## Simulation study on light propagation in an isotropic turbulence field of the mixed layer |

Optics Express, Vol. 22, Issue 6, pp. 7194-7209 (2014)

http://dx.doi.org/10.1364/OE.22.007194

Acrobat PDF (1835 KB)

### Abstract

Water tank experiments and numerical simulations are employed to investigate the characteristics of light propagation in the convective boundary layer (CBL). The CBL, namely the mixed layer (ML), was simulated in the water tank. A laser beam was set to horizontally go through the water tank, and the image of two-dimensional (2D) light intensity fluctuation formed on the receiving plate perpendicular to the light path was recorded by CCD. The spatial spectra of both horizontal and vertical light intensity fluctuations were analyzed, and the vertical distribution profile of the scintillation index (SI) in the ML was obtained. The experimental results indicate that 2D light intensity fluctuation was isotropically distributed in the cross section perpendicular to the light beam in the ML. Based on the measured temperature fluctuations along the light path at different heights, together with the relationship between temperature and refractive index, the refractive index fluctuation spectra and the corresponding turbulence parameters were derived. The obtained parameters were applied in a numerical model to simulate light propagation in the isotropic turbulence field. The calculated results successfully reproduce the characteristics of light intensity fluctuation observed in the experiments.

© 2014 Optical Society of America

## 1. Introduction

2. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. **88**(03), 541–562 (1978). [CrossRef]

4. 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]

5. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express **18**(20), 21269–21283 (2010). [CrossRef] [PubMed]

6. R. Z. Rao, S. P. Wang, X. C. Liu, and Z. B. Gong, “Turbulence spectrum effect on wave temporal-frequency spectra for light propagating through the atmosphere,” J. Opt. Soc. Am. A **16**(11), 2755–2762 (1999). [CrossRef]

8. M. Kelly and J. C. Wyngaard, “Two-dimensional spectra in the atmospheric boundary layer,” J. Atmos. Sci. **63**(11), 3066–3070 (2006). [CrossRef]

12. J. W. Deardorff, G. E. Willis, and B. H. Stockton, “Laboratory studies of the entrainment zone of a convectively mixed layer,” J. Fluid Mech. **100**(01), 41–64 (1980). [CrossRef]

14. 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]

15. A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. **20**(7), 705–714 (1977). [CrossRef]

16. V. A. Kulikov, M. S. Andreeva, A. V. e. Koryabin, and V. I. Shmalhausen, “Method of estimation of turbulence characteristic scales,” Appl. Opt. **51**(36), 8505–8515 (2012). [CrossRef] [PubMed]

18. J. Zhang and Z. Y. Zeng, “Statistical properties of optical turbulence in a convective tank: experimental results,” J. Opt. **3**(4), 236–241 (2001). [CrossRef]

19. Z. B. Gong, Y. J. Wang, and Y. Wu, “Finite temporal measurements of the statistical characteristics of the atmospheric coherence length,” Appl. Opt. **37**(21), 4541–4543 (1998). [CrossRef] [PubMed]

## 2. Theory

### 2.1 Turbulence spectra in water

15. A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. **20**(7), 705–714 (1977). [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**(01), 113–133 (1959). [CrossRef]

*T*denotes temperature,

*D*are the molecular viscosity coefficient and the diffusion coefficient.

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]

23. C. H. Gibson and W. H. Schwarz, “The universal equilibrium spectra of turbulent velocity and scalar fields,” J. Fluid Mech. **16**(03), 365–384 (1963). [CrossRef]

24. H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. **34**(03), 423–442 (1968). [CrossRef]

*n*, the refractive index of water, varies with temperature

*T*[25

25. 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. (3).

_{0}### 2.2 Light intensity fluctuation spectra

*L*in a weak turbulence field, in the plane perpendicular to the propagating direction, the spectral density of 2D light intensity fluctuation can be presented as following expression [1].where the subscript

*I*denotes light intensity,

*k*is the wavenumber of light wave.

4. 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. R. Frehlich, “Effects of global intermittency on laser propagation in the atmosphere,” Appl. Opt. **33**(24), 5764–5769 (1994). [CrossRef] [PubMed]

_{.}

## 3. Water tank simulation experiments

### 3.1 Experimental facilities

14. 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]

^{−1}to 100 K m

^{−1}as requested. After the water in the tank reached a steady state (about 10 minutes) and the temperature distribution stabilized, then the bottom heating was started to drive the thermal convection in the tank. With these, the whole process of ABL development was simulated.

### 3.2 Experimental methods

^{−1}and controlled by computer during experiments. To avoid disturbing the flow field through which the light beam goes, the horizontal distribution of temperature is measured 300 mm from the center of the light beam.

^{−1}. In order to make the conversion between gray and light intensity possible, we measured their corresponding relationship and found out there is a good linear relation between the two of them. However, inevitably CCD has certain dark-current-shot noise and detector saturation. According to the experiment results, the gray-scale caused by dark-current-shot noise has an average value of 7. Theoretical analysis has indicated that the larger the ratio of the mean value of image to the gray level due to the effect of the noise, the smaller the error caused by noise [27

27. R. J. Hill and J. H. Churnside, “Observational challenges of strong scintillations of irradiance,” J. Opt. Soc. Am. A **5**(3), 445–447 (1988). [CrossRef]

## 4. Numerical simulating method

28. 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]

*x*-axis (the light beam is set to travel in

*x*-axis). Turbulent flows in each part have independent influences on optical wave propagation. In each part, turbulent flow affects the lightwave by changing the phase rather than the amplitude of it. Phase fluctuations are represented by a thin phase screen

*θ*(

*y, z*). The thin phase screen, which is retrieved from two-dimensional phase spectrum, is caused by the refractive index fluctuation. The temperature fluctuations can be transferred into the refractive index fluctuation via Eq. (6), and correspondingly further into the refractive index fluctuation spectra via Eq. (3). The intensity of turbulence in water is 10

^{6}times larger than in atmosphere. Hence, we need to consider features of distributions of 2D light intensity fluctuation spectra (caused by turbulence) when adjusting the grid distance and the thickness of the phase screen. The simulated light intensity fluctuation spectra should cover most of the ranges of asymptotic theory spectra.

^{−1}under the condition of strong turbulence (C

_{n}

^{2}~5 × 10

^{−7}m

^{-2/3}) and thus makes us to set the grid size as 5 × 10

^{−2}mm. For each phase screen, 4096*4096 grids were taken. Therefore the side length of each phase screen is 204.8 mm, much larger than the length of the inner-scale (the fluctuation energy mainly concentrates near the inner-scale) and matches the scale of image of irradiance fluctuation collected in our experiments. The allocation of phase screen interval must follow the rules that the SI formed by the adjacent phase is less than 0.1. Here, we set the phase screen interval to be 20mm.

*θ*(

*y*,

*z*) = 0, and the other settings in the numerical simulation for the second part are not changed. The numerical simulations indicate that the light intensity distribution in the cross section perpendicular to the light beam varies with distance when the light beam travels in its last 500mm path, although there is no turbulence in this path (please see the next section). The reason is that the light beam becomes non-planar wave after it passes through the water tank, and some light does not travel in the original direction.

*β*>0.5, see the definition in Section 5). After our calculation, when numerical simulated SI reaches 0.5 without any modifications to the gray-scale image, if the average value is set and noises are increased, the SI will be reduced by 30 percent and the shape of the spatial spectra of irradiance will have little change. The water tank simulating results shows, the SI is usually under 0.5 except where near the outer surface of the oil tank. Generally, extremely small SI is highly unlikely because the refractive index of the water is usually very large.

## 5. Results and discussion

### 5.1 Mean temperature and one-dimensional temperature fluctuations

^{−1}. When heating at the bottom, convection begins and boundary layer develops. The temperature within the boundary layer is almost constant, and the boundary layer is well mixed, so called the CBL or the ML. The layer at the top of the CBL keeps strong inversion with turbulence, and is called the EZ. The dotted line in Fig. 3 represents averaged temperatures measured at different heights at the moment of 1991s, when the image shown in Fig. 6 was taken simultaneously.

29. S. J. Caughey and S. G. Palmer, “Some aspects of turbulence structure through the depth of the convective boundary layer,” Q. J. R. Meteorol. Soc. **105**(446), 811–827 (1979). [CrossRef]

^{−9}m

^{-2/3}, the outer-scale

*L*is 0.25 m, the inner-scale

_{0}*l*is 0.0021 m, and the power-law

_{0}*α*is 1.67. Using these fitted 3D spectra parameters, and applying Eq. (5), 1D horizontal power spectra can be obtained (please see the solid line in Fig. 5). In the same way, the parameters of the refractive index spectra at different heights can be derived. The results are listed in Table 1.These parameters are used in numerical simulations (see the next section). Table 1 shows that, all the power-laws of refractive index fluctuation spectra are 1.67 in the ML, which agrees with the ‘-5/3′ law, whereas the spectra in the EZ are steeper, which means more energy in low-frequency part and less energy in high-frequency.

31. J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Cote, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in convective boundary-layer,” J. Atmos. Sci. **33**(11), 2152–2169 (1976). [CrossRef]

### 5.2. Scintillation index and boundary layer structure

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

32. F. Beyrich and S. E. Gryning, “Estimation of the entrainment zone depth in a shallow convective boundary layer from sodar data,” J. Appl. Meteorol. Climatol. **37**(3), 255–268 (1998). [CrossRef]

33. T. Luo, R. Yuan, X. Wu, and S. Deng, “A new parameterization of temperature structure parameter in entraining convective boundary layer,” Opt. Commun. **281**(23), 5683–5686 (2008). [CrossRef]

### 5.3 Characteristics of spatial scintillation spectra

^{−1}respectively) are very close to the peak wavenumber of vertical fluctuation spectra (0.194, 0.216 and 0.237mm

^{−1}respectively). Figure 8 gives the vertical change of peak wavenumbers for both horizontal and vertical power spectra in the CBL. It shows that the peak wavenumber of the horizontal and vertical direction are very close to each other, barely changing with height in the ML. This result indicates that the turbulence in the ML is isotropic. The mean peak wavelength of the horizontal fluctuation spectra in the ML is 5.02mm with standard deviation of 0.66mm; the mean peak wavelength of vertical fluctuation spectra is 4.86mm with standard deviation of 0.49mm. However in the EZ, the wavenumbers of the two spectra separate from each other, implying that the turbulence is anisotropic. This situation will be discussed in another paper [20].

### 5.4 Comparison between numerical simulations and water tank simulations

*β*= 1.47) because the turbulence at this height (close to the heating surface) is very strong. But the irradiance measurement result cannot be obtained, since this part of the light beam is kept off before it enters the water tank so that the image of received light beam does not include the part below the height of 20mm (the purpose of this process is to avoid the light reflected from the tank bottom, which will disturb the light beam). Thus the comparison cannot be made.

*L*= 1.5m), and another case is that the light beam passes through the whole route (i.e.,

*L*= 2.0m). The results for the two ideal cases are also listed in Table 1. It can be seen that the process of adding noise and fixing mean value only influences the SIs of simulated image for strong fluctuation (at the heights of 10mm and 30mm), which decreases the calculated SIs. But for weak fluctuation this treatment does not influence the numerical simulation results. It can also be seen that the SIs are significantly increased from 1.5m to 2.0m although the light beam propagates through the route without turbulence. The reason is that the light beam exiting from the water tank becomes non-planar wave, and some light deviates from its original direction. The evidence can be seen in Fig. 6(a). The upper part of the light image is not disturbed since there is no turbulence in the route of this part, and this part of light maintains the plane wave during propagating in its whole route. Thus the edge of the light beam keeps its original shape as the edge of a round. However, the other part of the light image is distorted by the turbulence in the water tank so that it is full of bright lines, and the edge of this part is moved to an outer position. It can be seen in Fig. 6(a) that the edges of the two parts are not continuous and there is a step between them. That is to say, the width of the lower part of the light beam is enlarged by the turbulence in the water tank, since the direction of some light in this part is changed by the refraction effect of turbulence. Actually, the effect of the turbulence in the water tank is similar to an array of random lens. The plane wave becomes non-planar wave after passing through the lens, and the propagating direction of different part of the light becomes different. Thus for the non-planar wave travelling in the air, the width of the light beam continues to increase, and the irradiance varies with the distance.

*L*= 2.0m). The first part is in the water and full of turbulence, while the second part is in the air and can be regarded as in vacuum. Analysis of the experimental data indicates that the inner scale (~2.0mm) in the water tank is larger than the Fresnel length (~0.84mm). Under these conditions, the analytic formula

*L*= 2.0m. At the height of 30mm, the analytic SI is obviously larger than that from the numerical simulation, since the scintillation is almost saturated and the analytic formula based on the weak fluctuation theory is not accurate enough. At the height of 10mm, the difference between the analytic SI and the numerically simulated SI is too large, because the scintillation has already been saturated. In this situation, the analytic formula does not exist, and the comparison is meaningless. On the other hand, for the light propagation in the water tank, the turbulence is homogeneously distributed in the light route (

*L*= 1.5m). According to the weak fluctuation theory, the analytic formula of SI in this condition is expressed as

*L*= 1.5m. Due to lack of inner scale measurements in our previous experiments [33

33. T. Luo, R. Yuan, X. Wu, and S. Deng, “A new parameterization of temperature structure parameter in entraining convective boundary layer,” Opt. Commun. **281**(23), 5683–5686 (2008). [CrossRef]

15. A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. **20**(7), 705–714 (1977). [CrossRef]

17. A. Maccioni and J. C. Dainty, “Measurement of thermally induced optical turbulence in a water cell,” J. Mod. Opt. **44**(6), 1111–1126 (1997). [CrossRef]

## 6. Conclusion

## Appendix

*η*= 0 represents the position of a plane wave incident upon the turbulent medium, and the observation point is a distance

*L*away from

*η*= 0. The variable

*η*increases along the direction of wave propagation.

*L*=2.0m, and the integration limit is [0, 2] (unit: meter) and can be split into two parts, where the first part [0, 1.5] is full of turbulence with

*L*= 2.0m can be calculated by

*L*= 1.5m can be calculated by

## Acknowledgments

## References and links

1. | V. I. Tatarskii, |

2. | R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. |

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

4. | 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. |

5. | L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express |

6. | R. Z. Rao, S. P. Wang, X. C. Liu, and Z. B. Gong, “Turbulence spectrum effect on wave temporal-frequency spectra for light propagating through the atmosphere,” J. Opt. Soc. Am. A |

7. | R. B. Stull, |

8. | M. Kelly and J. C. Wyngaard, “Two-dimensional spectra in the atmospheric boundary layer,” J. Atmos. Sci. |

9. | R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media |

10. | D. Peng, Y. Xiuhua, Z. Yanan, Z. Ming, and L. Hanjun, “Influence of wind speed on free space optical communication performance for Gaussian beam propagation through non Kolmogorov strong turbulence,” J. Phys. Conf. Ser. |

11. | F. Beyrich, J. Bange, O. K. Hartogensis, S. Raasch, M. Braam, D. van Dinther, D. Graef, B. van Kesteren, A. C. van den Kroonenberg, B. Maronga, S. Martin, and A. F. Moene, “Towards a validation of scintillometer measurements: the LITFASS-2009 Experiment,” Boundary Layer Meteorol. |

12. | J. W. Deardorff, G. E. Willis, and B. H. Stockton, “Laboratory studies of the entrainment zone of a convectively mixed layer,” J. Fluid Mech. |

13. | M. F. Hibberd and B. L. Sawford, “Design criteria for water tank models of dispersion in the planetary convective boundary-layer,” Boundary Layer Meteorol. |

14. | 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. |

15. | A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. |

16. | V. A. Kulikov, M. S. Andreeva, A. V. e. Koryabin, and V. I. Shmalhausen, “Method of estimation of turbulence characteristic scales,” Appl. Opt. |

17. | A. Maccioni and J. C. Dainty, “Measurement of thermally induced optical turbulence in a water cell,” J. Mod. Opt. |

18. | J. Zhang and Z. Y. Zeng, “Statistical properties of optical turbulence in a convective tank: experimental results,” J. Opt. |

19. | Z. B. Gong, Y. J. Wang, and Y. Wu, “Finite temporal measurements of the statistical characteristics of the atmospheric coherence length,” Appl. Opt. |

20. | R. Yuan, School of Earth and Space Sciences, University of Science and Technology of China, Anhui, 230026, China, and J. Sun are preparing a manuscript to be called ” Simulation study on light propagation in an anisotropic turbulence field of the entrainment zone.” |

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. | C. H. Gibson and W. H. Schwarz, “The universal equilibrium spectra of turbulent velocity and scalar fields,” J. Fluid Mech. |

24. | H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. |

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

26. | R. Frehlich, “Effects of global intermittency on laser propagation in the atmosphere,” Appl. Opt. |

27. | R. J. Hill and J. H. Churnside, “Observational challenges of strong scintillations of irradiance,” J. Opt. Soc. Am. A |

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

29. | S. J. Caughey and S. G. Palmer, “Some aspects of turbulence structure through the depth of the convective boundary layer,” Q. J. R. Meteorol. Soc. |

30. | 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,” Chin. J. Atmos. Sci. |

31. | J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Cote, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in convective boundary-layer,” J. Atmos. Sci. |

32. | F. Beyrich and S. E. Gryning, “Estimation of the entrainment zone depth in a shallow convective boundary layer from sodar data,” J. Appl. Meteorol. Climatol. |

33. | T. Luo, R. Yuan, X. Wu, and S. Deng, “A new parameterization of temperature structure parameter in entraining convective boundary layer,” Opt. Commun. |

34. | P. Stoica and R. L. Moses, |

**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: February 6, 2014

Revised Manuscript: March 2, 2014

Manuscript Accepted: March 10, 2014

Published: March 19, 2014

**Citation**

Renmin Yuan, Jianning Sun, Tao Luo, Xuping Wu, Chen Wang, and Chao Lu, "Simulation study on light propagation in an isotropic turbulence field of the mixed layer," Opt. Express **22**, 7194-7209 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7194

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### References

- V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
- R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(03), 541–562 (1978). [CrossRef]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
- 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]
- L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010). [CrossRef] [PubMed]
- R. Z. Rao, S. P. Wang, X. C. Liu, Z. B. Gong, “Turbulence spectrum effect on wave temporal-frequency spectra for light propagating through the atmosphere,” J. Opt. Soc. Am. A 16(11), 2755–2762 (1999). [CrossRef]
- R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988).
- M. Kelly, J. C. Wyngaard, “Two-dimensional spectra in the atmospheric boundary layer,” J. Atmos. Sci. 63(11), 3066–3070 (2006). [CrossRef]
- R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2(3), 179–201 (1992). [CrossRef]
- D. Peng, Y. Xiuhua, Z. Yanan, Z. Ming, L. Hanjun, “Influence of wind speed on free space optical communication performance for Gaussian beam propagation through non Kolmogorov strong turbulence,” J. Phys. Conf. Ser. 276, 012056 (2011).
- F. Beyrich, J. Bange, O. K. Hartogensis, S. Raasch, M. Braam, D. van Dinther, D. Graef, B. van Kesteren, A. C. van den Kroonenberg, B. Maronga, S. Martin, A. F. Moene, “Towards a validation of scintillometer measurements: the LITFASS-2009 Experiment,” Boundary Layer Meteorol. 144, 83–112 (2012).
- J. W. Deardorff, G. E. Willis, B. H. Stockton, “Laboratory studies of the entrainment zone of a convectively mixed layer,” J. Fluid Mech. 100(01), 41–64 (1980). [CrossRef]
- M. F. Hibberd, B. L. Sawford, “Design criteria for water tank models of dispersion in the planetary convective boundary-layer,” Boundary Layer Meteorol. 67, 97–118 (1994).
- 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]
- A. S. Gurvich, M. A. Kallistratova, F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20(7), 705–714 (1977). [CrossRef]
- V. A. Kulikov, M. S. Andreeva, A. V. e. Koryabin, V. I. Shmalhausen, “Method of estimation of turbulence characteristic scales,” Appl. Opt. 51(36), 8505–8515 (2012). [CrossRef] [PubMed]
- A. Maccioni, J. C. Dainty, “Measurement of thermally induced optical turbulence in a water cell,” J. Mod. Opt. 44(6), 1111–1126 (1997). [CrossRef]
- J. Zhang, Z. Y. Zeng, “Statistical properties of optical turbulence in a convective tank: experimental results,” J. Opt. 3(4), 236–241 (2001). [CrossRef]
- Z. B. Gong, Y. J. Wang, Y. Wu, “Finite temporal measurements of the statistical characteristics of the atmospheric coherence length,” Appl. Opt. 37(21), 4541–4543 (1998). [CrossRef] [PubMed]
- R. Yuan, School of Earth and Space Sciences, University of Science and Technology of China, Anhui, 230026, China, and J. Sun are preparing a manuscript to be called ” Simulation study on light propagation in an anisotropic turbulence field of the entrainment zone.”
- 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(01), 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]
- C. H. Gibson, W. H. Schwarz, “The universal equilibrium spectra of turbulent velocity and scalar fields,” J. Fluid Mech. 16(03), 365–384 (1963). [CrossRef]
- H. L. Grant, B. A. Hughes, W. M. Vogel, A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34(03), 423–442 (1968). [CrossRef]
- 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. Frehlich, “Effects of global intermittency on laser propagation in the atmosphere,” Appl. Opt. 33(24), 5764–5769 (1994). [CrossRef] [PubMed]
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