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Generalized modified atmospheric spectral model for
optical wave propagating through
non-Kolmogorov turbulence
Bindang Xue,1,* Linyan Cui,1Wenfang Xue,2Xiangzhi Bai,1and Fugen Zhou1
1School of Astronautics, Beihang University, Beijing 100191, China
2Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China
*Corresponding author: xuebd@buaa.edu.cn
Received September 14, 2010; revised March 17, 2011; accepted March 18, 2011;
posted March 18, 2011 (Doc. ID 134399); published April 29, 2011
A new generalized modified atmospheric spectral model is derived theoretically for wave propagating through
non-Kolmogorov turbulence, which has been reported recently by increasing experimental evidence and theore-
tical investigation. The generalized, modified atmospheric spectrum considers finite turbulence inner and outer
scales and has a spectral power law value in the range of 3 to 5 instead of the standard power law value of 11=3.
When the inner scale and outer scale are set to zero and infinity, respectively, this spectral model is reduced to the
classical non-Kolmogorov spectrum. © 2011 Optical Society of America
OCIS codes: 010.1290, 010.1300, 010.1330.
1. INTRODUCTION
Turbulence remains an important unsolved problem in phys-
ics and engineering, such as in pipe flow, aeronautics, and
meteorology [1]. Atmospheric turbulence is one of the most
important examples of turbulence, which has a significance
degrading impact on the quality of imaging and laser commu-
nication systems [2–4]. Traditionally, the performances of
imaging and laser communication systems are estimated with
the assumption that the turbulence is of the Kolmogorov type
and the Kolmogorov spectral model is applied mainly for
mathematical simplicity [5]. Nonetheless, the Kolmogorov
spectral model is theoretically valid only in the inertial sub-
range. When it is used to estimate the performance of imaging
and laser communication systems, it is ordinarily extended to
all ranges by assuming the inner scale size is zero and the out-
er scale is infinity. Several mathematically convenient turbu-
lence spectra with specific inner and outer scale have been
proposed, such as the Tatarskii, von Karman, and exponential
[5] spectra. Strictly speaking, these spectral models have the
correct behaviors only in the inertial range and are commonly
used for theoretical studies on optical wave propagation. The
Hill spectrum [6,7], however, can characterize the high fre-
quency enhancement property in the spectrum measured
by the experimental data [1], but has an awkward form for
theoretical studies. Compared with other turbulence spectral
models, the modified atmospheric spectrum of Andrews [8]is
the only analytical spectrum featuring a high wave number
“bump”just prior to the onset of the dissipation range that
has been observed in experimental data and also has a
convenient analytical form for theoretical studies. However,
it was developed for Kolmogorov turbulence with a specific
power law value of 11=3, and cannot be directly applied in
non-Kolmogorov turbulence cases.
In the past decade, both the experimental data [9–12]
and theoretical investigations [13–15] have exhibited
non-Kolmogorov turbulence spectra in certain portions of
the atmosphere. Many applications [16–18] have proposed a
non-Kolmogorov spectrum with a variable power law value
in the range of 3 to 5 instead of the standard power law value
of 11=3for Kolmogorov turbulence. In addition, a variable
amplitude factor has been proposed instead of a constant
value of 0.033. This modification still has the same problem
as the Kolmogorov spectrum. To handle non-Kolmogorov at-
mospheric turbulence, some theoretical spectral models were
developed, such as the generalized von Karman spectrum [19]
and the generalized exponential spectrum [20], which consid-
er finite turbulence inner and outer scales and have general
spectral power law values in the range of 3 to 5 instead of
the standard power law value of 11=3. The turbulence spectra
considered in this study are listed in Table 1, which gives a
clear description of these spectra.
In this study, the modified atmospheric spectral model
[8] is generalized to apply in non-Kolmogorov atmospheric
turbulence. The generalized modified atmospheric spectrum
considers finite turbulence inner and outer scales and has a
general spectral power law value in the range of 3 to 5 instead
of the standard power law value of 11=3.
2. MODIFIED ATMOSPHERIC SPECTRUM
Modified atmospheric spectrum is a turbulence spectral mod-
el that considers the influence of finite inner and outer scales,
and is given by [8]
ΦnðκÞ¼0:033C2
nκ−11=31−exp−κ2
κ2
01þa1·κ
κl−b1
·κ
κl7=6exp−κ2
κ2
lð0≤k<∞Þ;ð1Þ
where C2
nrepresents the refractive-index structure parameter
for Kolmogorov turbulence and has the unit of m−2=3,κis the
912 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 Xue et al.
1084-7529/11/050912-05$15.00/0 © 2011 Optical Society of America
spatial wave number, a1¼1:802,b1¼0:254,κl¼3:3=l0,
κ0¼C0=L0,l0, and L0are the turbulence inner and outer
scale, C0is chosen differently depending on the application,
and, because the outer scale itself is not well defined, it is
difficult to assign any particular constant C0with outer scale
parameter κ0. In this study, we set C0¼4πjust as in [8].
In order to include both inner scale and outer scale’s ef-
fects, the generalized spectra recently adopted have the form
as [17,19,21–23]
Φnðκ;αÞ¼AðαÞ·^
C2
n·Fðk; l0;L
0;αÞð0≤κ<∞;3<α<5Þ;
ð2Þ
where AðαÞis a constant that maintains consistency between
the refractive-index structure function and its power spec-
trum and ^
C2
nis the generalized refractive-index structure con-
stant (with units of m3−α, when α¼11=3, with units of m−2=3).
Physically, it is a measure of the strength of the fluctuations in
the refractive index and the behavior of ^
C2
nat a point along the
propagation path can be deduced from the temperature struc-
ture function obtained from point measurements of the mean-
square temperature difference of two fine wire thermometers.
Fðk; l0;L
0;αÞis the function that includes the influence of
finite inner and/or outer scale, and has different forms for
different spectra [17,19,21–23].
To generalize the modified atmospheric spectral model for
non-Kolmogorov atmospheric turbulence, κ−11=3in Eq. (1)
should be replaced by κ−αand, by using Eq. (2), the general-
ized modified atmospheric spectrum becomes
Φnðκ;α;l
0;L
0Þ¼^
AðαÞ·^
C2
n·Fðk; l0;L
0;αÞ
¼^
AðαÞ·^
C2
n·κ−α·fðk; l0;L
0;αÞ
ð0≤κ<∞;3<α<5Þ;ð3Þ
fðκ;l
0;L
0;αÞ¼1−exp−κ2
κ2
01þa1·κ
κl−b1
·κ
κl7=6exp−κ2
κ2
l;ð4Þ
where ^
AðαÞhas the same meaning as AðαÞ,κl¼cðαÞ=l0,
κ0¼C0=L0, and cðαÞis the scaling constant.
In the next section, the expression forms of ^
AðαÞand cðαÞ
will be derived.
3. REFRACTIVE-INDEX STRUCTURE
FUNCTION FOR GENERALIZED MODIFIED
ATMOSPHERIC SPECTRUM MODEL
The refractive-index structure function DnðR; αÞdescribes the
behavior of the correlations of turbulence refractive-index
field fluctuations between two given points separated by a dis-
tance R. For three-dimensional Kolmogorov turbulence, the
relationship between DnðRÞand ΦnðκÞcan be described as [5]
DnðRÞ¼8πZ∞
0
κ2·ΦnðkÞ·1−sin κR
κRdκ:ð5Þ
For non-Kolmogorov turbulence, the relationship between
DnðRÞand ΦnðκÞis
DnðR; αÞ¼8πZ∞
0
κ2·Φnðk; αÞ·1−sin κR
κRdκ:ð6Þ
Substituting Eq. (3) into Eq. (6), and setting L0to infinity [this
will be explained after Eq. (13)], gives following expression:
DnðR; αÞ¼8πZ∞
0
κ2−α·^
AðαÞ·^
C2
n
·1þa1·κ
κl−b1·κ
κl7=6exp−κ2
κ2
l
·1−sin κR
κRdκð7Þ
By expanding 1−sin κR
κRwithin a Maclaurin series [24],
1−sin κR
κR¼X
∞
n¼1
ð−1Þn−1
ð2nþ1Þ!κ2nR2n;ð8Þ
and inserting it into Eq. (7), then interchanging the order of
series summation and integration, Eq. (7) becomes
Dn1ðR; αÞ¼8π·^
AðαÞ·^
C2
n·X
∞
n¼1
ð−1Þn−1
ð2nþ1Þ!R2nZ∞
0
κ2−αþ2n
×1þa1·κ
κl−b1·κ
κl7=6exp−κ2
κ2
ldκ:ð9Þ
Using the gamma function ΓðxÞand hypergeometric function
1F1ða;b;zÞ[24] yields
Table 1. List of Some Atmospheric Turbulence
Spectraa
Type of Turbulence Styles of
Spectra
Spectra Inner
Scale
Outer
Scale
Kolmogorov turbulence (with
power law value of 11=3)
Most
simple
spectrum
Kolmogorov
spectrum
××
Analytical
spectra
Tatarskii
spectrum
p×
von Karman
spectrum
pp
exponential
spectrum
pp
Physical
spectra
Hill spectrum pp
modified
atmospheric
spectrum
pp
Non-Kolmogorov turbulence
(with power law value in the
range of 3 to 5)
Most
simple
spectrum
non-
Kolmogorov
spectrum
××
Analytical
spectra
generalized
von Karman
spectrum
pp
generalized
exponential
spectrum
pp
a“p”and “×”represent with or without considering the influence of a finite
inner/outer scale, respectively.
Xue et al. Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. A 913
ΓðxÞ¼Z∞
0
κx−1·e−κdκðκ>0;x> 0Þ;
1F1ða;b;zÞ¼X
∞
n¼0
ðaÞn·zn
ðbÞn·n!;ð10Þ
where ðaÞnis the Pochhammer symbol and has the form
ðaÞn¼ΓðaþnÞ
ΓðaÞ¼aðaþ1Þ…ðaþn−1Þ:ð11Þ
As a result, the refractive-index structure function in Eq. (9)
becomes
DnðR; αÞ¼4π^
AðαÞ^
C2
nκ3−α
lΓ−α
2þ3
2
×1−1F1−α
2þ3
2;3
2;−R2κ2
l
4þa1·Γ−α
2þ2
×1−1F1−α
2þ2;3
2;−R2κ2
l
4−b1·Γ−α
2þ25
12
×1−1F1−α
2þ25
12 ;3
2;−R2κ2
l
4:ð12Þ
For statistically homogeneous, isotropic, non-Kolmogorov
atmospheric turbulence, the related refractive-index structure
function is given by [8,19]
DnðR; αÞ¼^
C2
nlα−5
0R2;0≤R≪l0
^
C2
nRα−3;l
0≪R≪L0
:ð13Þ
It should be mentioned that, because the random field of
index refractive fluctuation is nonisotropic for scale sizes
larger than the outer scale L0, no general description of the
refractive-index structure function can be predicted for
R>L
0. Consequently, in the derivation of refractive-index
structure functions for a generalized modified atmospheric
spectrum, L0is set to infinity for calculation purposes [8,19].
By using Eqs. (12) and (13), the unknown ^
AðαÞand cðαÞin
Eq. (3) will be derived next.
A. Expression Derivation of ^
Aα
When l0≪R≪L0, then R2κ2
l
4¼R2c2ðαÞ
4l2
0
≫1,1F1ða;b;−xÞin
Eq. (12) can be expanded approximately for big arguments
and is given by [24]
1F1ða;b;−xÞ≈ΓðbÞ
Γðb−aÞx−aðx≫1Þ:ð14Þ
Substituting Eq. (14) into Eq. (12), DnðR; αÞ[see Eq. (12)]
becomes
DnðR; αÞ≈−4π^
AðαÞ^
C2
nΓ−α
2þ3
2
×Γð3=2Þ
Γðα=2Þ1
2α−3
ðRÞα−3ðl0≪R≪L0Þ:ð15Þ
By using Eqs. (15) and (13), and considering the properties of
the gamma function [24],
Γðαþ1Þ¼αΓðαÞ;Γð1−αÞΓðαÞ¼ π
sinðπαÞ;
ΓðαÞΓðαþ1=2Þ¼21−2αffiffiffi
π
pΓð2αÞ:ð16Þ
As a result, ^
AðαÞcan be expressed with the form
^
AðαÞ¼Γðα−1Þ
4π2sinðα−3Þπ
2;ð17Þ
which has the same form in non-Kolmogorov spectral
models [16,17].
B. Expression Derivation of cα
When 0≤R≪l0, then R2κ2
l
4¼R2c2ðαÞ
4l2
0
≪1,1F1ða;b;xÞin Eq. (12)
can be expanded for small arguments and is given by [24]
1F1ða;b;xÞ≈X
1
n¼0
ðaÞn·zn
ðbÞn·n!¼1þa
bxðx≪1Þ:ð18Þ
By substituting Eq. (18) into Eq. (12), DnðR; αÞbecomes
DnðR; αÞ≈π^
AðαÞ^
C2
nκ5−α
lR2·Γ−α
2þ3
23−α
3þa1
·Γ−α
2þ24−α
3−b1
·Γ−α
2þ25
1225 −6α
18 ð0≤R≪l0Þ:ð19Þ
Using Eqs. (19) and (13), the expression of cðαÞcan be
derived:
cðαÞ¼π^
AðαÞΓ−α
2þ3
23−α
3þa1
·Γ−α
2þ24−α
3−b1
·Γ−α
2þ25
1225 −6α
18 1
α−5:ð20Þ
4. GENERALIZED MODIFIED ATMOSPHERIC
SPECTRUM MODEL
By substituting Eqs. (17) and (20) into Eq. (3), we can obtain
the expression of the generalized modified atmospheric
spectral model. Figures 1(a) and 1(b) show ^
AðαÞand cðαÞ
as functions of α. When α¼11=3,^
Að11=3Þ¼0:033 and
cð11=3Þ¼3:25ð≈3:3Þ, Eq. (3) reduces to the modified atmo-
spheric spectrum [see Eq. (1)].
By setting the inner scale to zero and the outer scale to
infinity, the generalized atmospheric spectrum becomes
Φ0
nðκ;α;l
0;L
0Þ¼^
AðαÞ·^
C2
n·κ−αð0≤κ<∞;3<α<5Þ;
ð21Þ
where ^
AðαÞhas the same form as in the non-Kolmogorov spec-
tral model [21–23]. The generalized atmospheric spectrum
reduces to the non-Kolmogorov spectrum for the particular
case of zero inner scale and infinite outer scale.
914 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 Xue et al.
Comparing the non-Kolmogorov spectrum [21–23]to
Eq. (3) with zero inner and infinite outer scales gives
Φnðκ;α;l
0;L
0Þ
Φ0
nðκ;α;l
0;L
0Þ¼fðκ;l
0;L
0;αÞ;ð22Þ
where fðκ;l
0;L
0;αÞhas the same form as Eq. (4). Fig. 2shows
that the high frequency enhancement characteristics in the
generalized modified atmospheric spectra for power law
values of 10=3,11=3, and 3.9 are similar to the Kolmogorov
turbulence case (α¼11=3)[8].
Figure 2shows that the generalized modified atmospheric
spectrum is very close to the non-Kolmogorov spectrum in
inertial range for the inertial subrange (1=L0≪κ≪1=l0) with
an obvious high wave number “bump”just prior to the
dissipative range (κ≫1=l0) and then decreases markedly.
As shown in Fig. 2, a nonzero inner scale reduces values
of the spectrum at high wave numbers (κ>cðαÞ=l0) over
that predicted by the non-Kolmogorov spectrum, that is,
fðκ;l
0;L
0;αÞ<1. At low wave numbers (κ<C
0=L0), a similar
reduction in values of the spectrum is caused by the presence
of a finite outer scale.
The various power laws produce different effects on the
form of spectra. As αincreases, cðαÞdecreases, as seen in
Fig. 1(b),soκl¼cðαÞ=l0also decreases. Because the “bump”
position depends on the value of κl, it shifts to a smaller spatial
wave number position.
5. DISCUSSION AND CONCLUSIONS
In the modified atmospheric spectral model [8], the ½1þa1·
ðκ
κlÞ−b1·ðκ
κlÞ7=6component of Eq. (1) characterizes the high
Fig. 1. AðαÞand cðαÞas functions of α. (a) AðαÞ; (b) cðαÞ.
Fig. 2. (Color online) Scaled generalized modified atmospheric spectrum as a function of spatial wave number with a logarithmic scale (L0¼2m,
l0¼1mm).
Xue et al. Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. A 915
frequency enhancement in Kolmogorov turbulence, and the
coefficients a1,b1, and 7=6present in the modified spectrum
were chosen by fitting a curve to the Hill numerical spectrum
[7], which has the form
ΦnðκÞ¼0:033C2
nκ−11=3fexpð−1:29κ2l2
0Þþ1:45
× exp½−0:97ðln κl0−0:452Þ2g ð0≤k<∞Þ;ð23Þ
where the part of f·gthat describes the “bump”character
of the spectrum has a complex form. In fact, the Hill spectrum
is the solution to a second-order linear homogeneous differ-
ential equation:
d
dκκ14=3½ð13:9κηÞ3:8þ1−0:175 d
dκΦnðκÞ¼14:1κ4η4=3ΦnðκÞ:
ð24Þ
It is derived from the temperature differential equation [6]
d
dκHðκÞd
dκΦTðκÞ¼2Dκ4ΦTðκÞ;ð25Þ
where HðκÞ¼ð3=11Þβ−1ε1=3κ14=3½ðκ=κþÞ2bþ1−1=ð3bÞ.κþ¼
0:072=ηand b¼1:9.κþparameterizes the position and b
parameterizes the width of the transition between the inertial
convective range and the viscous-convective range. The
choice of these two parameters arises from comparing the
model with the experimental data for Kolmogorov turbulence
case.
In this study, when we analyze the spectral enhancement
of the generalized modified spectrum in non-Kolmogorov
turbulence, we assume κþand bare unchanged. Although
Kolmogorov turbulence can be regarded as a special case
of non-Kolmogorov turbulence, physically, κþand bshould
be reevaluated, and these two parameters should take differ-
ent values for different α(3<α<5). Accordingly, the coeffi-
cients a1,b1, and 7=6in the generalized modified atmospheric
spectrum model should depend on experimental results.
However, with the limitations imposed by experimental
measurements, it is challenging to acquire sufficient data in
the troposphere and stratosphere to refine these parameters.
Consequently, the theoretical results presented in this study
will need to be justified by future experimental data.
ACKNOWLEDGMENTS
This work is partly supported by the United Fund Foundation
of the Civil Aviation, the National Natural Science Foundation
of China (NSFC, No. 60832011), and the Aeronautical Science
Foundation of China (20080151009).
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