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]. Turbulent fluctuations of both velocity and temperature fields show anisotropy at the top of CBL [2]. The anisotropic situation can be often observed at the tropopause and stratosphere [3–6]. 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, 8].

Since the theory of locally isotropic turbulence was introduced [9], the theory has been extensively applied to light propagating [10], pollution dispersion [11] and many other fields [12, 13]. 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, 5], which are mainly divided into two categories: one focuses on the correlation moment for turbulent variables [1], another focuses on modified Kolmogorov's spectra [3, 5, 14, 15]. 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]), 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] and can hardly yield the 2D spectral distribution [16].

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, 17–19]. The isotropic turbulence eddies (temperature fluctuation) are generated at the bottom of CBL [7]. 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]. The turbulence can be considered isotropic along horizontal direction, but not the case in vertical direction [14]. 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]. Usually power spectrum was adopted to study the relationships between the temperature fluctuations and the turbulence effects [10] 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 coefficient$C_{aniso}$, 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 $C_{aniso}$ = 1. Contrast to earlier researchers, for example Gurvich et al [3], 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]. 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 as${\Phi }_T(\stackrel{\rightharpoonup }{q})$, where $\stackrel{\rightharpoonup }{q}$is 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 density${\Phi }_T(\stackrel{\rightharpoonup }{q})$. Thus, $\stackrel{\rightharpoonup }{q}$has a relationship with conventionally-used wavenumber $\stackrel{\rightharpoonup }{\kappa }({\kappa }_x,{\kappa }_y,{\kappa }_z)$, shown as:

We replace conventionally-used $\stackrel{\rightharpoonup }{\kappa }$ with $\stackrel{\rightharpoonup }{q}$ to express 2D spectral density of temperature fluctuation${\Phi }_T({\kappa }_y,{\kappa }_z)$ to makes it easier to analyze 2D light intensity fluctuation, because the integration with wavenumber $({\kappa }_y,{\kappa }_z)$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]:

Similar to the isotropic situation [7], P_r = 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, C_θ = 2.8, Kolmogorov microscale η_k = (υ³/ε)^1/4, and ε is the viscous dissipation rate. η_k has a linearrelation with the inner-scale l₀ [22], which is shown as l₀/η_k = 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/η_k and q≥1/η_k. For all spectral space, the value of ${\varphi }_T(q)$is considered similar to isotropic situation [7] as follows: Function ${\varphi }_T(q)$in Eq. (3’) has a maximum value when q≈1/η_k, decreases with the decreasing of q in the range of q≤1/η_k, and leaves crosspoint q = q_m when intersecting with${\varphi }_T(q)=1$. ${\varphi }_T(q)=1$, when wavenumber q is smaller than crosspoint q_m; and for those q larger than q_m, Eq. (3’) is adopted to decide the value of${\varphi }_T(q)$. In this way, the value of the turbulence spectrum is assured to change continuously with the wavenumber q. The results show that q_m≈0.16/η_k which fits the requirement of q<<1/η_k. By doing so, the power-law will be bigger than α when q_m≤q<1/η_k, which is a good match with experimental results for isotropic situation [7].

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]:

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

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],

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 spectrum${\Phi }_n(q)$, has the same statistical characteristics as those described in Eqs. (1)–(5). Accordingly, the anisotropic coefficient$C_{aniso}$, the refractive index structure constant$C_n^2$, spectral power-low$\alpha$, 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]. 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]. 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].

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).

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

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] 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]. 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 C_aniso, the refractive index structure constant C_n², spectral power-low$\alpha$, outer-scale L₀ and inner-scale l₀ are identical over the turbulent path. The identical C_aniso 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], except that the light beam did not transport through lower part of the CBL in this study. Here, β is defined as

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, 28] and can be explained in the Table 1 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].

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

View Table

Figure 2 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 σ_I²). 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 λ_H to the vertical one λ_V. The peak wavelength ratio (λ_H/λ_V) 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.

 

Fig. 2 1D horizontal and vertical spectra of light intensity fluctuation at the height of 196mm.

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Fig. 3 Peak wavelengths vary with heights (a) and peak wavelength ratios with heights (b).

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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] 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] where typical parameters in water-tank were fixed as C_n² = 6.0 × 10⁻⁷m^-2/3, l₀ = 4 × 10⁻³m, L₀ = 0.18m, α = 2.3, and the anisotropy coefficient C_aniso = 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 λ_H is smaller than the λ_V.

 

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).

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More numerical simulation cases indicate that the λ_H/λ_V increases with the increasing of anisotropy coefficient, as shown in Fig. 5.There is a linear relationship between the λ_H/λ_V and the anisotropy coefficients, as given by the numerical simulation results

 

Fig. 5 The relationship between the peak-wavelength ratios and the anisotropic coefficients.

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Based on the results in Fig. 3(b), the C_aniso is about 1 in the ML, and the C_aniso is larger than 1in the EZ. The maximum value for C_aniso 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 C_aniso. 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 C_aniso 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 C_aniso and assume the turbulence field is isotropic, namely, C_aniso = 1. For an anisotropic turbulent field, this method may give a wrong result [7].

 

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.

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Fig. 7 Measured (dot) and fitted (line) temperature fluctuation spectrum at the height of 190mm in Fig. 6.

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For the 1D refractive index spectrum shown in Fig. 7, different C_aniso will give the different spectra parameters according to Eq. (5). For example, the spectra parameters at 190mm with C_aniso = 1 will be the refractive index structure constant C_n² = 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 C_aniso is 3. The spectra parameters at 190mm with C_aniso = 3 will be C_n² = 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 C_aniso with the superscript * is just assumed as 1 for comparison; and in another row, the C_aniso 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 C_aniso 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:

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]. 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.