/K~ + K~J we finally obtain

6.3 Solution of the equations of geometrical optics by

using spectral expansions

The relation (6.34') between the spectral densities FS(KJO) and !n(K) is equivalent to

the relation (6.28) between the structure functions of the phase fluctuations and the refrac­

tive ind.ex fluctuations. The relation (6.34') can be obtained from Eq. (6.16) by still an-

other method) which does not require the introduction of structure functions. As was shown in

Chapter lJ the locally isotropic random field n1

(;) can be represented in the form of the fol­

lowing stochastic integral

00

n1 (x)yJz) J J (1- ( 6. 35) -oo

102

Since n is a real quantity} we have 1

We shall look for an expansion of the phase fluctuation field s1 (;) which has the same form)

i.e.

sl(xJy)z) (6.37)

-oo

The random amplitudes da(K2JK3)x) satisfy the relation

Substituting the expansions (6.35) and (6.37) in Eq. (6.16)) we obtain

00

aS1 (xJOJO) dX + JJ

-oo

103

---

00

= knl (X J 0 J 0) + k J J (6.39) - 00

Setting y = z = 0) we have

Subtracting this equation from (6.39)) we obtain the equation

(6.4o) - 00

satisfied for arbitrary y and z. Equating the integrand to zero) we obtain the relation

We integrate this equation with respect to x from 0 to L. Since da(K2)K3)o)

no fluctuations at the "input" to the turbulent region)) we have [c]

L

dcr(K2JKyL) = k J dxdv(K2JKyx).

0

We multiply Eq. (6.41) by its complex conjugate equation

L

J 0

dX 1 d V* ( K I K I X 1 ) 2) y )

104

0 (there are

(6.41)

written for the point (K2,K3) and average.

(6.38), we obtain

Taking into account the relations (6.36) and

whence

L J dx 1 Fn(K2,K3,x - x 1 )5(K2 - K2)5(K3 - K3)dK2dK3dK2dK3

0 0

L

FS(K2,Ky0) - k2 J dx

0

L

J dx 1Fn(K2,Kyx - x')

0

( 6.42)

The function Fn(K2,K3,x- x 1 ) is even with respect to x- x'. Applying Eq. (6.26), we obtain

L

2k2 J (L - x)Fn(K2,Kyx)dx •

0

As shown in Chapter l, the function F (K ,K ,x) falLs off rapidlyforx > l/K. n 2 3

( 6.4 3)

Therefore, only

the region x ~ l/K contributes substantially to (6.43). If l/K << L, the chief contribution

to the integral (6.43) is obtained for x << L. In this region,L - x N L and

L

FS(K2,Ky0) .... 2k2L J Fn(K2,Kyx)dx"'

0

00

0

105

Applying Eq. ( 6. 32), we obtain the previous relation

(6.44)

from which follows also the equivalent relation (6.28). This spectral method of solving the

problem is equivalent to the method based on structure or correlation functions, but is in

many respects more convenient.

We now apply the spectral method of solution to determine the amplitude fluctuations.

Above we obtained the relation (6.20), which relates the fluctuations of the "level"

X= log A/A0

to the refractive index fluctuations nl(;). We shall look for the locally iso­

tropic random field X(;) in the form of an expansion

00

X(x,y,z) = X(x,o,o) + !!~ - 00

According to the general formula (l.46)

o(K - K' )o(K - K1) X

2 2 3 3

(6.45)

(6.46)

where FA(K2

,K3

, jxj) is the two-dimensional spectral density of the field of fluctuations of the

level. Substituting the expansions (6.35) and (6.45) into Eq. (6.20), we obtain

- 00

l06

(6.47)

0 0 -oo

Setting y = z = 0 in (6.47), we obtain the relation

L X

X(L,O,O) -~ f dx J d~ J J(K~ + K~)dv(K2,Ky~). 0 0 -co

Subtracting this equation from Eq. (6.47), we find

-oo

L X fr 2 2 ~ i(K2y + K3Z)J l J J =2 dx d~ (K2

+ K3

) l- e dv(K2,K3,~), (6.48)

0 0 - 00

whence L X

da(K2, KyL) l J J 2 2

=2 dx d~(K2 + K3)dv(K2,K3,~). (6.49)

0 0

We multiply Eq. (6.49) by its complex conjugate, written for K2,K3,L, i.e.

L x'

da*(K2,K3,L) = ~ J dx' fds'(K22

+ K32

)dv*(K2,K3, ~') 0 0

and average. Taking into account Eqs. (6.46) and (6.36), we obtain

l07

L L x x'

FA(K2,Ky0) = ff J dx J dx 1 J d~ J d~ 1 K4Fn(K2,K3'~- ~'), 0 0 0 0

2 2 2 where K denotes the ~uantity K2 + K

3•

E~. (6.50), which relates the two-dimensional spectral density of the amplitude fluctua-

tions of the wave to that of the refractive index fluctuations, can be simplified considerably.

Consider the expression

x x'

J d~ J d~'Fn(K2,K)'~- ~ 1 ). 0 0

As was shown, the fUnction Fn(K2,K3,~-~') is appreciably different from zero only in the

region ~~-~ 1 I ~ 1/K, adjacent to the line~=~~. Therefore, in (6.51) we can replace the

rectangular region of integration by a s~uare region with side e~ual to the smaller of the

numbers x,x', which we denote by /; thus we have

x x' ')'

J d~ J d~ 1 F n ( K 2, K )' ~ - ~ 1 ) "' J d~

0 0 0

')'

J d~ 1 Fn(K2,K3'~- ~ 1 ). 0

(6.52)

Since the function Fn(K2,K3,~-~1 ) is even with respect to ~-~ 1 , then, applying E~. (6.26),

we obtain the expression

')'

2 J (1- ~)Fn(K2,K3,~)d~. (6.53)

0

for the integral (6.51). In most of the region of integration with respect to x and x', the

108

quantity y is of order L. For values of K2

,K3

satisfying the ine~uality 1/K << L, we can

simplify the integral (6.53) further. The function Fn(K2,K3,~) is appreciably different from

zero for ~ ~ l/K << L. Since 1 ~ L, we have y - ~ N 1 in the region which contributes appreci-

ably to the integral, and the integral (6.53) takes the form

')' 00

2/ J Fn(K2,K3' Ud~ "' 2/ J Fn(K2,K3' ~)d~ 0 0

where I is the three-dimensional spectral density of the refractive index fluctuations. Sub­n

stituting (6.54) into (6.50), we obtain

L

Jdx 0

L

J dx' min(x,x').

0

(6.55)

The integral figuring in (6.55) can be calculated in an elementary fashion and e~uals L3/3.

Thus, the relation

is valid for K >> 1/L. We have taken into account that

and

109

Eq. (6.56) relates the two-dimensional spectral density of the amplitude fluctuations of

the wave to the three-dimensional spectral density of the refractive index fluctuations. It

follows from this formula that the amplitude fluctuations of the wave do not depend on its

frequency and are proportional to the cube of the distance traversed by the wave in the in-

homogeneous medium. In a similar way) it follows from Eq. (6.44) that the phase fluctuations

are proportional to the square of the frequency and to the distance traversed by the wave in

the inhomogeneous medium. These results do not depend on the form of the spectral or struc-

ture (correlation) function of the refractive index inhomogeneities.

6.4 Amplitude and phase fluctuations o~ a wave propagating

in a locally isotropic turbulent flow

We use Eqs. (6.44) and (6.56) to calculate the amplitude and phase fluctuations of a wave

propagating in a locally isotropic turbulent flow. As was shown above (see page 58 )J in this

case the structure function of the refractive index has the form

1 02 r2/3 for r >> t

0J

n

D (r) n 2 c2 t2/3(E-) for r << t • n 0 t 0

0

(6.57)

2 -11/3 I The spectral density~ (K) corresponding to (6.57) equals 0.033 C K for K << 1 t and ~ n · o

quickly falls off to zero for K N 1/t • The way! (K) falls off for K N 1/t is related to o n o

the form of the structure function Dn(r) in the region r N t0

and at present has not yet been

ascertained exactly. We can adduce different spectral functions !n(K) which correspond to the

form (6.57) of the structure function for small and large r. One such function was introduced

on page 48J i.e.

02 K-11/3 n

110

where Km is connected with t0

by the relation

K t m o

5.48. (6.59)

Substituting the spectral density (6.58) into Eqs. (6.44) and (6.56) for the spectral densities

of the amplitude and phase fluctuations) we obtain

{ :·2l k~C~K-ll/3

{ :.Ol7 L3C~rol/3

for K > K • m

(6.60)

(6.61)

The spectral density of the phase fluctuations has a non-integrable singularity at zero.

Consequently) the field of phase fluctuations in the plane x = L is a locally isotropic ran-

dom field and is characterized by a structure function rather than by a correlation function.

Using the formula (see page 23)

00

D(P) 4n J [1 - J (KP )] F( KJ 0 )KdK, 0

( 6.62)

0

we obtain

K m

D8

(P) 4n(0.2l)k~C2 J [1 - J (Kp)] K -B/3dK n 0

( 6 .. 63)

0

for the structure function n8

(P). For KmP << 1} 1- J0

(KP) N K2

P2/4 over the whole region

of variation of KJ and the integral reduces to the formula

( p << t ) • 0

(6.64)

111

(We have used the relation (6.59) to express Km in terms of t0.) For KmP >>lJ the integra­

tion in ( 6.63) can be extended to ooJ as a result of which we obtain [a]

(6.65)

A formula similar to (6.65) was first obtained by Krasilnikov [42J44J.

We now turn to the amplitude fluctuations. The spectral density (6.61) of the amplitude

fluctuations is finite for K = 0. Therefore) the field of amplitude fluctuations in the plane

x ~ L is homogeneous and isotropic and the fluctuations have a correlation function BA(P).

Using the formula (see page 24)

00

BA (p) 2:rr f J0

(KP)FA(KJO)KdKJ (6.66)

0

we obtain K m

BA( P) 2:rr(O.Ol7)L3c2 n J J ( K p) K 4 I 3 dK .

0 (6.67)

0

The value of the correlation function BA(P) at P = 0 gives the mean square fluctuation of the

logari tbm of the wave amplitude 1

i.e.

A 2 (log -;:-)

A 2 (log-;:-)

0

0

Km

2:rr(O.Ol7)L3C~ J K4

/3dK)

0

(6.68)

(the value of K is expressed in terms oft). ThusJ the mean square fluctuation of the m o

logarithm of the amplitude depends on the dimensions of the smallest inhomogeneities of the

refractive index I~J (on the inner scale of turbulence t ) and is proportional to the 0

112

2 characteristic Cn of the structure function of the refractive index fluctuations. According

to (6.67) and (6.68), the correlation function of the fluctuations of the logarithm of the

amplitude of the wave in the plane x = L, normalized to unity, is equal to

(6.69)

0

where ~ = K/Km· The function (6.69) is shown in Fig. 8. The correlation distance of the

amplitude fluctuations in the plane x = L agrees in order of magnitude with the inner scale

of turbulence t0

•

Fig. 8 The correlation coefficient of fluctuations of the

logarithm of the amplitude in the plane x = L under

the condition {i.L << t 0

[f] •

6.5 A consequence of the law of conservation of energy

It is appropriate to indicate an important property of the function B ( p). It follows A

from Eq. (6.61) that FA(O,O) = 0. Since the correlation function is the Fourier transform

of its spectral density, we have

113

00

J J cos(K2TJ + K3t)BA( ,, 0dTJdL

- 00

and it follows from the equality F(O,O) 0 that

00 00

- 00 0

Eq. (6.70) is a consequence of the law of energy conservation which itself follows from

Eq. (6.17) t

(6.17)

In fact, suppose that the region T containing the refractive index fluctuations is bounded

by the planes x = 0, x = L and some "lateral" surface located at a finite distance from the

origin of coordinates, and suppose that n1

= 0 outside of T. Moreover, suppose that the

region T is imbedded in the region T1

bounded by the planes x = -€ and x = L + €. We inte­

grate Eq. (6.17) (which can be written in the form div grad s1 + 2k ~ = 0) over the region

T1 • Applying Gauss' theorem, we obtain

1 gradn s1 do + 2k. J J Xdydz - 2k. J J Xdydz o, (6.71) x=L+E x=-€

where dais an element of surface bounding the region T. Since the quantity s1 is constant

outside the region T (see Eq. ( 6.16)), then grad s1 = 0 on the boundary of the region T1, and

the surface integral vanishes, i.e.

grad s do = o. n J..

114

However, the values of X on the planes x = L and x = L + € coincide. Since there are no

amplitude fluctuations at the "input" to the region occupied by the inhomogeneities, we have

J J Xdydz 0.

X=-€

Then it follows from Eq. (6.71) that

o. -oo

We multiply Eq. ( 6. 72) by (L, y', z') and average. As a result we obtain the relation

00 00 J J X(L,y,z)X(L,y',z') dydz J J BA(TJ, 0dTJdt = 0 (6.73) -oo -co

(TJ = y- y', t = z- z'),which agrees with (6.70). Thus, the relation (6.70) must be satis-

fied independently of the form of the structure function or the spectrum of the refractive

index fluctuations. It follows from Eq. (6.73) that the correlation function of the fluctua-

tions of any quantity X which satisfies a conservation law of the type (6.72) must change its

sign at least once.

6.6 Amplitude and phase fluctuations of sound waves

We now consider the problem of amplitude and phase fluctuations of sound waves. As was

shown in Chapter 5, the amplitude II of the acoustic wave potential satisfies the equation

2 ~

6 II+~ II+ 2i ~~·'VII= 0. 2 c c

c

115

Dividing this equation by TI and taking into account the identity

l::.TI 2 IT = !::. log TI + ( \l log TI) ,

we obtain the equation

2 2 w;:;: !::. log TI + ('V log TI) + w2 + 2i • \l log TI = 0. c c

(6.75) c

iS We set TI = Ae or log TI = log A+ iS. Substituting this expression in (6.75) and equating

the real and imaginary parts to zero, we obtain two equations

o,

~

2w u !::. S + 2 \l log A • \1 S + 7 c • \l log A = 0.

Since I'V sl N k = 2~/l , then as l ~ 0 we can neglect the term!::. A/A in Eq. (6.76)

(see page 94). Thus, Eq. ( 6. 76) takes the form

2 w2 -7

(\7 S) = -2

- 2 ~ ~.\J S. c c

c

(6.77)

The velocity of sound c figuring in Eqs. (6.77) and (6.78) is a function of the temper-

ature T. Suppose the temperature T undergoes fluctuations T' about a mean temperature T0 ,

i.e. T = T + T'. Since the velocity of sound in the air is proportional to JT, we have 0

l l T' 2 ... 2 (l- T"), (6.79) or c c 0

0

where c = c(T ) is the mean value of the velocity of sound. 0 0

We shall regard T'/T and 0

~/c0 as quantities of the first order of smallness, and we set log A= log A0

+X , S = S0 + s1,

where X and s1

are the fluctuations of the logarithmic amplitude and phase of the wave.

Substituting these values of log A, S and c in Eqs. (6.78) and (6.77), we obtain

ll6

2 (\7 s ) + ( 2 v s + v s ) . v s =

0 0 l l

2 T' k (l - -)

T

+ 2w(l c

0

0

-7

~) u T c

0 0

2w(l c

0

-7

~) u T c

0 0

• ('V log A + \l X) 0

(6.80)

o, (6.81)

where k = w/c . 0

The quantities S0

and log A0

of the zeroth order of smallness satisfy the

equations

!::. S 0

+ 2 \7 log A0

• \1 S 0

0.

Assuming that the unperturbed wave is plane, we set A 0

~

\] so km,

~

canst and

(6.82)

where m is a unit vector in the direction of propagation of the unperturbed wave. Equating

to/zero the terms of the first order of smallness in (6.80) and (6.8lL we obtain the equa­

tions

2 \1 s 0

2\lS 0

\l s k2 ~ - 2k2 ;:;: • ; . l = - T c 0 0

• \1 x + !::. s1

= o,

for the validity of which it is necessary that the conditions

ll7

(6.83)

(6.84)

(6.85)

be satisfied (see page 95). Eq. (6.84) agrees with Eq. (6.15), which was obtained for electro­

magnetic waves (recall that V log A = 0). Eq. (6.83) can be written in the form 0

v s 0

2( T' ~ ~). . v sl = k - ~ - m • c 0 0

This equation agrees with Eq. (6.14) if we set

T' nl = - 2T"" -

0

~

~ u m•-=

c 0

m.u. ~ ~

c 0

(6.86)

All subsequent results can be obtained from the corresponding formulas for electromagnetic

waves if in them we take n1

to be the expression (6.86).

By using (6.86h the structure function Dn(r) of the refractive index can be expressed in

terms of the temperature structure function DT(r) = C~r2/ 3 [g] and the wind velocity structure

'function D ik ( r), i.e.

D (r) n

But the cross correlation function of the fluctuations of temperature and wind velocity

vanishes in a locally isotropic field (see note [c] to Chapter 5), so that the last term

drops out of the equation. Since

(6.87)

where~ is a unit vector directed along;, we have

(6.88)

118

~ ~

where cos ~ = m•n.

Thus we have

and

D (r) n

In a locally isotropic turbulent flow, D rr

r 2 2 ] CT 1 C 2 2

- + - ...X. ( 4 - cos ~) r /3. 4T2 3 c2

0 0

C2 2/3 4 c2 r2/3. v r ani Dtt = 3 v

(6.89)

Thus, it follows that the structure function of the refractive index of sound waves depends

on the angle between the direction of wave propagation and the direction of the line joining

the observation points. The expression (6.89) goes under the integral sign in Eq. (6.28) and

in the analogous formula for the correlation function of the amplitude fluctuations of the

wave. However, in most of the region of integration cos ~ ~ 1 for p << L. Therefore, we can

write approximately

D (r) n (6.90)

Thus, the amplitude and phase fluctuations of a plane sound wave are approximately described

by the same final formulas as the corresponding fluctuations of an electromagnetic wave, if

we take c2 to be the expression

n

(6.91)

An exact calculation based on Eq. (6.89) gives the same formula (6.65) for the structure func­

tion of the phase fluctuations of a sound wave as for the fluctuations of an electromagnetic

wave, but with a value of the numerical coefficient which is changed by a few percent [46J .

The expression for the amplitude fluctuations of a sound wave agrees with Eq. (6.68) to an

even higher degree of accuracy.

119

6.7 Limits of applicability of geometrical optics

The theory of amplitude and phase fluctuations of electromagnetic and acoustic waves.

which we have just considered was based on the equations of geometrical optics, which are

valid when the condition

A.<< t 0

(6.92)

is satisfied. However it is easy to see that in some cases this condition is not sufficient

for the solution obtained on the basis of geometrical optics to remain valid when diffraction

effects are taken into account. We can convince ourselves of this by using the follo~dng

simple argument [4 7] .

Let an obstacle with geometrical dimensions t be located on the propagation path of a

plane wave. At a distance L from this obstacle we obtain its image (shadow) with the same

dimensions t. At the same time, diffraction of the wave by the obstacle will occur. The

angle of divergence of the diffracted (scattered) wave will be of order eN A.jt. At a dis­

tance L from the obstacle the size of the diffracted bundle will be of order BL N A.L/t.

Clearly, in order for the geometrical shadow of the obstacle not to be appreciably changed,

it is necessary for the relation ~ << t or J'AL << t to hold. When there is a whole set of

obstacles with different geometrical sizes, it is obviously necessary that this relation be

satisfied for the smallest obstacles,which have the size t0

• Applying a similar argument to

the problem under consideration, we convince ourselves that the solutions we have obtained

are valid only in the case where the inequality

)IT,<< t (6.93) 0

is satisfied, where t0

is the inner scale of the turbulence. In other words, the theory of

amplitude and phase fluctuations based on the equations of geometrical optics is valid only

for limited distances L satisfying the condition

L << L cr

120

A more detailed analysis shows that the conditions (6.92) and (6.93) are sufficient for agree­

ment of the amplitudes and phases of the solutions obtained by using the equations of geometri­

cal optics and those obtained by using the wave equation [48,49]. We shall also arrive at the

same conclusion in Chapter 7.

121

Chapter 7

CALCULATION OF AMPLITUDE AND PRASE FLUCTUATIONS

OF A PLANE MONOCHROMATIC WAVE FROM THE WAVE EQUATION

USING THE METHODS OF "SMALL" AND t'SMOOTH" PERTURBATIONS

7.1 Solution of the wave equation by the method of small perturbations

As we have already seen in Chapter 6J even when the condition A << t is metJ the solution 0

of the problem of amplitude and phase fluctuations of a wave which we obtained using the equa-

tions of geometrical optics becomes unsuitable for large distances LJ which exceed the critical

distance L = t2/A. At large distances one can no longer neglect the diffraction of the wave cr o

by refractive index inhomogeneities) regardless of the smallness of the diffraction angle. In

order to take account of diffraction effects in solving the problem of parameter fluctuations

of a wave traversing an inhomogeneous medium, it is necessary to start from the wave equation

22-+-6 u + k n (r)u = o. (7.1)

Setting

(7.2)

as in Chapter 6, and assuming that ln1 (;)I << 1, we apply the method of small perturbations to

solve Eq. (7.1). To do soJ we look for a solution in the form of the sum of an unperturbed

wave u0

J which satisfies the equation 6 u0

+ k2u

0 = OJ and a small perturbation u1J i.e.

(7.3)

Substituting (7.2) and (7.3) into Eq. (7.1)J and taking into account that 6 u + k2u =OJ we

0 0

obtain

122

o.

The last term in the equation is of order n~ and can be omitted. If we assume that I u11 <<I u

0 I,

or more precisely that lu1/u0

j $ n1, then in Eq. (7.4) we can also neglect the term n1(;)u

1•

Then we obtain the equation

which is valid when the condition

which expresses the smallness of the fluctuations of the fieldJ is satisfied.

Let the unperturbed wave u0

have the form

u 0

A e 0

iS 0

(7-5)

(7.6)

(7-7)

where A0

and S0

are its amplitude and phase. To find the amplitude and phase of the perturbed

iS wave u = u

0 + u1, we set u = Ae • Then log u =log A+ iS and by (7.3) and (7.7)J we have

and

log u = log A+ iS log(u0

+ u1

)

log

ul log A+ iS = log A

0 + iS

0 + u-

o

123

Separating the real and imaginary parts of the last equation) we find

A ul log A= X= Re u

0 0

s - s 0

7.2 The equations of the method of smooth perturbations

(7.8)

(7-9)

The equation (7.5) and the formulas (7.8) and (7.9) just obtained are valid in the case

of small amplitude and phase fluctuations) i.e. when lxl << l and lsll << l. These conditions

are much more stringent than the condition AI~ sll << 2~) which was needed in order to apply

the method of small perturbations to the equations of geometrical optics. In order to avoid

this restriction) it is natural to try to apply the method of small perturbations to the equa-

tion

which contains only derivatives of log u) rather than directly to Eq. (7.l) [a].

We set log u = log A + iS = t ( Re t = log A) Im t = S) • Then we have

( )2 2 ...... 2

6 t + ~ t + k ( l + nl ( r) ) = 0.

We then set t = t + t ; t satisfies the equation 0 l 0

2 2 6 t + (~ t ) + k = o.

0 0

Substituting t = t0

+ tl in Eq. (7.ll) and taking into account (7.l2)) we obtain

o.

l24

(7.l0)

(7.ll)

(7.l2)

(7.l3)

( ) . 2 2 ...... .

In Eq. 7-l3 we can oiDlt the term k nl(r) which is of the second order of smallness. In the·

case where 1~ tll << ~~ t0

IJ or more precisely where ~~ tll ~ nll~ t0

IJ we can also neglect

the term(~ tl) 2 in (7.l3). Finally we obtain the ~quation

2 ...... 6 t l + 2 ~ t

0 • ~ t l + 2k nl ( r)

which is valid when the conditions

(7.l4)

(7.l5)

are met. Since 1~ t0

l ~ k = 2~/A ) the second condition (7.l5) can be written in the form

and expresses the smallness of the change of tl over distances of the order of a wavelength. -t

By using the substitution tl = e 0 w) Eq. (7.l4) can be reduced to the form

to Since e u) Eq. (7.l6) coincides with Eq. (7.5) obtained by the method of small perturba-

o -t tions. Consequently) w = ul and tl = e 0w = ul/u

0• We now find expressions for the amplitude

and phase fluctuations of the wave. Since

we have

Therefore

log A+ iS and

A ul log A = X = Re tl = Re u- )

0 0

s - s 0

(7.l7)

l25

Eq. (7.16) and the ~ormulas (7.17) and (7.18) agree ~ormally with Eq. (7.5) and the ~ormulas

(7.8) and (7.9). However, ~or the validity o~ these expressions the conditions A.IV s1 J << 2n

and A. IV XJ << 1 have to be met, rather than requiring the smallness o~ the perturbations X and

s themselves. As we shall see below, the inequality JxJ < Js1 1 is usually satis~ied. There­

~ore, when the inequality :>-.IV s1 1 << 2n is satis~ied, the inequality A.jV Xi << l is satis~ied

also. Thus, to apply the method o~ solving the wave equation presented above, the

conditions [b J

(7.19)

must be met; these conditions are the same as those ~or applying the method o~ small pertur-

bations to the equations o~ geometrical optics. As is well known, the solution o~ Eq. (7.16)

has the ~orm

(The integration in (7.20) extends over the region where n1(;) is di~~erent ~rom zero.)

the quantity 1jr 1 = w(;) /u0 (;), we obtain

ikl; -;,I _e _____ dV' •

(7.20)

For

(7.21)

The ~unction (7.21) is the general solution o~ Eq. (7.14) ~or any ~ction 1jr0

which satis~ies

the equation V 1Jr + (V 1Jr )2 + k

2 = o. 0 0

We now consider the problem o~ ~luctuations o~ aplane monochromatic wave, con~ining our-

selves to the case where the wavelength A. is small compared to the inner scale o~ turbulence

t . We locate the origin o~ coordinates on the boundary o~ the region occupied by the re~rac­o

tive index inhomogeneities, and we direct the x-axis along the direction o~ propagation o~ the

126

incident wave. Then u (;) 0

A eikx and Eq. (7.21) takes the ~orm 0

(-+ 1 ) -ik(x-x') n

1 r e

eikl; - ;, I I;- ;'I

dV'. (7.22)

In the case where A.<< t , Eq. (7.22) can be greatly simpli~ied. In this case, the angle o~ 0

scattering o~ the waves by re~ractive index inhomogeneities is o~ order no greater than

e :A./t and is thus small. There~ore, the value o~ 1)r1(;) can only be appreciably a~~ected by

0 0

the inhomogeneities included in a cone with vertex at the observation point, with axis direct-

ed towards the wave source, and with angular aperture e 0

= A./ t 0

<< l. In most o~ this region

(, 2 2 1

lx-x'I>>V\y-y') +(z-z').

There~ore we have

(x - x') 1+ (y- y')2 + (z- z')2

(x - x') 2

~ (y - y' )2 + (z - z' )2]

"'(x-x')l+ 2 2(x-x')

(x - x') + (y- y')2 + (z - z')2

2(x-x')

eikl;_;, I Substituting this expansion in and retaining only the ~irst term o~ the expansion in

the denominator o~ (7.22), we obtain the approximate ~ormula

~k ( y - y I ) 2 + ( z - z I ) 2) exp ~ 2(x- x')

dV'. x - x'

127

It is not hard to show that the function (7.23) is the exact solution of the equation

(7.24)

2 I 2 obtained from Eq. (7.14) by omitting the term d t 1 dx . We note that by retaining only the

first two terms of the expansion

~ ~ 2/ 4; 3 k I r - r 1 I = k(x - x 1 ) + kp 2(x - x 1) + kp 8(x - x 1 ) + ...

we change the phase of (7.21) by an amount of order kp4/(x- x')3. Since p ,... 8L "" A.L/ t and

0

(x- x') "'Lin the important region of integration) then the error permitted here is small if

(7.25)

Since A.<< t ) the quantity t 4/A.3 is much larger than the distance L t2

/A.J which deter-o o cr o

mines the limits of applicability of geometrical optics.

7. 3 Solution of the equations of the method of "smooth" perturbations

by using spectral expansions

We shall begin with Eq. (7.24) in order to solve the problem of amplitude and·phase fluc­

tuations of a sufficiently short (A.<< t ) plane wave. We use a method of solving Eq. (7.24) 0

which is based on the use of spectral expansions (just like the way we solved the equations of

geometrical optics in Chapter 6). In a turbulent medium) n1 (;) is a locally isotropic random

field. To represent n1(;), we apply the representation (6.35):

(7.26)

-oo

128

we shall look for the same kind of expansion for t 1(;)J i.e.

tl (-;.) (7.27)

-oo

Substituting the expansions (7.26) and (7.27) into Eq. (7.24)) we obtain

00

JJ

(7.28)

Setting y = z = 0 in this equation and writing K~ + K~ = K2

) we obtain

JJ o. (7. 29)

We subtract Eq. (7.29) from Eq. (7.28) 1 i.e.

129

-oo

o. (7-30)

It follows from Eq. (7.30) that the random amplitudes dv(K2,K3,x) and d~(K2,K3,x) are related

by the differential equation

o. (7-31)

The solution of this equation which goes to zero for x = 0 (the fluctuations of the field

vanish at the boundary of the region filled with the refractive index inhomogeneities) has the

form

X

ik J dx 1 exp[~ (7-32) 0

(The integration in (7.32) is carried out with respect to x'; see note [c] to Chapter 6.)

We now find the relations between the spectral amplitudes of the field s1 of the phase

fluctuations of the wave and the field log(A/ A ) of the fluctuations of logari tbmic amplitude 0

of the wave. Using the formula X= Re ~l and the expansion (7 .27), we obtain

130

Changing variables from K2 ,K3 to -K2,-K

3 in the last integral, we find

+ J J [l -e i(Kif + K3z)J dij>(K2, Kyx) : dcp*( -K2 , -Kyx)

-oo

In a completely analogous way, we obtain for S = Im ~ the formula 1 1

Denoting the spectral amplitudes of the random fields X~) and s1(r) by da(K

2,K

3,x) and

da(K2,K3,x) ( as in Chapter 6) and using (7.33) and (7-34), we obtain

131

(7-33)

dcp(K2JKyx) + dcp*(-K2J-K3

Jx) da(K2JK

3Jx) 2 (7-35)

dcp(K2JKyx) - dcp*(-K ,-K Jx) da(K 2)K

3,x) 2 3 (7.36) 2i

Substituting the expression (7.32) for dcp(K2,K3

Jx) into these formulas) we obtain

(7·37)

0

(7.38)

0

The physical meaning of Eq. (7.32) or of the equivalent Eqs. (7.37) and (7.38) is trans-

parent. Inhomogeneities of the wave field characterized by the wave number K (i.e. by geo-

metrical dimensions t = 2rr./K) are "made up" by the superposition of refractive index inhomo-

geneities dv(K2

,K3,x') characterized by the same wave number K (i.e. having the same geo­

metrical dimensions t = 2rr./K). Moreover, the refractive index inhomogeneities with dimensions

t which are located at a distance x - x' from the observation point appear with weight

sin(rr.A2jt2) or cos(rr.A2/t2)J where A2 = A(x- x') is the square of the radius of the first

Fresnel zone. In other words, the weight of a refractive index inhomogeneity depends on the

relation between its dimensions and the dimensions of the Fresnel zone. Using the relations

(7.37) and (7.38) between the random spectral amplitudes of the refractive index and of the

fluctuations X and s1

J we can find the relations between the spectral densities of the corre­

sponding structure or correlation functions. We multiply Eq. (7.37) by its complex conjugate

* da (K;)K3)x)) i.e.

132

X I , 2( ")J da*(K2JK3Jx) = k J dx" sin Kx- x dv*(K' K1 x") - 2k 2) y •

0

Averaging) we obtain

X

J d " . IK2(x - x' )J

X Sln~~ 2k X

0 0

(7.39)

But according to the general formula (6.36), we have

(7 .4o)

and

(7 .41)

where Fn(K2)K3)x' - x") is the two-dimensional spectral function of the refractive index and

FA(K 2,K3)o) is the two-dimensional spectral density of the structure or correlation function

of the fluctuations of X in the plane x = const. Substituting (7.40) and (7.41) into Eq. (7.39),

133

we obtain

0 0

(7.42)

Similarly) from Eq. (7.38) we can obtain the relation

fK2

(x - x' )l Jl(2(x - x" )J X cos[ 2k J cos[ 2k j

0 0

(7 .43)

The relations (7.42) and (7.43) can be greatly simplified. First of all we note that

Fn(K2)Kyx' - x") = Fn(K2JK3

J x"- x'). Then,in (7.42) and (7.43) we introduce new variables

of integrations= x' _ x" and 2'fl = x' + x". The integration with respect to 'f1 can be carried

out explicitly, since Fn does not depend on 'Tl· As a result, we arrive at the formulas

L

J 0

134

(Here we denote the coordinate x of the observation point by L.) As has already been repeated­

ly pointed out, the function Fn(K2,K3,s) falls off very rapidly to zero forKs~ 1. Therefore,

the important contribution to the values of the int·egrals (7.44) and (7.45) occurs for s ;$ ~. 1 K

2s K In the region s .$ K , we have k .$ k We assumed above that the wavelength )... is much less

than the inner scale of turbulence t0

• But t - 1/K , where Km is the largest wave number for o m

which F (K,g) still differs from zero. Therefore we have 1/k << 1/K and K/k < K /k << 1. n m m

Thus, K2

g/2k << 1 in the important region of integration and we can write

2 2 2 cos K g - 1, sin U ... K g ~ 2k ~

sin K2

( 2L - g) . K~ 2k N Sln k .

we shall be interested in the structure (or correlation) functions of X and s1

only for values

of the arguments which are small compared to L. This means that in (7.44) and (7.45) we con-

sider only values of K which satisfy the condition 1/K << L. Since g ;$ 1/K in the important

region of integration, then within this region we have g <<L. Taking all these simplifica-

tions into account, we obtain

L e k3 . K\) FA (K2, K3

, 0) J F n ( K 2, K )' s ) dl; , L - K2 Sln k 0

(7.46)

L e k3 ~ FS(K2,Ky 0) J L + K2 sin T Fn(K2,Kys)dl; •

0

(7.47)

Since the function Fn(K2,K3,g) falls off rapidly to zero for large t;, the integration in

(7.46) and (7.47) can be extended to infinity, without appreciably changing the values of the

integrals. Since

00

J 0

where I (K) is the three-dimensional spectral density of the refractive index structure func­n

135

tion (see Eq. (1.53)). Eqs. (7.46) and (7.47) take the form

(7.48)

In the case where the refractive index field is a locally isotropic random field) we have

Therefore, recalling that

we have~ (O,K 7KA) =} (K), and Eqs. (7.48) and (7.49) finally become

~n 2 _; n

(7.50)

(7-51)

Eqs. (7.50) and (7.51) relate the two-dimensional spectral density of the structure

(correlation) functions of the amplitude and phase fluctuations of the wave in the plane x = L

to the three-dimensional spectral density of the structure (correlation) function of the

refractive index. UsingEqs. (1.50) and (1.51) we can go from the spectral densities FA(K,O)

and F8

(K7

0) to the structure (correlation) functions of the amplitude and phase of the wave

in the plane x = L, i.e.

136

4n J [1- J0

(KP)]FA(K,O)KdK) (7 -52)

0

js1(L,y,z) - S1 (L,y' ,z') j2

= 4n J I}- J0

(KP)]F8

(K,O)KdK, (7-53) 0

where

7.4 Qualitative analysis of the solutions

Using Eqs. (7.50) and (7.51), we can draw some general conclusions about the character of

the amplitude and phase fluctuations of the wave. It follows from Eqs. (7.50) and (7.51) that

the phase fluctuations are always larger than the fluctuations of logarithmic amplitude [c].

As K--+ 0, we have

Therefore FA ( 0, 0) = 0 if !n( K) goes to infinity at K

existence of the correlation function

00

2n J J0

(Kp)FA(K,O)KdK,

0

-4 0 no faster than K • This implies the

(7-54)

of the amplitude fluctuation~which satisfies the equation (seep. 114)

00

(7-55)

Eq. (7.55) is a consequence of the law of energy conservation (see the analogous equation (6.70)).

137

We now consider the general character of the behavior of the correlation function of the

amplitude fluctuations. It follows from Eq. (7.50) that the two-dimensional spectral density

of the correlation function BA(p) is the product of two functions: the three-dimensional spec­

tral density~ (K) of the structure function of the refractive index fluctuations and the n 2

function f1 - ~) sin2 K k L. In the general case, ~ (K) has the form shown in Fig. 9·

\_ K L n

1.0

\ \ \ \

--~---------------\

Fig. 9 General form of the spectral density of the refractive index fluctuations.

In the region of small scales which are much less than t , i.e. forK >> 2n/t , the func-o 0

tion ~ (K) is equal to zero or is negligibly small. For smaller values of K, lying between n

2n/L and 2n/t (L is the outer scale of turbulence), ~n(K) grows asK decreases (since the 0 0 0

refractive index fluctuations which are larger in size are related to the large scale inhomo-

geneities). In the case where the refractive index fluctuations obey the "two-thirds law",

~ (K) is proportional to K-ll/3 in this region, but in the general case its appearance in this n

region can be different. ForK < 2n/L, the growth of~ (K) slows down; this is connected with o n

the fact that the refractive index fluctuations are finite. Strictly speaking, in this range

";!;" ~ ~ of sizes the function ~JK) loses its meaning, since the structure function Dn(r1,r2) then

depends on each of the arguments ;1

and ;2

separately, and it is impossible to represent it in

the form of a spectral expansion of the type ( l.ltl). The function f(K) = f;_ - ~ \sin K:L is 1 K4L2 K~ f-ik r;;t \. K L)

approximately equal to 6 - 2- fork<< 1. For K =VI; ='J fr- , the functJ.on f(K) is equal k

138

unity, and for large values of K, f(K) approaches unity, undergoing smaller and smaller oscil­

lations. The characteristic scale of this

1 K4

L2

function is the quantity K ,.. ~ • For K << K , 0 lfL 0

f(K) "" b 7 ' and for K >> K0 , f(K) ... 1. Depend~ng on the size of the parameter K = ~ , 0/):L

the following relative positions of the points K , 2n and 2n 0 ~ L are possible:

0 0

1) 2)

We consider first the case where _!_ >>~ V'fL to

curves ~n(K) and f(K) is shown in Fig •. 10.

In this case the relative position of the

1.0

\ \ \

--4--------------\

Ko K

Fig. 10 Relative position of the curves ~n(K) and (1 - ~ sin K~) ·K~ k

in the case Iff, << t • 0

F 1 I 1 K4

L2

or al values of K < 2n t0

, the function f(K) is approximately equal to b __ 2_

k fore, we have

and

There-

(7.56)

(7-57)

since in this case l + K~ sinK~ ~ 2. Eqs. (7.56) and (7.57) are valid when the condition

JIT, <<to is met, and agree with Eqs. (6.44) and (6.56), which were obtained in Chapter 6 by

using the equations of geometrical optics. In the case under consideration (see Fig. 10) the

product of the functions ~n(K) and f(K) has a maximum near the point 2n/t and is equal to zero 0

139

) 2 /o Thus, l'n the case where r:C:L << t

0, the spectrum of' the (or negligibly small f'or K > n -v

0 • V .1\..I..J

the ampll·tude fluctuations is concentrated near the point 2n/t 0 (i.e., correlation function of'

in this case refractive index inhomogeneities with scales of' the order to have the greatest

· ) It follows f'rom general properties of' the Fourier influence on the amplitude f'luctuatlons ·

transform that the correlation function BA(P) has a characteristic scale (correlation distance)

of' order t0

• This general conclusion is illustrated by the example given in Chapter 6 (see

Fig. 8). 1 1 1

we now consider the case where the system of' inequalities 10

<<)IE<< t0

or

t << J):.L << L holds. In this case the relative position of' the curves !n(K) and f'(K) is 0 0

shown in Fig. 11.

f(K)

1.0

ol_~~~----------~--------------~2~~~~~o~~K 2~/Lo Ko

Fig. 11 Relative position of' the curves ! n( K) and ( 1 -_!_sinK~) K~ k

l·n the case t << JIL << L • 0 0

The product of' !n(K) and f'(K) has a maximum near the point 2n/JL'i. and goes to zero f'or

2n/t . The behavior of' the function !JK) f'or K < 2n/t o has almost no ef'f'ect on the char-K ~ o

( ) 2 ( )~ ( ) · e in this region the function f'(K) is near acter of' the function FA K, 0 == nk-Lf' K ~n K ' Slnc

zero. rvT th spectrum of' the correlation function of' the

Thus, in the case where t0

<< yAL << L0 , e

t 2 I f);L ( · the ref'racti ve index amplitude fluctuations is concentrated near the poin n VA~ l.e.,

rvT make the largest contribution to the amplitude fluctuationS geneities with scales of' order yAL

of' the wave) . It follows f'rom this that the correlation function of' the amplitude fluctuations

in the plane x = L has a characteristic scale (correlation distance) of' order ~· Below we

1 f' h correlation function. shall give a concrete examp e o sue a

140

Finally, we consider the third case, where the condition_!_<<~ or y0(L >>L is met. j5:L L

0 o

The relative position of' the curves ~n(K) and f'(K) in this case is shown in Fig. 12. As can

be seen f'rom the figure, in the case under consideration the chief' contribution to the spectrum

of the correlation function of' the amplitude fluctuations is made by large scale inhomogeneities

in the interval (L0

, vr:C:L) · However, it must be pointed out at once that in the range of' scales

exceeding L0

, the refractive index field is not a locally homogeneous and isotropic random field~

Therefore, strictly speaking, f'or such scales the structure function of' the refractive index

field depends on the coordinates of' both points of' observation, and f'or it one cannot define a

spectral density !n(K) of' one argument, even a vector argument. The same thing obviously

applies as well to the spectral densities FA(K,O) and F8

(K,O) of' the amplitude and phase fluc­

tuations of' the wave, since the latter are proportional to !n(K). The functions FA(K,O) and

F (K,O) have meaning only in the region K > 2n/L • Therefore, one can speak of' the structure s ~ 0

functions DA(p) and D8(p) of' the amplitude and phase fluctuations of' the wave only f'or p SL0

•

For large values of' p, this function begins to depend not only on the distance p between the

observation points, but also on the location of' these points in the plane x == L.

f (K) \ \ -~----------------

\ ~

Fig. 12 The relative position of' the curves

!n(K) and (1 - ~2 sin k~) in the case j5:.L >> L • K L K o

Let us examine the f'orm of' the spectral densities FA(K,O) and F (K,O) f'or K > 2n/L • In ~ s 0

this region the function f'(K) == 1 - ~ sin ~ ~ l and l + ~ sin K~ ~ 1. Therefore, we have K L k K2L k

nk~ I (K) n (7-58)

Thus, for ~ >>L0

, the two-dimensional spectral densities of the structure functions of the

amplitude and phase fluctuations of the wave are equal to one another in the range K > 2~/L "rJ 0

and proportional to the three-dimensional spectral density of the structure function of the

refractive index fluctuations. Consequently, in the region P $ L0 , the structure functions

DA(p) and DS(p) are equal to

DA(p) "' 4~2k~ I [l - J0

(KP)] In (K)KdK • (7-59)

0

In this case, the fluctuations of the amplitude and phase of the wave are proportional to the

distance L traversed by the wave in the turbulent medium.

In the region p ~L0, the form of the structure functions DA and DS depends in an essen­

tial way on the character of the largest scale components of the turbulence, which are not

homogeneous and isotropic and therefore cannot be universal. We can only assert that, since

FA(o,o) ~ o, the mean square amplitude fluctuation of the wave is always finite and does not

depend on refractive index inhomogeneities with scales larger than JIL . If we assume that the random refractive index field is statistically homogeneous and iso-

tropic for all scales (such an assumption is made in many papers, despite the fact that it is

not adequately justified, because it considerably simplifies the solution of the problem),

we can draw further conclusions about the character of the amplitude and phase fluctuations

the wave. In this case, the fields of the amplitude and phase fluctuations of the wave

plane x = L are also homogeneous and isotropic, so that they have correlation functions

00

BA(p) 2~ I J0

(KP)FA(K,O)KdK , (7.60)

0

BS(p) 2~ I J0

(KP)FS(K,O)KdK •

0

Substituting the expressions (7.50) and (7.51) for FA(K,O) and FS(K,O) into Eqs. (7.60) and

(7.61), we obtain

00

2<~;, J J0

(KP)(l - K;, sin K;) :Ji:,fK)KdK • (7 .62)

0

As can be seen from Fig. 12,

!n(K) over almost all of the

spectrum where K < 2n/ .[)I. .

for If£ >> L , the function ( 1-=F ~ sin K~) f!\ ( ) . . ° K~ k .l.n K coincJ..des with

regJ..on of integration, with the exception f 0 a small piece of the

egr over the segment 0 < K < 2n/ JI:L is small Since the int al

compared to the integral over the whole range of K' we have approximately

00

2n~~ J J o(Kp) !n(K)KdK • (7.64) 0

However, it should be noted that if we discard the quantity~ sin K~ • E ( 6 ) K

2_ k J..n q. 7• 2 , we

obtain the expression nk~ ! (K) f t L n or he spectral density of the correlation function of the

' an expressJ..on whJ..ch does not go to zero at K = o. There-amplitude fluctuations of the wave · .

fore, the expression forB (p) hi h A w c was obtained using En (7 64) '11 t · ':1.• • WJ.. no J..n general satisf

the relation ( 7 55) whi h · Y • ' c J..S a consequence of the law of conservation of energy; this can

sometimes lead to physically meaningless conclusions.

In the case being considered, the functions B (r) n and !n(K) are connected by the relation

(1.25), i.e.

Eq. 7.64) and changing the order of integration, we find Substituting this expression into (

00 00

Bn (r)rdr J J 0

(Kp)sin(Kr)dK •

0

Taking into consideration that [53]

00

J 0

J (Kp)sin(Kr)dK 0 rl

/r.-2-_-p-2

2 2 for r < p

we obtain the formula

00

k~ J Bn (jp2 + x

2) ilx

0

( JfL >> L ) ) 0

which relates the correlation functions of the amplitude and phase fluctuations of the wave in

the plane x = L to the correlation function of the refractive index fluctuations. Setting

p = 0 in Eq_. (7.65)) we obtain an expression for the mean sq_uare amplitude and phase fluctua-

tions of the wavet

00

x2 = s~ = k

2L J B (x)dx •

n

(7.66)

0

The q_uanti ty

00 00

l J l J B (x)dx ) Ln=m

B (x)dx =-n 2 n

n 0 nl 0

(7.67)

which agrees in order of magnitude with the outer scale of turbulence) is called the integral

scale of turbulence [a]. Substituting (7 .67) into (7 .66L we obtain the formula

X2- s2- 2 2 - l- nl k LLn·

It follows from Eq_. ( 7. 68) that for JfL >> L ) the size of the amplitude and phase fluctuati 0

of the wave is determined by two parameters of the turbulence: n~ ) the size of the mean sq_uare

refractive index fluctuations) and Ln) the integral scale of the turbulence.

144

p l u e and phase fluctuations of the wave in the plane The correlation distance of the am l't d

ra sea e of these fluctuations) i.e. x ::: L can be characterized by the "integ 1 1 "

00 00

J JJ 0 0 0

Going over to polar coordinates in (7.69)) we find

Finally) setting B (o) A

B (r) n DO) rdr . n

(7.69)

(7.70)

(7.71)

Since Bn(r) is appreciably different from zero only in t he interval (OJL )J we have n

J 0

so that

L ""L ""L A n 0

(7-72)

Thus) for .jfL >> LoJ the correlation distance of the ampl' t d l u e and phase fluctuations in the

plane x L = agrees in order of magnitude with the outer scale of turbulence [ e J • Summarizing the results of our q_ualitative analysis of the character of the amplitude and

phase fluctuations of the wave) we come to the conclusion that the following cases can occur J

depending on the size of the parameter ,ji:L :

l) J1L << t

0

• The amplitude fluctuations of the wave do not depend on the frequency and

grow with the distance like L3, while the phase fluctuations are proportional to the square of

the frequency and to the distance L. The correlation distance of the amplitude fluctuations of

the wave in the plane x == L is of order t 0

t << jiT. << L • The form of the correlation function of the amplitude fluctuations 2) 0 0

depends on the concrete form of the spectral density !n(K) of the refractive index fluctuations.

The correlation distance of the amplitude fluctuations of the wave in the plane x ~ L is of

order .j}:L . The fields of amplitude and phase fluctuations in the plane x = L are not

3) J}:L >> L • 0

locally isotropic random field$. For P ;:; L0

, we have D A( p) ~ n8(p). The amplitude and phase

fluctuations are proportional to the square of the frequency and to the distance L.

4) J1L >> L0

and the field of refractive index fluctuations is homogeneous and isotropic

Here the correlation fUnctions of the amplitude and phase fluctuations of the wave in the plane

x = L coincide. The correlation distance of the amplitude and phase fluctuations of the wave

in this plane agrees with L0

in order of magnitude.

In all the cases considered) the largest wave numbers which participate in the spectral

expansions of the amplitude and phase fluctuations of the wave are of order 2</t • It fall~ 0

from this that the correlation and structure fUnctions of the amplitude and phase fluctuations

of the wave change quite slowly over a distance of order t 0 • In particular, the fUnction n8 (p)

has a square law character in a region of order t0

near the origin, and in the same region BA(p

2/ 2 has the form BA(o)(l- a P t 0 + ••· ).

Eqs. (7.50) and (7.51) obtained above relate the spectral densities of the amplitude and

phase fluctuations of the wave to the spectral density of the refractive index fluctuations.

tions of logarithmic amplitude and the structure function of the phase

Using these formulas, we can obtain relations relating the correlation fUnction of the fl

to the structure function of the refractive index fluctuations. Such relations have been

obtained in J54, 55 J , but their form is much more complicated (the formulas contain double

grals) than the form of Eqs. (7.50) and (7.51). This fact greatly hampers

these formulas both for the purposes of qualitative study of solutions and for practical cal-

culations with various concrete correlation functions. Eqs. (7.50) and (7.51) for the spe

densities FA(K)O) and F8

(K)O) are much more convenient.

146

We now consider an example. L t e the field of refractive index fluctuations be statisti-

cally homogeneous and isotropic and let it be described by the correlation function [f]

B (r) n (7. 73)

The spectral density corresponding to the function (7.73) . lS equal to (see page 18)

n2 a3 2 2 l e -K a /4

8rr. .;;. (7-74)

The functions (7.73) and (7 74) • are simple enough to permit complete lation functions of the ampl"t d calculation of the carre-

l u e and phase fluctuations. However) one should remark that

these functions are h c aracterized by only one scale a) which can be fication as both the inner and th regarded with equal jus ti-

e outer sc 1 f t a e 0 he turbulence [gl . Th obtained by using thes f U erefore, results

e unctions are in many respects not sufficiently general. we established in Ch t Moreover) as

ap er l) the refractive index fluctuations in a turbulent quit t atmosphere are

e accura ely described by the "two-thirds law" ' and not by the correlation function (

Therefore, the present example is f 1·13). 0 a purely illustrative h b c aracter and the formulas obtained

elow can be used only to the extent that their form does l ti not depend on the form of the corre-

a on function of the refractive index (for e 1 2 xamp e) the dependence of X and . 1J:L on L for J1L << a

VA~ >>a has a universal h t c arac er which d J oes not depend on B (r)).

Substituting the fun t. ( 4 n c lon 7·7 ) into Eqs. (7.50) and (7.51). , we obtain

l

8 v;c

The mean square fluctuation

2 3 2 n1 a k L r;_ -~ sin K2J:\

~ K~ k]

of logarithmic amplitude is equal to (see (7-54))

(7-75)

~-== 2n I F (K,O)KdK == A

0

00

~ k . K\) exp ~ 4=}ili< · J;i 2 a3k~ I T nl - K~ s~n k

0

( 77 ) can be calculated in The integral figuring in 7·

4L 2 ( arc tan D) where D == 2 · 2 1 - D ' ka a

Thus we have

2 v:;. 2 2 arc tan D) X == 2 nl ak 1( 1 - D 7

and completely analogously

2 j;. 2 2 ( arc tan D) Sl == 2 nl ak 1 1 + D •

an elementary way and eq_uals

(7~77)

(7. 79)

r 51] without using spectral obtained in the paper of Obukhov L

EI1S. (7.78) and (7.79) were ( lled wa~re ':1. • • ( 8 ) d (7. 79) the so-ca .v

The q_uanti ty D f~guring ~n 7 .. 7 an sions of the fluctuations. t F r:lL (the radius of the firs re

. tional to the sq_uare of the ratio of V Al.J

parameter) ~s propor 1

of this par i f the inhomogeneities). Depending on the va ue

zone) to a (the average s ze o

the following limiting cases can occur: 2

a) D << l or 1 << ~~ == 1 cr'

1 1- D2 and Eq_s. (7-78) and (7.79) take Then - arc tan D "" 3

D

the form

"2 8#. 2 13 X == -3- nl a3 '

82 == v; 2 ak21 • 1

nl

148

These formulas, which are valid for L << 1cr' correspond to geometrical optics.

b) In the opposite case, where D >> 1 or L >> L cr' the q_uanti ty ~ arc tan D << 1 and

which corresponds to the case considered above, where {IT, >> 10

•

Using the relations

00

J 0

(7.80)

(7.81)

(7.82)

we can also determine the correlation functions of the amplitude and phase fluctuations of the

wave. The integrals appearing in (7.81) and (7.82) can be expressed in terms of the integral

exponential function Ei(z), so that Eq_s. (7.81) and (7.82) take the form

(7.83)

(7.84)

Eq_s. (7.83) and (7.84) were ootained "by Chernov [56]. AsP_,,.. O, these expressions reduce

We must "bear in mind that as z ~ O, the function Ei ( z) has a to Eq_s. (7.78) and (7.79). logarithmic singularity with Im Ei(z) ~ arg z. An analysis of Eqs. (7.83) and ([.84) leads

to the conclusion that "both for D << 1 and for D >> 1 the correlation distance of the ampli­

tude and phase fluctuations of the wave is of order a C?6J. However, this result is caused "by

the special choice of the form of the correlation function of the refractive index fluctuations,

i.e. the Gaussian curve (7.73). Af3 we have shown aoove, when the condition J"):L << t is met, 0

agrees in order of

the correlation distance of the amplitude fluctuations in the plane X= L

it is of while in the intermediate

magnitude with t o' in the case where .j):L >> L '

order L0

, 0

are of the same order of magni-

case it eq_uals JAL. However, since the values of t andL 0 0

correlation function, i.e. t N L ,.. a [g] then in the present case the 0 0 tude for a Gaussian

correlation distance of the amplitude fluctuations of the wave in the plane x = L is of order

a for any value of the parameter Jll., (i.e. "both for D << 1 and for D >> 1) •

7·5 Amplitude and phase fluctuations of a wave

propagating in a locally isotropic turbulent medium

We now consider another, much more realistic example. Let the refractive index field "be

locally homogeneous and let it "be described "by the structure function

\ c2 r2/3 for t << r << L

n 0 0

D (r)

(7 .85)

n c2 t2/3 (.E..)2 for r << t n 0 t 0

0

In the region of values of r exceeding L , the form of the structure function D (r) is not o n

universal. However, this indeterminacy does not affect the correlation function of the ampli

tude fluctuations. In fact, as we have already seen aoove (see Figs. 10,11), in the case

where Vii<< L , the behavior of the spectral density:<!\ (K) forK < 2n/L has almost no i o n o

ence on the calculation of the correlation function of the amplitude fluctuations of the wave

Thus, if we restrict ourselves to values of L which satisfy the condition JIL << L , we csn 0

150

specify the form of the function D (r) n for r ;2:, Lo in an aroi trary way (

(

it is only necessary

that Dn r) does not grow faster than r for r > L ) • In t. o par lcular, we can assume that for

r ~ 10

, Dn ( r) preserves the same form as for t << ·r o << Lo' i.e., we can omit the condition

in the case of the structure function

Inhomogeneities with dimensions

r <<Loin Eq_. (7.85). A similar situation also occurs

of the phase fluctuations of the wave. much larger than p

have little effect on the value of D ( ) s p • u the condition Therefore, if p is restricted hy

p << Lo' the form of the structure function D (r) for n r .(j Lo has no important "bearing, and we

can also omit the condition r << L i E ( 0

n q_ • 7 • 85 ) •

As we have seen aoove the stru t ' c ure function (7 85 ) • can "be associated with th e spectral

density

where Km = 5.48/t0

•

expressions

FA(K,O)

F8(K,O)

forK < K m

forK > K m

(7.86)

Substituting (7.86) into Eq_s. (7.50) and (7 51) ht • ' we ou ain the following

\ :·033rtC~~ E- K~ sinK~ K-ll/3 for K < K , m

for K > K , m

\ :·033rtC~~ E +_!_sin/~ K-ll/3 for K < K ., K21 k m

for K > K • m

(7.88)

We ~hall find the correlation d an structure functions corresponding to these spectral densities

{IT. << to' K2L/k < K!_L/k << l in the sep t ara e cases where Jii, << t d . F:T

0 an v :A.L >> t • For

and Eq_s. (7.87) and (7.88) reduce to Eq_s. (6.61) ando(6.6o), which correspond to geometrical

The correlation and structure functions optics. of the amplitude and phase fluctuations of

151

the wave in this case were studied in Chapter 6.

We now consider the case where JiL >> t . In this case) in order to calculate the st 0

ture functions of the amplitude and phase) it is necessary to use the unsimplified formulas

(7.87) and (7.88). For DA(p) we obtain the expression

K

4n2 (o.033)C~k2L Jm I} - J0(Ko)] ~- :;. sin KJ) K-8/

3 dK •

0

First we consider the behavior of the function DA(p) for P << t 0 • In this case KmP << 1 and

the approximate eq_uality 1 - J0

(KP) ... ~ K2P2 is valid in the whole range of integration ( OJKm)

Conseq_uently) for p << t 0 we have

K2L/k m

DA(p) = ~ -n2(0.033)C!kl3/6L5/6p2 I 0

(1 - ~) x-5/6 dx •

X

2 Since we are considering the case where J1L >> t ) then K L/k >> 1 and the integral appe o m

in (7 .90) is approximately eq_ual to 6(K~/k) 1/6 [h]. Conseq_uentlyJ for P << t 0

2 p )

i.e.) for small pJ the structure function of the amplitude fluctuations has a parabolic char­

acter) and therefore for p << t0

J the correlation function of the amplitude fluctuations of

the wave has the form BA(p) ... BA(o)(l - ~p ) (see page 146). For p >> t 0 J the integration 2

(7.89) can be extended to infinity. This only influences the form of the function DA(p) for

small pJ but we have considered this case separately (Eq_. (7.91)). We obtain the formula

(7·

152

for the correlation function BA(p). First we find the mean sq_uare fluctuation of logarithmic

amplitude) i.e.

(7-93)

Calculating the integral appearing in this formula [ i]) we obtain

( J'IL >> t ) • 0

(7-94)

As we have seen above (see page 146)J the dependence of the 2 q_uantity X on L has the form

X2 "' L3 in the case \/):L << t0

and the form X2

... L in the case /):L >> L • For t << \(5:.L << L - 0 0 0)

the exponent ~ in the formula x2 ... L ~ has an intermediate valueJ determined by the form of the

structure function of the refractive index fluctuat~ons. If ( ) ~ ... Dn r "' r ) the quantity ~ has the

value (~ + 3)/2; for 0 < ~ < 1) we have 1.5 < ~ < 2.

The correlation coefficient bA(p) = BA(P)/BA(o) of the amplitude fluctuations of the wave

• an • J can be found by numerical integra-in the plane x = LJ determined by Eqs. (7 92) d (7 94)

tion [j]. ... of an infinitely By integrating along the contour consisting of the c~rcumference

... appearing in 7.92) large circle and the rays Re K = Im KJ Im K = OJ we can reduce the ~ntegral (

to a form sui table for numerical integration. Th f ( ) e unction bA P obtained as a result of the

/ e corre ation distance of the amplitude fluctua-numerical integration is shown in Fig. 1A. Th 1

tions of the wave agrees · d f . r:;:;-J.n or er o magnitude with the radius yAL of the first Fresnel zone.

As was to be expected (see Eq_. (7.55))J the function bA(p) takes on negative values) since

the relation

must be satisfied [k J .

153

Fig. 13 l tion Coefficient of the fluctuations The corre a of logarithmic amplitude in the plane x = L, with

the condition t << ~ << Lo • 0

h fluctuations of the Calculation of the structure function of the p ase

we now turn to the

(7.52) and (7-53), we obtain the formula wave. Adding Eq_s •

00

DA(p) + DS(p)::::: 8rc~~ J [!_- Jo(KP)] l_n(K)KdK.

0

It follows from this that

2 ~ 2 D (p) + D ( p) = 8rc ( o.o33)k en

A S 0

(7 86) Let p << t 0

or KmP << l. in the case where l_n(K) is determined by Eq_. • •

l 2 2 l - Jo(KP) ~ 4 K p and

(7 ·95)

(7 .96)

Then

( p << t ) 0

(7-97)

(K is expressed in terms oft by using the relation K t = 5.48). ForK p >> 1, the integra-m o m o m

tion in ( 7. 96) can be extended to infinity and then (see note [ cij to Chapter 6)

(7.98)

Eq_s. (7.97) and (7.98) allow us to find the function D8

(p) if we know the function DA(p). For

small values of P (Eq_. (7.91)), DA(p) = 1.72 c! k~ t~1/3 p2• Substituting this value of DA(p)

into Eq_. (7.97), we obtain

For p >> t0

, we can find DA(p) from the relation

( p << t ) . 0

Substituting this expression into (7.98) and using Eq_. (7.94), we obtain

(7-99)

(7.100)

For P >> ..[f.L, and actually even for p ,... {):£ , the second term in (7 .100) is small compared

to the first and

( p ~ {):£). (7.101)

For values of p small compared to VIT.,, the second term is of the same order of magnitude as

the first. If we use the asymptotic expansion of bA(p) for t0

<< p <<JIL given in note [j]J

then we obtain a formula of the form (7 .. 101), but with a numerical coefficient which is half

as large, i.e •

155

(t << p << ,fii, ) . 0

(7 .102)

Eg_. (7 .101), which is valid under the condition \I')J: >> t , agrees with Eg_. (6.65) which is 0

valid when the condition J):-1 << t is met. Thus, the character of the phase fluctuations of 0

the wave does not change when L goes through the critical value Lcr N t~jA • However, in these

two cases the amplitude fluctuations are described by different formulas.

7.6 Relation between amplitude and phase fluctuations

and wave scattering

As we have seen above, Eg_. (7.21), which determines the size of the field fluctuations,

can be obtained both by using the eg_uation

which determines the wave scattering, and by using the eg_uation

o,

which describes the amplitude and phase fluctuations of the wave. In deriving Eg_. (7.21) by

the first method, we added together the incident wave u0 and the wave u1 scattered by the

inhomogeneities of the medium. The scattered waves, arriving at the observation point with

random values of amplitude and phase, are added to the "unperturbed" wave and produce ampli

and phase fluctuations of the total wave. The relation between the amplitude and phase

tuations of the wave and the scattering of the waves can be pursued in more detail. As we

2 shown above, X = BA(O), the mean sg_uare amplitude fluctuation of the wave, can be expressed

in terms of the function !JK) by using Eg_s. (7.50) and (7.54), i.e.

(7.103)

ec lOll o e volume V into The function !n(K) also determines the effect·ive scattering cross s t' f th

the solid angle dD (see page 68), i.e.

dcr( e) (7.104)

where we have omitted the factor sin2Xwhich depends on the polarization of the incident wave.

Introducing dcr0,the effective scattering cross section of the unit volume, we have

dcr (e) = 2~k4~ (2k sin ~)dD o ~n 2 •

We make the change of variable

K 2k sin~ 2

(7.105)

(7-106)

in the integral (7.103). The variable Kin Eg_. (7.10~) ranges from 0 to ro. At th _; e same time,

the g_uanti ty 2k sin ~ can only take values from o to 2k for real e. The value of the function

!n(K) is zero or negligibly small for K > K • m Therefore, in the case K >> K considered in m

this section, the upper limit of integration in (7.10~) can be _; replaced by any number which is

v c:.. corresponding to the value e = n./2. Substituting much larger than Km' in particular by 2k/ 12,

(7.106) into (7.103), we obtain the formula

n./2

[

sin(4kL sin2

!!..) J 1 - 2 ! (2k

2 e n 4kL sin 2

(7.107) J . e)k2 . Sln 2 slnede •

0

157

can eliminate the function !n from Eq. (7 .107) by expres­If now we use Eq. (7.105), then we

sing it in terms of da0/dn ,i.e.

2 X = nL

n/2 [ J 1 0

sin( 4k1 sin2 ~) l da (e) --%n- sine

2 e 4k1 sin 2

de • (7 .108)

( )/ 1 t the effective Scatterl·ng cross section at the angle e into The quantity dao e dn is equa 0

We denote by dal(e) the effective scattering cross section the solid angle dn = sin e de d~.

e de bounded by the cones of aperture e and e +de. Since into the solid angle dn1 = 2n sin

da (e) does not depend on~' then da0/d~ = da1(e)/2n and da 0 (e)sinede/d~sinede = da1 (e)/2n. 0

Eq. (7 .lo8) takes the final form

[

l _ sin(4kL sin2 ~) l

4 . 2 e k1 Sln 2

(7 .109)

. The amplitude fluctuations The expression just obtained has a simple physical meanlng.

superposition of the scattered waves (the quantity dal(e) in the of the wave are caused by the

integrand) • The factor

sin(4k1 sin2 ~)

1--------4 . 2 e k1 Sln 2

. f the Fresnel zone and the dimensions of the s depends on the ratio between the dimenslons o . -

above, the quantity t(e), the size of the lnh~mo tering inhomogeneities. In fact' as shown 1\.

a1 t t(e) = i-./(2 sin ~2), i.e. 2 sin~= 178' . at the angle e, is equ 0 v\~} geneities which scatter

Therefore, the quantity

. 2 e 2n 1-.2 :X.L

4kL s1n - = - 1 -- = 2n --2 :x. t2(e) t2(e)

iS proportional to the square of the ratio of the radius of the first Fresnel zone to the

size of t(e). The function

1-sin( 2ni-.L/ t 2( e))

2n:X.L/t2

(e)

has a maximum for t (e) "' {):.1 Thus, in forming the amplitude fluctuations of the wave,

"fullest" use is made of the energy scattered by the inhomogeneities whose dimensions are

close to the radius of the first Fresnel zone. In the case where all the inhomogeneities are

"large" compared to /i.L , they act like coherent scatterers, and the effects of the different

Fresnel zones spanned by the inhomogeneities cancel one another out to a considerable extent.

In this case

sin(4k1 sin2 ~)

1 t2

(e) . ~ 2n.AL~ - Sln ---2n.A1 t2(e)

1 2._2 . 4 e "' 6 • 16k L Sln 2 , 1--------

and (7 .109) reduces to the formula

(7.110) 0

which corresponds to geometrical optics. In the opposite case, where the radius of the first

Fresnel zone is much larger than the largest scale refractive index inhomogeneities, there are

a large number of incoherent scatterers inside each zone, and their effects add up like energy.

In this case /i.L/t(e) >> 1, and Eq. (7.109) takes the especially simple fon;n.

159

n/2

X2 _! L J ( ) 1 -2

da1

e = 2 aL •

0

Here a is the effective scattering cross section of a unit volume or the scattering coeffici

(a has the dimensions of cm-1 ). This quantity determines the attenuation of the wave due to

scattering in going a unit distance.

Eq. (7 .111) and the similar formula for phase fluctuations of the wave can be obtained

without recourse to the expression (7.104) defining the effective scattering cross section

To do this) we divide the whole region traversed by the wave in the inhomogeneous medium into

volume elements Vk' whose linear dimensions are much larger than the correlation distance 10

the refractive index fluctuations. The field at a point M is the sum of the field u 0

incident wave and of the fields ~ scattered by the volumes Vk in the direction of the point

i.e.

U=U 0

+ 2: ~. k

(7.

We represent the field u in the form u = A exp(iS ) and the fields u in the form ~ exp( 0 0 0 0 K -k

Since the volumes Vk are separated from one another by distances which are much larger than

the waves ~ scattered by them will be statistically independent. We assume that the fluc­

tuations of the field are small) i.e. that [t]

Then we have

log u = log u +log (1 + 2: :) "' log U 0 + 'L: : · 0 \.:: 0 0

160

log A = log A + 0 ~ ~ cos(S - s )

0 k 0 )

s So + ~ ~ sin(Sk - S ) • 0 0

this we obtain

Ai~ ---2- cos(S. - S ) cos(Sk _ s )

A l o o 0

(7.113)

(7 .114)

(7.115)

(7.116)

for the mean square fluctuations. Because of the statistical independence of the waves scat-

tered by the different volumes V l t k' on Y he terms with i = k are different from zero in the

double sums. Therefore we have

{ 2 ~ cos (s - s ) A k o

0

2

s~ = ~ ~ sin2(sk - s ) •

A o 0

(7.117)

It can be h s own that the quanti ties A_ and Sk - So -k are statistically independent [6o]. There-

fore

~ 2 Ak cos (sk

161

equal to 1/2 when its argument is uniformly dis­since the mean square value of the cosine is

tributed in the interval (0,2~). Similarly

so that

(7 .118)

------ modulus of the scattered field produced at 2 (e ) represents the mean square The quantity 1k k V (the angle of scattering is denoted by ek). The flux

the point M by the scattering volume k __ 2 2 where r is the distance from the

of scattered energy is proportional to the quantity Ak_ r dn,

center of the volume Vk to the point M. The density of the energy flux of the primary wave uo

which we denote by Vkdao(ek), is equal to incident on vk,

so that

. · (7 118) we obtain Substituting this expresslon ln • '

t in this formula, we have Going from summation to integra ion

2 2 lf x = sl = 2 v

162

(7 .119)

(7 .120)

If we locate the origin of spherical coordinates at the observation point M, the polar angle

2 will equal the scattering angle e and dV = r dndr. Therefore

(7.121)

0

and we again arrive at Eq. (7.111). Since we assumed that the field at the point M is propor-

tional to ~ and does not depend on the dimensions of the scattering volume Vk' we have hereby

assumed that the linear dimensions of the volume Vk are much smaller than the radius of the

first Fresnel zone, i.e. that L0

<< Vk1

/ 3 << ~ . Therefore, Eq. (7.21), and Eq. (7.111) as

well, is valid when the condition ~ >> L0

is met.

With this we finish our study of the problem of parameter fluctuations of a plane mono-

chromatic wave. In Part IV, we shall consider some applications of the theory presented above

(calculation of the frequency spectrum of the fluctuations, dependence of the fluctuations on

parameters of the receiving apparatus) and we shall compare the theory with experimental data.

Chapter 8

PARAMETER FLUCTUATIONS OF A WAVE PROPAGATING IN A TURBULENT :MEDIUM

WITH SMOOTHLY VARYING CHARACTERISTICS

Until now we have considered the case of a locally isotropic ref~active index field}

it has been assumed everywhere that the structure C~ does not vary along the entire

path of the wave. However} in practice one must almost always deal with inhomogeneous turbu-

lence. As already noted} the "two-thirds la~' obtained in Part I is valid only in the case

where the distance between the two observation points does not exceed the outer scale of tur-

bulence L0

} i.e.} it is valid in a region with dimensions of the order L0 • If we place our

of observation points in another region with dimensions of the order L~J which does

sect the first region} then the 11 two-thirds lav' holds for it as well} but this time

another value c• 2 of the structure constant. Therefore} we can assume n

function of the coordinates} which changes appreciably only in distances of the order L0

(see Chapter 3)· Thus

I~ ~ I 2 ~ ~ where rl - r2

<< L0

, and Cn(r) changes appreciabl:y only when r changes by an amount of

L 0

• For small values of I-; 1 - -; 2 I J we have

as before. The concept of the spectral density ~n(K) corresponding to (8.1) and (8.2) was intro

in Chapter 3· We can immediately write down an expression for !n(K)J by changing the canst

c2 in Eg_. (7 .. 86) to the function c2(-;). Then we have [a]

n n 164

for K < K m

for K > K m

(8.3)

(8.4)

wo- r s law J but some other correla-In the more general case J where we shall not use the "t thi d 11

tion or structure function} we shall always assume that il\ (K};) can be ~n represented in the form

( 8. 3)' but with another function ! do)(/() • In analogy to ( 8. 3 L we can also write the two-~ -+-

dimensional spectral density F (K K \x' nl r' + r" n 2} y - x J--2- ) (see Chapter 3):

As before} the functions Fn and ~n are connected by the relation (l.53). In particular J we have

00

(8.6)

e problem of amplitude and phase fluctuations of a plane We now consider how to solve th

wave propagating in a medium with a smoothly varying 11 intensity' of turbulence. To solve this

problem we shall start 'thE ( 4) Wl <l• 7.2 and we shall use the same kind of two-dimensional spec-

tral expansions of the f re racti ve index fluctuations as in 1 so ving the problem for a homogeneous

medium. In deriving Eg_s. (7.26)-(7.43)} we never used the t' ( as sump 1on that F K K x 1 x11 ) de-n 2} 7._} }

pen~ only on xl - xll • Th _.; erefore Eg_s. (7.42)-(7.43) continue to hold in the case where

Fn (K2}K3

} x 1 } x") has the form ( 8.5)} i.e.

0 0

= ~2 J J [cos ~· - cos K2(\- ~ ~ c~(j:)fn (K, I' I )d~d' '

(8.7)

D

" 2~ _ ~rl + ;n The region of integration D is a rhombus with where S = X I - X

11 J 21") = X

1 + X J r - "

vertices at the points (o,o), (1,L/2), (o,L), (-L,L/2). Similarly, we have

X

D

integrands are even with respect to s, we need carry out the integra~ Bearing in mind that the

tion only in the right hand half of the region D. Then we obtain

L

J L/2

2(1 -T))

c!(;)dll J 0

l66

the important region of integration with respect to g, we have Kg~ l (since fn(K,g) ~ 0

2 Ks >>l; see page 23). Therefore,K s/k $K/k <<lin this region. Consequently,

cos K2s/2k N l and the inner integrals in (8.9) are approximately equal to

21") 2

J ~ - cos K ( 1k- TJ )j f(K,g)ds n 0

and

2(1 - T))

[1 - cos K2(\- ~)j J f (K,s)ds • n

0

larger part of the region of integration both 2T) and 2(L - T)) have the order

Therefore, we have approximately

co

J f n ( K, s ) ds = 1( In ( 0 ) ( K) 0

2(1 - T))

J 0

i.e., both inner integrals in ( 8. 9) are approximately equal to

Thus we obtain

I c!(?) :2,[ 0 l(K) sin2 r2(~- ~)}~ '

0

(8.lo)

! c~(;) !n(o)(K) cost2(~- ~)}~ •

0

f E (8 lo) that the contribution to FA(K

2,K

3,o) of inhomogeneities located near

It follows rom q. • · [b l since the integrand vanishes for T) = L.

the observation point lS zero _ J,

1 t. n function of the amplitude fluctuations of the wave in the

We now consider the corre a lO

_ L According to the general formula (1.51), we have plane x - •

00

B A ( p) 2n J J o (K p )FA (K' O)KdK '

0

i.e.

. 2l~2(L - 1) )] KdK Sln L 2k. }

0

( 8.13)

1 the problem of amplitude fluctuations of the wave. Consider the

Eqs. (8.12) and (8 .. 13) so ve (o) -+

~ the refractive index fluctuations Obey the t•two-thirds law". In this case !n (K)

case -wuere

is given by Eq. (8.4) and

For K~/k << 1 (geometrical optics)' we obtain m

( 8.13)

168

L 2 2 2! 2 ~ X = 4n (0.033)k cn(r)dT)

K m / 4 2

J K-11 3 K (L - TJ) KdK = 4k2

0 0

L

= 7.37 t~7/3 J c!(;)(L- T))2

dT). (8 .. 14) 0

.. (We have used the relation K t = 5.48, see page 49 ) ,. For K~/k >> 1, i.e. /):1 >> t , the . m o m o

integration in ( 8.13) can be extended to infinity, &ld then [ c]

L

x2 = 0.56 k7/6 J c!(;)(L- x) 5/6dx • (8.15)

0

c! = const, Eqs. (8.14) and (8.15) imply Eqs. (6.68) and (7.94) for homogeneous

It is convenient to change Eqs. (8.14) and (8.15) somewhat, by locating the origin

of coordinates at the observation point. Then we have

L

x2 = 7.37 t ~7 13 J c!(;)x

2dx ( j):L << t )

0 (8.16)

0

and

L

x2 = o.56 k 716 J c!c; )x5/6 dx ( j):L >> t ) •

0 (8.17)

0

When the wave source is located at infinity, we can set L = oo in Eqs. (8.16) and (8.17). Then,

:l.n estimating the quantity /):£ , it suffices to take the distance in which c2

essentially falls n

to zero instead of L.

We now consider the phase fluctuations of the wave. Adding (8.10) and (8.ll), we obtain

and

L

DA(p) + D8(p) = 8n~2 J 0

2(-+ C r)d'l) , n

c!(r)dT) J [;l- J0

(KP)] !n(o)(K)KdK •

0

Eq. (8.19) replaces Eq. (7.95) for the case of homogeneous turbulence.

by Eq. (8.4), we obtain an expression similar to (7.98), i.e.

L 2-T

C (r)dx .. n J

0

( 8.19)

It follows from (8.,20) (see the derivation of Eqs. (7.101) and (7.102)), that in the region

J}:L >> t , we have 0

L

J 0

L

J 0

( t << p << If.£ ) ' 0

(p ~ Vi£) .

The integration in Eqs. (8.16) - (8 .. 22) is carried out along the 11ray

1 from the observation

point to the source. Comparing Eqs. (8.16) and (8.17) with Eqs. (8.21) and (8.22), we can

easily discover an important difference between them. In fact, all the inhomogeneities,

less of their distance from the observation point, have the same effect on the phase

tions [d]. However, the inhomogeneities which are furthest away from the observation point

have the greatest effect on the anrpli tude fluctuations of the wave [b J .

170

Let us consider an example. In analyzing the twinkling and quivering of stellar

we encounter a very n if onun or.m distribution in height of the refractive index fluctua-

tions. Usually, the strongest fluctuations are observed near the surface of the earth, and

the fluctuations become weaker as the height increases., In this case, the lower layers of the

t;ttmosphere will make the largest contribution to the integral 00

which determines D8(p). However, higher layers will make the basic contribution to the inte-

gral 00

amplitude fluctuations of the wave. For example, let

Then (assuming that Ji:Z: >> t ) we h o 0 ' ave

If C= is given in the form

0

5/6 x dx _ 0.56 n 02 7/6 11/6 2---- k z

1 + x 2sin~ no o 12

171

(8.23)

( 8.24)

(8.25)

(8.26)

or

~ 4 2 7/6 ll/6 X ). cno k zo

2.91 ~ ~ P513 c2 z = 4.57 c2 k2 p5/3 z Ds(P) no 0 no 0

the same structure, but they have ~uite dif­E~s . ( 8. 24)' ( 8 • 27) and E~s • ( 8. 25) ' ( 8. 28) have

ferent values of the numerical coefficients.

172

Chapter 9

AMPLITUDE FLUCTUATIONS OF A SPHERICAL WAVE

In addition to the problem of the amplitude and phase fluctuations of a plane monochro-

considered in Chapter 8, in many cases it is of interest to consider the problem of

a spherical wave [60]. To solve this problem, we shall start with E~. (7 .21),

the solution of Eg_. (7 .14) for the case of an arbitrary unperturbed wave, i.e.

2 --?

6 ~ l + 2 \1 ~ o" \1 ~ l + 2k nl ( r) 0 '

u (~) represent a spherical wave propagating from the origin of coordinates, i.e. 0

Q is some constant. Then we have

eik(r' -r)

r'

(9.1)

( 9.2)

Chapter 8, we assume that l << t0

, where t0

is the inner scale of the turbulence. Then,

contribution is made to the integral (9.4) only by the region of integration con-

a cone of aperture e N l/t << 1, whose vertex lies at the observation point and 0

assume to be the x-axis) is directed from the wave source to the observa-

Inside of this cone, we have

173

]x'] >> ]y1], ]z'] and ]x-x'] >> ]y-y'], ]z-z'].

In this case, the expansions

y12 + zr2 ~~ ~] (y- y1)2 + (z- zt)2 r 1 ,... x 1 + - , r - r' N x - xt + -'-"-'~....::.--t-__,......-l---~-.L--

2x' 2(x - x')

are valid. In addition to the values of the field at the point ; = (x,o,o), we shall also be

interested in values of the field at points for which ;;2

+ z2

is not zero but

than the distance x to the wave source. Therefore, the field u = ~ eikr of the incident o r

can also be represented in the form

Substituting all these expansions into EQ. (9.4), we obtain

nl (X I J y I J z I ) X

f 2 2 2

2xx'(yy1 + zz')- x 1 (y + z) 2 2 2 l

- X (y' + z' ) r dx'dy 1dzt • (9 .. 5)

X x'(x-x')

We assume that the random field n1

(x' ,y',z') is a homogeneous and isotropic random fie

and can be represented in the form of a stochastic Fourier-Stieltjes integralt

nl (X I J y I J z I )

00 J J exp~ ~2Y' + K3z) Jdv(K2,K3,x') ,

-oo

where the QUantities dv(K2

,K3,x 1 ) satisfy the previous relation (7.40), i.e ..

(9.7)

(We have assumed that the field n1 (;) is homogeneous and isotropl' c 1· n order to simplify the

aolution of the problem. However in wh t f 11 p ' a o ows, it will not be difficult to extend the

solution obtained to the case of a locally homogeneous and isotropic field n1 (;) as well.) We

o expans1on, i.e. sball look for the function tl(x,y,z) in the form of the same kind f .

00

(9.8)

• Q• • , we obtain Substituting the expansions (9.6) and (9 8) into E (9 5)

00

J J expG 02Y + K3~]dcp(K2,K3,x) = ~x x -oo

X J dV' X

x 1(x-x1)

00

X J J expG 0~Y~ + K3ZJ Jdv(K2,K3,x'). (9-9) -oo

175

-i(K y + K z) . 1 2 3 and integrate with respect to Y and z

we multiply this equatlon by 2 e 4n oo ( / ) J exp( iA.z )dz "" 5(A.) and that the limits -oo and oo • Bearing in mind that 1 2n

-oo

equality

00

dK2dJ(3 J J 5(K2 - K2)5(K3 - K:3)Ckp(K2,K:3,x)

-oo

holds [a], we obtain

00

If X

X X

X

00

J J expG 02Y' + K3z9 Jdv(K2,K:3,x') •

-oo

r d ' ·n (9 10) can be extended between The integration with respect to the variables Y an z l •

regl·on where the values of y' and z' are large, the int limits -oo and oo , since in the Changing the order oscillates rapidly and the integral over this region is near zero.

gration in ( 9 .lO) and also changing x to L (the distance to the wave source), we obtain

dx' X

x 1 (L- x') -oo

176

00

x J J J J exp G (K2Y' + K3z 1

- K2y - K3z) J X

(9 .. 11)

inner fourfold integral can be simply evaluated by contour integration in the complex plane

iS easily seen to equal

8n3L(L - x1

) tiL(L x') (K~ + K~j ~ L~ ~ ------ exp ---------'::.. 5 K 1

- K - 5 K 1

ikx' 2k_xl 2 2 x' 3 - K L~

3 x') (9.12)

tituting (9.12) in Eq. (9.11) and carrying out the integration with respect to x2 and K:3,

a simple formula connecting the spectral amplitudes of the field fluctuations and

e of the refractive index fluctuations:

(9.13)

(9.13) is similar to Eq. (7.32) for a plane wave [b] and has a simple physical mean-

Field · h ·t· h t · db th b (i.e. by the dimensions t"" 2n) ln omogenel les c arac erlze y e wave num er K K

inhomogeneities of the medium with characteristic wave number K ~' or with X

X t' = t L . These inhomogeneities are at the distance x from the wave source. The

x/L takes into account the magnification of the dimensions of the image due to illumina­

by a divergent ray bundle. The quantity L(L - x)K2/2kx in the argument of the exponential

· a1 2/ 2 2 A.x(L - x) equ to n A t' , where A = L is the square of the radius of the first Fresnel

177

, I

zone for a spherical wave, and t• is the size of the inhomogeneities in the x-plane which

t Using relations similar to (7.35) and duce field inhomogeneities of size •

find the spectral amplitudes of the quantities X= log(A/Ao)and Sl = S- So.

9..S before, by da and dO', we obtain

~ -k ! dx cos t(L -X:~ + K~)}v ~2 ~ ' K3 ~ ' ~ •

0

th 1 tion (structure) fUnctions of the We now turn to the spectral expansions of e corre a

quantities X and s. To do this, we form the expression

Whi ch contains cosines instead of sines. Using the and the analogous expression for dO'dO'* ,

2 2 2 , K'2 = K' 2 + K' 2, we obtain relation (9.7), and writing K = K2 + K3 2 3

178

L L

da(K2,K3'L)da*(K2, K3,L) = k 2Ll.j. J,[ dxl dx2

0 0

X

X

(9.17)

from the relation (9.17) that the fields of X and S are not statistically homogene-

plane X= L (for these fields to be statistically homogeneous, the factors o(K2 - K2h 5(K

3 - K3) would have to appear in the right hand side of (9.17). It is clear that these fields

be homogeneous on the sphere r = L. The departure from statistical homogeneity is rela-

ted in the first place to the fact that we are examining the field in the x = L plane and not

on the sphere r = L, and in the second place to the fact that the unperturbed wave appears in

our case in the form

so that the direction y = 0, z = 0 is singled out.

We now calculate the mean square fluctuation x2 of the logari tbmic amplitude. Since

X(L,y,z) JJ

then, setting y = z o, we obtain the formula

179

X(L,o,o) JJ -oo

Consequently, we have

2 X (1,0,0) = X(L,O,O)X*(L,O,O)

00

(9.18

-oo

Using the expression (9.17), we find

X

(9.

Taking into account that

we carry out the integration with respect to K2 and K3, obtaining

180

Since

2 . L(L - x1 )K

Sln 2kx l

X

(9.20)

in the case of isotropic turbulence, then in Eq. (9.20) we can go over to polar coordinates in

the (K2,K3) plane and carry out the integration over the angular variable, on which the inte­

grand does not depend. As a result, we obtain the formula

X

(9.21)

We designate the inner integral in (9.21) by P(K), i.e.

(9.22)

181

We make the change of' variables x1

- x2

= ~) x1 == ~ in this integral. Then we have

d L(L - ~)K ~sin 2k ~ ~

2

J~ F (J.<L n \ ~ )

~ - L

L(~ - ~)(L - ~ + ~)K2 X sin d~ •

2k~2

Consider now the inner integral

Q==

~

J F (~L n \~ J

L(~ - ~)(L - ~ + ~)K2 sin d~ .

2k~2 ~ - L

Since the function Fn(KJ \~\) is appreciably different f'rom zero only f'or K~ $ lJ then only

part of' the region of' integration where \~\KL/~ ~ 1 or \~\ $ ~/KL contributes

the integral (9.24). Since KL >> l (the chief' influence on the values of' the fluctuation

at the observation point is due to inhomogeneities with dimensions of' the order J1L J

KL "'JLli.. >> 1 (see page 140), then \~\<<~in the important region of' integration.

f'ore, we have ~ - ~ ..., ~ and L - ~ + ~ N L - ~· It can be shown that the discarded terms in

the argument of' the sine are of' order no larger than v0J1 << 1. Therefore Eq. (9.24) takes

the f'orm

Q==

2 L(L - ~)K

sin----2k~

Since the :f'unction Fn ~~ , I<~ :fallS o:f:f quickly to zero even :for I< I << ~' the intea:r>at~iolll. ( 9. 25) can be extended between the limits -oo and oo • Taking into account that (see page

00

J Fn ~~ ' 1<0 d<

182

expression [c J

( 9· 26)

(9.27)

the change of' variables KL/n = K' in (9 27 ) . . , • J we obtaln

(9.28)

tuting this expression in Eq ( 9 21 ) h • • ) we ave

00

X2

= 411:~2L J dK J (9.29) o K

the order of' integration with respect to K and K' l'n (9.29) and simultaneously relabel-

the variables of' integration K and K'' we obtain

K

J . 2 [J,K 1 ( K - K I )l Sln [ 2k jdK' . (9.30)

0

The inner integral in (9.AO) ..~ reduces to the Fresnel integrals

X X

C(x) I s(x) J 0 0

and equals

ffnk [ K~ ( r:;r{\ . K~ ( r:;r{\ J V -y:- Los Tk C \j_ ~) + s1n Tk S \}/_ ~) ,

so that

x2

= 2n~2L l ~ l - ~ Gos ~ C ~) +

+ sin~ S ~) J ) ~JK)KdK •

Eq. (9.31) expresses the mean square fluctuation of' the logarithmic amplitude of' a spherical

wave in the case :A << t in terms of' the spectral density~ (K) of' the refractive index f'luc-o n

tuations [d]. A similar f'ormula exists f'or the mean square phase fluctuation of' the wave)

namely

However) we note that while Eq. (9.31) can be extended to the case of' locally isotropic +,·~h,,,.~ 4

lence (since the term in curly brackets in (9.31) goes to zero like K as K ~ 0)) Eq. (9.32)

valid only in the case of' homogeneous and isotropic turbulence.

Consider the case where the relation VXL << t holds. In this case) 0

holds f'or all values of' K f'or which ~n(K) is dif'f'erent f'rom zero.. Making a series expansion

the integrand of' ( 9. 31) in powers of' j K2L/2rck ) we obtain the f'ormula

184

0

comparison) we give here the f'ormula which dete~~nes x2 f' 1 ,rvr ... J.J..L.L. or a p ane wave when v:AL << t . 0

Chapter 6) we obtained the f'or.mula

which it f'ollows that

00

F (K)O)KdK = ! 1(~3 J n 3

0

~9o,mparing Eqs. (9.33) and (9.34)) we convince ourselves that when VXf, << .e,0

) the mean square

~!uctuation of' the logarithmic amplitude of' a spherical wave is ten times smaller than the

corresponding quantity f'or a plane wave J and that this ratio does not depend on the f'or.m of'

spectral density of' the refractive index fluctuations (or on the f'orm of' its correlation

o.r structure function) [e].

We now consider the case where the correlation funct1· 0 n of' the

tions exists and has the f'in' t i t al 1 ( 44 1 e n egr sea e Ln see page 1 ):

00

2 2 B (r)dr = ~

n B~ \OJ n

00

J 0

refractive index f'luctua-

li'or Vf1 >> t ) we l t th 0

can neg ec e rapidly oscillating function in the integrals (9.31) and

(9·32) (see the similar example f'or a plane wave on pages 143-144 )) obtaining the f'ormula

(9.35)

which coincides with Eq. (7.68) for a plane wave. Thus) for )T:-L >> t ) the mean square 0

amplitude fluctuations of a plane wave and of a spherical wave are equal to each other.

We now consider two concrete examples.

1. Let the field of refractive index fluctuations be homogeneous and isotropic and let

it be described by the correlation function

The mean square fluctuation of the logarithmic amplitude of the wave for J'f1 << a can be

found by using Eq. (7.78) for a plane wave. Bearing in mind that for a spherical wave the

quantity x2 is 10 times smaller than (7.78)) we obtain

This expression agrees with the quantity found in Bergmann's paper Q+5] by using the euu.o.vJ_ur!l

of geometrical optics. If VfL >> aJ the mean square fluctuation of the logarithmic

of the spherical wave agrees with the corresponding expression for a plane wave. Using Eq.

(7.80)) we obtain

X2 - j;( 2 k2aL - 2 nl

2 2 A more detailed investigation of the expressions for X and s1 is carried out in the papers

~8J6JJ for the special case where the correlation function of the refractive index fluctua­

tions has the form of the Gaussian curve (9.36). However) we shall not give the expressions

appearing in these papers J which express X2

and S~ for an arbitrary value of the ratio JIT. / because of their excessive complexity. (In the limiting cases of small and large values of

the parameter JIT, /a) these formulas agree with the relations (9.37) and (9.38) ·)

the applicability of Eqs. (9.37) and (9.38) in practice is quite doubtful) since the correla-

tion function of the refractive index fluctuations does not have the form of a Gaussian curve

under actual atmospheric conditions.

186

We now consider a much more realistic example) when the field of refractive index

ions is locally isotropic and is described by the "two -thirds law". .As we have already

the spectral density!n(K) corresponding to th;i.s law can be taken to be

~n(K)

for K < K m

for K > K m

(9.39)

the size of the amplitude fluctuations separately for J):.L << t and ,/IT.. >> t . - 0 0

case we can use the expression x2 = 2.46 c2 1 3 t-7/3 which is valid for a plane n o

Dividing this expression by lOJ we obtain

X2 2 c2 13 o-7/3 :::: o. 5 v n o (9.40)

VXf, >> t0

(and at the same time V'f1 << 10

) we have to use the general formula (9-31) 1

x2 ~ 2n

2(o.033)c! k~ jm K-ll/3 ~l - );l;. ~os ~ c ~) +

+ sin ~ S (li1) J l KdK •

the upper limit of integration in (9.41) can be replaced by infinity. Making

change of variables V K2L/2rek = x) we obtain the expression

00

+ sin "~2 S(x)D <ix •

2 re~ c(x) J -8/3

X

0

( 9.42)

By numerical integration7 we find the integral in (9.42) to be equal to 0.90. ThusJ for

J"'f1 >> t 7 we have 0

The expression (9.43) differs from the corresponding expression (7.94) for a plane wave

by the numerical coefficient. The mean square fluctuation of the logarithmic amplitude

spherical wave in the case {IT., >> t 0

is approximately 2 .. 4 times smaller than the correspond~

ing quantity for a plane wave.

188

Part IV.

EXPERIMENTAL DATA ON PARAMETER FLUCTUATIONS OF LIGHT

AND SOUND WAVES PROPAGATING IN TRE ATMOSPHERE

Chapter 10

EMPIRICAL DATA ON FLUCTUATIONS OF TEMPERATURE AND WIND

VELOCITY IN TRE LAYER OF THE ATMOSPHERE

NEAR TRE EARTR AND IN TRE LOWER TROPOSHERE

Experimental investigations of the fluctuations of meteorological fields have been

initiated comparatively recently7 so that there is a lack of detailed data7 with the excep-

of some investigations devoted to the study of fluctuations of wind velocity and tem-

in the layer of the atmosphere near the earth [}7 7 36 7 37 7 627 6~. It is charac-

teristic of turbulence in the layer of the atmosphere near the earth that the turbulent

regime is strongly influenced by the earth's surface; therefore such turbulence has its own

special peculiarities. The layer of air several tens of meters thick lying near the earth's

surface is a turbulent boundary layer 1}4 7 64 7 65]. In the simplest case 7 where the air

moves over a plane surface 7 its mean velocity u is a function of the height. In the case

where we can neglect the effect of the buoyancy forces on the motion (the buoyancy forces

appear when the mean air temperature depends on the height) 7 the wind velocity varies with

the height according to the logarithmic law 1}_4 7 647 65]

v* z u( z) = ""K log z J

0

(10.1)

which is valid for z >> z0

• Here v* is a constant with the dimensions of velocity7 K is a

constant approximately equal to o.4 7 and z0

is a height determined by the roughness of the

underlying surface.

Eq. (10.1) is valid up to heights of the order of several tens of meters (30-50 m);

for large values of z 7 the growth of u( z) slows down. Within the logarithmic boundary layer

189

t f th turbulence like the rate of energy dissipation E, of the atmosphere) characteris ics o e

the coefficient of turbulent diffusion K, etc.J also depend on the height. To a first

ti K and E (see pages 29J 41) are given by the formulas [64J 65] approximation, the quanti es

(10.2)

2 2/3 2 CE 2/3, for the structure In Part IJ we obtained the expression Drr( r) = Cv r , where Cv

Substl.·tuting Eq. (10.2) into this last formula) we obtain function of the wind velocity.

1 of the atmosphere, the structure constant Cv falls ThusJ in the logarithmic boundary ayer

-1/3 [ l off with height like z aJ•

We can also write a similar expression for the concentration fluctuations of a censer-

In Part I we obtained the following formula for the structure vative passive additive ~.

function of ~:

andL 0

Substituting the expressions (10.3) and (10.1), we obtain the formula

(10.

it fil the mean concentration~ of a passive In the case of a logarithmic wind vela c Y pro e J

conservative additive is alSo distributed according to a logarithmic law [64J 65] I

190

~( z) canst + ~* log ~ z ) 0

(10.6)

where ~* is a constant with the same dimensions as·~- Substituting (10.6) into Eq. (10.5))

we obtain the expression

2 similar to Eq. (10.4) for Cv.

( 10. 7)

Eqs. (10.6) and (10.7) can be applied to describe the form of the mean temperature pro-

· file and the character of the temperature fluctuations in the layer of the atmosphere near

the earth. However) we should remark at once that in the case where the mean temperature

of the air varies with height) in particular when

T(z) z canst + T* log z- ,

0

(10.8)

Eq. (lO.l)J which describes the wind profile) becomes inapplicable. However, when the ver-

tical gradients of the mean temperature have small valuesJ the correction to Eq. (10.1) is

also small, and to a first approximation we can disregard it. In this case) we have approxi­

mately DT(r) = C~ r 2/3J where

2

C2 = 2 4/3 T* T a K ~ • ( 10. 9)

z

The quantities c! and c! defined by Eqs. (10.4) and (10.7) depend on z and change appreciably

when z is changed by an amount of the same order of magnitude as the value of z itself.

Therefore) in the boundary layer, the "two-thirds law" holds for distances r which are restric-

ted by the condition

r << z (10.10)

(see page 50). For large values of rJ the structure functions Dv(r) and DT(r) grow more

slowly than r 2/3 [96, 67].

191

In experimental investigations of the microstructure of the fields of wind velocity,

temperature, humidity, etc., in the atmosphere, one must use very sensitive, low inertia

instruments. Ordinarily, hot wire anemometers are used to measure wind velocity

[i7, 68], and resistance thermometers are used to measure temperature

There still do not exist sufficiently low inertia humidity detectors, which satisfy the ne

sary requirements (high sensitivity, small working volume) for measuring turbulent fluctuat

A hot wire anemometer is a thin wire (usually of platinum), which is heated by an electrical

current to temperatures of several hundred degrees centigrade. The heat exchange

and consequently its temperature, depends on the velocity of the wind flowing past it; this

allows one to relate the electrical resistance of the wire to the velocity of

on it [b]. The inertia of the hot wire anemometer is very small (for a wire of diameter

it does not exceed 0.1 sec [68] ), and its dimensions are of the order of one or two

To measure the structure function of the wind velocity, two hot wire anemometers are

opposite arms of a Wheatstone bridge, so that the current through the galvanometer is a funct

of the difference of the wind velocities at the points where the anemometers are located.

a detailed description of the apparatus, see the papers [17, 68].

Measurements of wind velocity fluctuations in the atmosphere made by both Soviet [)-7]

and foreign workers [69] have confirmed the "two-thirds law" to a sufficient degree of accu-

racy. In Fig. 14, we show the empirical structure functions obtained by Obukhov at various

heights above the earth's surface [).7]; the curves correspond to the "two-thirds law". The

dependence of the structure constant C on height, expressed by Eq. (10.4), agrees satisfac-v

torily with the experimental data, where, according to Obukhov' s data, the constant C equals

1.2. Measurements performed by Townsend [70], lead to the value JC = 1 .. 4 [ c]. Thus, the

formulas Drr(r) = C~ r 2/ 3 and C~ = Cv;(K2z2)-l/3 have been confirmed experimentally. This

allows us to make quantitative estimates of the fluctuations of wind velocity using simple

measurements of the profile of the mean wind speed in the layer of the atmosphere near the

earth. Measuring the mean values u1

and u2 of the wind speed at two heights z1 and z2 wi

the layer of the atmosphere near the earth and applying Eq. (10.1), we can determine the

quantity v*:

(10.11)

192

the quantity Cv can be determined from the formula

c v (10.12)

K .v 0. 4 and JC ,., 1. 4 [ d J • In the layer of the atmosphere near the earth, c · v lS equal

a few cgs units in order of magnitude.

l~vl/v 0.08r------,-------.--------<'}..-...,.~

0~------~-------L------~~~ 0 20 40 60 .e,cm

Fig. 14 Empirical structure functions of the wind field in the layer of the atmosphere near the earth [).7] •

We now consider measurements of temperature fluctuations in the layer of the atmosphere

near the earth. The difference between the temmeratures at two ~·~ points can be measured by using

a pair of low-inertia resistance thermometers (platinum wires a few tens of microns in dia­

meter), included in the i ·t c rcul of an unbalanced Wheatstone bridge. The voltage across the

galvanometer arm, which is proportional to the temperature difference of the detectors, is

amplified and then subjected to statistical analysl·s. (F or a discussion of the apparatus, see

the :papers '36, ""7, 68l.) F' 15 h L / ~ lg. s ows the empirical structure function of the temperature

193

field obtained by Krecbmer; the curve corresponds to the "two-thirds law'.

Fig. 15 Empirical structure functions of the tempera­ture field in the layer of the atmosphere near the earth.

Numerous measurements of the structure functions of the temperature field in the layer

of the atmosphere near the earth's surface have been made by the author of this book [)7].

The measurements confirmed the "two-thirds law" for the temperature field and allowed the

intensity of the temperature fluctuations to be related to the mean temperature profile. Fi

l6 shows the experimentally obtained dependence o~ the quantity CT on K2/3z-l/3T* (see Eq.

(10.9)); each point of the graph was obtained as a result of measuring the structure fun

DT( r) for four values of r) beginning with r = 3 em and ending with r = l m.

half of the graph corresponds to unstable stratification of the atmosphere) i.e. to

of the mean temperature with height) while the left hand side corresponds to stable

cation (temperature inversion)) i.e. to an increase of the mean temperature with height.

is evident from the graph) for unstable stratification the empirical dependence of CT on

K2/3z-l/3T* corresponds to Eq. (10.9)) and the coefficient a turns out to be equal to 2.4o.

For stable stratification (temperature inversion)) the growth of CT lags behind the growth

K2/3z-l/3T*) which is a consequence of the influence of the temperature stratification on

regime (violation of the condition that the additive be passive ). However) even

case we can determine the empirical dependence of c 2/3 -l/3 T on K z T* (see Fig. 16)

indicates the empirical law obtained by analyzing th e experimental data by the

squares [e]).

X

-0.20

X

X

X --------X -----X X

X 0

Fig. 16 Dependence of c th h T) e c aracteristic of the tempera-

ture microfluctuations) on meteorological conditions.

• ) August 1954; 6) March-April 1955; 0 ) June-July 1955

(l.5 m); 0 ) June-July 1955 (22 )· m J X ) July-September) ( 1955)

(The values of CT plotted in the lower left hand quadrant

are positive. )

195

The graph in Fig. 16, or Eq. (10.9) in the case of unstable stratification, allows

to make quantitative estimates of the size of the temperature fluctuations by using

tively simple measurements of the mean temperature profile in the layer of the atmosphere

the earth. By measuring the values T1

and T2 at two heights z1 and z 2 and applying Eq. (lo

we can determine T*:

Then the quantity CT can be determined in the case of unstable stratification by using the

formula

1.4

or by using the graph (Fig .. 16) in the case of stable stratification. It is clear from the

figure that the size of CT in the layer of the atmosphere near the earth varies from zero

(for isothermal stratification of the atmosphere) to values of the order of 0.2 deg cm-l/3.

Fig. 17 shows the monthly-averaged diurnal trend of the quantity CT (for August 1955); this

curve can also be used to estimate the size of CT.

In addition to measurements of the temperature structure function in the layer of the

atmosphere near the earth, measurements have alSo been made of the temperature fluctuations

in the lower troposphere up to heights of the order of 500-700 m (on tethered balloons) [6o]

These measurements have alSo confirmed the "two-thirds law". The values of CT so obtained

-l/3 lie in the range 0 - 0.03 deg em • At nighttime, appreciable temperature fluctuations

(CT N O.Ol - 0.03 deg cm-l/3) are observed only in the inversion layer near the earth, whi

usually extends from the level of the earth up to heights of the order of hundreds of met

During the day when the stratification is unstable, temperature fluctuations in the lower

troposphere are usually observed up to greater heights [f] •

196

0.16

0.12

0.04

-0.04

-0.08

X - K2/3 Z-1/3 T *

o-Cr

Fig. 17 Diurnal trend of CT in the layer of atmos­

phere near the earth (August 1955).

197

Chapter 11

EXPERIMENTAL DATA ON THE AMPLITUDE AND PRASE FLUCTUATIONS

OF SOUND WAVES PROPAGATING IN THE LAYER OF THE ATMOSPHERE NEAR THE EARTH

Beginning in 1941, Krasilnikov and his coworkers performed a series of experiments to

study the amplitude and phase fluctuations of a sound wave propagating in the layer of the

atmosphere near the earth. The time structure function @J(t+~) - S(t)]2

of the phase

tuations and the mean value [log( A/ A )] 2 of the fluctuations of logarithmic amplitude 0

sound wave were measured in Krasilnikov's experiments. First we consider the phase fl

tions of the wave. In the case where the inhomogeneities in the distribution of wind ve

ity and temperature do not have time to change appreciably in the time ~, we can assume

they are merely convected (without "evolution") by the mean wind [a].

the wind is perpendicular to the direction of propagation of the sound

is v, then the value S(t+~) of the phase at the point M coincides with the value

t of the phase at the point which is a distance v~ away from M. Thus we have

According to Eq. (7.101)

for t0

<< p. Thus, the relation

must be satisfied, i.e., the phase variability is proportional to the structure constant Cn

to the sound frequency, to the square root of the distance traversed by the sound wave,

to the time interval raised to the power 5/6. Fig. 18 shows the dependence of aS on 1,

198

by Krasilnikov and Ivanov-Shyts [71], while Fig. 19 shows the dependence of aS on

sound frequency is 3000 Kcps, the distance L = 22, 45 and 67 m, v = 5 m/sec, ~ = o.o4,

Fig. 18 Dependence of the phase fluctuations of

of a sound wave on distance. (The quantity J1 is plotted as abcissa, and the average value of

aS for various ~t is plotted as ordinate.

Fig. 19 Dependence of the fluctuations of the phase differ­

ence aS = j [}3(t+~) - S(t)] 2 on ~. (The quantity Cv~)5/6 is plotted as abcissa, and the average of aS for various

values of Lis plotted as ordinate).

199

--------;

As can be seen from the figures) the dependence of aS on L and v~

with Eq. (11.2). Ultrasonic experiments performed at frequencies up to 50 Kcps also

satisfactory agreement between the experimental and theoretical results [72]. Thus)

dependence ( 11. 2) has been confirmed experimentally over a frequency range from 1 to

Fig. 20 shows the dependence of the quantity a A= j [log(A/A0

)].2 on the distance

the data are referred to the distance 22m). The dependence of aA on Lis satisfactorily

approximated by the formula aA = AL~J where~- 0.8. Note that we ought to have~ N 0.92)

according to Eq. (7.94). (In the experiments under consideration -j"i:£ >> .t .) Thus) the 0

experiments of Krasilnikov and Ivanov-Shyts agree satisfactorily with the theoretical

(7 .94).

Fig. 20 Dependence of the fluctuations of logarith­mic amplitude of a sound wave on the distance. (The ratio of the fluctuations at the distance L to the fluctuations at the distance 22 mJ averaged over nearby frequencies) is plotted as the ordinate.)

Using the results of measurements of the quantities aS and aA) we can estimate the

tity en appearing in Eqs. (11.2) and (7.94)J which characterizes the intensity of the fluc­

tuations of the sound velocity. If we use the formula

e n

to find e with the values a = 46° = 0.8 radJ k = 58 m-1

(f = 3 Kcps)J L = 67 m) v = 5 m/ n) S

~ = 0.2 sec then e turns out to be equal to 0.0010 m-l/3• This same quantity) determined ) n

200

the relation [b]

the value a A = 0.44 and the same values of k and LJ turns out to be equal to 0.0016 m-l/3

taken from the paper [71] ) • If we bear in mind that the values of aS and a A

obtained as a result of analyzing phase and amplitude fluctuation records of different

the agreement we obtain between the values of en must be regarded as satisfactory.

III we obtained Eq. (6.91)) which relates the quantity en for acoustic waves to the

and e2

determining the fluctuations of temperature and wind speed) i.e. v

(11.3)

mean sound velocity [c]. Using Eqs. (10.12) and (10.14)) which express eT

of the mean values of the temperature and wind speed at two heights z1

and z2

the layer of the atmosphere near the earth) we obtain the formula

(11.4)

used the values T = 290°e and c0

= 34om/sec. Here 6T = T(z2 ) - T(z1

) and

= v(z2) - v(zl) are expressed in °e and m/sec) respectively~ The value en= 0.0010 m-l/3

above corresponds to a velocity difference bv for the heights z2

= 8 m and z1

= 4 m

[d]J which represents a typical value. Thus) Eqs. (11.2) and (7.94) give the

for the order of magnitude both of the amplitude fluctuations and of the phase

the wave.

Quite.similar measurements of the amplitude fluctuations of a sound wave were made by

The measurements of the fluctuations of acoustic amplitude were accom-

measurements of the profiles of mean temperature and mean velocity) which

permitted the calculation of e by using Eq. (11.4). Fig. 21 shows the dependence of n

0 A = J [log( A/ A0)]

2 on L obtained by Sucbkov (for a frequency of 76 Kcps) . It is clear from the

figure that the experimental results are well described by the theoretical formula (7.94)) i.e.

201

(In all the experiments, the condition JX1 >> t was satisfied.) 0

CJA

0.20

0

0.15

0

0.10

0

0

0

0

OL-------L-----~------~------~------~---0 2 3 4 5 L,m

Fig. 21 Dependence of' the fluctuations of' logarithmic amplitude

of' an ultrasonic wave on the distance (f' = 76 Kcps).

Sucbkov carried out 28 series of measurements of' the dependence of' the quantity

on frequencies from 3 to 76 Kcps. The experimental data were approximated by the formula

a A= PiLa,. The mean value of a, for acoustic waves (18 series) was equal to 1.1, while the

value of' a, for ultrasonic waves (30~76 Kcps) was 0.95. The values of' a, obtained are close

the theoretical value of' a,= 11/12 = 0.92. Sucbkov calculated the value of' the quantity

using Eqs. (11.4) and (11.5) and measurements of' the profiles of' temperature and wind

Fig. 22 shows a comparison of' the values of' aA (denoted by a ) obtained as a ac

measurement and the values of' aA(denoted by amet) calculated from Eqs. (11.4) and (11.5) by

using simultaneous measurements of' the profiles of' mean temperature and mean wind velocity.

202

correlation coefficient between the quantities log a and log a equals 0.90 (97 points ac met

plotted in the figure). In calculating Cv' the constant JC (see p. 190) was taken to

If' we take C == 1.2 (see p. 192), then the whole group of' points is translated up-

and the regression line does not go through the origin of' coordinates. ThusJ measurements

a A lead to the same value jC == 1.4 as obtained by Townsend using a wind tunnel.

-1.5 -1.0 -0.5

0

0 <ID

0 0

0 0 0

0o ocf3 0 ~

0

oo

0 0

-0.5 0

oro 000

0

0

0 0

0 00 0

-1.0 0 0

0 0 0

0 0 0

0

0

0

Fig. 22 Comparison of' measured values of' the amplitude fluctuations of sound waves

(3 Kcps < f < 76 Kcps) with values calculated by using measurements of the

profiles of' wind velocity and temperature.

203

-1.5

log O"ac

Sucbkov also made measurements of the time autocorrelation fUnction of the amplitude

tuations of a sound wave. In the case where the direction of the wind is perpendicular to

direction of propagation of the sound and the correlation time is considerably less than

[a], the relation

log A(tA+ T) log ¥I= BA(vT) 0 0

is approximately valid. The function BA(p) for ~ >> t 0 was calculated above (see Fig.

The correlation distance of the amplitude fluctuations is equal to v'AL in order of magni

It follows from ( 11.6) that the correlation time of the amplitude fluctuations is of

JfL I v. Fig. 23 shows correlation functions obtained by Sucbkov for the amplitude

tions, where the vT/ J):i is plotted aS abcisSa

0.5

Fig. 23 Empirical autocorrelation functions of the fluctuations of

logarithmic amplitude. (1, L = 4 m; 2, L = 8 m; 3, L = 16 m)

204

fferent curves correspond to different distances between the transmitter and the receiver

If we plot the quantities log(A/A0

)log(A1 /A0

) = f(T) in natural units, i.e.

of T, then the curves obtained for different L have a different appearance. If

curves in units of T0

= v0(L I v , all three curves come closer together, especially

values of vTI v0(L.

experiments are in good agreement with the fluctuation theory presented in

The comparison of measured and calculated values of the quantity a illustrates the A

ty of making quantitative estimates of the size of the amplitude fluctuations of sound

by using simple measurements of the wind velocity and temperature profiles in the atmos-

205

Chapter 12

EXPERIMENTAL INVESTIGATION OF THE SCINTILLATION

OF TERRESTRIAL LIGHT SOURCES

Introductory remarks

An investigation of the scintillation of a terrestrial light source was carried out

during 1956 and 1957 at the Institute of Atmospheric Physics of the Academy of Sciences

of the USSR [74,75]· Experiments in the layer of the atmosphere near the earth are very

attractive, since in such experiments, in addition to measurements of the amount of scin·

tillation of the light source, one can simultaneously make measurements of the refractive

2 index fluctuations (i.e. determine the size of C ); moreover, one can make measurements n

for different and accurately known values of L. Thus, terrestrial experiments can give

much more complete data than stellar scintillation experiments, data which can easily be

compared with the theory of the phenomenon.

A portion of steppe with a regular profile was selected for making the experiment;

this guaranteed homogeneity of the turbulent regime along the entire propagation path of

the ray. (The light was propagated in the horizontal direction at an approximately uni-

form height above the underlying surface.) The light source could be moved to different

points, located at distances of 250, 500, 1000 and 2000 meters from a fixed point. At

distances closer than 250 m, the effect of scintillation was lower than the noise charac-

terizing the apparatus which was used, and therefore measurements were not made at such

distances. It was difficult to use distances greater than 2000 m, because of irregular-

ities of the profile of the terrain. The average height of the ray above the

surface was 1.5 m for operation of distances of 250 and 500 m, 2 m for operation at a

distance of 1000 m, and 5 m at a distance of 2000 m.

A 30 watt incandescent lamp was used as a primary light source. The light from it

was focused on a diaphragm 0.5 mm in diameter by using a light-concentrating objective.

Behind the diaphragm was placed a light chopper rotating at 100 revolutions per second,

206

contained 150 slits. The diaphragm was located at the focus of the exit objec-

a focal distance of 250 mm and diameter of 100 mm) of the light source, out of

emanated a weakly divergent bundle of light, modulated with a frequency of 15,000 cps.

tion of the light, with subsequent resonant amplification of the signal at the

frequency, made it possible to avoid the influence of extraneous, unmodulated

sources, and also simplified the receiving apparatus (the need for using a de amplifi-

(Fig. 24) consisted of two type FEU-19 photomultipliers; the light

tubes had first passed through two diaphragms 1 t oca ed in the plane per-

ray and then through a system of prisms. The distance p between the

could be varied over a range from 0.5 to 50 em. Th d' t e lame er of the receiving

was equal to 2 mm, which completely eliminated the effect of 11 objective-aver­

The ac components of the out-nut voltages of the h t r p o omultipliers,

tudes of which were proportional to I(M1

) and I(M2), where I(M) is the instan­

through the diaphragm located at the point M, were

by tuned amplifiers with pass bands of about 2000 cps, and then detected. Volt-

"~ a e e ec or outputs. proportional to I(M1

) and I(M2), ~·=re formed t th d t t In

ers there was a special tracking system which assured that the relation vl = v2

fied (with a constant averaging time of 100 sec). Aft bt er su racting out the de

, voitages V' = V - V and Vf - v -l l 1 2 - 2 - V2 were formed, proportional to the light

I(Ml)- I(M1 ) and I'(M2 ) = I(M2)- I(M2), respectively. The

subjected to automatic statistical analysis by using a special equip-

measured (in identical units): the probability distribution of

ions I 1 (~), the mean square fluctuation [I' (M1 ) J 2, the mean value I (M1 ), the

function I'(M1

)I'(M2

) = B(M1

,M2), and the frequency spectrum of the fluctuations

in the frequency range from 0.05 to 1000 cps. At the same time that the measurements

scintillation of the terrestrial light source were made, meteorological measurements

:made along the propagation path, which allowed the quantity c! to be calculated. Tem­

profiles were measured in the layer from 0. 5 to 12 m, as well as profiles of the

velocity and wind direction in the same range. By using these measurements, it was

207

possible to determine the turbulence parameters EJ K and T*. Since the experiment was

carried out over a very level portion of steppe and since the turbulent regime was iden­

tical over different parts of the propagation path) meteorological measurements were set

up only at one point.

We now give the basic results of the measurements.

Light source

1st diaphragm 1

2nd diaphragm I

Amplifier and detector

Amplifier and detector

Frequency analyzer

Amplitude analyzer

Squarer

Subtracting circuit

Fig. 24 Block diagram of apparatus for measuring

the scintillation of a terrestrial light source.

12.1 The probability distribution function of the

fluctuations of light intensity

It follows from the theory of the phenomenon that the logarithm of the amplitude of

the light wave is expressed in terms of the refractive index fluctuations along the pro-

pagation path by using an integral of the type

log(A/A ) 0

J J J F(;' )n' (;' )dV'.

D

208

integral we can subdivide the whole region of integration D into a large number of

Di with linear dimensions of the order of the outer scale of turbulence L0

J which

of the experiment are of the order of the height of the ray above the

There is no correlation between the fluctuations of n(~') in these regions. There-

we obtain the formula

log( A/ A ) 0

expresses log( A/ A0

) as the sum of a large number of uncorrelated terms. Because of

central limit theorem) the quantity log(A/A ) must be distributed according to a nor­o

Since log(I/I ) = 2 log(A/A )J the quantity log(I/I ) must also be distributed 0 0 0

J and the quantity I must have a log normal distribution.

--~

experiment gives good confirmation of this fact. In Fig. 25 the quantity! -l [F(I)]

as abcissaJ where ! - 1 (x) is the function which is the inverse of

!(x) l

=--J2n

X

J -oo

2 exp(-t /2)dt

F(I) is the empirical distribution function of I) while the quantity log(I/I ) is plotted 0

In these coordinates the log normal law is indicated by a straight line [b J. lOO empirical distribution functions F(I) were analyzed. All of them are in

agreement with the hypothesis of a normal distribution of the quantity log I.

Using the hypothesis that the quantity log I has a normal distribution) we can relate

2 - 2 -experimentally measured quantities a1

= (I - I) and I to the quantity

in the theory. We can easily convince ourselves that they are connected by

209

2 (J

This formula was used to further analyze the experimental data.

F(I)

Fig. 25 Probability distribution of the intensity

fluctuations of light on a log normal scale.

12.2 Dependence of the amount of scintillation

on the distance ~d on the meteorological conditions

As already noted above) in the atmosphere the quantity t0

is a few millimeters in order

of magnitude. Therefore) the parameter Lcr = t~/A ) which determines the limit of applica­

bility of geometrical optics} is 100 meters in order of magnitude. Consequently) in our

210

\rne:riJ:lleilvS the condition L > Lcr was satisfied) and geometrical optics was not applied

the calculations. As follows from (11.5)) in the case under consideration) e12

expressed by the formula

2 (J

2 experiments described) simultar1eous measurements of cJ at different distar1ces

light source ar1d the receiver were not made. During the time necessary for

the light source from one point to ar1other ar1d for aiming the light source

receiver) the meteorological conditions had time to char1ge substar1tially. There­

in order to compare values of e12

obtained at different distar1ces) it is first neces-

them to identical meteorological conditions. The simplest way of making

a reduction is to average the values of e12 pertaining to one distar1ce over all the

this gives a value e12

(L) which pertains to the average meteorological con-

2 The averaged values of cJ are given in Table 1.

TABLE l

L) meters 2 \/cJ!v

Number of (J

Measurements av

2000 0.420 0.65 75 1000 0.128 0.36 171

500 0.027 0.16 78 250 0.0078 o.o88 50

Amore accurate reduction of the values of e12 to identical meteorological conditions

also made. For every distance L) the measured values of cJ were compared with simul­

taneously measured values of the vertical temperature gradient dT/dz) or more precisely}

of the quar1tity CT = a(Kz)2

/ 3(dT/dzL defining the intensity of the temperature fluctua­

tions in the layer of the atmosphere near the earth (see Chapter 10). For each distar1ce

the dependence between e1 and CT = a(Kz) 2/3(dT/dz) was approximated by the formula cJ = AC~ )

Where A and a were found for each distance by the method of least squares (in logarithmic

units). The values of a found for different distances L turned out to be quite close

211

together. The average for the four distances was ~ = 0.2. ThusJ for all the distances)

the dependence of a on CT can be approximated by the same formula

a== K(L)C0 '2

. T

The values of K(L) (see Table 2) can now be determined for each distance as the regression

coefficient of the values of a on cg· 2 (the ~uantity CT is expressed in degrees per cml/3).

The results given in Tables 1 and 2 are in good agreement with the theoretical dependence

a oc:= Lll/12.

TABLE 2

LJ meters ..... 2000 1000 500 250

------·---------.------~--·-~----K (L) 1.3 0.86 0.32 0.14

If we approximate the data of Table l (Fig. 26) by the formula a2 = const Ln and

the values of n and the constant by the method of least s~uaresJ then for n we obtain the

value 1.96J which is very close to the theoretical value of ll/6 = 1.83. A similar value

of nJ determined from the data of Table 2J turns out to be e~ual to 2.1. This value of n

is also close to the theoretical value of ll/6. ThUS) we can regard the dependence of the

amount of scintillation of light on the distance as agreeing satisfactorily with the theo­

retical formula a2

oc= Lll/6 •

12.3 The correlation function of the fluctuations of light

intensity in the plane perpendicular to the ray

As already noted [c]J for-Ji.L >> t (for light this is practically always the case), 0

the correlation distance of the fluctuations of light intensity is of order vf:[:E , and the

correlation function of the intensity fluctuations depends on the argument P/ JX1 • experiments which were carried out) this similarity hypothesis was immediately verified.

Measurements of the correlation coefficient R were made for different values of v'ALJ corre~

212

(L, km)

11/s

0.42 -2-

Fig. 26 Distance dependence of the logarithmic

intensity fluctuations of light.

Fig. 27 Empirical correlation function of the flucuations of light intensity.

213

p/JIT

sponding to L = 2000, 1000 and 500 meters. However, the distances P between the diaphr

were set in such a way that the quantity P/yf:[L always took the identical values 0.25, o.

1, 2, 4 and 8. The measurements of R had a rather large scatter, caused by insufficient

accuracy of measurement. However, the large number of measurements of R greatly decreased

the error, so that the mean values of R obtained for the same P/ V'"'iL but different V'fL agree very satisfactorily with each other. Table 3 gives the quantities R obtained for

different values of L, and also the average data for all L.

TABLE 3

L = 2000 m L = 1000 m L = 500 m Averages for all L

JiL = 3.2 em JfL = 2.2 em v'fL = 1 .. 6 em _e__ i'i.L

I

' 5% confidence

R n R n R n R n intervals

0.25 0.58 8 o.46 15 - - 0.50 23 0.05

0.5 0.27 9 0.31 19 0.27 12 0.29 4o 0.05

1.0 0.09 11 0.10 18 0.16 15 0.12 43 0.06

2 -0 .. 05 7 -0.05 15 -0.07 14 -0.055 36 0.08

4 -0.08 6 -0.09 13 -0.03 14 -0.062 33 0.08

8 -o.o8 7 -0.03 14 -0 .. 13 9 -0.072 30 o.o6

Fig. 27 gives a graph of the data of Table 3· The values of R obtained for different

1 are indicated by different signs. It is clear from the figure that the difference

values of R obtained for different v'fL lies within the limits of accuracy of the meas11re-,

ments. (The vertical lines in the figure represent 5 percent confidence limits [a].)

results obtained substantiate quite satisfactorily the theoretical conclusion that the

correlation function of the fluctuations depends on pjvf[L and that the correlation dis­

tance of the intensity fluctuations is of order J):.L. Thus, all attempts to determine

"the average size of the inhomogeneities"in terms of the correlation distance of the flue-

tuations of light intensity are doomed to failure, since from these measurements one can

only infer the quantity ~ ..

214

12.4 Frequency spectra of the fluctuations of the logarithm

of the light intensity (theory)

presenting the results of measurements of the frequency spectrum of the inten-

intensity of light in the observation plane x = L. Let the mean velocity of motion

index inhomogeneities be constant and equal to ~ along the entire wave

We assume from the beginning that the refractive index inhomogeneities

the process of convection. Below, we shall

conditions which must be satisfied if such an approach to the problem is not to

appreciable errors.

resolve the velocity of motion~ of the inhomogeneities into two components, i.e.

+ ~t' where ~n is perpendicular to the direction of wave propagation ~d ~t is

it. It is easy to convince oneself that convection of the inhomogeneities

the propagation direction does not lead to appreciable changes of the field I(y,z),

ded only that the angle ex, between the wind velocity ~ and the direction of wave propa-

satisfies the inequality ex,>> JfJL [e]. Therefore, we can assume that the field

(y0,z

0) at the time t

0 + ~ coincides with the field at the point (y

0 - vy~'

v ~) at the time t • Using this relation, we can express the time autocorrelation z 0

on RA(T) of the fluctuations of logarithmic amplitude at the point (y ,z ) in terms 0 0

space correlation function BA(P):

(12. 2)

above, the transverse correlation distance of the amplitude fluctuations of the

of order JIL. It follows from (12.2) that the correlation time of the field is of

T = V'f.E/v • o n

the condition which when satisfied allows us to regard the refractive

inhomogeneities as "frozen-in11 .. It is clear that for this to be the case, it is

cient that the inhomogeneities of size JXL, which are chiefly responsible for producing

215

the amplitude fluctuations of the wave, should not have time to change appreciably during

the time 't" • Af3 shown above (see Chapter 2)) the "lifetime" of an inhomogeneity of size 0

is equal to 't"t "'t/vt"' t/(Et)l/3. Fort-.. Jll we obtain 't" "' j):.L ( € j'i.L) -1/3.

.fi.L This quantity must be large compared to 1"0 , whence v n >> ( E J)JJ1/3. But v n is in order of

magnitude equal to the velocity of the flow as a whole and can be expressed in terms of the

outer scale of turbulence L0

, i.e. vn "' (EL0

)1

/ 3 . Therefore, the ''frozen-in" condition can

be used in the case where jii << L . 0

(This condition is practically always satisfied for

light propagating in the atmosphere.)

We now calculate the frequency (time) spectrum of the amplitude fluctuations of the

wave. Denoting the spectral density of the fluctuations by W(f), we have by definition

vn is in order of magnitude equal to the velocity of the flow as a whole and can be

· t f th t sc le of turbulence L i e v "' ( EL )1

/ 3 . Therefore, the "frozen-ln erms o e ou er a o' • · n 0

in11 condition can be used in the case where .Jff, << L • (This condition is practically 0

always satisfied for light propagating in the atmosphere.)

We now calculate the frequency (time) spectrum of the amplitude fluctuations of the

wave. Denoting the spectral density of the fluctuations by W(f), we have by definition

or

00

W(f) =4! cos(2nf't")RA('t")d't"

0

W(f)

00

4 J cos ( 2nf't") B A ( v n 't") d 't" •

0

Using the expression [g]

00

2n J FA(K,O)J0

(KP)KdK

0

and changing the order of integration, we obtain the formula

W(f) anJ 0

J J (Kv 't")cos(2nf't")d't". o n

0

The inner integral is the well known discontinuous Weber integral [?3]:

216

J J (Kv 't") cos( 2nf't" )d't" = o n 0

W(f) 8n

W(f) = 8n v

n

00

J FA(K,O)

2nf v n

J 0

1

0

KdK

J 2 2 2 2 K v - 4n f n

2 2 2 2 for K v < 4n f • n

K'vn' this expression finally reduces to

dK. (12.4)

• (12.4) relates the frequency (time) spectrum of the amplitude fluctuations of the wave

the two-dimensional spectral density FA(K,O) of the amplitude fluctuations. For com­

theory with experimental data it is convenient to consider the dimensionless

fW(f) U(f) = -

00---

J W(f)df

0

satisfies the condition of being normalized in logarithmic units,

217

i.e.

J U(f) dlog f

0

4) x2 We use Eq. (7.87) for FA(K,O) and Eq. (7-9 for

DO

J W(f)df, expressions which are 0

2 valid for the case Cn = canst. Then for u( f) we obtain the expression [h]

where

2 2 J -1116 U(f) = f'W(f) = l.35D J rl - sin(t + n ) (t2 + s-22) dt '

l t2 + n2 -x2

o

n = flf 0

and v

n

As follows from Eq. (12.5), the quantity f'W(f)IX2

is a function of n = flf0

, which does

not change its appearance when vn and 1 are changed. (In logarithmic units) change of vn

or 1 corresponds to translation of the curve U(f) along the horizontal axis.) The DO

f0W(f) I J w(f)df is shown in Fig. 28, and the normalized function U(f) obtained by numeri

0

cal integration of ( 12. 5) is shown in Fig. 31 [i] ·

t0 W(f)/X 2

-0.5

Fig. 28 Theoretical shape of the frequency spectrum

of the fluctuations of logarithmic amplitude for constant wind velocity.

218

12.5 Frequency spectrum of fluctuations of light intensity

(experimental results)

The frequency spectrum of the fluctuations of light flux was measured by using a ire­

analyzer consisting of 30 filters, each with a bandwidth of one half octave

f = J2), arranged in a bank one half octave apart, from 0.05 to 1160 cps. lower

cintillation spectra were analyzed, obtained at distances L of 1000 and 2000 meters.

quantity vn' the component of the mean wind velocity perpendicular to the ray) was cal­

by using synchronous meteorological measurements. The measurements at each distance

divided into .3 groups depending on the size of vnJ namely

1 < v < 2 mlsec, 2 < v < 3 mlsecJ and 3 < v < 4 mlsec. n n n

spectral densities W(f) of the fluctuations were obtained for each group (the aver­

carried out in logarithmic units). Then the "normalized" spectral denai ties DO

W(f)df were calculated. Fig. 29 gives the quantities U(f) = fW(f) I J W(f)df 0

fTTE~s-ooncling to the different wind velocities vd which are the averages for the given

of measurements; the abcissas are measured in logarithmic units.

is clear from the figure that when the mean wind velocity is increased) the curves

are shifted in the high-frequency direction. We can find the frequencies fm corre-

g to the maximum of the curve U(f); fm is defined as one half the sum of the frequency

for which u( f) = ~[u( f)] max" Table 4 gives the values of the mean wind velocity v n

groups of the quantities fm and fm.Jff, I vn.

TABLE 4

L = 1000 m 1 = 2000 m

mlsec -1 1.46 2.18 ).46 1.61 2.59 ).51 vn' ........ fm' cps ............. 20 25.6 45.7 18.1 25o6 39.8

f /lLiv ••••••••••• m n 0.31 0.26 0.)0 0 .. 35 0.)1 0 .. )6

219

tW(t)/x2

0.4

0.3

0.2

0.1

L=2000m

--o- Vn = 1.6 m/sec -o-- Vn =2.6 m/sec --t::r- Vn =3.5m/sec

fW(f)/X 2

0.4

0.3

0.2

0.1

L=1000m

-o- Vn =1.5 m/sec --o-- Vn =2.2 m/sec

-b:- Vn =3.5 m/sec

Fig. 29 Empirical frequency spectrum of' fluctuations of' light intensity

for different wind velocities (aJ L = 2000 m; bJ L = 1000 m).

220

I'Tl'l-le quantity f' ,[ff, I v is approximately constant) with mean value equal to 0.32. ;.~.~" m n

J the frequencies f' are connected with v and J'XL by the relation m n

v f' = 0 . 32 --.E. • m a (12.7)

note that a calculation based on the hypothesis of' 11 f'rozen-in11 turbulence) leads to the

on f' = 0.55 v I JI:LJ which differs from (12.7) by a numerical coefficient. However) m n

theoretical relation between the spatial correlation distance R0

j}3A(R0

) = ~ and f'mJ

R = o.44 0

v n f'

m

satisfactorily} since according to the experimental dataJ R = 1.5 ~J which 0

with (12.7) leads to the formula

v R

0 = 0.48 f'n

m

(12 .. 8)

(12.9)

gives a more detailed verification of' the similarity hypothesis expressed by

In Fig. 30 all the frequency spectra represented in Fig. 29 are reduced to the

l mlsec and L = 1000 m. As is clear from the figure} the spectra which are

function U( f) depends only on the argument f' JXf, I v } i.e. n

f'W(f')

J w( f')df'

0

(12.10)

A theoretical calculation of' the fUnction appearing in the right hand side of' (12.10)

made above J by using the hypothesis of "f'rozen-in11 turbulence. Fig.. 31 gives a com-

of' the theoretical curve and the experimental data obtained by averaging the graphs

• 30 [j]. It is clear from the figure that the theoretical curve is 11 narrower11 than

221

~ I

Fig. 30 Reduction of the frequency spectra

to v = l m/sec and L = 1000 m. n

fW{f)/X 2

0.5

0.4

0.1

0 <:j 0 00 0

C'J C'J <:j

c) c) c) c)

Fig. 31 Comparison of the empirical spectrum of fluctuations of light

intensity with the theoretical spectrum ( lJ theoretical curve;

2} experimental curve) •

222

~~1c~~~~--tal curve; this is evidently related to the fact that it is assumed in the

the wind velocity is constant along the entire propagation path.

conclusion} we state the basic results of the experiment:

The fluctuations of light intensity caused by atmospheric turbulence have a log

distribution·

The dependence of cr2

= [log(I/I )] 2

on L is found to be in satisfactory agreement 0

the theory of the phenomenon} which leads to the formula cr 2 oc:: Lll/6•

Direct measurements confirm the theoretical conclusion that the correlation func-

fluctuations of light intensity depends on pj JIL and that the correlation

of order .j"XL.

It is confirmed that the frequency spectrum of the fluctuations of light intensity

on f VX£ / v J and good agreement is observed between the intervals of time corre­n

and space correlation.

223

Chapter 13

TWINKLING AND QUIVERING OF STEIJ..AR D1AGES IN TELESCOPES

The first experiments concerned with the study of fluctuations of intensity and angle

arrival of light waves were carried out while investigating the twinkling and quivering

of stellar images in telescopes. Recently) interest in this problem has increasedj this

is explained both by the requirements of observational astronomy and by the close rela-

tion which exists between these phenomena and certain features of radio propagation in the

troposphere. Here we shall not give a detailed exposition of all the known facts) nor

shall we present the numerous theories which describe the phenomena of twinkling and

quivering of stellar images in telescopes; we confine ourselves merely to a short account

of the basic facts and their interpretation.

When we make an observation in a telescope) we see the diffraction image

the form of a luminous core and a series of concentric rings. However) it is hardly the

case that such an image is seen all the time. Usually the stellar image does not remain

fixed in the field of vision) but rather experiences irregular displacements in all pos-

sible directions) which are called "quivering". At the same time) some of the diffraction

rings are missing or are smeared out. Under especially unfavorable observational condi-

tions) we see a "dancing" irregular "patch" J which in no way recalls the

of the star. Simultaneously) one also observes "twinkling" of the star) i.e. irregular

changes in its brightness. The astronomical "seeing" (i.e. diffraction image) and the

quivering of the image are intimately related (since both of these effects are

phase fluctuations of the wave). When the 11 seeing" is bad) one usually observes consider-

able 11 quivering" of the images.

A large number of experimental papers are devoted to the study of the phenomenon of

quivering of images) a review of which is contained in the papers of Kolchinski [8o)81] •

This author arrives at the basic conclusion that the mean square fluctuation of the

of arrival of the light from the star is directly proportional to the secant of the

224

i.e.

---2 2 ( 6a.) = A sec e. (13.1)

quantity A is a few tenths of an angular second in order of magnitude) and depends on

meteorological conditions. Fig. 32 gives the results of observations of the quivering

$tars) performed at the Central Astronomical Observatory of the Academy of Sciences of

USSR at Goloseyev r81J • The rms al f t ~ v ues o he fluctuations of the propagation direc-

of the wave is plotted along the vertical · 8XlS) while sec e is plotted along the hori-

0

0

0 0

0

0 <o 0 0 0

0

00 0

0

0

Flg. 32 Dependence of the amount of quivering of

stellar images on the zenith distance.

225

~.rsece

0

The points in Fig. 32 have a large scatter) caused by the fact that the graph compri

results of observations made under different meteorological conditions. In order to sho~

the dependence of the quantity (6 ~) 2 on sec e) we must average the quantities (6 ~) 2

belong to neighboring values of sec e. One also obtains a similar result by constructing

regression line) whose equation has the form (with a logarithmic scale):

~ 2 log (6 ~) = log A + 2p log sec e.

The quanti ties A2 and p) found by the method of least squares) turn out to be equal to

A= 0.35" and p = o.47. This value of pis in good agreement with Eq. (13.1).

The theoretical law (13.1) was first established by Krasilnikov [82]. Suppose that

two interferometer slits are located at the points A and B at a distance b from each other.

If the surface of the wave front is parallel to AB) then the phases of the oscillations at

A and B are identical. Rotating the wave front by the angle b,a, << 1 produces a phase dif-

ference b.S between the oscillations at A and B which is equal to 6S = kb 6a, • It follows

from this that the quantity ( .6. a,) 2

can be expressed in terms of ( 6 S) 2

:::: DS (b) by using the

formula

(13.3)

If b > Vi£ ) then DS(b) is given by Eq. (8.22L and

00

J (13.4)

0

where the integration in (13.4) is carried out along the "ray" directed toward the light

source. We assume that the quantity c2 depends only on the height z above the n

face. Setting x = z sec e) we obtain from (13.4):

00

( .6.a,) 2 = 2.91 b -l/3 sec e J 0

2 C (z)dz. n

226

copic observations of the quivering of stars) the role of the quantity b is played

telescope. In general) when b is changed to D) the value of the

in (13.5) can change a little. However) the character of Eq. (13.5)

the same. It follows from (13.5) that the quantity (.6.~) 2 is proportional to sec e)

agrees with (13.1). The size of (6~) 2 decreases slowly as the diameter D of the tele-

is increased.

to note that c2(z) usually takes its largest values in the lower

n

of the atmosphere) which lie near the earth~s surface. Therefore) the largest con-

to the integral

J 0

2 C (z)dz n

lower layers of the atmosphere) which also play the basic role in the pheno-

"seeingn and quivering of stellar images.

quantity b,a, has a Gaussian distribution. This conclusion is

agreement with the fact mentioned in Chapter 12 to the effect that the quantity

has a Gaussian distribution) since) as follows from general considerations)

and s1 (the fluctuations of logarithmic amplitude and phase of the wave) must

the same distribution law.

We now turn to the problem of the twinkling of stars (fluctuations of the light inten-

In practice) extensive measurements of fluctuations of light intensity are made much

easily than measurements of the "quivering" of stellar images) so that there exist a

n'l.lJIIber of experimental papers on this problem [83-86) 78]. By placing a photoelectric

ce in the focal plane of the telescope) the light flux can be transformed into an elec-

cal voltage) which is extremely suitable for statistical analysis [84)86)75]. As a

of numerous observations) it has been established that the size of the fluctuations

light flux passing through the diaphragm of the telescope) depends significantly on

dimensions of the diaphragm) the zenith distance of the light source and its angular

ions) and the meteorological conditions. The dimensions of the diaphragm of the tele­

have a very great influence both on the size of the fluctuations (see Fig. 33) and

the way they depend on the zenith distance.

227

-~p~2

(log Po)

0.2

Fig. 33 Empirical dependence of the amount

of twinkling on the diameter of the telescope·

diaphragm (1, winter; 2, sUIIIIIler) [a,8Li] •

Figs. 34 and 35 show how two samples of the quantity "~ ~ (P - P)2

/ P2

(whe>e P is

light flux through the telescope objective), obtained for different values of the diameter

of the telescope diaphragm, depend on sec e. The slope of the curves log ap =

Of e) as Well as the behavior of the curve log ap = f(log sec e) for for small values

values of e ,. depends strongly on the diameter of the diaphragm. Therefore, before

ing to a further study of the experimental data and their interpretation,

theoretical role of the dimensions of the telescope diaphragm.

log <Jp

0.4

Fig. 34

()~ ~\

_.;\s'?}v

0 0

Dependence of the amount of twink­ling on the zenith distance whe~ the telescope diaphragm has a dla­meter of three inches.

228

log <Jp [>, ,.

0.2 ~\

\_r-:.~?Jrv ,>

0

0

-0.6 L__L _ _j__..L--_..1~--1---:::-;:--;:-::-:

0

Fig. 35 Dependence of the amount of ling on the zenith distance the telescope diaphragm haS a meter of 12.5 inches.

A photocell placed in the focal plane of the telescope responds to the entire light

p through the telescope diaphragm. If I(y,z) is the intensity of the light wave on

surface of the objective (the energy flow density), then

( 13.6)

z is the surface of the objective. The q_uantity I has a log normal probability dis­

(see Chapter 12). If jE ~ vfAL, i.e., if the diameter of the objective is

correlation distance of the fluctuations of the light intensity, and conse-

if changes in the light flux through different parts of the objective take place

taneously, then the q_uanti ty P also has a log normal distribution law. But if L: >> J...L,

if a large number of uncorrelated inhomogeneities of the light field can be found

the limits of the objective, then by the central limit theorem, the q_uantity P

However, the telescopes used in practice usually have

such that no more than 2 to 4 uncorrelated field inhomogeneities fit inside of

example, for D = 4o em and ~ = 10 em (see below), ~~L = 4. In this case the

P is still very close to the log normal law. The experimental data

fifteen inch telescope, confirm this conclusion. Thus,

a sufficiently high degree of accuracy, we can assume that P, just like I, has a

normal distribution. We now find the parameters of this distribution.

The q_uantity I can be represented in the form

I ~ I0

exp ~log t;] ~ I 0 exp [2X(y, z)], (13.7)

X(y,z) = log A(~,z) is the logarithmic amplitude of the light wave, distributed 0

to the normal law. Thus we have

P = I0 J J e2X(y, z) dydz. (13.8)

L:

229

To determine the important quantity ~og :0

)

2 , where log P

0 = log P, it is su:f'fi cient

to consider the first and second moments M1

and M2 of the quantity P. M1 is defined by

the equality

"""2X 2X2 2 (,, A) 2

But e = e J where X = 'C_og Ao J which is valid for any normally distributed quan-

tity X. Thus we have

We now calculate M2:

2X2

e

JJJJ 2: 2:

2X(y,z)+2X(y'Jz') d d d 'd 1 e y z y z .

If we assume that the two-dimensional distribution of the quantity X is also normal) then

it is easy to show that

or

which is equivalent to (13.12); here BA(;- ; 1) is the correlation function of the flue

tions of logarithmic amplitude) considered above. To derive (13.12)J it is sufficient to

consider the characteristic function of the two-dimensional normal distribution. Thus we

have

230

ff{f 2: 2:

haB a log normal distribution) then the quantity ~og :0

)

2 can be expressed in terms

formula

(13.15)

To evaluate this expression) we introduce the spectral expansion of the correlation

of the intensity fluctuations of the wave:

00

J,[ -oo

the formula

4 ~ -T

B (r -r ) the quantity e A 1 2

231

(13.16)

can be expressed in terms of F (K K ) by I 2) 3

Then we have

~og ~)2

X JJJJ 2:: 2::

We introduce the function

1 2::

If

( 13.18)

which describes the Fraunhofer diffraction of the diaphragm 2:: [b]. Then (13.18) takes the

form

~og :J2 1 +-1-(1)2

( 13.20)

As follows from Eq. (13.19)J the function VI:(K2JK3

) is appreciably different from zero only

for K ~E) where D is the dimension of the diaphragm 2::. Thus} the small-scale components

the field (with dimensions less than D) do not contribute to the fluctuations of P.

we consider the case where the telescope diaphragm is a circle of radius R. ThenJ

is easily seen

1 VI: = ----:2

rrR

R

J 0

pdp (13.21)

0

232

\j K~ + K~ • Substituting this expression in Eq. (13.20)J introducing the coor­

K2 = K cos cp J K3

= K sin cp and integrating with respect to cp J we obtain the formula

(13.22)

is .ssumed that F1

(K2,K

3) ~ F

1 (! K~ + K~)-] In Eq. (13.22) we can go over from the

ral density FI(K) of the fluctuations to the correlation function BI(p) of the flue­

Inverting Eq. (13.16) and taking into account the isotropy of the fluctuations

plane x = canst} we obtain

00

FI(K) = ~rr J BI( p)J 0 (KP) pdp. (13.23)

0

stitute this expression in (13.22) and change the order of integration:

(13.24)

inner integral can be calculated [c] and turns out to be equal to

cos ....e.. - ..E.. 2R 2R for P < 2RJ

0 for P > 2R.

233

Consequently, we have

(_,log :o\2 = log { 1 + 24- 2

. ) rtR (I) j Bl ( p) [arc co a ;. -

0

-~ A}dp) ~ log (arc cos x- x.£7)x~},

where D = 2R. Noting that

1

~6 J Grc cos x - x /1 - x2) xdx == 1_,

0

we rewrite this formula in the form

But it follows from (13.13) that

Thus we have [ d]

~og :;/ log

{ 6 J

l 4BA (Dx) r: 1"

0

e \_arc cos x - x

Setting D = 0 here and recalling that 4BA(o) = 4X2 = o2

, we obtain~og :~ o2

• For

234

Gag :0

)

2 < o

2. We introduce the quantity G = :2 Eag :

0

)

2, which characterizes

decrease in twinkling due to the averaging action of the objective. Furthermore, setting

1 G = 2 log

a (13.28) Grc COS X - X J 1 - X

2) X~ } •

For bA(p) we take the fUnction represented in Fig. 13, which is applicable in the case

}i:L >> t and c2 = const. Numerical integration of the expression (13.28) for a2 = 4 o n

to the results shown in Fig. 36. As was to be expected, the function G

argument D / J):L , i.e.

that

ED j ~D )-7/3 G --,a oc:. --

Vff. /):L

~ and a ~ 0. As can be seen from the figure, G is small compared to unity when

Thus_, in the case where the diameter of the telescope diaphragm exceeds the

J'AL of the fluctuations, the fluctuations of the total light flux through

scope diaphragm are weakened considerably. This is explained by the fact that for

several field 11 inhomogeneities11 with different signs can be found within the limits

telescope diaphragm, and therefore they partially compensate one another.

235

Fig. 36 The theoretical dependence of' the relative decrease

in the size of' the total light flux through an objective on the

diameter of' the objective, under the condition i 0 << Vi:L << L0 •

We now compare this dependence with the data observed by the

Semi-annual averages of the frequency spectra of the fluctuations of the total light flux

through the telescope diaphragm are given below (page 247), for different sizes of the 2 - 2 I -2

diaphragm. Integrating these spectra, we can find the quantity ap = (P - P) P as a

function of the diameter of the diapragm [a]. The values of ap obtained in this way are

given in Table 5·

TABLE 5

Diameter ap of the

Diaphragm) Winter Summer inches

1 o.476 0.373 3 o.346 0.250 6 0.189 0.160

12 .. 5 0.098 o .. o8o

AS shown above, the quantity [}og(PIP0

)]2

is connected with a~ by the relation

is valid in the case where the quantity log(PIP0

) has a normal distribution law. We

use this formula to go over from the values of a~ just obtained to values of [}og(PIP0[! 2

•

result of these calculations, we obtain the following values (Table 6).

Diameter of the

Diaphragm, inches

1 3 6

12.5

TABLE 6

Winter

0.205 0.110 0.035 0.0096

Summer

0.130 0.061 0.026 o .. oo64

Choosing an appropriate value of the parameter ~' we can achieve very good agreement

the values of [lo~(PIP0 )] 2 = f(D) just obtained and the theoretical dependence. The

of Table 6 agree best with the curve in Fig. 36 (for a2 ~ o) when Ji:L = 3.6 inches

measurements) and YI:L = 3.2 inches (summer measurements). In Fig. 37 we compare

[log(PIP0

)]2

(see Table 6) with the values of G(D lv'At) calculated for

values of v'AL and a. Even small changes of the parameter y0J: destroy the linear

(}.og(PIP0

)]2

and G(D I v'AL) . Thus, a comparison of the measured values

and G(D I y0J:) allows us to determine the important parameter ,tr£ . Direct measurements of' the correlation distance of the fluctuations of light intensity

by Keller Q18] by using two telescopes which could be moved apart) gave a value of

of' the order of' 3.5 inches.

237

l I

(log f )2 0

Fig. 37 Comparison of the data of Figs. 33 and 36 (1, winterj 2, summer).

(To the left and right of the points joined by the straight line

are points corresponding to the values of -JIT, changed by l em.

These points no longer lie on straight lines.)

we now turn to the dependence of the amount of twinkling on the star's zenith angle e

It follows from Eq. (8.17) that

00

a2 = 4 ?f = 2.24 k 716 j c~Cr) x516 dx,

0

where the integration in (13.29) is carried out along the ray directed toward the light

source. Assuming that c2 depends only on the height z above the earth's surface, we carry n

out the change of variables x = z sec e in (13.29), where e is the star's zenith angle.

00

2.24 k 7 16(sec e )1116 J 2 rJ =

0

(13-30) gives the mean square fluctuation of the light intensity. In order to obtain

mean square fluctuation of the light flux P through a telescope diaphragm of diameter

must multiply the right hand side of (13.20) by the function G(D I JIL ) which depends

Since L = H0

sec e, where H0

is the order of magnitude of the thick­

layer in which appreciable refractive index fluctuations occur, the

on G (n I J):i) also depends on e • Thus the formula

~og ~) 2

(13-31)

lead to different types of dependence on sec e for different relations between D and

• For example, in the case D >> y')JI 0

tituting this expression in (13.31), we obtain

Eog ~y oc D-7 /3 H~/6 sec3e j 0

(13-32)

, if we observe the dependence of the amount of twinkling on the zenith angle using a

diameter telescope (D >> ~ ), we should obtain the dependence !}og(P IP )] 2 oc: sec3e.

0 0

239

As shown above, the quantity ~ is of the order of 3 to 4 inches, i.e. 8 to 10 em. 0

fore, when the diameter of the telescope diaphragm is of the order of 40 em (15 inches),

2 3 can already expect the dependence crp~sec e (for values of e which are not very large, the

relation Jiii:.. sec e << D is still valid). 0

In Fig. 38 we show (in logarithmic units) the function cr~ = f(sec e) obtained by

Butler [89]. For e < 60°, the experimental data are well approximated by the formula

cr <>= sec3 e. The Perkins Observatory [84] obtained the same kind of dependence, using a

12.5 inch diameter telescope (see Fig. 35).

log o-p

0

0

0 0

1.2

Fig. 38 Dependence of the amount of twinkling on

the zenith distance, obtained by using a

15 inch diameter telescope.

0

1.4 log sec e

For small. values of D << ~ the function G(D/ y'AL) changes inappreciably when

sec e is changed. In this case, the d d f 1-i ( / J 2 epen ence o ~og P P0

) on sec e is determined by

the factor (sec e)11

/6 . For intermediate values of the ratio D/ J'AH the function

0 }

l}og(P/P0

)]2= f(sec e) can be approximated for small e by the formula [log(PjP

0)]

2 = A(sec e

where 11/6 < ~ < 3. Fig. 34 Ehows the function cr~ = f( sec e) obtained by the Perkins Obser-

vatory using a 3 inch diameter telescope; for small e, ~p2 i ti u s sa sfactorily approximated by

the formula crp2

oc:::: (sec e) 1•8 . Thi d d · · s epen ence lS 1n good agreement with the value~= 11/6.

which ought to be t d · r:;:-;;-expec e forD << yA.H0

• According to the data of [84], ~ = 1 .. 8 for

D = l inch, ~ = 2 for D = 3 inches, ~ = 2.4 for D = 6 inche~ and N --Q ~ 3 for D = 12 inches.

240

. 39 shows in logarithmic units the function [iog(P/P0

)]2

= f(sec e), obtained by

[88] using a 250 mm diameter telescope. The solid line indicates the theoretical

calculated from Eq. (13.31) for JiiC = 9 em. 0

logJiog ~ )2. 0

0

0

0~--------~--------~--------~--------~-logsecB 0.2 0.4 0.6 0.8

Fig. 39 The dependence of the quantity log

on the zenith distance, obtained by using a 250 mm

diameter telescope.

Thus, for zenith angles which are not very large, the dependence of the amount of twink-

on the zenith distance, obtained as a result of observations, is well explained by Eq. 2

It should be noted that for large zenith angles, the quantity crp depends strongly

azimuth, which greatly increases the scatter of the points in graphs of the type of Fig. 2

The agreement which we obtain between Eq. (13-31) and the function crp = f(sec e)

a result of observations with various values of D, assuming that the quantity

to 10 em, once again confirms this estimate for J5Jf:. Below, we shall obtain a 0

estimates which also agree with the first estimate.

As follows from Eq. (13.31), the lower layers of the atmosphere, where the quantity C~ largest, do not have an important effect on the amount of twinkling, since the product

/6 2 Cn(z) is small for small z. Therefore, higher layers of the atmosphere, where the func-

241

tion z5/6c2(z) takes larger values, play the chie~ role in the phenomenon of twinkling. n

Moreover, the height at which this ~ction achieves its maximum can serve as a more pre-

cise definition of the quantity H0

•

We now consider the problem o~ the frequency spectrum of the fluctuations. Just as

in the calculation of the size of the fluctuations, here we must also take into account the

averaging action of the telescope objective. AB we have already shown above, the small-

scale components of the fluctuation ~ield, with dimensions no greater than R (the radius

the telescope diaphragm) do not contribute appreciably to the quantity [log(P/P0 )]2

.

fore, it can be stated that the high frequency components of the fluctuations of the light

flux will also be considerably weakened. To calculate the frequency spectrum of the flue-

tuations o~ the total light flux through the telescope diaphragm, we muBt calculate the time

autocorrelation ~ction of the ~luctuations o~ this quantity. As was already shown in

Chapter 12, in considering the time behavior o~ the fluctuations, one may take into account

only the motion of the refractive index inhomogeneities in the direction perpendicular to

the ray (if the angle~ between the direction of the ray and the wind velocity is not too

small). AB proved in deriving Eq. (12.2), the field at the point (L,y,z) at the time

can be regarded as being the same as the field at the point (L,y-v ~,z-v ~) at the timet. y z

Using this argument, we can write a generalization o~ Eq. (13.6)

in the form

P(t) J J I ( Y I ' z ' ) dy I dz t

L:

P(t + ~) JJ I(y- v ~,z - v ~)dydz. y z

L:

(r3.33)

Averaging these equations, subtracting them from the unaveraged equations and dividing by

P = I L:, we obtain

242

P(t) - p 1 L: JJ

L:

P(t + ~) - p ------=! p L: JJ

L:

dy1 dz 1 ,

I(y- vy~,z - vz~) - r dydz.

tiplying these expressions together and averaging, we obtain the ratio 0~ the time auto­

ion ~ction o~ the ~luctuations o~ the total light ~lux through the diaphragm 0~

objective to its mean value, i.e.

~( ~) = 21- 2

L: (I) (13.34)

expansion (13.16) and change the order of integration in the expression

obtained:

co

J J F I(K2,K3)dK2dK3 X

-co

X JJJJ L: L:

the de~inition (13.19) o~ the ~ction V , we obtain L:

co

~(~) = (1~2 JJ -co

(13.35)

We now find the time spectral density of the fluctuations, i.e.

00 00

WP(f) == 4 J cos(2:rrf-r)~(-r)d-r = 2 J e-2:rrif-r~('r)d-r. 0 -oo

Substituting the expression (13.35) into the right hand side of this formula and bearing in

mind that

00

we obtain

00

JJ

( 13. 36)

We now consider the case where the diaphragm has the form of a circle of radius R = D/

Using the expression (l).2l) ~or Vl: and bearing in mind that F1 (K2,K3) ~ F1 ~K~ + K~), introduce new variables K

2 = K coB cp, K

3 = K sin cp. Moreover, writing vy = vn cos cp0 ,

vz == vn sin cp0

, we obtain

Joo [ 2J1 (KR)J2 FI K) ~ KdK w (f)=~

p (f)2 0

Bearing in mind that

2:rr

J ! o [2:rrf - KV cos( cp - cp )] dcp = n o 0

244

2rt

J o [2:rrf - KV cos( cp - cp )] dcp • n o 0

2

v 2 2 4 2..p2 K V - :rl ..1. n

0

2 2 2 2 for 4:rr f < K v , n

2 2 2 2 for 4:rr f > K v , n

obtain the formula

2:rrf v n

(13.37)

the change of variables J K2 2

4 2f2 , vn - :rr = K vn' this formula finally reduces to the

X R J K2 + 4:rr

2:r2 jv2 n

J K2 + 4:rr2f2 /v2 n

X

r~· (13.38)

':I. corresponding to the 11 two-We consider the case where F1

(K) is given by En. (7.87),

rds lawn for the refractive index fluctuations [:r]:

(13.39)

X ~ 2 0 -11/6[ 2 41( f + --2-

v n

th fun t . w (f) it l's more convenient to consider the function normalized Instead of e c lOll p )

with respect to ~2 J which characterizes the change in the frequency spectrum as a result 2 ~7/6 11/6

of the averaging action of the objective. Dividing (13.40) by ~ = 1.23Cn L and

introducing the quantities

we obtain

v lk f = ...E:.v ~ o 2n L

, n = f/f ) 0

p

X l23 1 ( P J t 2 + n2) J 2

p J t 2 + r}

and t

dt .

It follows from this expression that the dimensionless quantity

only on two dimensionless parameters: the parameter n = )2!rAL f/v considered in n

12 and P = )2n/AL R) the ratio of the radius of the diaphragm of the telescope to

correlation distance JIT, of the intensity fluctuations) i.e.

fWp(f) --2 -=F(n,p).

~

function F(n,o) coincides with the function (12.5) considered in Chapter 12.

In Fig. 4o we show) for various values of p) the function f0WP(f)/~2 obtained by numeri­

integration of Eq. (13.42). It is clear from the figure that when p is increased) the

frequency components drop out of the function WP(f); these components are related to

small scale components of the fluctuations of I) whose dimensions are less than R.

p=O 0.2

0.6

1.0

1.4---------

1.8------

p=2.5-----

-0.5 0 0.5 log (f/f0 )

Fig. 4o Theoretical form of the frequency spectra of fluctuations of the loga­

rithm of the total light flux through a telescope) as a function of

246 the diameter of the telescope.

To compare the theoretical function Wp(f) with experimental data} we have chosen the

frequency spectra} obtained at the Perkins Observatory) of the intensity fluctuations of

the light flux through a telescope with different diaphragm sizes (Figs. 41 to 44). As is

clear from the figures} the experimental data agrees qualitatively with the functions

wp(f) calculated above.

Fig. 41 Frequency spectrum of the fluctuations of

total light flux through a telescope with

a diaphragm of diameter 1 inch (lJ winter;

2, summer).

JWp

0.05

0.04

0.03

0.02

0.01

0 1 f, cps

Fig. 42 Frequency spectrum of the fluctuations of

total light flux through a telescope with

a diaphragm of diameter 3 inches (1, winter;

2} summer).

248

Fig. 43 Frequency spectrum of the fluctuations of

total light flux through a telescope with

2

a diaphragm of diameter 6 inches (1, winter;

2, summer).

Fig. 44 Frequency spectrum of the fluctuations of

total light flux through a telescope with

a diaphragm of diameter 12.5 inches

(1} winter; 2} summer).

500 t, cps

500 f, cps

v, cps

Fig. 45 Dependence of the quasifrequency of

twinkling on the zenith distance.

250

It should be noted that the experimental data presented in Figs. 41 to 44 represent

s of Wp(f) which are averaged over a whole season and which pertain to different values

i.e. to different f0

• It is easy to see that when the functions WP(f) pertaining to

rent values of f0

are averaged, we obtain a spectrum which is wider in comparison to

Therefore, we ought not to expect quantitative agreement between the theoretical

and average frequency spectra (of the type in Figs. 41 to 44). At the s~e time, the

2 renee in wind speed is not reflected in the size of ap, which is the integral of WP(f).

In some papers, instead of detailed measurements of the frequency spectrum of fluctua-

light intensity, cruder estimates are made of the characteristic frequencies of

(For example, the average number of intersections of the function P(t) with its

or the average number of maxima of P(t) is determined~ Fig 45 shows the depen­

obtained by Zhuk.ova @8] for the average number (per second) v of maxima of the func­

P(t) as a function of sec e. The observations were made during the course of one

, when the wind velocity was 5 m/sec at the earth's surface and 36m/sec at a height

We can see a regular decrease of the quantity v as a sec e increases. The

f(sec e) shown in Fig. 45, which is well approximated by the curve

=canst (sec e)-1/ 2 (the solid curve), can be simply explained from the point of view of

developed above. It can be shown (see [9o]) that the number of maxima of the

is determined by the smallest refractive index inhomogeneities. However, when

wind speed is appreciable and when the dimensions of these inhomogeneities are small,

"wiggles11 which they produce in the curve P(t) have a very fine structure and cannot be

by the recording device [gJ. The number of strong maxima of the curve under con­

actually calculated in [88], is determined by the dimensions and

of motion of the "running shadowsn, which have dimensions of the order of )J...H sec e 0

the dimensions of the correlation distance of the intensity fluctuations). Therefore

average rate of the maxima, is determined by the relation

v canst v

n

yAH sec e 0

(13.43)

is the component of the wind velocity normal to the ray. It follows from Eq. (13.43)

quantity v is inversely proportional to J sec e , which is in good agreement with the

251

curve of Fig. 45 [h]. Thus) the curve in Fig. 45 is one of the direct corroborations of

the fact that the quantity ~ is the correlation distance of the fluctuations of light

intensity.

Starting from Eq. (13.43)) we can once again estimate the quantity ~· Since the

constant in (13.43) is of order unity) then taking vn ~ 20 m/sec and v = 70 cpsJ we obtain

f:lH N canst x 30 em. This value of ~ agrees in order of magnitude with the estimate VAno o

obtained previously.

We now briefly discuss the dependence of the amount of twinkling on the angular dimen-

sions of the wave source. It is well known that the stars twinkle more than the planets.

This fact can easily be explained with the help of the following simple considerations.

The various points of the surface of a planet are incoherent sources of light. As

is well known ~2]J the intensity of the total field of incoherent sources is equal to the

sum of the intensities of the separate sources. Denote by i(eJ~) the density of the light

flux in the direction (eJ~). Then the intensity of the total field is

I ( 13.44)

where the integration extends over the solid angle subtended by the planet's disk. Aver--

aging (13.44)) we obtain

I= J ~dD. Q

It is natural to assume that i = canst within the limits of the solid angle Q. Then

Y ~ i QJ where the angular dimension of the planet is Q. The fluctuations of the quantity

I are given by the relation

I - I J [i(e)~) - i ]dnJ (13.46)

Q

252

the mean square intensity fluctuations are given by

(I - r)2

= J J [i(eJ~) - i] ~(e' )~') - Ddndn'. Q Q

It is convenient to evaluate the size of the fluctuations by using the relative quan-

(13.48) -2

(I - I) 1 (1)2 ~ Q2 If

Q Q

it denotes the quantity i- i. We introduce the correlation coefficient bi(v) of the

tions of the flux density i 1 for two directions ( e1J ~l) and ( e2 ) ~2 ) which make an

each other:

b. (v) l

i'(elJ~l)i'(e2J~2)

( i 1) 2

(I - r)2 ~ 1

(r)2 = (i)2 n2

is obvious that

a point source. Therefore) the function

253

represents the relative decrease in twinkling of a planet of angular size Q compared with

a point source. In order to evaluate the function Kl(n)J we can use the following argument.

Suppose that a point source of light is located at the observation point and that

is a circular objective with areaS= n12 at the boundary of the refracting atmosphere

the thickness of the refracting atmosphere). Then the dependence of the fluctuations of the

total light flux through the objective on its dimensions is expressed by the same function

Kl(n). On the other handJ this dependence is expressed by the function G(D /vfJ(L) cal­

culated above (see Fig. 36).

Instead of the solid angle QJ it is convenient to introduce the angle r subtended by

the planet's diameter. Then the diameter D of the imaginary objective will be equal to /L

and the function Kl(n) = K(A) takes the form

where ~ is a numerical coefficient of order unity [i]. As is well known) decrease

ling because of the finiteness of the angular dimensions of the light source is an effect

which can already be observed for sources with angular dimensions of the order of

l" = 0.5 X lo-5 radians. This means that the argument of the function (l3.52) is already

of order unity for such a value of 1 (see the curve in Fig. 36). Using thisJ we can make

still another estimate of the quantity ~. From the relation 0.5 x l0-5 IFC]'5:. -.. lJ 0 0

we obtain /FC]'5:."' 2 X l05 • Setting A = 0.5 X l0-4 em) we obtain ·15Jf .-. lO em) which is o vAno

in good agreement with all previous estimates.

2 In conclusion) we make a numerical estimate of the parameter en which characterizes

the atmospheric refractive index fluctuations for visible light. In order to be able to

2 estimate en from data on the twinkling and quivering of stellar images in telescopes) it is

necessary to specify somehow the profile of the quantity e2 (z). According to some data on n

2 measurements of en in the range of centimeter radio waves) this quantity falls off with

-2 z Therefore) we specify the profile of e2(z) in the form n

Eqs. (8.27) and (8.28)) we obtain

(--2 4 e2 -l/3 6a,) = .6 R b • no o

(l3.53)

(l3-55)

to experimental data) [log(P/P0

)J 2 = 0.205 forD~ 0 and e = 0 (see Table 6).

to Kolchinski 1 s data) at zenith J ( 6 a,) 2 = 0. 35" = l. 7 X lO - 6 radians (this figure

a telescope of diameter b = 4o em). Using the indicated data) and assum­

A = 0.5 microns) we can obtain eno = 7 x l0-9 em-l/3 from Eqs. (l3.54) and

55)) regarded as a system of equations in e2 and H • nO o

It should be noted that the value of enO which is obtained is practically independent

2( ) 2 2 how the profile of e z is specified. For example) specifying e (z) = e exp(-z/H )J n n nO o

obtain eno= 3·7 X lo-9 em-l/3• Thus) the indicated estimate of the ord.er of the quantity

is reliable enough.

It is well known that the air's refractive index fluctuations in the range of visible

are mainly due to temperature fluctuations. en is connected with the characteristic

of the temperature fluctuations (e figures in the 11 two-thirds law" (T -T )2= e2 r 2/3 T l 2 T l2

the temperature field) by the relation

6 -6 e :::: 9 X lO p e n T2 TJ (l3-56)

255

where T is expressed in degrees K) CT in degrees am-l/3 and p in millibars. Using this

formula) we can estimate the quantity CT. For z = 5 em) for example) we obtain CT =

degrees em-l/3 ~]. The estimates of' c! which we obtain agree in order of' magnitude with

data based on measurements of' the intensity of' scattering of' UHF waves) propagating beyond

the horizon (see Chapter 4)) where for heights of' a few kilometers) one obtains a value of

c ,.. 5 to 10 x 10-3 degrees em -l/3.. Of' course) it should be noted that the estimates given T

are in tae nature of' a rough guide) which enables us to determine only orders of' magnitude.

Much better results could be achieved by analyzing measurements of' the twinkling and quiver-

ing of' stars together with simultaneous aerological measurements) like those which were

made in the layer of' the atmosphere near the earth. Such measurements would also enable us

to evaluate more reliably the roles played by different layers of' the atmosphere in the

phenomena of twinkling and quivering of' stars and other distant sources of' radiation) and

would enable us to solve a series of' problems connected with long distance propagation of

UHF radio waves in the troposphere.

By comparing the curves in Figs. 34) 35) 38 and 39 with each other) we can observe

still another characteristic feature of' the dependence Q.og(P/P0B~ f'(sec e)· Beginning

with values of' e "' 60°) the power law growth of' these functions slows down. In doing so)

the curves [log(P/P0

)]2= f'(sec e) "saturate" for values of' the diameter D of the telescope

diaphragm which are large compared to ~) while for small D there even occurs a decrease

in the twinkling of' light as the zenith distance increases (Fig .. 34). This circumstance)

which is at first glance extraordinarily strange) was recently explained in the paper [99] •

The issue involved here is that all the data given in Figs. 34) 35) 38 and 39 pertain to

twinkling not of' monochromatic but of polychromatic light. However) in point of fact)

because of different atmospheric refraction for rays of different wavelengths) the rays of

different colors arriving at the same observation point traverse different paths in the

atmosphere. The distance between rays of different wavelengths increases as the

tance of the light source increases) and for e N 60° is of the order of' 10 em at the bound­

ary of the refracting atmosphere. The fluctuations of' light intensity are mainly produced

by·the atmospheric inhomogeneities with sizes of order ~) located within the paraboloid

p2 ·= :x.z sec e surrounding the ray) which has a diameter of the order ~ "'10 em. Conse­

quantly) if' the distance between the rays exceeds VAR0

as a result of refractive differ-

esJ then the twinkling in different parts of' the spectrum is uncorrelated. Therefore)

large e) the total intensity of' polychromatic light experiences smaller relative flue-

ons than the intensity of' monochromatic light. Moreover, if' due to this "chromatic"

ctJ the twinkling of' the polychromatic light decreases more rapidly than it increases

to growth of' ·1 = H sec e (as is the case for small D) where) as e increases) 0

P/P0)]

2 grows comparatively slowly)) then the total effect of' the twinkling decreases

grows.

Detailed calculations given in ~9] enable one to completely explain the character of'

experimental curves in Figs. ~) 35) 38 and 39) both for large and for small values of' e.

257

* APPENDIX

Addendum to ChaRter 5

( ) and (5 .3) of the text follow from more exact relations if one neglects Eqs. 5.1

t . f ~, d T' Treatments retaining such terms have terms involving the spatial deriva lves o u an •

~ J and Krai chnan Iii] . for the case T' = 0, and by Batchelor [iii], been given by Lighthill ~ l ,

for the general case. For T' = 0, these authors find

(A)

where p' and w are the density variation and particle velocity associated with a weak ~

· d' f an density p and turbulent velocity u'. In con-sound wave propagating lll a me lum o me 0

trast, one obtains from Eq. (5.3)

(B)

ht 'd of Eq. (A) which is retained in Eq. (B) and the part which The part of the rig Sl e

t . to the scattering in the first Born approximation which, is neglected give contribu lOllS

in general, are of the same order of magnitude. This is because the scattering arises prin-

t f · comparable to the acoustic wave length. cipally from interaction with eddy-struc ures o SlZe

In particular, the angular dependence of the scattering is strongly affected. Eq. (A)

implies a zero in scattered intensity at 90° which is lost in Eq. (B).

When the time dependence of~~ is neglected, and when the turbulence is isotropic, one

finds from Eq. (A) the differential cross-section

* See Translator~s Preface.

--------------------------------~~~~~~~-------------------------~·~

4 -2 ( sin 2e) 2

e 2~k Vc . e E(2k sin 2)dD, 4 Sln 2 .

da(e) (C)

is to be contrasted with Eq. ( 5. 23) of the text, for the case IT = 0. ( Cf. [i J, Eq.

The notation in Eq. (c) agrees with that in Eq. (5.23) of the

is identical with E(k)/4~k2 of [i] and with E(k)/2 of [ii].) When the time

of 'i!• is taken into account l}i], there result deviations from Eq. (C) at very

scattering angles. A recent investigation of a time-dependent case has been given

Chapter 4).

There is an analogous change in the angular dependence of the scattering when the

involving spatial derivatives of T1 , which are neglected in deriving Eq. (5.23) of

are reinstated. Under conditions which are plausible for atmospheric scattering,

~ii] finds that the angular dependence of da(e) is given by cos2e, for the case

0, in contrast to Eq. (5.23).

It should be emphasized that all the corrections discussed above are negligible when

sound wave length is very small compared to scales in which there is appreciable tur-

t excitation. In this case, however, the Born approximation no longer provides a

d description of the sound propagation.

REFERENCES

M.J. Lighthill, Proc. Cambridge Phil. Soc., 49, 531-551 (1953).

R.H. Kraichnan, J. Acoust. Soc. Am., 25, 1096-1104 (l953)j erratum, 28, 314 (1956).

G.K. Batchelor, Symposium on Naval Hydrodynamics (Editor, F.S. Sherman), Ch. XVI

(Publication 515, National Academy of Sciences - National Research Council, Washing-

ton, 1957).

R.H. Lyon, J. Acoust. Soc. Am., 2!, 1176-1182 (1959).

259

* NOTES AND REMARKS

Part I

Chapter 1

~ (p.)) A detailed exposition of the topics discussed in this section can be found

in the papers of Yaglom [)._, 4, 5] and Obukhov [?} 3] .

b (p.3) Here and everywhere afterwards} the overbar denotes averaging over the whole

set of realizations of the function f(t); in the applications} this averaging is very often

replaced by time averaging or space averaging.

c (p.3) The asterisk denotes the complex conjugate. (T)

d (p.4) In addition to this definition of stationarity (stationarity in the wide

sense)Jthere is also another definition (stationarity in the narrow sense)} namely} f(t) is

called stationary if the distribution function (1.1) is invariant with respect to all shifts

of the set of points t1Jt

2J•••JtN by the same amount -r. However} in practice} functions

which are stationary in the wide sense are almost always stationary in the narrow sense as

well} so that we need not distinguish between these two definitions. Below we shall need

only the definition of stationarity in the wide sense; therefore} in the text of this book

we shall always omit the explanatory phrase "in the wide sense".

~ (p.4) In cases where f(t) ¢ OJ one can always introduce the new random function

F(t) f(t) - f) for which F(t) = o.

o(·) denotes the Dirac delta function. (T)

Since Bf(-r) = Bf(-'T), then W(w) = W(-w)} and the expansion (1.6) can alSo be

written in the forms

00 00 I cos(w-r)W(w)dw = 2 I cos(w-r)W(w)dw •

-oo 0

* See Translator's Preface. 260

We note that it can easily be shown that the mean value of f(t) must be a

ar function of time ~n the case of a function with stationary first increments). Thus}

assumption that f(t) is a fun ti 'th t c on Wl s ationary -increments is valid only for time

during which the law of change of the mean value f(t) can be considered to be

However} this leads to a much larger range of applicability than the

of stationarity} according to which the mean value cannot change at all. In

ral} in cases where the assumption that the first increments are stationary is not auf-

further and assume that the l'ncrement~ ~ c o~ some higher order

[Lo] · In what follows} we shall assume that the first increments are sta-

(p.lO) In fact} in beginning the study of a random process which we are not sure

stationary} it is more appropriate to construct its structure function than

correlation function. Furthermore} the practical construction of the structure func-

is always more reliable} since errors in the determination of the mean value f(t) do

affect the value of D~(-r). In th h ~ e case w ere the constructed structure function turns

to be constant for large -r} we can find B~(~) ~ • as well by using eq. (1.14').

l (pp.l2}21) rgy o~ e ~ uc uations is Of course} in all actual cases} the ene ~ th ~l t

From this it is clear that in cases where the function w(w) becomes infinite at

the function does not have the physical meaning of energy.

k (p.l3) Combine the formulas

00

1 sin vn

I ( z) v

2m+v I ( ~) (E:t!r(m + v + 1)]

m=o

r(v)r(l - v) n = sin vn

261

to obtain

K (z) =! r(v)(~)-v- ~ r(l- v)(-2z)v + ••· , (lvl < l). v 2 2 2v

t (p.l3) To calculate the integral

00 J w-(p+l)(l- cos wt)dw,

0

one can start with the familiar expression for

00 J w~ e -cxw cos wt dw , ( ~ > -1) ,

0

(T)

and then apply the principle of analytic continuation with respect to ~· To obtain the

final result, one has to pass to the limit ~ ~ O.

m (p.l6) Here o(K1

- K2) = o(Klx - K2x)o(Kly - K2Y)o(K1z - K2z) and dK = dKxdKydKz.

~ (p.l7) Equivalently, differentiating

we obtain

00

V(K) ~ J Bf(r) cos Kr dr ,

0

dV(K) _ dK -

00

0

so that (1.27) follows from (1.25). (T)

£ (p. 23)

2rc J cos(x cos e)ae = 2rc J0

(x),

0

where J (x) is the Bessel function of the first kind of order zero. (T) 0

262

E (p.26) Use the formula

r(2z) l 2z ( l 2 r z)r(z + 2). 2j;(

(T)

Chapter 2

~ (p.)l) The value of this constant must be found experimentally. In this regard see

~ (p.33) Some considerations pertaining to the behavior of Drr and Dtt for large r

given at the end of Chapter ).

~ (p.38) If we assume that the eddies, even including infinitely large ones, are iso-

c, then from the incompressibility condition and some supplementary hypotheses we can

following expansion for E(K) for small K [16]:

E(K) 4 6 CK + O(K ) .

r, since under actual conditions the large scale eddies are inhomogeneous and aniso-

c, the applicability of this result to atmospheric turbulence is very doubtful.

In studying the structure functions Dtt and Drr in wind tunnels, one does

usually succeedin obtaining large enough Reynolds numbers to make the interval (t ,L) 0

~e recall that t0

N L/(Re)3/4 .]

Chapter 3

~ (p.40) The assumption that the temperature is passive is in general not true,

ce buoyancy forces are associ a ted with the temperature inhomogeneities [20] . However,

a given dynamical regime of turbulence, which already ~akes into account the action of

mean temperature profile, the fluctuating part of the temperature can be considered to

Recently, Obukhov [94] investigated the departures from the "two-

law" for a temperature field,which are connected with its lack of passivity. As a

result of an analysis of the influence of the buoyancy forces on the turbulent regimeJ he

concludes in this paper that in regions small compared to the characteristic dimension ~J

t t . o .... ey the same "two-thirds law" (see p. 46) as obeyed by passive the temperature flue ua 1ons 'U

additives. However, in the range of sizes (~,L0 ), the "two-thirds law" is violated. The

5/4 - -3/4 -3/2 ( ; i th dimension ~ is defined by the relation ~ = € N ~ where ~ = g T0 , g a e

T i th temperature. and the symbol N is explained on acceleration due to gravity, 0 S e mean ,

P• 44). In the free troposphere, ~can be several times smaller than L0 • (The ratio L0/~ depends on the meteorological conditions and turns out to be e~ual to L0/~ = (Ri)

3/2

J where

Ri is the Richardson number; see note [e] to Chapter 10.)

E. (p.4l) To justify E~. (3.4), one can give an argument similar to that made in deriv-

ing the formula ~ = _ D grad ~, by just replacing the process of transport of the property

~ due to molecular motion by the transport of ~ due to the chaotic motion of small parcels

of air.

~ (p.42) we recall that we consider the motion of the fluid to be incompressible,

we take div ~ = ovi/oxi = o, whence it follows that ovi/oxi = 0 and ovi/oxi = o.

~ (p.49) The relation (3.28) can be obtained from the Navier-Stokes e~uation in just

the same way as the relation (3.11) was obtained from the diffUSion e~uation. The

(3.28) is valid in the case of stationary turbulence and actually represents a condition

the turbulence to be stationary.

~ (p.51) For a more detailed account of the results of measurements of temperature

fluctuations in the atmosphere, see Part IV.

f (p. 52) We note that the constant D~0 )( 1;1 - ; 2 \) is defined only for 1;1 - ; 21 To co:struct the spectral expansion (3.34), we must specify the function Df( 1;1 - ; 2 1) in

~ i!;' ~ 1(~ ~ )) some reasonable fashion for large values of 1;1 - r 2 1· The function .':tf(K, 2 rl + r2

is obtained as a result has meaning only for \KI >>L~1, so that the way of specifying the

does not influence the function ihf in the range \KI >>L0-l. The situation iS function Df .'f.

just the same in deriving E~. (3.33).

~ (p.53) ThiS is true only approximately. Some detailS of the spectral distribution

C2 (For example, the r~uanti ty t

0 depends on the Reynolds number.)

can in general depend on f• ~

However, in the region L~l << K << t~1, a universal spectral density !~o)(K) can still be

defined. ~ exact mathematical theory of random processes with smoothly varying mean

264

eristics, has been given in [92] and in R.A. SilvermanJ "A matching theorem for locally

onary random processes", Co~ Pure Appl. Math., 12, 373 (1959). (T)]

E: (p.54) Actually, as shown in the reference ci t·ed in note [g] above, in the case of

random process with smoothly varying mean characteristics, there is no need for using

chastic Fourier-Stieltjes integrals, since in general ordinary individual Fourier trans-

of the sample functions of the process exist with probability one. (T)

This correlation between neighboring spectral components of the random field

is simply related to the space correlation properties of radiation scattered by f(;),

the latter is a random refractive index field. In this connection, see R.A. Silverman,

cattering of plane waves by locally homogeneous dielectric noise", Proc. Camb. Phil. Soc.,

530 (1958). (T)

Part II

Chapter 4

~ (p.59) In Part IV, time changes of the refractive index field are taken into account

the case of line-of-sight propagation. For the case of radio scattering, time changes

taken into account in several papers, e.g., R.A. Silverman, "Fading of radio waves scat-

by dielectric turbulence11, J. Appl. Phys., 28, 506 (1957); erratum, ibid., 28, 922

) and R.A. Silverman, "Remarks on the fading of scattered radio waves", IRE Trans.

ennas and Propagation, Vol. AP-6, 378 (1958). See also Appendix. (T)

b (p.60) b denotes the Laplace operator. (T)

~ (p.6l) Thus we neglect the fluctuations produced in the incident wave as a result

its propagation from the source of radiation to the scattering volume (see Part III).

d (p.6l) It follows from (4.13) that

outer scale of the turbulence. Thus it is not just the smallness of n1

justifies neglecting multiple scattering (i.e. the terms E2 + E3

+ ••• ); in fact, the

nature of the scattering medium also helps attenuate multiple scattering, for other­

we would have v2 instead of VL3 in this estimateo For a detailed analysis of multiple 0

scattering in a one-dimensional random medium, see I. Kay and R.A. Silverman, "Multiple

scattering by a random stack of dielectric slabs", Nuovo Cimento, Vol. 9, Serie X,

Supplemento No. 2, 626 (1958). (T)

~ (p.64)

f (p.64)

~ (p.67)

In detail, c2

= iC A • (k -l 0

~) = ikC1 A0 -~. (T)

The velocity of light is denoted by c. (T)

When the volume is a cube with side 2h, we have

sin l 2h sin l3h

nA2 nA3

For l = O, F(O) = (h/n) 3 for A. = n/h, F(l) vanishes, and for l

large A, F(l) oscillates and falls off rapidly. Ash~ oo, F(l) ~ o(l).

~ (p.68) Rigorous conditions for the validity of approximations like In(K) N !n(K)

as K ~ oo are given in the one-dimensional case by H.S. Shapiro and R.A. Silverman, "Some

spectral properties of weighted random processes", IRE Trans. Inform. Theory, Vol. 5, No. 3,

E: (p. 75) Incidentally, we note that Villars and Weisskopf were apparently unfamiliar

papers of Obukhov [3o] and Of Obukhov and Yaglom [!.3] on pressure fluctuations.

,.,"'·rP.i·'or·e, their way of deriving Eq. (4.39) is much more .complicated than the way we give.

(p.76) A similar expression for the effective scattering cross section of sound

in a turbulent flow was obtained by Blokhintsev [)2, 33] in 1946.

.R. (p. 77) A more detailed investigation shows that Eq. (4.47) is applicable only in

case where the quantity ')' is small compared with e. In the case where ')' > e, the change

e within the scattering volume begins to play an important role. In this case, the effec­

size of the scattering volume is determined by the angle e rather than y, and V N D3e2

•

this into account modifies the analysis of the experiment of Bullington et al.

in [31] and the conclusion drawn from it. (T)]

~ (p.80) A more detailed description of the results of measurements of CT will be

IV.

123 (1959). (T) Chapter 5

! (p.69) As is well known, an infinite sinusoidal diffraction grating produces dif-

fraction of a plane monochromatic wave only at one angle e (more accurately, at two equal

angles ±e), which satisfies a relation similar to (4.20). In the case of diffraction by

a sinusoidal diffraction grating of finite dimensions L0

, each of the diffracted bundles

has a spread~ e N l/L • This means that finite dimensional sinusoidal diffraction 0

gratings with neighboring periods can also participate in diffraction at the angle e, since

these lattices include the direction e because of the spread of their diffracted bundles.

~ (p.69) For example, for A = 10 em, e = 0.033 and H = 2 km, the size of t is

3m ± 0.5 em.

~ (p.69) In the majority of applications, the condition (4.21) is met satisfactorily.

Apparently, the size of 10

in the troposphere is of the order of 100 m.

~ (p.8l) See Appendix. (T)

b (p.83) We assume, therefore, that~'(;) and T'(;) do not depend on time. The actual

s of these quantities in time can be regarded as a change of the different realizations

random fields.

.£ (p.84) At first glance, Eqs. (5.13) to (5.15) differ from the corresponding Eq. (4.12)

electromagnetic waves by the presence of the factor k6 instead of k4

• However, this

is only apparent, since in (4.12), A0

represents the amplitude of the field E0

,

(5.13), A0

is the amplitude of the potential rr0

• However, the amplitude of the

c pressure or the acoustic velocity is proportional to kA0

, so that k6

A2 = k4

(kA )2

• 0 0

In fact, in the case of isotropic turbulence, the quantity ~'(;1)T'(;2 ) can

t (p.69) The structure of this kind of turbulence was described at the end of Chapter 3· ···~0·~~ne~nn the vector P = ;l - ; 2, i.e. has the form A(p)p. Since div ~ = 0, then

~v(A(p)p) = 3A + pA'(p) = 0 also, whence A= cjp3. Since for p = o, A(P)P must be finite, m (p.74) Actually, pressure fluctuations in a turbulent flow lead to much smaller

refractive index fluctuations than temperature and humidity fluctuations. The corresponding

estimates are easily carried out by using Eqs. (4.36) and (3.44). We do not consider here

the second paper by the same authors [23], because of its incompatabili ty with turbulence

266

it follows that C = 0, as was to be shown.

..::_ (p.89) In Part IV we shall study this matter in more detail.

f (p.90)

00

E(K) 1

:1( 3 J J J exp( -iK•-;)Bii (;)dV =

00 00

= + J Bii (r)r sin Kr dr = + J r sin KrrBrr +! --9:. (r2

B ~ dr = ~K ~K t r~ ~~

0 0

2 [00

l~~K I -r/t e r sin Kr dr - K J -r/t e

0

(T)

~ (p.90) This fact follows from the general expression for E(K) for small K which

is valid for homogeneous isotropic turbulence, i.e. E(K) = CK4

+ (see note [c] to

Chapter 2). However, this result is hardly applicable to atmospheric turbulence.

~ (p.l04) To avoid confusion, we agree that the differential of the variable which

integrated will be the first to appear after the integral sign.

d (p.ll2) We have used the formula

ch can be obtained from the familiar formula

00

J 0

analytic continuation with respect to q.

(- l < q < !) 2

e (p.ll2) A formula similar to (6.68) was first obtained by Krasilnikov [43,44].

, in this work, instead of the"inner scale11 of turbulence t , he uses a "smoothing 0

which is assumed to be proportional to the wavelength (without sufficient jus-

Chapter 6 This is the condition for the applicability of the geometrical optics

Then

0

and

o.

The last term of the equation is of order no greater than kEn1/ t 0

and is always much less

than the third term of the equation when kt 0 >~ 1. Since the term 2 grad (Eo grad log n)

in Eq. (6.1) is related to the change of polarization as the wave propagates, this effect

is small in the case A. << t • 0

£ (p.94) See pages 60 - 61.

268

-~~.4u-~~~~~lJ~onj see section 6.7. (T)

We assume here that the "two-thirds law" is satisfied for the temperature

than for the potential temperature Hj this is valid only in the layer of the

near the earth, where T and H are practically the same.

Chapter 7

§ (p.l24) This way of approximately solving the wave equation was proposed by Rytov

and was used by Obukhov [51] to solve the problem of amplitude and phase fluctuations.

£ (p.l26) The quantity cp = ~[Y' s1J is equal to the deviation of the direction of

the perturbed wave from the initial direction. Thus, the condition (7.19)

es a restriction on the size of the fluctuations of the propagation direction of the

( ) This assertion can be proved rigorously for monotone decreasing fUnctions ~ p.l37

ln(K) ·

d (p.l44) The quantity Ln is finite only when the function Bn(r) decreases sufficient-

ly rapidly as r ~ oo.

~ (p.l45 ) If we bear in mind that the spectral density of the correlation function of

th i Small in the region K < 2rc/ Jf.L) then we can con-the amplitude fluctuations of e wave s

elude that the correlation function BA(p) must undergo smooth oscillations of small size)

Because of these oscillations of the function BA(p)J the rela-with period of order J):L.

tion (7.55) is also satisfied.

! (p.l47) The function ( 7. 73) was used in the paper of Obukhov [5l] and in the

related papers of Chernov ~6) 57] and other authors ~8 J 6l] •

) If We define. as usual. the inner scale t0

of the turbulence as §£ (pp. l47 ,l50 ~ ~

t = 0

J-2B(O)/B"(O))

and the outer scale Ln of the turbulence as

Ln = B(o) J B(r)dr) 0

then) using ( 7. 73)) we obtain t 0

= a and Ln = ~ v;c a.

h (p.l52) It is easy to show that

A

pA-p J (l - si~ x)xp-ldx = l + o(A-p)

0

f'or 0 < p < l.

i (p.l53) In calculating the integral) we used the formula

j (l - si~ x)xa.--3dx = n[2I'(4 - a.) sin ~~-l (o<a,<2))

0

ch can be obtained from the well-known formula f'or

00 J xa,-4 e-f3x sin x dx (a, > 2)

0

analytic continuation with respect to a, and subsequent passage to the limit f3 ~ 0.

~more accurate value of' the numerical constant in (7.94) is 0.307. (T)]

l (pp.l53)l55) By using (7.92) we can obtain asymptotic expansions of' bA(p) for large

small values of the parameter p / v'U.

p >> '.(5::1) we have

recall that

For t << p << v'U) we have 0

! (p.l53) We note that the quantity Jll,) f'or which the correlation fUnction (7 .92)

a negative minimum) corresponds to the average size of the "running shadows" which

observes twinkling sources of' light.

i (p.l60) As we have seen above) this condition is not necessary (see page l26).

Chapter 8

~ (p.l64) Strictly speaking) the fUnction !n(K);) can be defined uniquely only in

region K >> l/L • 0

27l

~ (pp.l68,l70) AB is well known, if the observation point is located near the surface

of the lens, then the intensity of light at the point is just the same as in the absence

of the lens.

~ (p.l69) We have

00

x2 = 2l/6 rt

2(0.033)k7 16 J x-ll/6 sin2x dx.

0

To evaluate the integral, we use the formula

00 J x-(p+l) sin2x dx =

0

which is obtained from the formula

00

p-2 2 1(

sin ¥ r(p + l)

(O<p<2),

(l- cos w~)Aiwl-(p+l)dw = 2Art ~p - 00 sin rt~ r(p + l)

(o < p < 2))

(see example b on page l3 and note [ tJ to Chapter l) by setting ~ = l and w = 2x. ( T)

~ (p.l70) This remark can be illustrated by the following example. If a plane-parallel

slab is placed on the ray path, then the phase shift produced by it does not depend on the

coordinate of the slab.

Chapter 9

~ (p.l76) This e~uali ty ac~uires precise meaning after multiplying both sides by

f(K2,K3

) and integrating with respect to K2

and K3

•

£. (p.l77) E~. (7.32) can be obtained from (9.13) if we carry out the integration in

(9.13) on the segment from L - R to L and let L go to infinity, keeping R finite. This case

corresponds to an infinitely remote source of spherical waves.

272

~ (p*l83) We note that the transition from E~. (9.25) to (9.26) can only be carried

out for a spherical wave. Therefore, the subse~uent formulas do not go over to the corre­

spo~ding formulas for a plane wave (see [b J ) . ~ (p.l84) In general, it can not be asserted that the integrand in E~. (9.3l) is the

spectral density of the correlation function of the fluctuations of logarithmic amplitude.

! (p.l85) This effect can be explained with the help of the following simple consider­

Suppose that along the path of the plane wave, at a distance L from the observation

point,there is located a converging lens with a focal distance f which greatly exceeds L.

It can easily be calculated that as a result, the diameter of the bundle bounded by the con­

tour of the lens is reduced in the ratio l:(l + ~), as compared to the diameter of the same

bundle without the lens. The compression of the bundle leads to an increase of light inten-

12 21 sity in the ratio l:(l + ~) - l:(l + ;r)· Carrying out a similar calculation for the case

where the source of light is located at the distance L + a from the observation point (i.e.,

at the distance a from the lens), where a << f , we find that the relative compression of

L a a diameter of the bundle is e~ual to l + ~ "a+L . Since we always have "a+L < l, then

relative change of light intensity of a spherical wave is always less than the corre-

sponding quantity for a plane wave. Thus, the amplitude fluctuations of a spherical wave

must be less than the amplitude fluctuations of a plane wave.

Part IV

Chapter lO

~ (p.l90) Thus, fluctuations in the difference of velocities at two points which are a

fixed distance r from each other,decrease when the pair of points is translated upwards.

However, wind velocity fluctuations at one point do not depend on the height of the point

order of magnitude v * (see [)_ 4] ) •

E_ (p.l92) We note that this relation is non-linear, which makes working with the appa-

ratus much more difficult.

~ (p.l92) Later (see p.203) we cite a value of the constant Vc obtained from measure­

ments of amplitude fluctuations of sound waves. It is close to the value of l.4 obtained

by To'Wil.Bend.

273

~ (p.l93) We give preference to this value of C} since it agrees better with numerous

measurements of amplitude fluctuations of sound (see p.203 ).

~ (p.l95) By a more refined argument} one can arrive at the conclusion that for stable

stratification

f(Ri) } 2/3 -1/3 T

K Z *

where Ri is the (dimensionless) Richardson number; which characterizes the extent to which

the temperature stratification influences the turbulent regime. The Richardson number

Ri g(dT/C'Jz)

T(2JU:/2Jz) 2

depends both on the form of the temperature profile and on the form of the wind profile;

here g = 9.8 m/sec2•

! (p.l96) The information cited above concerning temperature fluctuations in the lower

troposphere is of a preliminary nature and needs further elaboration.

Chapter 11

~ (pp.l98,204) It can be shown that this condition is satisfied in the layer of the

atmosphere near the earth if ~ << Kz/v*. For z of the order of a few metersJ the quantity

Kz/v* is of the order of a few seconds.

b (p.201) See Eq. (7.94). (T)

.£ (p.201) Here Tis written instead ofT (cf. Eq. (6.91)). (T) 0

d (p.201) This i.s the order of magnitude obtained if we set z = 8 m (the value given

in (]l]L 6T = o in Eq. (11.4) .. (T)

Chapter 12

~ (p. 209) Another justification of this conclusion can be given} based on the

independence of the different spectral components of the turbulence. (However} note that

the finite size of D leads to correlation between neighboring .spectral components in the

harmonic analysis of the integral

ThUS J for the integral to be approximately normalJ we must require that the volume in wave

number .space which contributes most to the integral contain many "substantially uncorre-

lated subvolumes". This is tantamount to the requirement that D itself contain many "sub-

stantially uncorrelated subvolumes". ( T))

~ (p.209) In Fig. 25 the function log x is marked off along the horizontal axis}

while the function I -l(x) is marked off along the vertical axis. Thus} the points in Fig.

25 are actually a plot of I -l(F(I)) vs. log (I/I ). If a random variable has a normal 0

distribution (with mean zero and variance one} say)} then its empirical distribution func-

tion G(x)J which is itself a random variable depending on the sample used} converges uni-

formly in probability to

X

(1/ V2rt) J 2 exp(-t /2)dt

-oo

size increases.. Moreover) I -l ( I(x) ) = x} so that f -l( G(x)) is approxi-

Similarly} if a positive random variable ~ has a log normal distribution) then

its empirical distribution function F(x) converges (in the sense indicated) to the di.stri-

bution function

(1/ \}2;( ) log x

J - 00

2 exp(-t /2)dt}

since Prob(~ < x) = Prob(log ~<log x) and log ~is normally distributed. Thus} I -l(F(x))

is approximately log x and if'! -l(F(x)) is plotted against log (x/x0

)J the resulting curve

is approximately a straight line. (T)

275

~ (p.212) See pages 140, 153· (T)

d (p. 214) we note that the theoretical curve of R == f( PI yffL) has a zero for

p == o.8 JU., while the experimental curve has a zero for P = 1.5 ..[i:L • This discrepancy

can evidently be explained by the fact that in the experiment described we had a geometri­

cal bundle instead of a plane wave. It is easy to see that this ought to lead to an

increase in the correlation distance.

~ (p.215) As we convinced ourselves above, the chief contribution to the fluctuations

of I are produced by the inhomogeneities of order Vff, contained inside of the paraboloid

y2 + z2 = AX with vertex at the point of observation. Displacement of an inhomogeneity

along the axis of the paraboloid can appreciably affect the field I only in the case where

as a result of the displacement, the ratio between the size of' the inhomogeneity and the

diameter of the paraboloid changes appreciably. It is easy to see that such a displacement

is of order 1. At the same time) a displacement of' the inhomogeneity perpendicular to the

axis of' the paraboloid by an amount jiT. J which takes place in a time -r = jiT. I vn J also

appreciably changes the field I. The longitudinal displacement in the time -r is equal to

t::, x == -rv = ( vtlv ) \(i:L. If !:::. x << LJ i.e. if a. = v nlv t >> J}JL J then the longitudinal t n

displacement can be neglected.

! (p.216) We use the frequency f' instead of w and we make the expansion with respect

to positive frequencies; this simplified comparison of' the results of theory and experiment.

The relation inverse to (1.23) has the form

00 J COB ( 21tf "L") W ( f) df' •

0

g (p.216) Cf. Eq. (1.51) • (T)

~ (p.218) The condition J'U >> t0

has been used to set the upper limit of integra-

tion in ( 12.5) equal to oo • ( T)

~ (p.218) The function fW(f) I x2 has a maximum for f = 1.38 f 0 = 0.55 vn I -JXL. 1 (p.221) The positions of the maxima of the theoretical and experimental curves in

Fig. 31 have been deliberately made to coincide.

Chapter 13

~ (p.228J236) The dependence of the amount of twinkling on the size of the diaphragm

given in [84] has to be corrected) since this dependence was obtained by sUilliD.ing the ampli-

tudes of the fluctuations at different frequencies instead of' summing the squares of the

amplitudes. Table 5 was constructed by using the integrals (given in [84] J pp. 120-123) of

the squares of the frequency spectra shown in Figs. 42 - 45. The date presented in Fig. 33

was constructed from Table 6.

E. (p.232) In radiophysical applications V"£ is expressed in terms of' the function

which describes the directivity pattern of the antenna.

~ (p.233) For Re(v) >- ~) Re(fl + v + 2) > Re(A + l) > 0) the formula

00

A= J 0

00 1(

X JJ 0 0

is valid [53], where

a = PJ we obtain

But we have [? 3 J

v J (at)J (bt)J (ct) (b

2c)

--l:.fl;..._ __ v.:..__~_:V~- dt ::::: X

tv+A v;. r(v + ~)

"' 2v J (at)J (wt)sin ~ fl ~v A d~dt

w t

W::: J b 2 + c

2 - 2bc cos ~. Setting fl :::::

1(00

If 0 0

277

c ::::: RJ

J J (ux)J 1

(vx)dx = p p-

0

Consequently} A = 0 for P > 2R and

J 2 arc sin(PI2R)

for p < 2R.

for v < u J

for v > u •

p p cos 2R - 2R

~l v.L-4;2J

~ (p. 234) We note that this formula could also have been obtained immediately from

(13.15) by introducing the coordinates J 1 = R sin ~J z1 = R cos ~J J1 - J 2 = p cos ~J

z1

- z2

= p sin ~ and integrating with respect to ~J ~ and R.

~ (p.241) See also page 256.

! (p.245) The formula in question describes the spectrum of the fluctuations of light

intensity only in the case of small fluctuations) when we neglect the difference between cr2

and log (1 + cr2

).

§£ (p.251) The frequency of these "wiggles" is of the order of thousands of cycles per

second) whereas their amplitude is negligible. Therefore they do not register when P(t) is

recorded by using a relatively low frequency loop oscillograph.

E: (p.252) The quantity vn generally depends on e also) i.e. vn = v)l - sin2e cos 2~ }

where ~ is the angle between the azimuth of the star and the direction of the wind. HoweverJ

the data of Fig. 45 apparently attests to the fact that at the time of the observations the

quantity~ was close to 90°.

~ ( p. 254) It should be noted that the function G( D I JXL ) calculated above was com-

puted for the case of fluctuations of a plane wave) whereas in the case being considered we

have a homocentric bundle of incoherent waves. This difference can slightly modify the

function G(D I y:;:L ) . We can take account approximately of such a modification by introduc-

ing into the argument of the function some constant factor ~ of order unity.

1 (p.256) The estimates of the size of the temperature fluctuations made in Q?1]

were based on the erroneous idea that under atmospheric conditions both the case

JlH sec e << t 0 0

(for small e) and the case ylH sec e >>L (for large e) can occur. 0 0

279

l.

2.

3-

4.

5·

6.

7-

8.

9-

10.

ll.

12.

13.

*

* REFERENCES

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Wav-e Propagation. in. a T11rbu.le:n.t

Medi11:n1 V. I. Tatarski

This mathematical monograph, translated by R. A. Silverman, formerly of New York University's Institute of Mathematical Sciences,. describes the phenomena associated with the propagation of electromagnetic and acoustic waves through atmospheric turbulence. It approaches the subject from both the theoretical and experimental sides, placing special emphasis on recent Russian contributions-just those contribu­tions which are the least accessible in English. Applications to phase and amplitude fluctuations, the scintillation of stars, radio scattering, and other problems are stressed.

Part I covers some topics from the theory of random fields and turbu­lence theory, including statistical description. Part II, on the scattering of waves in the turbulent atmosphere, is supplemented with an appendix (scattering of acoustic radiation) by R. H. Kraichnan of New York University. Part III is a detailed presentation of line-of-sight propaga­tion of acoustic and electromagnetic waves through a turbulent medium. And Part IV is a comparison of theory with experimental data in­cluding the author's own important work with terrestrial light sources, multiple observations, etc.

Because of its incorporation of the results of research not easily available in English, this book is invaluable to specialists in radiophysics and atmospheric acoustics and optics. Additional notations by the translator, 103 bibliographical references, and 45 figures are all helpful extras.

"An important contribution. . . . The translation is readable both from the language and the technical points of view," George Weiss, Science.

Unaltered, unabridged reprint of the first translated edition (1961). Notes. Prefaces. xvi + 285pp. 7Ys x 10. 81840 Paperbound $3.25

A DOVER EDITION DESIGNED FOR YEARS OF USE!

We have made every effort to make this the best book possible. Our paper is opaque, with minimal show-through; it will not discolor or become brittle with age. Pages are sewn in signatures, in the method traditionally used for the best books, and will not drop out, as often happens with paperbacks held together with glue. Books open flat for easy reference. The binding will not crack or split. This is a permanent book.