1. Introduction

Turbulence in the atmosphere is a well-known negative factor that corrupts a propagating laser beam. Atmospheric turbulence is mainly a result of negative temperature gradient between the surface of the Earth and the atmosphere. Since the optical turbulence is a direct result of the negative temperature gradient, both negative (daytime) and positive (typically night) temperature gradients produce surface layer turbulence. Peak ${\text{C}}_{\text{n}}^2$ values may be comparable, although daytime ${\text{C}}_{\text{n}}^2$ is generally higher. During daytime, the Earth absorbs sun radiation and becomes hotter than the air, causing the air nearest the ground to be hotter than that above. The negative temperature gradient (with respect to the direction upward) results in convection currents in air, which at a stronger gradient eventually break into multiple turbulent convection eddies or cells of many different sizes (Fig. 1). Wind speed fluctuations could also contribute to the creation of eddies. Each of these factors create regions of high or low refractive index, which are randomly changing. The fluctuations of the refractive index of the atmosphere associated with eddies are known as optical turbulence. When a laser beam propagates through turbulent air, the phase front gets corrupted by encounters of random heterogeneities in the refractive index. Also, the magnitude, path distribution of the phase gradient and propagation distance will contribute to the degradation of the beam. As a result, the intensity of the beam at the position of the receiver fluctuates or scintillates, and the beam wanders and widens beyond the diffraction limit of the transmitter aperture. This significantly limits the performance of the systems using laser beams propagating through the atmosphere. Since the optical turbulence is a direct result of the negative temperature gradient, it seems natural to hypothesize that clouds blocking the sunlight and preventing the surface of the Earth from heating could decrease or eliminate the temperature gradient (Fig. 2) and thus mitigate the negative effects of the turbulence.

 

Fig. 1. Scattering of light rays in turbulent air near hot ground (negative temperature gradient).

Download Full Size | PDF

Optical turbulence can be characterized by the refractive index structure constant $C_n^2$ , which for homogeneous and isotropic turbulence can be defined as [1]

where n($\stackrel{\rightharpoonup }{r}$ ₁) and n($\stackrel{\rightharpoonup }{r}$ ₂) are the indices of refraction at points $\stackrel{\rightharpoonup }{r}$ ₁ and $\stackrel{\rightharpoonup }{r}$ ₂. ${\text{C}}_{\text{n}}^2$ has dimensions of m^-2/3 and varies from approximately 10^-17 to 10^-12 m^-2/3 in the atmospheric boundary layer. Equation (1) is valid when the distance |$\stackrel{\rightharpoonup }{r}$ ₁-$\stackrel{\rightharpoonup }{r}$ ₂| is within approximately 10 cm and 100 cm in the boundary layer[2].

 

Fig. 2. Build-up of a negative temperature gradient when the Sun heats the Earth (a) and elimination of the gradient when the sunlight is blocked by a cloud (b).

Download Full Size | PDF

In this paper, the hypothesis regarding cloud-cover mitigation of the turbulence is validated using a statistical analysis of a substantial experimental data base on $C_n^2$ , wind speed, solar irradiance, and other weather factors collected over a period of 10 months. Section 1 describes the experimental approach. Section 2 presents experimental results, data of statistical analysis, and discussion. Section 3 describes cloud cover mitigation and the effects during day time.

2. Experiment

The refractive index structure constant $C_n^2$ was measured with a LOA-004-AR long baseline, atmospheric turbulence sensor and optical anemometer from Optical Scientific, Inc.[2] It must be noted that this instrument is sensitive to crosswind wind speed, which is one measurement for which it is made. The measurements were made during 10 months starting in late Spring of 2004 at Redstone Arsenal, Huntsville, North Alabama (860 35’ west longitude, 340 44’ north latitude, 177 m above sea level). The site is a level grass-covered field on a marshy soil that is 100% moist during the spring and summer. The field was surrounded by a few 10-m high trees and one-story buildings clustered on one side of the receiver. The buildings were at least 10 m and parallel to the laser path. A distance of 500 m separated the scintillometer source and the receiver. The propagation path was oriented east-west before May 10, 2004 and north south after May 10 with prevailing winds coming from west or southwest. The height of the propagation path was 1 m above ground. The sensor used optical scintillation as the method of measuring optical turbulence in terms of $C_n^2$ . Scintillation is a general term, which describes changes in brightness of a light beam passing through the atmosphere. The schematic of the experimental system is presented in Fig. 3. Transmitter 1 emits a modulated beam of infrared light (0.94-µm wavelength). Dual channel receiver 2 detects this beam and reconverts it to an electric signal. Variations measured between the transmitted and detected signals caused by the scintillating air parcels provide the basis for the measurement of $C_n^2$ .

where k=2π/λ is the wave number (λ is the wavelength), d is the distance between the transmitter and the receiver, ${\sigma }_I^2$ is the intensity scintillation index

where 〈I ²〉 and 〈I〉 are the average square intensity and average intensity, respectively, of the light coming to both modules of the receiver simultaneously. $C_n^2$ processor 22 (see Fig. 3) of the instrument computes the refractive index structure constant using Eqs. (2) and (3). The dual modules of the receiver make it possible to measure cross wind speed. This is done by detecting the time difference between the two signals as the air parcels move across the beam path determined by the time-varying intensity log-amplitude covariance B_I (τ)

where Φ_n (κ) is the Kolmogorov spectrum of the refractive index structure constant (Φ _n (κ)=0.033$C_n^2$ κ ^-11/3), J ₀ is the Bessel function of the zeroth order, and V_c is the cross wind speed. V_c is extracted from the time derivative dB(0)/dτ after the refractive index structure constant has been computed [3]. Covariance processor 23 of the instrument performs the previous computation. The turbulence and covariance data are then processed in microprocessor 26 to compute averages, calculate engineering units, perform system diagnostics, and communicate with a computer. The system provided the data on $C_n^2$ and cross wind speed every 10 s.

 

Fig. 3. Schematic of the experimental scintillometer for the measurement of the refractive index structure constant.2 It consists of three major units: transmitter 1, receiver 2, and signal processing unit 3. Transmitter 1 includes LED 4, mirror 5, and LED modulator 6. Light emitted by LED 4 propagates through turbulent air 7 along two paths 8 and 9. The receiver consists of two identical sub-units. Each of them has mirrors 10 and 11, photodetectors 12 and 13, detector control units 14 and 15. Signals 16 and 17 from the receiver sub-units enter the processing unit 3. Signal processing unit 3 consists of two identical channels made of preamplifiers 18 and 19 and demodulators 20 and 21. Signals from demodulators 20 and 21 enter $C_n^2$ processor 22 and covariance processor 23. $C_n^2$ data is stored in unit 24, and covariance. The refractive index structure constant can be obtained from equation[2] data is stored in unit 25. Both sets of data are sent to digital processor 26, which sends the data to a computer.

Download Full Size | PDF

Simultaneously, every 10 s the data on solar irradiance was provided by a solar radiometer (Weathertronics star pyranometer model 3020). Additionally, every 15 min the following parameters were measured: peak wind speed and wind direction (with an R. M. Young wind monitor model 05103), relative humidity, atmospheric pressure, temperature, and the temperature gradient (the difference between two temperature readings at 3 m and 0.5 m above the ground respectively).

 

Fig. 4. (A) $C_n^2$ and (B) solar irradiance for May 11, 2004. Each data point was obtained by averaging over a 15 minute interval.

Download Full Size | PDF

3. Experimental results and discussion

$C_n^2$ data clearly show a well-defined peak during mid-day hours and minima near sunrise and sunset. As can be deduced from Fig. 4, during mid-day hours $C_n^2$ also drops whenever thick clouds block the sunlight (for instance, drops 1, 2, and 3 in Fig 4A). These preliminary results indicate that there might be strong a relation between $C_n^2$ and solar irradiance controlled by cloud-cover. Figure 5 shows the raw data on the crosswind speed, solar irradiance, and $C_n^2$ with a frequency of one data point every 10 s in the middle of the day, when the level of solar radiation is near its maximum and does not significantly change. Both the crosswind speed and $C_n^2$ are measured by the scintillometer/anemometer. The drop in solar irradiance is primarily due to cloud cover. Cloud cover was based on the observers taking the data and is subjective in nature. The data in Fig. 5(a) represents the midday of June 21, 2004 when the sky was mostly clear with few scattered clouds that appeared between 1:35 pm and 2:10 pm. The data in Fig. 5(b) represents the midday of July 7, 2004 when scattered clouds were continuously passing across the sky. The data in Fig. 5(c) represents the midday of July 12, 2004 when there was almost continuous overcast accompanied by occasional precipitation. The data in Fig. 5 thus represent a wide variety of conditions, from clear sky to complete cloud-cover, and can be used for the analysis of the relationship between cloud-cover and $C_n^2$ .

It can be seen from Fig. 5 that the drop in $C_n^2$ frequently occurs when there is a drop in solar irradiance associated with cloud-cover. Similar effects could be seen in the data obtained by D.L. Hutt [4]. In order to conduct a detailed analysis of this relationship, log of $C_n^2$ was plotted versus log of solar irradiance as is shown in Fig. 6. The plot in Fig. 6(a), corresponding to June 21, shows most of the data points clustered in a narrow region of high $C_n^2$ and high solar irradiance.

There are a certain number of points widely scattered in the direction of low irradiance associated with scattered clouds. However, a relationship cannot be clearly seen from these widely scattered points. The only conclusion that can be derived here is that the refractive index structure constant remains high when the solar irradiance is relatively constant and high (clear sky). The plot in Fig. 6(b) (July 7) corresponds to mixed conditions of clear sky with a few scattered, thick clouds. A certain number of data points still cluster near the “clear sky” region. But now a significant number of data points spread towards the low irradiance-low $C_n^2$ region. It becomes reasonable to perform a linear regression fitting of the data (straight line in Fig. 6(b)). The data points to some extent tend to cluster along the fitting line. The Pearson’s correlation coefficient r calculated for this data set is 0.75. This is not a proof of strong correlation, but rather of positive association between the solar irradiance and the refractive index structure constant. Figure 6(c) (July 12) corresponds to a mixture of scattered and mainly continuous cloud conditions. The data points are evenly distributed in a wide range of solar irradiances. The correlation factor here is 0.74 indicating a positive association between the irradiance and $C_n^2$ . As can be seen from Fig. 5(c), the last fragment of data record (from 11:35 am till 12:00 pm) for July 12 corresponds to an almost continuous cloud-cover with a slowly varying density. The data plot for this time interval is presented in Fig. 6(d), which consists of points taken from Fig. 5(c) belonging to the time interval 11:35–12:00. The spread of the data points with respect to the fitting line is much narrower now, comparing to that in Fig. 6(b). Correspondingly, the correlation coefficient is now 0.93. This indicates a strong correlation between the solar irradiance determined by the cloud cover and $C_n^2$ . In order to check for some possible relation, $C_n^2$ was plotted against crosswind speed for July 12 (Fig. 7). The data points are very widely spread with respect to the linear regression line, and the correlation factor is only 0.105. Crosswind is only a small portion of the total wind speed, which changes direction often enough to create an anti-correlation. This implies that within the limits of the observed values of crosswind speed the refractive index structure factor does not depend on crosswind.

 

Fig. 5. (A) Crosswind, (B) solar irradiance, and (C) $C_n^2$ for (a) June 21, (b) July 7, and (c) July 12, 2004. Data was taken in ten second intervals. cloud cover as varied from10% to 90% based on estimations by observer.

Download Full Size | PDF

 

Fig. 6. Log [$C_n^2$ ] plotted versus log [Solar irradiance] for (a) June 21, (b) July 7, and (c), (d) July 12 2004. The plots are obtained from raw data presented in Fig. 3. Data points in plot (d) correspond to time between 11:35 am and 12:00 pm when there was a continuous cloud-cover with variable density. Pearson’s factor r of correlation between log [$C_n^2$ ] and log [Solar irradiance] is 0.75 for plot (b) 0.74 for plot (c), and 0.93 for plot (d). Cloud cover was between 10% and 90% based on the estimations of the observer.

Download Full Size | PDF

Similar results (with less data points though can be derived from the plots of $C_n^2$ versus peak wind speed (Fig. 8), relative humidity (Fig. 9), atmospheric pressure (Fig. 10), and direction of the wind (Fig. 11). Peak wind speed is the highest wind speed in a 15 minute interval. There is a weak positive association of $C_n^2$ with the temperature (Fig. 12) and unexpectedly weak negative association with the temperature gradient (Fig. 13). The latter result is not in full agreement with the initial hypothesis that $C_n^2$ was linked to solar irradiance through the temperature gradient. Possible explanation is an improper placement of the temperature sensors for the evaluation of the temperature gradient. The top sensor is placed at an elevation of 3 m and the bottom sensor is at 0.5 m above the ground. Apparently, the temperature gradient between these points is not sensitive enough to the irradiance. This issue has been previously discussed in the literature [2] and the recommendation was derived for the lower sensor to be placed below the surface. Unfortunately, that was not possible in the setting at which the experiments were conducted.

 

Fig. 7. Log [$C_n^2$ ] plotted versus the speed of the crosswind for July 12 2004. The plot is obtained from raw data presented in Fig. 3. Correlation factor is 0.105.

Download Full Size | PDF

 

Fig. 8. Log [$C_n^2$ ] plotted versus the peak wind speed for July 12 2004. The data points were taken at a rate of one point per every 15 min.

Download Full Size | PDF

 

Fig. 9. Log [$C_n^2$ ] plotted versus the relative humidity for July 12 2004. The data points were taken at a rate of one point per every 15 min.

Download Full Size | PDF

 

Fig. 10. Log [$C_n^2$ ] plotted versus the static atmospheric pressure for July 12 2004. The data points were taken at a rate of one point per every 15 min.

Download Full Size | PDF

 

Fig. 11. Log [$C_n^2$ ] plotted versus the direction of the wind for July 12 2004. The data points were taken at a rate of one point per every 15 min.

Download Full Size | PDF

 

Fig. 12. Log [$C_n^2$ ] plotted versus the temperature for 7 July 2004. The data points were taken at a rate of one point per every 15 min. Correlation factor is 0.62.

Download Full Size | PDF

 

Fig. 13. Log [$C_n^2$ ] plotted versus the temperature difference for 7 July 2004. The data points were taken at a rate of one point per every 15 min. Correlation factor is -0.59.

Download Full Size | PDF

However, one might question the generality of these conclusions. Moreover, some authors report on strong correlation between the structure constant and Cross wind speed [5,6,7,8,9]. In order to respond to these concerns, the correlation coefficients between the irradiance and the structure constant and also between the cross wind speed, total wind speed, peak wind speed and the structure constant were calculated for the data taken during the midday at various weather conditions for 48 days over an extended period of time between April 2004 and January 2005 (see the Table 1). Each day, the correlation between $C_n^2$ and the irradiance was higher than that between $C_n^2$ and the cross wind speed, total wind speed and peak wind speed. The average correlation factor in the former case was 0.802 while in the latter case it was just 0.127. One explanation is that the wind direction changed significantly enough to reduce the cross wind component so that the correlation between $C_n^2$ and cross wind was reduced. The correlation coefficient for the total wind speed and peak wind speed were consistently greater than that for crosswind speed. When the wind direction was consistent, the correlation coefficients for crosswind, total wind and peak wind speeds were consistent. The wind direction on 26 October 2004 was consistent enough to produce a correlation coefficient of 0.50. When compared with days when the correlation coefficient for cross wind was the lowest, the wind direction on October 26 was relatively constant. Figure 11 shows wind direction changing relatively often throughout the day with respect to the north south line of measurements that were taken. Figure 8 shows the peak wind speed taken in intervals of 15 minutes showing a relatively good correlation. In order to get good correlations with cross wind speeds, measurements must be taken in areas where the wind direction is more of a constant. Based on this, the conclusion can be made that in the location where the experiments were conducted (northern Alabama) the optical turbulence in the near-surface atmospheric layer (1 m above ground) significantly diminished at cloud-cover conditions and changed just slightly during the variation of the total wind speed. Figure 14 provides even more convincing proof of strong correlation between the solar irradiance and the refractive index structure factor. The data correspond to June 28, 2004 when the cloud-cover came along with rain. Despite some scattering of the laser beam by the raindrops, there is still a strong correlation between the irradiance and $C_n^2$ . The correlation factor here is r≈0.96.

 

Fig. 14. Log [$C_n^2$ ] plotted versus log [Solar irradiance] for 28 June 2004. The data points were taken at a rate of one point per every 15 min. There was rain and overcast through the day. Correlation factor is 0.95.

Download Full Size | PDF

4. Cloud-cover mitigation and the effects of time of day

In the previous section the correlation between the cloud-cover controlled solar irradiance and $C_n^2$ was analyzed for a short period of time during the midday when the solar irradiance is at its maximum and stays relatively constant, assuming clear sky. The interpretation of the experimental data was made that the irradiance is the major factor influencing $C_n^2$ . The data points in the graph of $C_n^2$ versus the irradiance, instead of clustering near a straight line like in Fig. 14, make a loop consisting of two halves shown in Fig. 15 (see also, for example, Fig. 6 in Ref. [5]). The reason for the variations of $C_n^2$ is due to the drop of the irradiance caused by cloud cover, which is mixed with variations in ground conduction, latent and sensible heat. This creates a time lag between solar insolation and the other energy budget terms such as ground conduction, latent and sensible heat. The main source of radiation during the day and the night is sunlight and heat stored in the soil respectively. $C_n^2$ at sunrise and sunset will not correlate with solar irradiance because the energy budget is not dominated by any one term of the energy budget that was mentioned. The temperature of the ground does not equal the temperature of the air until 1.5 hours after sunrise and 0.5 hours before sunset because there is a phase shift associated with the thermal mass thermal conductivity and emissivity of the earth’s surface. [10, 11] When 24 hour data at Huntsville was compared with data taken in the desert, one notable difference was seen in that $C_n^2$ values in the desert were higher. The presence of a marsh as opposed to a desert would produce a significant difference. The overall shape of the graphs for both desert and marshy land is similar in nature. A simple direct correlation from sunrise to sunset using solar irradiance, and $C_n^2$ was not possible. However, there was a model presented in Ref. [12]. showing that it is possible to use environmental information and derive an accurate model. One piece of environmental information that cannot be used is cross wind speed which has no correlation to solar irradiance. The average correlation for cross wind speed, total wind speed and peak wind speed were 0.13, 0.39 and 0.42 respectively. An attempt was made to separate effects of cloud-cover from the effects of the position of the sun. The envelopes of the data on the irradiance and on $C_n^2$ taken for the entire day were manually derived. Then the drops of the irradiance and $C_n^2$ associated with the cloud-cover were calculated as differences between the envelopes and the data. The plot of the drop of $C_n^2$ versus the drop of irradiance for May 11, 2004 is shown in Fig. 16. The correlation between the irradiance and $C_n^2$ remains relatively weak (the correlation coefficient is about 0.5).

Table 1. Cumulative data on the Pearsons correlation coefficients between the structure constant and the irradiance and between the structure constant and the crosswind speed for the midday during ten months of observation

View Table

 

Fig. 15. Log [$C_n^2$ ] plotted versus log [Solar irradiance] for May 11, 2004 for the whole day (as shown in Fig. 4).

Download Full Size | PDF

 

Fig. 16. The drop of Log [$C_n^2$ ] versus the drop of log [Solar irradiance] for May 11, 2004 for the whole day.

Download Full Size | PDF

5. Conclusions

Statistical analysis of a substantial experimental database shows that a continuous cloud-cover that blocks the sunlight is a significant factor that mitigates optical turbulence. Analysis of the $C_n^2$ data and its associated meteorological data show correlations proportional to their influence on $C_n^2$ . However when crosswind is compared, there is very little daily correlation. The correlations for the total wind speed and peak wind speed show increasing correlations, which indicate that peak wind speed is an important factor in creating a model for $C_n^2$ . This shows that cloud cover on the order of 10% or greater reduces divergence and extends the range for laser beam propagation. This factor must be accounted for in the planning of utilization of lasers for specific military applications, such as laser assisted aiming, laser guided and high power laser weaponry. Since there has already been work in creating accurate models the need to develop a better model for predicting cloud-cover mitigation effects over an extended period of time during the day, at different seasons and latitudes was not a priority.