## 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

Theoretically, two-dimensional (2D) or three-dimensional (3D) spectra are usually adopted to describe the behavior of light propagation in turbulent mediums, for example, the refractive index spectra for weak-fluctuation [1], the precise spectra model [2], and von Karman spectral form of refractive index fluctuation spectra [3]. Based on the turbulent medium wave propagation equation, Tatarskii deduced the 2D expressions of light intensity and phase spectra in the cross-section perpendicular to the light path under weak-fluctuation conditions [1]. Under strong fluctuation conditions, theoretical expressions for high-frequency and low-frequency parts of light intensity fluctuation are also written in 2D form [4]. Recently, theoretical studies tend to describe light propagation in non-Kolmgorov turbulence in the form of 2D or 3D spectra [5]. However, experimental measurements are usually carried out at one fixed position to obtain the time series of temperature fluctuation (which can be converted to refractive index fluctuation) or light scintillation [6]. The one-dimensional (1D) spatial series can be derived from the time series based on the Taylor Frozen Hypothesis [7]. And the 2D and 3D spectra are usually generated under the isotropy assumption [8].

Several key parameters, such as the structure constant, inner-scale, and outer-scale of turbulence, which are closely related to characteristics of 3D spectra, are needed to describe the behavior of light propagation. These parameters are usually obtained under the assumption of isotropy and via a series of time-space conversion [9, 10]. Based on this assumption, Large Aperture Scintillometers (LAS) have been widely used to measure the heat flux and the wind speed [9, 11]. For the 2D distributions of light intensity, if the real 2D fluctuations are measured by experiments, we can directly compare them with the theoretical 2D distributions of light intensity and then obtain the required parameters by spectra analysis.

Light propagation experiments are usually conducted in the atmosphere boundary layer (ABL) where atmospheric turbulent flows mainly happen. The ABL is located in the lowest part of the troposphere, overlying the earth’s surface. One of the most important features of the ABL is that it is always turbulent. Turbulence is usually generated by convection in the daytime, so the ABL with convection is often called the convective boundary layer (CBL) and the turbulent convection results in a nearly homogeneous mixed layer (ML) [7]. Optical waves propagating in the ABL must be influenced by turbulence and there are therefore turbulent effects, such as beam drift, flickering and jittering. Limited by measurement methods and observation conditions, so far such research is mostly carried out in the near-surface layer and it is very difficult to expand to the whole ABL. That is why water tank experiments are often used to study the fundamental characteristics of turbulent flows in the ABL. According to similarity theory, water tank experiments can be used to study the dynamics of origination and development of the ABL and the behavior of turbulence in the real ABL [12–14]. The water-tank experiments are also suitable for studying the effects of turbulence on light propagation. For example, they are used to investigate the characteristics of light intensity distributions under strong-fluctuation conditions [15], and to estimate the inner scale [16–18] and the coherent length [19] of turbulence by measuring the arrival-angle fluctuation of light. However, in the previous studies, the measurements usually merely provide time series of light signals, and the turbulence field is usually set to be steady. That is to say, the results in the previous works are usually obtained under ideal conditions. The advantage of the water-tank experiment in this study is that the 2D distribution of light intensity fluctuation can be obtained, which can help us determine whether the turbulence is isotropic or not. Further, the results are more convincing for evaluating the suitability of theoretical descriptions of light propagation in the real atmosphere.

In this paper, we attempt to analyze the characteristics of 2D light intensity distributions by water tank experiments. The laser beam transmitted from a collimated light system was set to pass through the simulated ML and created the image with fluctuating intensity on the receiving screen. The CCDs serve to record the 2D gray-scale images which are proportional to the light intensity. Meanwhile, numerical methods were used to study the characteristics of light propagation in turbulent mediums. Images of 2D light intensity fluctuations were obtained to compare and verify with water tank observations. In fact, water tank experiments can effectively simulate the different characteristics of the ML and the entrainment zone (EZ). We pay great attention to the turbulence features and features of spatial light scintillation spectra in the mixed layer in this paper, and leave it to another paper to discuss those in the EZ [20].

The paper is organized as follows. In Section 2, we will introduce the theoretical models of turbulence spectra and light intensity fluctuation spectra. Section 3 presents the set-up of the water tank simulations and the relevant experimental measurements. Numerical simulation methods are presented in Section 4. Section 5 gives the results of the experiments; Section 6 presents conclusions.

## 2. Theory

In this section, we will introduce the formula for turbulence spectra in the water, and light intensity fluctuation spectra of a plane wave after it propagates in a turbulent medium for some distance.

#### 2.1 Turbulence spectra in water

The temperature spectrum in water has been given by Batchelor [15, 21],

where*T*denotes temperature, $\kappa $is the wavenumber, $\alpha $is named spectral power-law, $K(\alpha )=\frac{\Gamma (\alpha +1)}{4{\pi}^{2}}\mathrm{sin}[(\alpha -1)\frac{\pi}{2}]$. When turbulent flows satisfy the hypothesis of local homogeneous isotropy and the power-law $\alpha $ of turbulence spectrum in the inertial range equals 5/3, $K(\alpha )=0.033$. ${C}_{T}^{2}$is the temperature structure constant. In order to describe temperature fluctuations in dissipation range, a component ${C}_{n}^{2}\ne 0$is introduced into Eq. (1). The values of ${\phi}_{T}(\kappa )$vary from the convective inertial range ($\kappa <<1/{\eta}_{k}$) to the dissipation range ($\kappa \ge 1/{\eta}_{k}$), presented as follows,

${P}_{r}=\upsilon /D$is the Prandtl number, which is 7.04 for water, where$\upsilon $ and *D* are the molecular viscosity coefficient and the diffusion coefficient. $a\approx 2$, ${C}_{\theta}$ = 2.8, Kolmogorov microscale ${\eta}_{k}={({\upsilon}^{3}/\epsilon )}^{1/4}$,$\epsilon $ is the viscous dissipation rate. ${\eta}_{k}$has a linear relation with the inner-scale ${l}_{0}$ [22], which is shown as ${l}_{0}/{\eta}_{k}$ = 1.34,$\gamma $ = 0.72.

Equations (2) and (2′) only consider the wavenumber range of $\kappa <<1/{\eta}_{k}$ and$\kappa \ge 1/{\eta}_{k}$, which cover area 1 and 3 in Fig. 1.The value of ${\phi}_{T}(\kappa )$ in area 2 is considered as follows: Function${\phi}_{T}(\kappa )$in Eq. (2′) has a maximum value when$\kappa \approx 1/{\eta}_{k}$, decreases with the decreasing of $\kappa $in the range of$\kappa \le 1/{\eta}_{k}$, and leaves crosspoint $\kappa ={\kappa}_{m}$when intersecting with${\phi}_{T}(\kappa )=1$. ${\phi}_{T}(\kappa )=1$, when wavenumber $\kappa $ is smaller than crosspoint ${\kappa}_{m}$; and for those $\kappa $ larger than ${\kappa}_{m}$, Eq. (2′) is adopted to decide the value of ${\phi}_{T}(\kappa )$ (See the curve with hollow points in Fig. 1). In this way, the value of the turbulence spectrum is assured to change continuously with the wavenumber. The results show that ${\kappa}_{m}\approx 0.16/{\eta}_{k}$which fits the requirement of$\kappa <<1/{\eta}_{k}$. By doing so, the power-law will be bigger than $\alpha $when${\kappa}_{m}\le \kappa <1/{\eta}_{k}$, which is a good match with experimental results [23, 24].

Although the outer-scale has little influence on light scintillation, it should be taken into account when dealing with real turbulence spectra. Here we use this following Eq. (3) to show the effects of the outer-scale [3],

Only 1D temperature spectrum can we obtain from the measurements and the transformation from 1D spectrum${E}_{1}({\kappa}_{1})$ to 3D spectrum ${\Phi}_{T}(\kappa )$ can be deduced as follow:

We are going to use Eq. (5) to convert 1D spectrum to 3D spectrum in this paper.

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* [25],

There is an approximately linear relationship between the refractive index of water and the temperature, and it can thus be seen that the temperature spectrum has the same shape as the refractive index spectrum. Therefore the refractive index spectrum ${\Phi}_{n}(\kappa )$ has the same statistical characteristics described in Eqs. (1)–(5). Accordingly, the refractive index structure constant${C}_{n}^{2}$, spectral power-low$\alpha $, outer-scale *L _{0}* and inner-scale

*l*can be computed by Eq. (3).

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

In order to carry out numerical simulations, we need to carefully and properly set the minimum distance between grids and the number of grids, so that 2D light intensity fluctuations simulated will contain most of the energy. Here we give the theoretical expression of 2D scintillation fluctuation spectrum. After the lightwave propagates a distance *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].

*I*denotes light intensity, $\stackrel{\rightharpoonup}{\kappa}$ is the wavenumber of light intensity fluctuation spectrum,

*k*is the wavenumber of light wave.

However, Eq. (7) is inadequate to describe strong light intensity fluctuations. Turbulence tends to be quite strong in the water. Hence we should adopt asymptotic theory to describe scintillation fluctuation spectrum. The high-frequency part of scintillation spectrum can be approximately shown as [4]:

The low-frequency part of the asymptotic scintillation spectrum can be shown as:

_{.}

Based on Eqs. (8) and (10), we will decide the settings of numerical simulation grids in Section 4.

## 3. Water tank simulation experiments

#### 3.1 Experimental facilities

Our experiment was carried out in a rectangular water tank (see Fig. 2). The height, width and length of this water tank are 600mm, 1500mm, and 1500mm respectively. The height of the simulated boundary layer is no more than 300mm and thus aspect ratio maintains over 3. There are 10mm transparent glass plates surrounding the water tank. In order to reduce heat loss, we attach a patch of 30mm thick sponge to the outside of the glass plate (only keeping a few parts clear for optical measuring). A 1450mm × 1450mm × 60mm oil tank for heating is located at the bottom of the water tank. The oil tank is made of 2mm thick steel plates and filled with high-insulating and low-expanding transformer oil. There are 39 electric heating tubes (each tube has the maximum heating power of 1 KW) installed in the oil box to first heat the oil, and then the bottom of the tank. The heating tubes are capable of dissipating a maximum of 39 KW. With this indirect heating mechanism, the tank bottom is heated uniformly. Although the oil is low-expanding, it is still expanded to some extent during the experiments (the heat duration is about 50 minutes). An extra box is linked to one corner of the oil tank to contain the expanded oil. There are 25mm gaps between the lateral side of the water tank and the oil tank. In order to insulate the lateral heating (which may lead to organized upward motions along the lateral side of water tank), the gaps are filled with sponge. Thus the water is heated only by the upward face of the oil tank. This set-up can meet the requirement of similarities and accurately simulate the ABL [13, 14].

The de-gassed and de-ionized water was used as the working fluid, which permitted sufficient observation time under the conditions of large heating rates. To prepare for an experiment, the tank was first filled with 200mm room-temperature water. Then a plastic-foam plate, with many regularly distributed holes in it, was laid above the water. The hotter water was carefully added onto the floating plate to generate a stably stratified distribution in the tank. The temperature gradient can extend to the bottom of the tank by vertical exchange stimulated by the downward motion of the hot water. The inversion strength can be varied from 10 K m^{−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

Ten thermocouples are mounted at different heights on a metal pole with one end fixed on a horizontally moving vehicle (see Fig. 2). The vehicle moves horizontally to measure the horizontal distributions of temperature at different heights along the light beam direction. The speed of the car is set to 20mm s^{−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.

The main optical instruments used in our experiments are a set of collimated beam system and CCD for optical acquisition. The collimated beam system made twice beam expanding and one time collimation so that the thin beam emitted from He-Ne laser finally turned into 200mm diameter circular collimated beam, which can be approximately viewed as a plane wave. The collimated beam 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 screen 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 air. The image was captured by CCD and collected by computer whose sampling frequency is up to 30 Hz. The obtained image is saved as a gray-scale photograph, of which gray level is 256 and photograph resolution is 1024 × 1280 pixels (see Fig. 6). The spatial resolution of the photograph is about 5 pixels mm^{−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]. But if the mean value of image is too large, some pixels will appear to be saturate. Through continuously attempts during the experiments, it turns out that the ratio of saturated pixel is less than 1% when we adjusted the CCD aperture to make the mean gray around 80. All these parameters will be applied to calibrate the numerical simulated results so that we can later compare the numerical simulations with the results from water tank simulations. More details can be viewed in next sections.

## 4. Numerical simulating method

The algorithm in Martin et al. [28] was adopted to build up a numerical model. This algorithm separates the entire propagation path into many independent parts in *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.

According to asymptotic theory (see Eqs. (8) and (10)), high-frequency energy can be distributed to 20mm^{−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.

The numerical simulation is carried out by means of the above mentioned method to compare results with measurements in the water tank. The real case in the water tank measurement is that the whole path can be taken as two parts, the first in the water tank and the second in the in the air where there is almost no turbulence. In order to agree with the real situation, the numerical simulation treats the light propagation in the last 500mm (for simplicity) as in vacuum. In this situation, the phase screen is set as *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.

The spatial distribution of numerical simulated light intensity was converted into gray-scale image so as to quantitatively compare numerical simulated results with water tank simulated ones. The average gray-scale was fixed at 80 and meanwhile was added an average value of 7 as random noise. These settings will have certain effects on extremely weak fluctuations and extremely strong ones (namely the scintillation number *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

To analyze the characteristics of light propagation in the ML and their relations with temperature field, we measured the horizontal temperature distributions at different heights. Based on the temperature distribution, characteristics of the horizontal refractive index can be obtained and then applied to numerical simulation. Ultimately, we are able to compare numerical results with the measurement results of light propagation.

The temperature fluctuations from each sensor for every run were averaged and the mean
temperatures with heights at several moments can be obtained, as shown in Fig. 3.Nine lines in Fig. 3 were measured at the moments
of 0s, 458s, 755s, 984s, 1370s, 1665s, 1991s, 2210s and 2591s. The 0s line is the initial
temperature profile. The inversion in the tank was about 70 Km^{−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.

Figure 4 shows temperature variation at 7 different heights at the moment of 1991s. For the lower 4 levels (located in the ML), the temperature fluctuations decrease with height. But at the level of 110mm (in the EZ), the temperature exhibits larger fluctuations. The measured results coincide with the observations in real atmosphere, that temperature fluctuation decreases with height in the ML yet has large value in the EZ [29, 30]. Therefore the thickness of the CBL is around 110mm.

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 5 shows the
results at 30mm height (see the dots). According to the fact that the turbulence in the ML is
isotropic, we can adopt Eq. (5) to fit the
parameters of 3D refractive index fluctuation spectra. The refractive index structure constant
equals 7.1 × 10^{−9} m^{-2/3}, the outer-scale
*L _{0}* is 0.25 m, the inner-scale

*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.

Since the 1D refractive index fluctuation spectrum comes from the 3D one based on the isotropy assumption, the results shown in Fig. 5 demonstrate the fact that, the turbulence in the mixed layer simulated in the water tank is isotropic and the random variations of temperature as well as refractive index variations are in conformity with the “-5/3” law in the inertial sub-range. It has been well known that the temperature fluctuation in the real ABL meets the “-5/3” law in the inertial sub-range under unstable conditions [31]. Continuous heating at the bottom drives convective motion in the water tank. When the Rayleigh number or the Reynolds number exceeds the critical values, the turbulence is fully developed and reaches the state of isotropy. The Rayleigh number or the Reynolds numbers in the atmospheric CBL are several orders higher than those in water tank. The results in Fig. 5 imply that the turbulence in the real atmosphere, especially in the CBL, can be regarded as isotropic.

#### 5.2. Scintillation index and boundary layer structure

Spectral analysis requires a certain length of data sequence. But the photograph recorded by the CCD is approximately round in shape. In order to obtain the characteristics of turbulence field from surface to the ABL top, we adjusted the height of the collimated light system to set the center of the light beam 40mm above the surface. As shown in Fig. 6(a), the part of photograph with bright streaks and dark areas, which are caused by scintillation effect of turbulence on the light propagation, is wide enough. In the lower part of the photograph, bright streaks and dark areas are homogeneously distributed (this part is corresponding to the mixed layer, in which the average temperature is constant with height and the turbulence is isotropic), while in the upper part of the photograph, a series of horizontal bright streaks appears (this part is corresponding to the EZ, in which the average temperature increase with height and the turbulence is anisotropic). However, on the top only a few bright streaks sparsely remain, which are the several convective cells reaching the maximum height. In the above region, there is no turbulence and consequently no irradiance fluctuation.

Figure 6(b) shows a curve of normalized SI varies with height. The SI $\beta $, is defined as:

where $\u3008\u3009$denotes spatial average,*I*is the instantaneous value of light intensity (in our experiment, it’s the gray-scale value of CCD image). $\beta $slowly decreases with increasing height, reaches the minimum value at the height of 90mm, then gradually climbs to the maximum value at the height of 121mm and finally falls drastically. The structure of the CBL can be detected by the vertical change of SI [32, 33]. Below the height is the ML where lies the minimum$\beta $and above it is the EZ. The 121mm height corresponding to the maximum value is the level of the CBL top, which is very close to the location where the largest temperature fluctuation lies in Fig. 4. Judging by the temperature profile (see Fig. 3), the EZ is stably stratified. In this paper we only focus on the behavior of light propagation in the ML. As for the characteristics of light propagation in the EZ, we will exhibit the results in another paper [20].

#### 5.3 Characteristics of spatial scintillation spectra

Usually 256 pixel sequence length is used for calculating 1D power spectra, which equals 50mm in physical space (and is about two times the distance between temperature probes). Characteristics of a specific location cannot be revealed if the data sequence is excessively long. But if the data sequence is too short, the sample representative will be lacking and fluctuation will be too large. In order to get the smoother spectra curve and detect the peak location more easily, we apply the auto regression (AR) spectra method [34] to attain spatial spectra. Figure 7 gives the 1D light intensity fluctuation power spectra along horizontal and vertical directions at different heights. The x-coordinate is the spatial wave number and y-coordinate is the normalized power spectral density (multiplied by wavenumber and divided by variance). The central heights are located at 31.2mm, 58.3mm, 85.4mm and 121.6mm respectively. In Figs. 7(a)–7(c), the heights are located in the ML, and the horizontal and vertical spectrum almost overlaps each other, which means that the turbulence in the ML is isotropic. However, Fig. 7(d) shows that the spectral in horizontal and vertical directions are located in different places, which indicates that the turbulence in the EZ is anisotropic.

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. 7. In Figs.
7(a)–7(c), the corresponding peak
wavenumber of horizontal fluctuation spectra (0.173, 0.194 and 0.162 mm^{−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

Numerical simulations are carried out by using the parameters of refractive index spectra
derived from the measurements in the water tank. Figure
9(a) shows a numerically simulated light intensity fluctuation image at 30mm height.
Figure 9(b) shows the corresponding normalized light
intensity fluctuation spectrum from water tank experiment and numerical simulation. Figure 9(a) exhibits a very similar pattern to the measured
image shown in Fig. 6. Figure 9(b) indicates that there is a good agreement between the measured and the
simulated spectra. For the purpose of comparing to the measurement results, the SIs from
numerical simulations at the 7 different heights are also listed in Table 1. At the lowest height (10mm above the tank bottom), the result from
numerical simulation indicates that the irradiance fluctuations are in saturation regime
(*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.

As shown in the last but one column in Table 1, the SIs obtained by numerical method, in which the average gray scale and the noise are considered, are fairly close to those calculated from water tank measurements in the ML (at the levels of 30mm, 50mm, 70mm and 90mm). In order to know the influence of average gray scale and noise on the numerically simulated results, as well as the change of SI of the light beam going through the part of the route in the atmosphere, the SIs of two ideal cases are calculated, in which the average gray scale and the noise are not considered. One case is that the light beam only passes through the water (i.e., *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.

The light propagating route in this study can be considered as inhomogeneous (*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 $\beta =\text{77}\text{.2}\cdot {l}_{\text{0}}^{\text{-7/3}}\cdot {C}_{n}^{2}$ can be derived from the weak fluctuation theory on light propagation in the heterogeneously distributed turbulent medium [1] (the derivation is given in Appendix). It can be used to calculate the SIs at different heights in the water tank experiment. The analytic values of SI at the heights of 10mm, 30mm, 50mm, 70mm and 90mm are 7.1, 0.97, 0.36, 0.28 and 0.20 respectively. At the relatively higher levels (50mm, 70mm and 90mm), the analytic SIs are in good agreement with those from the numerical simulations of the ideal case for *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 $\beta =\text{3}3.18\cdot {l}_{\text{0}}^{\text{-7/3}}\cdot {C}_{n}^{2}$ (see the Appendix). The analytic SIs at the different heights are 3.05, 0.42, 0.15, 0.12 and 0.09 respectively. Except for the lowest height where the scintillation is in saturation, the analytic SI at each level in the ML agrees well with the one from the numerical simulation of the ideal case for *L* = 1.5m. Due to lack of inner scale measurements in our previous experiments [33], the formula $\beta =1.23\cdot {C}_{n}^{2}\cdot {k}^{7/6}\cdot {L}^{11/6}$ was applied empirically. Now we think this formula is not suitable for the water tank experiments.

The agreement between the measured and theoretically calculated SIs suggests that the experimental methods used in this study are reasonable. Even for the situation that the fluctuation is relatively large but the scintillation is not in saturation, e.g., at the height of 30mm, the SI from numerical simulations can still agree well with that from the water tank experiments. So the agreement between the results from numerical simulations and water tank measurements also implies that the parabolic approximation to the wave equation of light propagation is applicable. The case of the EZ is not same as that of the ML and will be discussed in another paper [20].

When the beam propagation covers only a short distance, the inner-scale is much larger than Fresnel-scale, and the main scale of the light intensity fluctuation spectra is turbulence inner-scale. Further analysis based on numerical simulations has been conducted. The results show that, the peak wavelength of 1D power spectrum is controlled by the inner-scale of turbulence, and the former is about 2.5 times of the latter. According to the detected peak wavenumber at different heights in the ML (as shown in Fig. 8), the average value of inner-scale of the horizontal light intensity fluctuation spectra is 2.01mm and the value of the vertical is 1.94mm. On average, the inner-scale of turbulence in the water tank is about 2.0 mm, which is very close to other measurements in the water tank experiments [15–17].

## 6. Conclusion

By using water tank experiments and numerical simulations, we investigate the characteristics of 2D intensity fluctuation of light propagation in the ML. Some conclusions are listed as follows:

(1) 2D fluctuation fields of plane wave propagation in turbulence can be obtained. When a plane wave propagates in the ML, 2D fluctuation fields appear to be isotropic in the cross section perpendicular to the light path.

(2) Using parameters of turbulence spectra attained by measuring the temperature directly, numerical simulations have been carried out. Consistent features between these numerical simulations and real measurements in the water tank are identified.

(3) Via both numerical and water tank simulation methods, the inner-scale of turbulence in the water tank, which is 2.0mm, has been determined.

This paper mainly discusses the influences of turbulence on light propagation in the ML. Since the turbulence field in the ML shows isotropic features, knowing the related parameters, the existing theoretical model can accurately describe the scintillation characteristics of light in turbulent mediums. However, in the EZ (at the top of the CBL), anisotropy exists. This leaves a problem: how to build a suitable theoretical model to describe the scintillation characteristics of light propagation in the EZ. We will investigate this problem and give our answers in another paper [20].

## Appendix

Based on Tatarskii, logarithmic amplitude fluctuation for the light propagating through a path, in which the turbulence is heterogeneously distributed, can be calculated by

*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.

In this study, *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 ${C}_{n}^{2}\ne 0$ and the second part [1.5, 2] has no turbulence with${C}_{n}^{2}=0$. In the first part [0, 1.5], ${C}_{n}^{2}$ is constant with distance since the turbulence in the mixed layer is isotropic and homogeneous. So Eq. (12) can be calculated as

Then, the SI at *L* = 2.0m can be calculated by

If light propagation only in the water tank is considered, namely, the observation point is at the edge of the water tank, the integration limit is just [0, 1.5], and Eq. (12) can be directly integrated to as

*L*= 1.5m can be calculated by

## 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.

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