Physics of wave_propagation_in_a_turbulent_medium

Physics of Fluctuations of Waves
in Turbulent Medium

V. I. Tatarskii
Zel Technologies & NOAA/PSD
vtatarskii@hotmail.com

Abstract
Semi-qualitative description of basic phenomena
in wave propagation in a medium with
fluctuating parameters is considered. In
particular, the fluctuations of phase, spatial and
temporal phase differences, angle of arrival,
intensity, their correlations and spectra are
analyzed for waves in turbulent atmosphere.
The simple consideration, based on descriptive
geometric optics and its diffractive limitations,
allows obtaining all basic relations without
complicated mathematics.

Structure of fully developed turbulence
For very high Reynolds numbers Re = UL/ν, where U is a
velocity of flow, L is a scale of flow (for instance, diameter of
jet or pipe, an elevation above a plane boundary of flow), and ν
is the kinematical viscosity, the structure of flow is turbulent. This
means that the values of velocity, density, pressure, temperature, and
refractive index chaotically vary from one point to another.

Here, color schematically represents the value of departure of
parameter (say, refractive index) from its mean value. As a rule, the
larger is some inhomogeneity, the more intense it is.

Turbulent flow is usually described statistically. If n is the
refractive index, the most important for wave propagation in
turbulent medium characteristic is so called structure function

Drr
,   nr
  nr 

 2

This function is related to the correlation function

Brr  nrnr 
,   

by the formula

Dr   n 2 r   n 2 r   2nrnr 
,r     

Why the structure function is more useful than the correlation
function for turbulent medium description?

Let us consider two points A and B in a turbulent flow

A

B

If the scale of inhomogeneity is large in comparison with the
distance r between points A and B (blue inhomogeneity at the
previous slide), it contributes almost the same amount of n in both
points, i.e., does not contribute much to the difference n(A) – n(B).
If the scale of inhomogeneity is small in comparison with the
distance R between points A and B, such inhomogeneity is
relatively weak and contributes only a small amount to the
difference n(A) – n(B). Thus, the main contribution is caused by
inhomogeneities, which scale is of the order of distance R between
these points. Because of this, the value of the structure function D
is a measure of the intensity of inhomogeneities having scale R.

For very large Reynolds numbers the structure function D depends
only on the distance R and is independent of the mutual orientation
of points A and B. It was found by A.N. Kolmogorov and A.M.
Obukhov in 1941 that the function D has the following shape:
Struct.
Funct.

100

10

1

0.1
Distance
0.1 1 10 100 1000


C expk 2 l 2 
Sk  , Dx  2 1  coskxSkdk
k 2  4 L
2 2  56
0

There are 3 ranges in this figure: quadratic (left), 2/3 slope
(middle) and constant (right). These ranges are separated by the
scales l (inner scale, 5 mm in this example) and L (outer scale,
10 m in this example).

Rather often all essential transverse scales of electromagnetic
problem, such as radius of the first Fresnel zone, the base of optical
or microwave interferometer, diameter of optical or microwave
aperture are much larger than the inner scale of turbulence l and
much less than the outer scale L. In such situation it is possible to
neglect the effect of these scales and consider the idealized model,
for which l = 0 and L = Infinity. In this case, we may use the
simple model Dr   C 2 r 23
Struc.
Funct.

100
50

10
5

1
0.5
Distance
0.1 1 10 100 1000

The model of power structure function (power spectrum) corresponds to
considering turbulence as a fractal set.

Phase Fluctuations
We start considering of wave propagation in a turbulent medium
with fluctuations of phase. Fluctuations of Φ are important in such
practical problems as optical measuring of distances, transmitting
high-accurate time, accuracy of large-base interferometers.

The geometric optics approximation is an adequate tool for this
problem.

We consider the ray intersecting inhomogeneities of refractive
index. If deviations of refractive index from unity are small, it is
possible to neglect refraction effect and consider the ray as a
straight line, because curvature of ray is the second order effect.

After passing a single inhomogeneity having scale Δx and deviation
of refractive index from unity equal to Δn, the ray obtains an
additional phase shift equal to ΔΦ = k Δx Δn, where k = 2π/λ is a
wave number. After passing N inhomogeneities the total phase shift
is equal to
  k x1 n 1  k x2 n 2    k xN n N
The mean value of Φ is 0, because mean value of Δn is zero. For the
mean square of Φ we obtain:
N
 2   k2 xj  2
 n j 
2
 k 2  x l x j   n l  n j  
j1 lj

But the second sum vanishes because the fluctuations in different
volumes are uncorrelated. Thus,
N
 2   k2 xj  2
 n j 
2

j1

Which Δx we must choose? We already mentioned that the
larger is inhomogeneity, the stronger is fluctuation Δn. Thus,
the main contribution is provided by the largest possible
inhomogeneities, having the size of the order of outer scale L of
turbulence. Thus, we must choose Δx = L. All terms in the last
sum are equal and we may write

  n j 
2
 2   k 2 L 2  2 N, 2
But the number of inhomogeneities N is equal to the ratio of
total distance X to L. Thus,

 2   k 2  2 LX
We may present the last formula in a little different form, if we
substitute
 2  C 2 L 23
where C is the constant, entering in the 2/3 law for refractive
index structure function.

This formula has the form

 2   MC 2 k 2 L 53 X
We inserted in the last formula some unknown numerical
coefficient M, because all previous reasoning was performed only
with the accuracy of indefinite coefficient. The value of M can be
obtained only by more rigorous theory.

Phase differences fluctuations
Fluctuations of spatial or temporal phase differences are important
for many practical problems: measurements of angle of wave
arrival, resolution of images, interferometry.
Let us consider two parallel rays in a turbulent medium separated at
distance ρ.

The most
Effective
inhomogeneity
Large inhomogeneity
provides the same
phase shift to both
ρ
rays and does not The most
contribute to phase effective
difference inhomogeneity

Contribution to
phase difference
from these inhomogeneities
is small

Let us consider contribution to the phase difference provided by
inhomogeneities of different scales. If the size of some
inhomogeneity is small in comparison with the distance ρ, it may
contribute only to phase shift along a single ray. Thus, such
inhomogeneity contributes to a phase difference. The most
important contribution will be provided by the largest
inhomogeneities of such type, i.e., by inhomogeneities of size
about ρ. The inhomogeneities having the size much larger than ρ,
are stronger, but they provide the same contribution to the phase
shifts for both rays. Thus, contribution to the phase difference
from such inhomogeneities will be small. Therefore, the most
important contribution to the phase difference are due to the
inhomogeneities of the size about ρ.
According to the 2/3 law, Dr   C 2 r 23 the deviation of n from
unity is of the order of
n j ~  j Cr 13 , where  j   0, 2
j  1,  i  j   0 for i  j

Now we can calculate the total contribution of essential
inhomogeneities to the phase difference. We must choose r = ρ in
the last formula and obtain
   N
j1
kn j   N
j1
k j C13  Ck43  N
j1
j

The mean value of ΔΦ = 0, and for the mean square of ΔΦ we have
 2  Ck43  j1 jl  j  l   C 2 k 2 83 j1  2  C 2 k 2 83 N
2 N N N
j

The number of essential inhomogeneities N = X/ρ, where X is the
total distance from the source of wave to the receiver and ρ is the
longitudinal scale of essential inhomogeneities, which for isotropic
turbulence is equal to its transverse scale. Thus,
 2  KC 2 k 2 53 X
where some unknown numerical coefficient K was introduced.

Let us find such transverse distance ρ, for which the variance
 2  C 2 k 2 53 X  1. This value is called “radius of
coherence.” We find
0  1  65 1 35
C 2 k 2 X 35 C k 65 X
Coherence radius plays an important role in the problem of
resolution of telescopes and other optical devices. Only if 0  D
where D is diameter of aperture, there exists a possibility of coherent
summation of waves in the focal plane. If 0  D, different parts of
aperture send incoherent waves to the focal plane, and it is
impossible to achieve diffraction limit of the lens resolution.
The important parameter is a ratio of coherence radius to the radius
of the first Fresnel zone. For this ratio it is easy to find
0 1

X C 2 k 76 X 116  610

Angle of arrival fluctuations
The angle of arrival is related to the phase difference. If we measure
the phase difference by interferometer having the base ρ and the
Phase shift angle between the wave vector of
incident wave and the normal to
ρ the base of interferometer is γ, the
γ phase shift δ = kρ sin γ appears.
Wave front Thus, for small γ,

 
k
Thus, fluctuations of angle of arrival and phase differences are
determined by the formula
 2
 2    KC 2 13 X
k 2 2

Dependence of   on ρ is shown in the following plot in the
2

semi-logarithmic scale
2

KC2 X
2.5

2

1.5

1

0.5

0.1 0.2 0.5 1 2 5 10

Decreasing of  2  with increasing ρ is caused by the effect of
averaging fluctuations over the interferometer base (or aperture of
telescope).

Formula for   shows unlimited increasing of fluctuations of γ
2

while ρ tends to zero. This result is incorrect, because the 2/3 law
is valid only for r > l. If ρ becomes less than l, the 2/3 law
changes for
Dr   C 2 l 23 r2
l2
If we repeat the derivation of  2  for this case, we obtain the
following result for the case ρ < l:
 2   KC 2 l 13 X
We may suggest the interpolating formula, working for all ρ:

  
2 KC 2 X
  l
2 2  16

The plot of this function is presented in the following Figure:
2
l2 3
KC2 X

1

0.8

0.6

0.4

0.2

0.01 0.1 1 10 100 l

Variance of angle of arrival fluctuations in entire range of ρ.
The case ρ < l corresponds to the aperture less than inner
scale of turbulence.

Temporal correlation and spectrum of
angle fluctuations

Let us consider a temporal fluctuations of the angle of arrival.
The angle of arrival in some moment t is determined by the pair
of rays and inhomogeneities located at these rays. At the moment
t + τ all inhomogeneities will be shifted to a new position and
instead of them a new inhomogeneities will cover our two rays.

1
t
2

3
t+τ
4

At the moment t these inhomogeneities were located at the positions
3 and 4, opposite to a wind direction.

Let us consider the temporal correlation function of the angle of
arrival γ. We have
1  2 3  4  1   2  3   4 
1  , 2  ,  1  2  
k k k 2 2

We can use the algebraic identity
a  bc  d  1 a  d 2  b  c 2  a  c 2  b  d 2
2
and present the correlation function in the form
 1   4  2   2   3  2   1   3  2   2   4  2
 1  2  
2k 2 2

But we already determined the variance of the phase differences for
an arbitrary separation between two rays:
KC 2 Xk 2 2
  2
 k 2 2  2  
2  l 2  16

For the rays 1 and 4 we must instead of ρ substitute ρ +Vτ, where V
is the transverse component of wind. For the rays 2 and 3 we must
substitute instead of ρ the value Vτ – ρ, and for pairs 1, 3 and 2, 4
we substitute instead of ρ the value Vτ. The formula for
autocorrelation function of γ takes the form

2 V   2 V   2 V 2
 1  2   B    KC 2X 16
 16
2 16
2 V    l 2
2
V    l 2
2
V 2  l 2

This function for KC^2X=1, V=500 cm/s, ρ = 5 cm, and l = 0.5 cm is
shown in the following Figure:

B

1

0.8

0.6

0.4

0.2

,s
0.1 0.2 0.3 0.4 0.5

The auto-correlation function of the angle of arrival fluctuations.

The spectrum of γ is determined by the formula

Q   cosB  d
0
Q

1  23

0.001

6
1. 10

9
1. 10

1
,s
0.001 0.1 10 1000

Zeroes in the spectrum are caused by the presence of difference in
the definition of γ (for the case of interferometer). In the real spectra
these zeroes will be filled in because of wind fluctuations. The
straight line in the spectrum corresponds to  23 dependence.

Intensity fluctuations
In geometric optics, the product of intensity by the cross-section of
beam (ray tube) is constant. Intensity is determined by the cross-
section.

The small cross-section
The large intensity

The large cross-section
The small intensity

The shape of a beam is determined by distribution of refractive
index in space.

Inhomogeneities of refractive index play role of random lenses
focusing or defocusing light.

Initial Final intensity
intensity
Less intensity

Negative lens Positive lens

It is known that for a spherical lens the focal distance F is equal to
ratio of the curvature radius ® to (n-1), where n is the refractive
index of lens material. For turbulent inhomogeneities the curvature
radius is of the order of scale R of inhomogeneity. Thus,

F R
nR   1

It follows from 2/3 law that nR   1  C R 13 Thus,
F R 23
C
Let us calculate the intensity change after passing a single
inhomogeneity.
δR
α
α
R
F

X

R
We have   F  CR
13 and l  X  CR 13 X

Thus, the change of the intensity is determined by the relation

IR 2  I  IR  R 2 , or I  I R
R
Using the obtained formula for δR we obtain

I  CX
I R 23
It is clear from this formula that the smaller is the scale of
inhomogeneity, the larger is the changing of intensity. Thus, the
most important contribution to the change of intensity is provided
by the smallest possible inhomogeneities of the order of inner
scale l of turbulence. Thus, we must set R = l in the last formula.
We also introduce a random number ξ, which accounts that the
sign of fluctuation of refractive index is random. Thus, for the
contribution of a single j-th inhomogeneity to intensity
fluctuation we obtain I j CX
 j
I l 23

The total change of intensity is determined by the sum
N
I  CX
I l 23
 j
j1

For the mean square of relative fluctuations of intensity we obtain
N N
I
  j  m   l 43
2
 C2X2 C2X2 N
I l 43 j1 m1

The last step is to substitute N = X/l. The result is
I 2
C2X3
 73
I l
This formula was obtained by geometric optics approach and is
valid if the geometric optics is true for this problem.

The effect of diffraction at inhomogeneities leads to spreading of
all rays. At the distance X from the inhomogeneity the sharp
boundary of ray tube spreads to the size X

X

l X
X

If X  l it is possible to neglect a diffraction and use the
geometric optics result. But if X  l, diffraction compensates
the focusing effect, and the inhomogeneity of the scale l does not
produce change of intensity. Thus, the minimal scale of
inhomogeneities, which still may cause the intensity fluctuations, is
l  X . Thus, in this case we must replace the inner scale of
turbulence l in the formula for intensity fluctuations for X .

The resulting formula has the form
I 2
 C2X3  C 2 X 116  C 2 k 76 X 116
I X
73
 76
Here, k = 2π/λ is the wave number.
It is possible to write a simple interpolation formula which
provides transition from geometric optics case to diffraction case:
I 2
 C2X3
I
l 143  X 73

More rigorous theory provides some numerical coefficients in
above formulae and another type of transition from geometric
optics range to diffraction range of distances.

2
I
10

1

0.1

0.01

0.001 X, m
10 20 50 100 200 500 1000

Dependence of intensity fluctuations on distance for λ = 0.63 μ,
l = 5 mm, C2  10 16 m23 . Transition from geometric optics regime
to diffraction regime takes place at the distance X = 40 m.

Saturated Intensity Fluctuations
Comparison of the experimental values of II 2   2 with
I
the theoretically predicted value  2  C 2 k 76 X 116 shows that there
I0
is a good agreement between them if  2  1. But in the region
I0
where  I0  1 the experimental value of I
2  2 does not increase
while  2 increases and remains approximately constant.
I0

The region  2  1 is called the region of strong or saturated
I0
fluctuations. It starts at the distance X 0  C 1211 k 711 where the
theoretically predicted value of  I0 becomes unity.
2

Let us consider the ratio of coherence radius to the radius of the
first Fresnel zone. This value was found above:
0 1

X C 2 k 76 X 116  610
It is clear from this formula that in the region of weak fluctuations,
i.e.,  2  1 , the radius of coherence is large in comparison with
I0
the radius of the firs Fresnel zone, while it is small in comparison
with X in the region of strong fluctuations. This is clear from
the following plot.

The radius of coherence 0 (red) decreases and the radius of the
first Fresnel zone X (blue) increases while the distance X
increases. At some distance X 0  C 1211 k 711 the coherence
radius becomes less than the radius of the first Fresnel zone.

0, X , cm
2
1.75
1.5
1.25
1
0.75
0.5 Weak fluctuations Saturated (strong) fluctuations

0.25
X, m
500 1000 1500 2000

Previously we found that the most important for the intensity
changes inhomogeneities have size of the order of X . Such
inhomogeneity caused focusing or defocusing of a beam. This
focusing (defocusing) is possible because the inhomogeneity acts
similarly to a lens, which coherently summarizes all wave field at
its surface. Such situation is possible only if the field, incident at a
lens, is coherent, i.e., if 0  X . But in the region 0  X ,
where the coherence radius is small in comparison with the scale
of lens, different parts of lens transmit (radiate) incoherent waves.
Because of this, such lens is unable to focus / defocus radiation.


Wave front

It is convenient to call the wave field for which 0  X as
degenerated wave field. The highly degenerated field ( 0  X )
can not be focused and for such field it is impossible to obtain a
sharp image in the focal plane.
If we return to the intensity fluctuations, we may conclude that
only the inhomogeneities located in the initial part X  X 0 of the
propagation path may produce intensity fluctuations. All
inhomogeneities, which are located in the region of strong
fluctuations, can not focus or defocus the wave and because of this
they have no (or have very little) influence on  I . But these
2

inhomogeneities continue to contribute to decreasing of radius of
coherence.
This qualitative picture explains the phenomenon of strong
fluctuations, but the corresponding rigorous theory is rather
complicated and is based either on the theory of random Markov
fields or diagram technique.

For more detailed information concerning the discussed problem, it
is useful to refer to the following publications (the simplest are listed
prior to more complicated).

1. Tatarskii V.I. Review of Scintillation Phenomena. In Wave propagation in
Random Media (Scintillation). Edited by V.I. Tatarskii, A. Ishimaru, V.U.
Zavorotny. Copublished by SPIE Press and IOP, 1993.
2. S.M. Rytov, Yu.A. Kravtsov, V.I. Tatarskii. Principles of Statistical
Radiophysics. vol. 4. Wave Propagation Through Random Media. Springer-
Verlag, 1989.
3. V.I. Tatarskii. The effects of the turbulent atmosphere on wave propagation.
Translated from the Russian by the Israel Program for Scientific Translations,
Jerusalem, 1971. Available from the U.S. Dept. of Comm., Nat. Tech. Inf. Serv.,
Springfield, VA, 22151
4. V.I. Tatarskii and V.U. Zavorotniy. Strong Fluctuations in Light Propagation in a
Randomly Inhomogeneous Medium. In Progress in Optics, vol. XVIII, edited by
E. Wolf, North-Holland Publishing Company, Amsterdam - New York - Oxford,
1980.

This presentation may be downloaded from the website

http://home.comcast.net/~v.tatarskii/vit.htm

Physics of wave_propagation_in_a_turbulent_medium

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