## Simulation study on light propagation in an anisotropic turbulence field of entrainment zone |

Optics Express, Vol. 22, Issue 11, pp. 13427-13437 (2014)

http://dx.doi.org/10.1364/OE.22.013427

Acrobat PDF (2169 KB)

### Abstract

The convective atmospheric boundary layer was modeled in the water tank. In the entrainment zone (EZ), which is at the top of the convective boundary layer (CBL), the turbulence is anisotropic. An anisotropy coefficient was introduced in the presented anisotropic turbulence model. A laser beam was set to horizontally go through the EZ modeled in the water tank. The image of two-dimensional (2D) light intensity fluctuation was formed on the receiving plate perpendicular to the light path and was recorded by the CCD. The spatial spectra of both horizontal and vertical light intensity fluctuations were analyzed. Results indicate that the light intensity fluctuation in the EZ exhibits strong anisotropic characteristics. Numerical simulation shows there is a linear relationship between the anisotropy coefficients and the ratio of horizontal to vertical fluctuation spectra peak wavelength. By using the measured temperature fluctuations along the light path at different heights, together with the relationship between temperature and refractive index, the one-dimensional (1D) refractive index fluctuation spectra were derived. The anisotropy coefficients were estimated from the 2D light intensity fluctuation spectra modeled by the water tank. Then the turbulence parameters can be obtained using the 1D refractive index fluctuation spectra and the corresponding anisotropy coefficients. These parameters were used in numerical simulation of light propagation. The results of numerical simulations show this approach can reproduce the anisotropic features of light intensity fluctuations in the EZ modeled by the water tank experiment.

© 2014 Optical Society of America

## 1. Introduction

1. A. Consortini, L. Ronchi, and L. Stefanutti, “Investigation of atmospheric turbulence by narrow laser beams,” Appl. Opt. **9**(11), 2543–2547 (1970). [CrossRef] [PubMed]

2. M. Antonelli, A. Lanotte, and A. Mazzino, “Anisotropies and universality of buoyancy-dominated turbulent fluctuations: A large-eddy simulation study,” J. Atmos. Sci. **64**(7), 2642–2656 (2007). [CrossRef]

7. R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, and C. Lu, “Simulation study on light propagation in an isotropic turbulence field of the mixed layer,” Opt. Express **22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

1. A. Consortini, L. Ronchi, and L. Stefanutti, “Investigation of atmospheric turbulence by narrow laser beams,” Appl. Opt. **9**(11), 2543–2547 (1970). [CrossRef] [PubMed]

15. R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antennas Propag. **34**(2), 258–261 (1986). [CrossRef]

16. A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media **11**(3), 163–181 (2001). [CrossRef]

## 2. Theory

17. A. S. Gurvich and I. P. Chunchuzov, “Three-dimensional spectrum of temperature fluctuations in stably stratified atmosphere,” Ann. Geophys. **26**(7), 2037–2042 (2008). [CrossRef]

7. R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, and C. Lu, “Simulation study on light propagation in an isotropic turbulence field of the mixed layer,” Opt. Express **22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

16. A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media **11**(3), 163–181 (2001). [CrossRef]

20. S. Kida and J. C. R. Hunt, “Interaction between different scales of turbulence over short times,” J. Fluid Mech. **201**(1), 411–445 (1989). [CrossRef]

21. G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity,” J. Fluid Mech. **5**(1), 113–133 (1959). [CrossRef]

7. R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, and C. Lu, “Simulation study on light propagation in an isotropic turbulence field of the mixed layer,” Opt. Express **22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

*P*is the Prandtl number, which is 7.04 for water, where

_{r}= υ/D*υ*and

*D*are respectively the molecular viscosity coefficient and the diffusion coefficient. In Eq. (3’),

*a*≈2,

*C*= 2.8, Kolmogorov microscale

_{θ}*η*(

_{k}=*υ*/

^{3}*ε*)

^{1/}

*, and*

^{4}*ε*is the viscous dissipation rate.

*η*has a linearrelation with the inner-scale

_{k}*l*[22

_{0}22. R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. A **68**(7), 892–899 (1978). [CrossRef]

*l*= 1.34. Due to the assumption that the anisotropic 3D temperature fluctuation spectrum (Eq. (3’)) has its corresponding isotropic physical variations, the inner-scale variable of anisotropic turbulence is similar to that of isotropic turbulence field.

_{0}/η_{k}*q*<<1/

*η*and

_{k}*q*≥1/

*η*. For all spectral space, the value of

_{k}**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

*q*≈1/

*η*, decreases with the decreasing of

_{k}*q*in the range of

*q*≤1/

*η*, and leaves crosspoint

_{k}*q = q*when intersecting with

_{m}*q*is smaller than crosspoint

*q*; and for those

_{m}*q*larger than

*q*, Eq. (3’) is adopted to decide the value of

_{m}*q*. The results show that

*q*≈0.16/

_{m}*η*which fits the requirement of

_{k}*q*<<1/

*η*. By doing so, the power-law will be bigger than

_{k}*α*when

*q*≤q<1/

_{m}*η*, which is a good match with experimental results for isotropic situation [7

_{k}**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

*E*(

_{T}*κ*) and

_{y}*E*(

_{T}*κ*) to 3D spectra Φ

_{z}*in anisotropic turbulence field is conducted as follow:*

_{T}*n*, the refractive index of water, varies with temperature

*T*[24

24. H. M. Dobbins and E. R. Peck, “Change of refractive index of water as a function of temperature,” J. Opt. Soc. Am. A **63**(3), 318–320 (1973). [CrossRef]

*L*and inner-scale

_{0}*l*can be computed by Eq. (2).

_{0}## 3. Water tank simulation experiments and numerical simulation method

^{−1}. The size and settings of the water tank can reasonably meet the requirement to simulating atmospheric boundary layer [25

25. R. Yuan, X. Wu, T. Luo, H. Liu, and J. Sun, “A review of water tank modeling of the convective atmospheric boundary layer,” J. Wind Eng. Ind. Aerodyn. **99**(10), 1099–1114 (2011). [CrossRef]

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

26. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. **27**(11), 2111–2126 (1988). [CrossRef] [PubMed]

*q*is determined by Eq. (1). The intensity of turbulence in water is 10

^{6}times larger than that in the atmosphere, hence, turbulence features need to be considered when setting up the grid size so that 2D light intensity fluctuations simulated will contain most of the energy. After considering this factor, the grid size of the phase screen is set as 5 × 10

^{−2}mm. The interscreen distance was set as 0.02mm [7

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

27. J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, and F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. **21**(6), 929–948 (1986). [CrossRef]

26. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. **27**(11), 2111–2126 (1988). [CrossRef] [PubMed]

26. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. **27**(11), 2111–2126 (1988). [CrossRef] [PubMed]

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

*q*in Eq. (1) in order to form a thin phase screen. It is should be pointed out that, the dependent phase screens means the interscreen distance is larger than the correlation length of the irregularities [26

**27**(11), 2111–2126 (1988). [CrossRef] [PubMed]

*C*, the refractive index structure constant

_{aniso}*C*, spectral power-low

_{n}^{2}*L*and inner-scale

_{0}*l*are identical over the turbulent path. The identical

_{0}*C*means that, the SDEs with a surface of equal spectral density value have the identical long axis and short axis for all phase screens, or the SDEs maintain same poses over the turbulent path.

_{aniso}## 4. Results and discussion

### 4.1 Characteristics of light intensity fluctuation in the EZ

*β*) with height as that in [7

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

*β*is defined aswhere

*I*is the instantaneous value of light intensity (in our experiment, it’s the gray-scale value of CCD image).

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

*κ*and divided by the light intensity variance

*σ*). The horizontal spectrum differs from the vertical one. The former has more energy in the low frequency and less energy in the high frequency. The typical scale of turbulence, which is represented by the peak wavenumber or the peak wavelength (they are reciprocal), can be detected from the power spectra. The peak wavenumber is the wavenumber corresponding to the maximum value of normalized spectral density as shown in Fig. 2. In Fig. 2, the peak wavenumber of horizontal spectrum is smaller than that of the vertical spectrum, which indicates the horizontal peak wavelength is larger than the vertical one. That is to say, it is more difficult for turbulence to develop along the vertical direction due to the existence of a relatively strong temperature inversion. A peak wavelength ratio can be defined as the ratio of the horizontal spectrum peak wavelength

_{I}^{2}*λ*to the vertical one

_{H}*λ*. The peak wavelength ratio (

_{V}*λ*/

_{H}*λ*) in Fig. 2 is 6.5. The curve of peak wavelength of horizontal and vertical spectra varying with height was presented in Fig. 3(a), and the curve of peak wavelength ratio changing with height is shown in Fig. 3(b). The peak wavelength ratio is very close to 1 in the ML (See Fig. 3(a)), indicating that the turbulence field the in the ML is isotropic. On the other hand, the peak wavelength ratio in the EZ is larger than 1, implying that the turbulence field in the EZ should be anisotropic.

_{V}### 4.2 Peak wavelength ratio and anisotropy coefficient

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

*C*= 6.0 × 10

_{n}^{2}^{−7}m

^{-2/3},

*l*= 4 × 10

_{0}^{−3}m,

*L*= 0.18m,

_{0}*α*= 2.3, and the anisotropy coefficient

*C*= 3.0. The results are presented in the Fig. 4, in which Fig. 4(a) is the numerical simulated photograph for light intensity fluctuation and Fig. 4(b) is the horizontal and vertical light intensity fluctuation spectra. Figure 4(a) shows a very similar pattern to that in Fig. 1(a), which is horizontally distributed bright streaks at the top of boundary layer; Fig. 4(b) shows that the

_{aniso}*λ*is smaller than the

_{H}*λ*.

_{V}*λ*/

_{H}*λ*increases with the increasing of anisotropy coefficient, as shown in Fig. 5.There is a linear relationship between the

_{V}*λ*/

_{H}*λ*and the anisotropy coefficients, as given by the numerical simulation results

_{V}### 4.3 Comparison between numerical simulation and water-tank simulations

*C*. At the same moment as Fig. 1, temperature fluctuations measured at 110mm, 127mm, 150mm, 170mm and 190mm were shown in the Fig. 6. The temperature fluctuations are relatively gentle at height of 110mm, 127mm and 150mm in the ML while more drastic at the height of 170mm and 190mm in the EZ. Based on the measured temperature fluctuations and the relations between the refractive index and temperature (Eq. (6)), 1D horizontal power spectra can be calculated. Figure 7 shows the 1D horizontal power spectra at 190mm as the round-dots. If the anisotropic coefficient

_{aniso}*C*is known, Eq. (5) can be used to obtain the 3D refractive index spectrum, and then Eq. (2) can be used to obtain the turbulence parameters. However, we often have no idea for the anisotropic coefficient

_{aniso}*C*and assume the turbulence field is isotropic, namely,

_{aniso}*C*= 1. For an anisotropic turbulent field, this method may give a wrong result [7

_{aniso}**22**(6), 7194–7209 (2014). [CrossRef] [PubMed]

*C*will give the different spectra parameters according to Eq. (5). For example, the spectra parameters at 190mm with

_{aniso}*C*= 1 will be the refractive index structure constant C

_{aniso}_{n}

^{2}= 1.7 × 10

^{−7}m

^{-2/3}, outer-scale

*L*= 0.31m, inner-scale

_{0}*l*= 0.0069m and power-law

_{0}*α*= 2.3. However, after analyzing the photograph recorded, the peak wavelength ratio is about 6 at the height of 190mm, and its corresponding

*C*is 3. The spectra parameters at 190mm with

_{aniso}*C*= 3 will be

_{aniso}*C*= 6.0 × 10

_{n}^{2}^{−7}m

^{-2/3},

*L*= 0.18m,

_{0}*l*= 4 × 10

_{0}^{−3}m, and

*α*= 2.3. The two sets of turbulence parameters give the same 1D spectrum, which are the solid line shown in Fig. 7. The two sets of numerical simulation were designed to obtain the SI = 0.03 for isotropy and SI = 0.67 for anisotropy. It will be seen in the following Table 1 that anisotropic turbulence method shows agreement with the real situation.

*C*with the superscript * is just assumed as 1 for comparison; and in another row, the

_{aniso}*C*is 3, attained from Fig. 3. Then the turbulence parameters can be calculated respectively and numerical simulations can be carried out to obtain two SIs. For anisotropic situations, if turbulence fields were considered as isotropic, the SIs from numerical simulation was far less than the water tank measurements. However, when the

_{aniso}*C*from the image was introduced, the numerical model could produce similar SIs to the water tank measurements (The detailed results have been elaborated in the last paragraph). There is similar result for the height of 170mm. Therefore, for anisotropic turbulence, the isotropic assumptions will produce huge difference between the theoretical predictions and the real measurements. The new anisotropic turbulence spectra model developed in this paper could significantly improve the theoretical predictions.

_{aniso}## 6. Conclusion

- (1) When plane wave transmits through the top of the CBL (ie, the EZ), 2D fluctuation field in the cross section perpendicular to the light path exhibits an obvious feature of anisotropy.
- (2) Applicable anisotropic turbulence spectra were proposed by inducing an anisotropic coefficient, together with refractive index fluctuation spectra converted from temperature measurements. The numerically simulated SIs and light intensity fluctuation spectra are all in great accordance with the real measured light image.
- (3) Anisotropy coefficients are determined by using 2D measurements of light intensity fluctuation. Anisotropy coefficients in the EZ are ranging from 1 to 4.

## Acknowledgments

## References and links

1. | A. Consortini, L. Ronchi, and L. Stefanutti, “Investigation of atmospheric turbulence by narrow laser beams,” Appl. Opt. |

2. | M. Antonelli, A. Lanotte, and A. Mazzino, “Anisotropies and universality of buoyancy-dominated turbulent fluctuations: A large-eddy simulation study,” J. Atmos. Sci. |

3. | A. S. Gurvich and I. P. Chunchuzov, “Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations,” J. Geophys. Res. |

4. | V. Kan, V. F. Sofieva, and F. Dalaudier, “Anisotropy of small-scale stratospheric irregularities retrieved from scintillations of a double star alpha-Cru observed by GOMOS/ENVISAT,” Atmos. Meas. Tech. |

5. | V. F. Sofieva, F. Dalaudier, and J. Vernin, “Using stellar scintillation for studies of turbulence in the Earth's atmosphere,” Philos. Trans. R. Soc. A |

6. | V. F. Sofieva, A. S. Gurvich, F. Dalaudier, and V. Kan, “Reconstruction of internal gravity wave and turbulence parameters in the stratosphere using GOMOS scintillation measurements,” J. Geophys. Res. |

7. | R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, and C. Lu, “Simulation study on light propagation in an isotropic turbulence field of the mixed layer,” Opt. Express |

8. | R. Yuan, J. Sun, K. Yao, Z. Zeng, and W. Jiang, “A laboratory simulation of the atmospheric boundary layer analyses of temperature structure in the entrainment zone (in Chinese),” Chin. J. Atmos. Sci. |

9. | A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds number,” Dokl. Akad. Nauk SSSR |

10. | V. I. Tatarskii, |

11. | J. P. L. C. Salazar and L. R. Collins, “Two-particle dispersion in isotropic turbulent flows,” Annu. Rev. Fluid Mech. |

12. | J. L. Lumley and A. M. Yaglom, “A century of turbulence,” Flow Turbul. Combust. |

13. | R. B. Stull, |

14. | A. S. Gurvich, “A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere,” Ann. Geophys. |

15. | R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antennas Propag. |

16. | A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media |

17. | A. S. Gurvich and I. P. Chunchuzov, “Three-dimensional spectrum of temperature fluctuations in stably stratified atmosphere,” Ann. Geophys. |

18. | E. S. Wheelon, |

19. | E. S. Wheelon, |

20. | S. Kida and J. C. R. Hunt, “Interaction between different scales of turbulence over short times,” J. Fluid Mech. |

21. | G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity,” J. Fluid Mech. |

22. | R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. A |

23. | L. C. Andrews and R. L. Phillips, |

24. | H. M. Dobbins and E. R. Peck, “Change of refractive index of water as a function of temperature,” J. Opt. Soc. Am. A |

25. | R. Yuan, X. Wu, T. Luo, H. Liu, and J. Sun, “A review of water tank modeling of the convective atmospheric boundary layer,” J. Wind Eng. Ind. Aerodyn. |

26. | J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. |

27. | J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, and F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. |

28. | V. A. Kulikov and V. I. Shmalhausen, “Thread-shaped intensity field after light propagation through the convective cell,” Cornell University Library, Atmospheric and oceanic physics, arXiv:1310.5273 (2013). |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: April 1, 2014

Revised Manuscript: May 16, 2014

Manuscript Accepted: May 17, 2014

Published: May 27, 2014

**Citation**

Renmin Yuan, Jianning Sun, Tao Luo, Xuping Wu, Chen Wang, and Yunfei Fu, "Simulation study on light propagation in an anisotropic turbulence field of entrainment zone," Opt. Express **22**, 13427-13437 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13427

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### References

- A. Consortini, L. Ronchi, L. Stefanutti, “Investigation of atmospheric turbulence by narrow laser beams,” Appl. Opt. 9(11), 2543–2547 (1970). [CrossRef] [PubMed]
- M. Antonelli, A. Lanotte, A. Mazzino, “Anisotropies and universality of buoyancy-dominated turbulent fluctuations: A large-eddy simulation study,” J. Atmos. Sci. 64(7), 2642–2656 (2007). [CrossRef]
- A. S. Gurvich, I. P. Chunchuzov, “Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations,” J. Geophys. Res. 108, 4166 (2003).
- V. Kan, V. F. Sofieva, F. Dalaudier, “Anisotropy of small-scale stratospheric irregularities retrieved from scintillations of a double star alpha-Cru observed by GOMOS/ENVISAT,” Atmos. Meas. Tech. 5(11), 2713–2722 (2012). [CrossRef]
- V. F. Sofieva, F. Dalaudier, J. Vernin, “Using stellar scintillation for studies of turbulence in the Earth's atmosphere,” Philos. Trans. R. Soc. A 371, 20120174 (2013).
- V. F. Sofieva, A. S. Gurvich, F. Dalaudier, V. Kan, “Reconstruction of internal gravity wave and turbulence parameters in the stratosphere using GOMOS scintillation measurements,” J. Geophys. Res. 112, D12113 (2007).
- R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, C. Lu, “Simulation study on light propagation in an isotropic turbulence field of the mixed layer,” Opt. Express 22(6), 7194–7209 (2014). [CrossRef] [PubMed]
- R. Yuan, J. Sun, K. Yao, Z. Zeng, W. Jiang, “A laboratory simulation of the atmospheric boundary layer analyses of temperature structure in the entrainment zone (in Chinese),” Chin. J. Atmos. Sci. 26, 773–780 (2002).
- A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds number,” Dokl. Akad. Nauk SSSR 30, 299–303 (1941).
- V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
- J. P. L. C. Salazar, L. R. Collins, “Two-particle dispersion in isotropic turbulent flows,” Annu. Rev. Fluid Mech. 41(1), 405–432 (2009).
- J. L. Lumley, A. M. Yaglom, “A century of turbulence,” Flow Turbul. Combust. 66, 241–286 (2001).
- R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988).
- A. S. Gurvich, “A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere,” Ann. Geophys. 15, 856–869 (1997).
- R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antennas Propag. 34(2), 258–261 (1986). [CrossRef]
- A. S. Gurvich, V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media 11(3), 163–181 (2001). [CrossRef]
- A. S. Gurvich, I. P. Chunchuzov, “Three-dimensional spectrum of temperature fluctuations in stably stratified atmosphere,” Ann. Geophys. 26(7), 2037–2042 (2008). [CrossRef]
- E. S. Wheelon, I. Geometrical Optics (Cambridge University, 2001).
- E. S. Wheelon, II. Weak Scattering (Cambridge University, 2003).
- S. Kida, J. C. R. Hunt, “Interaction between different scales of turbulence over short times,” J. Fluid Mech. 201(1), 411–445 (1989). [CrossRef]
- G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity,” J. Fluid Mech. 5(1), 113–133 (1959). [CrossRef]
- R. J. Hill, S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. A 68(7), 892–899 (1978). [CrossRef]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
- H. M. Dobbins, E. R. Peck, “Change of refractive index of water as a function of temperature,” J. Opt. Soc. Am. A 63(3), 318–320 (1973). [CrossRef]
- R. Yuan, X. Wu, T. Luo, H. Liu, J. Sun, “A review of water tank modeling of the convective atmospheric boundary layer,” J. Wind Eng. Ind. Aerodyn. 99(10), 1099–1114 (2011). [CrossRef]
- J. M. Martin, S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27(11), 2111–2126 (1988). [CrossRef] [PubMed]
- J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21(6), 929–948 (1986). [CrossRef]
- V. A. Kulikov and V. I. Shmalhausen, “Thread-shaped intensity field after light propagation through the convective cell,” Cornell University Library, Atmospheric and oceanic physics, arXiv:1310.5273 (2013).

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