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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13427–13437
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Simulation study on light propagation in an anisotropic turbulence field of entrainment zone

Renmin Yuan, Jianning Sun, Tao Luo, Xuping Wu, Chen Wang, and Yunfei Fu  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 13427-13437 (2014)
http://dx.doi.org/10.1364/OE.22.013427


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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

So far, isotropic turbulence field is often assumed for study of light propagation in turbulent media. To the contrary, a large number of observations in the near-surface layer and at the top of convective boundary layer (CBL) reveal characteristics of anisotropy. In the near-surface layer, after the narrow beam covers some distance horizontally, the dancing of the narrow beam along horizontal and vertical direction appears to have different statistical characteristics [1

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]

]. Turbulent fluctuations of both velocity and temperature fields show anisotropy at the top of CBL [2

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]

]. The anisotropic situation can be often observed at the tropopause and stratosphere [3

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. 108, 4166 (2003).

6

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. 112, D12113 (2007).

]. According to the light propagation experiments conducted in water tank, turbulent flows have obvious anisotropic characteristics at the top of the simulated atmospheric boundary layer [7

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]

, 8

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. 26, 773–780 (2002).

].

Since the theory of locally isotropic turbulence was introduced [9

9. 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).

], the theory has been extensively applied to light propagating [10

10. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

], pollution dispersion [11

11. J. P. L. C. Salazar and L. R. Collins, “Two-particle dispersion in isotropic turbulent flows,” Annu. Rev. Fluid Mech. 41(1), 405–432 (2009).

] and many other fields [12

12. J. L. Lumley and A. M. Yaglom, “A century of turbulence,” Flow Turbul. Combust. 66, 241–286 (2001).

, 13

13. R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988).

]. However, a large number of observations indicate that turbulence field often exhibits anisotropic features. Therefore, we need to understand the impacts of anisotropy on light propagation in the turbulence field. Some models have been drafted, which all consider turbulence as isotropic in the horizontal plane and anisotropic in the vertical direction. These models can describe the observed anisotropic characteristics to a certain extent [3

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. 108, 4166 (2003).

, 5

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 371, 20120174 (2013).

], which are mainly divided into two categories: one focuses on the correlation moment for turbulent variables [1

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]

], another focuses on modified Kolmogorov's spectra [3

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. 108, 4166 (2003).

, 5

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 371, 20120174 (2013).

, 14

14. A. S. Gurvich, “A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere,” Ann. Geophys. 15, 856–869 (1997).

, 15

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]

]. The measured temperature fluctuation was usually expressed as a sum of isotropic and anisotropic components. When the electromagnetic wave propagates in the turbulence field, if the anisotropic coefficient is large enough (for example, the anisotropy coefficient of 30 [3

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. 108, 4166 (2003).

]), the influences of isotropy and anisotropy can be detected by judging the light intensity fluctuation spectra along the horizontal direction. All these researches mentioned above came from both theoretical studies and experimental observations. Usually the experimental observations are limited to 1D measurements based on Taylor’s hypothesis [13

13. R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988).

] and can hardly yield the 2D spectral distribution [16

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]

].

In this paper, we managed to generate an anisotropic turbulence field by means of simulation in water-tank. To be specific, a temperature gradient was generated along vertical direction so that different characteristics were revealed in horizontal and vertical directions. We measured the temperature in turbulence field, which enable us to obtain features of refractive index distribution in turbulence field. In the meantime, light propagation experiments were conducted to analyze the different features of light intensity fluctuations along horizontal and vertical directions in anisotropic turbulence field. Anisotropy was quantitatively described by using the anisotropy coefficient introduced in turbulence spectra model.

The paper is organized as follows. In Section 2, we will introduce the anisotropic turbulence model by spectral method. Section 3 comes with settings of water-tank simulation and relevant experiment measurements. Numerical simulation method is also presented in Section 3. Section 4 gives the results of experiment. Section 5 shows conclusions and discussions.

2. Theory

Temperature inversion often occurs at the top of CBL [13

13. R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988).

, 17

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]

19

19. E. S. Wheelon, II. Weak Scattering (Cambridge University, 2003).

]. The isotropic turbulence eddies (temperature fluctuation) are generated at the bottom of CBL [7

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]

]. Due to the effects of buoyancy, the turbulence eddies can rise through the CBL. When reaching the level above the CBL top (i.e. the entrainment zone, EZ), the turbulent eddies are compressed in the vertical direction and extended in the horizontal direction due to the temperature inversion in the EZ [8

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. 26, 773–780 (2002).

]. The turbulence can be considered isotropic along horizontal direction, but not the case in vertical direction [14

14. A. S. Gurvich, “A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere,” Ann. Geophys. 15, 856–869 (1997).

]. In the vertical direction, turbulent eddies will have stronger fluctuations in shorter distance. Thus, horizontal correlation distance will be larger than the vertical one, which appears anisotropic to a certain degree [16

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]

]. Usually power spectrum was adopted to study the relationships between the temperature fluctuations and the turbulence effects [10

10. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

] for the influence of temperature fluctuations on light propagations. For the three-dimensional (3D) isotropic spectra, spectral density value is the same on a sphere surface. As for the 3D spectra in anisotropic turbulence field, points with the same density can be regarded as being on an ellipsoid surface, whose two horizontal axes are of the same value and the length of vertical axis is not equal to those of horizontal ones, and which is called the spectrum density ellipsoid (SDE). In most cases, the length of vertical axis of the ellipsoid for the spectrum domain is larger than horizontal axes. The coefficientCaniso, defined as an anisotropy coefficient, can be used to denote the ratio of the length in vertical axis to the counterpart in horizontal axis. The turbulence exhibits isotropic features when Caniso = 1. Contrast to earlier researchers, for example Gurvich et al [3

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. 108, 4166 (2003).

], we had different approaches to study anisotropic turbulence field: we do not separate the actual turbulence field into two parts, which are respectively isotropic and anisotropic components, but use a unified expression of spectrum to characterize the temperature fluctuation field (or refractive index fluctuation field). In fact, the two parts of turbulence fields may still has complicated interactions [20

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]

]. According to the experimental results, our theoretical method can reproduce the experimental results satisfactorily.

The 3D turbulence spectral density of temperature fluctuation can be denoted asΦT(q), where qis the vector of wavenumber, which represents the point on the surface of the SDE. All points on the surface of the SDE have the same spectral densityΦT(q). Thus, qhas a relationship with conventionally-used wavenumber κ(κx,κy,κz), shown as:

q=(κx*Caniso,κy*Caniso,κz/Caniso)
(1)

We replace conventionally-used κ with q to express 2D spectral density of temperature fluctuationΦT(κy,κz) to makes it easier to analyze 2D light intensity fluctuation, because the integration with wavenumber (κy,κz)has nothing to do with anisotropy coefficient.

Based on the consideration above, the 3D expression of anisotropic spectral density of temperature fluctuation in water can be written as the form of isotropic spectral density [21

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]

]:
ΦT(q)=K(α)CT2qα2ϕT(q)
(2)
where K(α)equals toΓ(α+1)4π2sin[(α1)π2] and α is named as spectral power-law. Similar to the isotropic turbulent fields,CT2is the temperature structure constant. In order to characterize the temperature fluctuations in dissipative range, we still adopt the method of isotropic turbulence and introduce the factorϕT(q)in Eq. (2). Besides, in this paper, the attenuation characteristics of anisotropic turbulence in the molecular dissipating range are considered the same as those of isotropic turbulence, which satisfies,

ϕT(q)=1q<<1/ηk
(3)
ϕT(q)=a2πK(α)Cθ(qηk)2/3exp[aPr(qηk)2]q1/ηk (3’)

Similar to the isotropic situation [7

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]

], Pr = υ/D is the Prandtl number, which is 7.04 for water, where υ and D are respectively the molecular viscosity coefficient and the diffusion coefficient. In Eq. (3’), a≈2, Cθ = 2.8, Kolmogorov microscale ηk = (υ3/ε)1/4, and ε is the viscous dissipation rate. ηk has a linearrelation with the inner-scale l0 [22

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]

], which is shown as l0k = 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.

Equations (3) and (3’) only consider the wavenumber range of q<<1/ηk and q≥1/ηk. For all spectral space, the value of ϕT(q)is considered similar to isotropic situation [7

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]

] as follows: Function ϕT(q)in Eq. (3’) has a maximum value when q≈1/ηk, decreases with the decreasing of q in the range of q≤1/ηk, and leaves crosspoint q = qm when intersecting withϕT(q)=1. ϕT(q)=1, when wavenumber q is smaller than crosspoint qm; and for those q larger than qm, Eq. (3’) is adopted to decide the value ofϕT(q). In this way, the value of the turbulence spectrum is assured to change continuously with the wavenumber q. The results show that qm≈0.16/ηk which fits the requirement of q<<1/ηk. By doing so, the power-law will be bigger than α when qm≤q<1/ηk, which is a good match with experimental results for isotropic situation [7

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]

].

Although outer-scale has little effect on optical scintillation, it should be taken into account when dealing with real turbulence spectra. Here we use the following equation to show the effect of outer-scale [23

23. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).

]:

ΦT(q)=K(α)CT2qα2ϕT(q)[1exp(q2L02/16π2)]
(4)

Only 1D temperature spectra can be observed in our experiment, thus a transformation from 1D horizontal and vertical fluctuation ET(κy) and ET(κz) to 3D spectra ΦT in anisotropic turbulence field is conducted as follow:

ET(κy)=2πkyCanisoΦT(q)qdq
(5)
ET(κz)=2πCanisokz/CanisoΦT(q)qdq (5’)

The logarithmic curves of horizontal and vertical fluctuation spectra, which are calculated by Eqs. (5) and (5’), differ in scales by the boundaries of inertia area and the dissipation zone. The differences depend on the anisotropic degree of turbulence field.

The characteristics of refractive index variation are needed in order to analyze the impact of turbulent flows on light propagation. Equation (6) shows that 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]

],

n=1.332156+[8.889(T20)0.161(T20)2]105
(6)

There is an approximately linear relationship between the refractive index of water and the temperature. Thus it can be assumed that the temperature spectrum has the same shape as the refractive index spectrum. Consequently, the refractive index spectrumΦn(q), has the same statistical characteristics as those described in Eqs. (1)(5). Accordingly, the anisotropic coefficientCaniso, the refractive index structure constantCn2, spectral power-lowα, outer-scale L0 and inner-scale l0 can be computed by Eq. (2).

3. Water tank simulation experiments and numerical simulation method

The experiment was carried out in a rectangular water-tank. The height, width and length of this water-tank are 60cm, 150cm and 150cm respectively. There are 10mm transparent glass plate surrounding the water-tank and a 6cm × 145cm × 145cm oil tank for heating at the bottom of the water-tank. The oil tank is made of 2mm thick steel plates and is filled with high-insulating and low-expanding transformer oil. There are 39 electrically heating tubes installed in the oil box to heat first the oil, and then the bottom of the tank. The heating tubes are capable of dissipating a maximum of 39KW. With this indirect heating mechanism, the tank bottom is heated more uniformly. To prepare for an experiment, the tank was filled with required temperature-inversion water. The inversion in the tank for this experiment was about 70Km−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]

]. During the experiment, the average temperature profile and horizontal temperature fluctuation at 10 different heights were measured. Light propagating experiments were conducted using collimated light system. A He-Ne laser was used as the light source, and the wavelength is 0.6328μm, so the Fresnel length is less than turbulence inner scale [7

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]

]. The collimated beam with a diameter of 200mm was led into one side of the water tank, through the turbulence field inside, exited from the other side and finally generated an image of the cross section of the laser beam on the receiving plate at about 500mm from the other side of the water tank. The total distance of the beam (counting from the entering side of the water tank) is 2000mm, from which the first 1500mm is inside the water tank and the rest 500mm in the air. More details could be found in [7

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]

].

To better observe EZ, the middle part of the beam was adjusted to pass through the EZ by adjusting the height of the collimated light system (see Fig. 1
Fig. 1 A photograph of cross-section of the collimated laser beam from measurement (a), and normalized variance varies with height (b).
).

In order to explain the light propagation experiments with the water tank, numerical simulations were conducted. Algorithm by [26

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]

] was adopted to build a numerical model. This algorithm divides the entire transmission path into many independent parts which all have unique effects on lightwave. For each part, turbulence affects lightwave by only changing the phase not the amplitude of it. We used many thin phase screens, which are perpendicular to the light path, to represent the phase fluctuation 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. (4) where q is determined by Eq. (1). The intensity of turbulence in water is 106 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

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]

], and 100 phase screens were used to simulate the propagation over the whole path. Some considerations for choice of the interscreen distance are given here. Firstly, the interscreen distance should be larger than the correlation length of the irregularities in order that each phase screen makes an independent random contribution to the wave phase. The correlation length of the irregularities in the water tank is about 0.005m [27

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]

]. Secondly, the interscreen distance should be short enough in order that the scattering can be ignored, for example, normalized variance of intensity fluctuation less than 0.1 [26

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]

], and then no intensity fluctuations are produced over the distance and the effect of the medium are purely an addition of phase. Thirdly, the Fresnel length over the interscreen distance is less than one grid size [26

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]

].

The real case in the water tank measurement is that the whole path can be divided into two parts, which are the first path in the water tank (the length is 1500mm and there are 75 phase screens) and the second one in the air with nearly no turbulence (the length is 500mm and there are 25 phase screens). In order to accord with the real situation, the numerical simulation treats the light propagation in the last 500mm as in vacuum for simplicity. Over the whole path, the grid size and the interscreen distance are same. The procedure is exactly the same as the one in the paper [7

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]

] except considering the anisotropic coefficient of refractive index fluctuations. The wave number in the expression of refractive index spectra was replaced by 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

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]

]. Although each phase screen makes an independent random contribution to the wave phase, the turbulence spectral Eq. (2) are used for all phase screens, and all parameters in Eq. (2) such as the anisotropic coefficient Caniso, the refractive index structure constant Cn2, spectral power-lowα, outer-scale L0 and inner-scale l0 are identical over the turbulent path. The identical Caniso 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.

4. Results and discussion

4.1 Characteristics of light intensity fluctuation in the EZ

Figure 1 shows the same CBL structure in terms of scintillation index (SI, denoted as β) with height as that in [7

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]

], except that the light beam did not transport through lower part of the CBL in this study. Here, β is defined as
β=I2I2I2
(7)
where denotes spatial average, I is the instantaneous value of light intensity (in our experiment, it’s the gray-scale value of CCD image).

The low part of the photograph in Fig. 1(a) shows homogeneously distributed bright streaks and dark areas, which are in accordance with other results showing the isotropic characteristics [7

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]

, 28

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).

] and can be explained in the Table 1

Table 1. Parameters of the refractive index fluctuations spectra computed from the fitted temperature spectra and the SI computed from both numerical and water tank simulation

table-icon
View This Table
using isotropic theory even if some parameters such as refractive index structure constants and inner scales are different from other experiments. The middle of the photograph in Fig. 1(a), which is corresponding to EZ, exhibits a few horizontally distributed bright and dark streaks of light intensity fluctuation. 1D horizontal and vertical spectra in the EZ are computed by the same method as mentioned in the paper [7

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]

].

Figure 2
Fig. 2 1D horizontal and vertical spectra of light intensity fluctuation at the height of 196mm.
shows the horizontal and vertical light intensity fluctuation spectra at the height of 196mm. In Fig. 2, the x-coordinate is the spatial wave number and y-coordinate is the normalized power spectral density (multiplied by wavenumber κ and divided by the light intensity variance σI2). 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 λH to the vertical one λV. The peak wavelength ratio (λH/λV) 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)
Fig. 3 Peak wavelengths vary with heights (a) and peak wavelength ratios with heights (b).
, 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.

4.2 Peak wavelength ratio and anisotropy coefficient

In order to analyze and explain the features of light propagation through the turbulent media in water-tank, numerical simulations were conducted using the anisotropic turbulence spectra given in Section 2 (see Eq. (4)). The detailed algorithms are the same as those in the paper [7

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]

] except for dealing with refractive index fluctuation spectrum (see Eq. (4)). For an individual example, the procedure will be exactly the same as that in paper [7

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]

] where typical parameters in water-tank were fixed as Cn2 = 6.0 × 10−7m-2/3, l0 = 4 × 10−3m, L0 = 0.18m, α = 2.3, and the anisotropy coefficient Caniso = 3.0. The results are presented in the Fig. 4
Fig. 4 Image of the light intensity fluctuation from numerical simulation (a) and corresponding 1D horizontal and vertical spectra of light intensity fluctuation from numerical simulation (b).
, 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 λH is smaller than the λV.

More numerical simulation cases indicate that the λH/λV increases with the increasing of anisotropy coefficient, as shown in Fig. 5
Fig. 5 The relationship between the peak-wavelength ratios and the anisotropic coefficients.
.There is a linear relationship between the λH/λV and the anisotropy coefficients, as given by the numerical simulation results

λHλV=1+2.5(Caniso1)
(8)

Based on the results in Fig. 3(b), the Caniso is about 1 in the ML, and the Caniso is larger than 1in the EZ. The maximum value for Caniso in Fig. 3(b) is about 4.

4.3 Comparison between numerical simulation and water-tank simulations

Numerical simulations are carried out by using the refractive index spectra parameters, as detailed in section 2. The 3D refractive index spectra were derived from the 1D spatial temperature fluctuations from the water-tank simulations and the anisotropic coefficient Caniso. At the same moment as Fig. 1, temperature fluctuations measured at 110mm, 127mm, 150mm, 170mm and 190mm were shown in the Fig. 6
Fig. 6 Temperature distribution at 5 different heights recorded at same moment as Fig. 1. The numbers 110mm, 127mm, etc. indicate the heights of sensors respectively.
. 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
Fig. 7 Measured (dot) and fitted (line) temperature fluctuation spectrum at the height of 190mm in Fig. 6.
shows the 1D horizontal power spectra at 190mm as the round-dots. If the anisotropic coefficient Caniso 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 Caniso and assume the turbulence field is isotropic, namely, Caniso = 1. For an anisotropic turbulent field, this method may give a wrong result [7

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]

].

For the 1D refractive index spectrum shown in Fig. 7, different Caniso will give the different spectra parameters according to Eq. (5). For example, the spectra parameters at 190mm with Caniso = 1 will be the refractive index structure constant Cn2 = 1.7 × 10−7m-2/3, outer-scale L0 = 0.31m, inner-scale l0 = 0.0069m and power-law α = 2.3. However, after analyzing the photograph recorded, the peak wavelength ratio is about 6 at the height of 190mm, and its corresponding Caniso is 3. The spectra parameters at 190mm with Caniso = 3 will be Cn2 = 6.0 × 10−7m-2/3, L0 = 0.18m, l0 = 4 × 10−3m, 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.

Table 1 summarized the SIs computed from numerical simulation and water tank results as listed in the column 7 and column 8 respectively. Based on the measured light intensity fluctuation, turbulence field at the heights 110mm-150mm shows good isotropic features and at the height of 170mm and 190mm, show anisotropic features. As shown in Table 1, for isotropic situations, namely at the heights of 110mm-150mm, numerical simulations could give similar SIs to those measured in water-tank.

For the height of 190mm, there are two rows of parameters. In one row, the Caniso with the superscript * is just assumed as 1 for comparison; and in another row, the Caniso 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 Caniso 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.

6. Conclusion

Combing water-tank simulation and numerical simulations, characteristics of the optical transmission in the anisotropic turbulence field were analyzed and the conclusions were briefed as follows:

  • (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.

The expression of anisotropic spectra given in this paper can well reproduce the anisotropic characteristics of 2D light image from experimental observations. However, the expression of anisotropic turbulence spectra in this paper is merely an empirical one. The definition of the anisotropic coefficient is conceptual. If treating the anisotropy coefficient as a constant, basically it means the anisotropic distributions of turbulent flows are spatially even. However, results in this paper show that the anisotropic coefficients increase with height in the EZ, which are about 1 to 4. The value of anisotropic coefficients in the water tank is much less than in the real atmosphere where the anisotropic coefficient is about 30 [3

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. 108, 4166 (2003).

]. The difference is not clear. Therefore, more work is needed to figure out the factors controlling the anisotropy coefficient to better understand the physical meanings of the anisotropic coefficient and the behavior of the turbulence. It will be left as our future work.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (40975006, 40975004, 41230419, 91337213 and 41075041). We also thank two anonymous reviewers for their constructive and helpful comments.

References and links

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]

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. 108, 4166 (2003).

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(11), 2713–2722 (2012). [CrossRef]

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 371, 20120174 (2013).

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. 112, D12113 (2007).

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]

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. 26, 773–780 (2002).

9.

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).

10.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

11.

J. P. L. C. Salazar and L. R. Collins, “Two-particle dispersion in isotropic turbulent flows,” Annu. Rev. Fluid Mech. 41(1), 405–432 (2009).

12.

J. L. Lumley and A. M. Yaglom, “A century of turbulence,” Flow Turbul. Combust. 66, 241–286 (2001).

13.

R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988).

14.

A. S. Gurvich, “A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere,” Ann. Geophys. 15, 856–869 (1997).

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]

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]

18.

E. S. Wheelon, I. Geometrical Optics (Cambridge University, 2001).

19.

E. S. Wheelon, II. Weak Scattering (Cambridge University, 2003).

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]

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.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).

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]

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]

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]

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]

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

  1. A. Consortini, L. Ronchi, L. Stefanutti, “Investigation of atmospheric turbulence by narrow laser beams,” Appl. Opt. 9(11), 2543–2547 (1970). [CrossRef] [PubMed]
  2. 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]
  3. 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).
  4. 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]
  5. 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).
  6. 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).
  7. 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]
  8. 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).
  9. 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).
  10. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  11. J. P. L. C. Salazar, L. R. Collins, “Two-particle dispersion in isotropic turbulent flows,” Annu. Rev. Fluid Mech. 41(1), 405–432 (2009).
  12. J. L. Lumley, A. M. Yaglom, “A century of turbulence,” Flow Turbul. Combust. 66, 241–286 (2001).
  13. R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988).
  14. A. S. Gurvich, “A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere,” Ann. Geophys. 15, 856–869 (1997).
  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, 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]
  17. A. S. Gurvich, I. P. Chunchuzov, “Three-dimensional spectrum of temperature fluctuations in stably stratified atmosphere,” Ann. Geophys. 26(7), 2037–2042 (2008). [CrossRef]
  18. E. S. Wheelon, I. Geometrical Optics (Cambridge University, 2001).
  19. E. S. Wheelon, II. Weak Scattering (Cambridge University, 2003).
  20. S. Kida, 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]
  22. 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]
  23. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
  24. 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]
  25. 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]
  26. 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]
  27. 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]
  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).

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