Optical waves and laser beams in the irregular atmosphere

355 Pages·2018·12.912 MB·English

by  Blaunstein, Nathan, Kopeika, Norman S

324 ◾ Index
Atmospheric turbulence (Continued) Bifurcation
energy cascade theory, 13–16 at breakdown of main vortex, 136
functions and parameters of, 16–19 of stability alternation, 125
Hill’s model, 22 Big alt-azimuthal telescope (BAT), 76–77, 161;
Kolmogorov spectrum, 19, 20 see also Coherent structures
modified vs. von Kármán normalized air motion in dome space, 161, 163
spectra, 23 dome with side heated by sun, 162
non-Kolmogorov turbulence, 23 temporal frequency spectra of temperature
optical turbulence, 17 fluctuations, 163
passive scalar, 20 turbulence formation, 164
refractive index, 17, 18 Boussinesq approximation, 161
spectral characteristics of, 19–23 BPA, see Backpropagation algorithm
spectrum, 13 Brewster angle, 36
Tatarskii spectrum, 22 Broadband laser radiation sources, 266
2/3 power law, 16
von Kármán spectrum, 21 C
Wiener–Khinchin theorem, 19, 20
Cauchy problem, 38, 206
B Cell, 141
Chaos in typical dynamic systems, 74,
Backpropagation algorithm (BPA), 279, 156, 166
280–281 Chaotization period, 77
Backscattering coefficients, 271 Characteristic of measurement path, 93–95
Backscattering efficiency factor, 271 Characteristics of radiation, effective, 220
Ballistic photons, 250 Chessboard-like structure, 128, 141
BAT, see Big alt-azimuthal telescope Closed local volumes, 73, 120
Beam Cloud cover models, 12; see also Hydrometeors
centroid, 199 Clouds, 11–12; see also Hydrometeors
of different profile, 199, 211 Cluster expansion of centered random function
Gaussian, 213 and Fourier transform, 41–42
ring, 224 Coherent structures, 74; see also Anisotropic
super-Gaussian, 213 boundary layer; Mountain boundary
Beam divergence vs. nonlinearity length, 203 layer; Semiempirical theory of
Beam parameters, effective, 220 turbulence; Turbulence
Beam radius, effective, 225–229 attenuation effect of refractive
Beam self-action, 195; see also Nonlinear fluctuations, 153
propagation of laser radiation attenuation of amplitude and phase
beam divergence vs. nonlinearity fluctuations, 150–154
length, 203 bifurcation of stability alternation, 125
feature of, 196 cell, 141
influence of atmosphere on, 196 coherent turbulence, 143–146
longitudinal scale of nonlinearity, 201 extension of coherent structure concept,
method of characteristics, 197–198 139–143
radiation brightness transfer Gerris Flow Solver, 160
equation, 197 incipient convective turbulence, 123–125
ray approximation, 197 incipient turbulence, 121, 124
relative effective beam intensity, 201 incoherent turbulence, 143–146
relative effective beam radius vs. jitter of astronomical images, 152, 153
dimensionless distance, 199 Kolmogorov turbulence, 143, 144
effect of thermal blooming, 195–196 lifetime of, 75
Benard cell, 129 lifetimes of turbulence, 154
formation and breakdown, 72 main results, 166–167
168 ◾ Optical Waves and Laser Beams in the Irregular Atmosphere
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176 ◾ Optical Waves and Laser Beams in the Irregular Atmosphere
Volume Turbulence (pcDVT), OSA, USA, Seattle, Washington, 2014, Paper PM4E.2,
ISBN: 978-1-55752-308-2, Control Number: 2041404.
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126 ◾ Optical Waves and Laser Beams in the Irregular Atmosphere
As was shown in Reference 4, the temporal frequency spectra of temperature
fluctuations W (f ) in open air are satisfactorily described by the von Kármán
T
model. Spectra of Kolmogorov developed turbulence have the extended inertial
range, in which W ∼ f−5/3 (see also Chapter 1). In this range, the energy is trans-
T
ferred from vortices with scales to smaller ones.
Figure 3.24 depicts the smoothed temporal frequency spectra of temperature
fluctuations W in a closed room and in open air.
T
As follows from Figure 3.24, in comparison with the open air, the spectra in the
closed room decrease much faster in the inertial range. In addition, in this range,
there are only some short frequency sections (located stepwise), in which the tur-
bulence can be considered as Kolmogorov (W ∼ f−5/3). These sections are observed
T
between steps of the spectral function at frequencies corresponding to local maxima
of the correlation function of fluctuations (or minima of the structure function). If
we smooth the steps, then the experimental spectra of undeveloped turbulence have
some characteristic parts with the fast power-law decrease. Thus, if W ∼ const in
T
the extended energy range, then with the increase of the frequency (in the inertial
range) first W ∼ f−8/3, and then W ∼ f−12/3. Consequently, in incipient turbu-
T T
lence, the energy transfer from large vortices to smaller ones is insignificant, that is,
vortices are slightly blurred. With the further increase of the frequency, in the vis-
cous range, where the spectral density is close to the noise level, the decrease of the
spectrum becomes more slowly, W ∼ f−2/3. The analogous behavior of the spectra
T
is also observed at other measurement points in the pavilion. At some points, the
stepwise form of the spectrum is even more pronounced.
To construct the theoretical model Φ (æ) of the temperature spectrum of
T
incipient turbulence, we can use the von Kármán model with the decrease in the
101 3
2
1
10–1
z)
H
2/
g
(deT10–3 12 –– OClpoesned a irroom
W
3 – ∼ f –5/3
4 – W (f), ν = 5/6
10–5 T 1
5 – W (f), ν = 1/3 5
T 2
4
0.01 0.1 1 10 100
f (Hz)
Figure 3.24 Smoothed temporal frequency spectra of temperature fluctuations
W in a closed room and in open air.
T
130 ◾ Optical Waves and Laser Beams in the Irregular Atmosphere
1.0 101
0.8 2/Hz) 100 1 2345678910
9 g
0.6 10 7 ) (de10–1
0.4 6 5 W(f10–2
τb() 4 10–3
0.2 8 0.01 0.1 1
3
f (Hz)
0.0
2
1
–0.2
–0.4
0 20 40 60 80 100 120
τ (s)
Figure 3.26 Correlation coefficient b and nonsmoothed frequency spectrum
T
W (top right inset) in the pavilion. Digits are numbers of maxima of b and W
T T T
corresponding to each other.
The selected spectrum W is calculated without smoothing with a rect-
T
angular spectral window. As known [164,165], this window acts as a slit
with the width ∼2/T (T = 120 s). This resolution is quite sufficient to find
maxima of the spectrum W being in one-to-one correspondence with the
T
maxima of b. The arguments of b and W maxima are connected by the
T T T
relation τ f = 1, k = 1, 2,… (the values of τ, f usually determine char-
k k 1 1
acteristic scales of decrease of the functions b and W ). From the relation
T T
2πR = υ τ = υ/f , we can easily reconstruct the diameter 2R of the main
1 1 1 1
energy-carrying vortex. In the pavilion, it is equal to 294.4 cm (υ = 9 cm/s,
f = 0.00973 Hz) at point 5.
1
From comparison of spectra W shown in Figures 3.24 and 3.26, we can
T
see that at the standard smoothing of the spectrum by a wide spectral win-
dow (variance of the smoothed spectrum in Figure 3.24 is 1% of the vari-
ance of the selected spectrum in Figure 3.26) actual maxima of the spectra
disappear. Therefore, to calculate the frequencies of spectral maxima (har-
monics), we should use data of Figure 3.26. However, a rectangular spec-
tral window has large side lobes leading to oscillations, especially at high
frequencies. We can dispose of these lobes by applying any of widely used
nonrectangular windows (the difference between them is small), for exam-
ple, the Welch window [166]. In comparison with the rectangular window,
this window decreases the variance approximately two times, increasing the
width of the frequency band also approximately two times. The increase of
the band is acceptable, because it turns out to be smaller than the average
width of the spectral maxima, which can be seen from the data for W in
T
Figure 3.26.
132 ◾ Optical Waves and Laser Beams in the Irregular Atmosphere
In this case, the oscillations themselves are usually called coherent (inphase).
It should be noted that the function f /(nf ) shown in Figure 3.27 is resistant
n 1
to variations of the level of the threshold filter. Variations of this function
appear to be i nsignificant at variations of the filter level. From the frequen-
cies f , using the relation 2πR = υ/f , n = 1, 2,…, following from the law
n n n
of c onservation of momentum of a liquid particle at remote edges of the
breaking down vortex and its daughter vortices (Heisenberg equality [1,2]),
we can calculate the corresponding vortex radii R (in cm):
n
R =147.21,24.52,18.39,13.38,11.32,8.66,7.36,2.23,2.16,1.91,,1.63,1.58,
n
1.35,1.30,1.26,1.23,1.16,1.13,1.11,1.08,1.02,0..98,0.97,0.94,0.91,0.90,
0.88,0.84,0.82,0.80,…

The vortex diameters are shown in Figure 3.28.
From the comparison of the data in Figure 3.28 and Table 3.1 (point 5,
ν = 5/6), we can see that the second frequency 6f corresponds approximately
1
to the outer scale LK (2R = 294 cm, 2R = 49 cm), while the frequencies
0 1 2
127f and 130f (at which the inertial range terminates) correspond to the
1 1
inner scale l (2R = 2.3 cm).
0 17
In the viscous range and in a part of the inertial range, vortices being
products of breakdown of large vortices (their frequencies are multiple to the
lower frequencies) are observed. For example,
f /f =11,15,20,24,25,…(n=8,11,16,21,22,…);
n 2
f /f =15,17,18,199,36,…(n=16,2 0, 21, 23 , 52 ,…).
n 3
800
R = R /β
n n/2
m) 100 fn L0
(cn Rn ~ n–2.22, β = δ
R
x 2 10 Rn ~ n–1, β = 2
e
rt
o
v
of 1 l
r 0
e
met ~n–1
Dia 0.1 ~n–2.22
0.01
1 10 100 1000 3000
n (number of harmonic)
Figure 3.28 Diameters of stable harmonics (vortices) 2R in the W fluctuation
n T
spectrum.
148 ◾ Optical Waves and Laser Beams in the Irregular Atmosphere
–0.00
D
uuu
–0.02
100
–0.04 Hz) ~ f –8/3
2g/10–2
e
d
(T10–4
–0.06 W DuTT
10–6
0.1 1 10 100
f (Hz)
–0.08
0 1 2
τ (s)
Figure 3.32 Third moments of the longitudinal difference of velocity and tem-
perature D (m3/s3), D (deg2m/s). Smoothed spectrum (bottom left inset).
uuu uTT
Coherent turbulence, summer daytime measurements in mountains at a height
of 680 m, July 2, 2007.
9
Cθ Cθ = 5.99
6
C = 2.93
3
3
C 1.9
0
1.0 1.5 2.0 2.5
τ (s)
Figure 3.33 Kolmogorov C and Obukhov C constants. Coherent turbulence
θ
measured in mountains at a height of 680 m, July 2 of 2007. Vertical lines show
the boundaries of the inertial range, and dashed lines are the average values of
the constants C and C .
θ
law) are accompanied by significant errors, as a rule. The data of our measure-
ments indicate the main cause for appearance of these errors. Variations of the
Kolmogorov and Obukhov constants within 100% (depending on the observation
point) lead to practically the same errors in determination of the characteristics
C2, C2, and C2.
T V n
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