The influence of optically active turbulence on the propagation of laser beams is investigated in clear ocean water over a path length of 8.75 m. The measurement apparatus is described and the effects of optical turbulence on the laser beam are presented. The index of refraction structure constant is extracted from the beam deflection and the results are compared to independently made measures of the turbulence strength () by a vertical microstructure profiler. Here we present values of taken from aboard the R/V Walton Smith during the Bahamas optical turbulence exercise (BOTEX) in the Tongue of the Ocean between June 30 and July 12, 2011, spanning a range from to . To the best of our knowledge, this is the first time such measurements are reported for the ocean.
© 2016 Optical Society of America
The ocean remains one of the least explored regions on our planet. This is primarily because of the poor transmission of light through water and the resulting limited range of imaging systems. Even in the clearest seawater, optical attenuation lengths on the order of only are encountered , with a corresponding Koschmieder visual range of about 80 m . By way of contrast, the visual range in the atmosphere is about 340 km  and virtually unlimited in the vacuum of space.
Despite these limitations, laser-based optical imaging, sensing, and communications systems are becoming more widespread for coastal and deepwater applications [4,5]. These systems, due to the high frequency and hence short wavelength as well as the fast propagation of light, can provide much higher resolution, wider bandwidth, and much lower latency than their acoustic counterparts.
Under most conditions, the visibility in natural bodies of water is limited by suspended particles. However, in clearer water, image resolution may be limited by inhomogeneities in the refractive index caused by turbulent mixing of media with different optical properties. Light scatters from these inhomogeneities, making the turbulence optically active. This is known as “optical turbulence.” In the ocean, this effect is particularly pronounced where gradients of water temperature and/or salinity exist, respectively known as thermocline and halocline. The degradative effects of optical turbulence on image formation in the ocean have been studied theoretically [6–11] and in laboratory environments, but few field experiments have been conducted [12–17] and little data exist on the effect of optical turbulence on laser beam propagation.
In order to study these effects in the field, an experimental apparatus was developed. This hardware projected a closely spaced grid of several focused laser beams toward a high-speed, large-area beam profiler over a vertical path of 8.75 m in naturally occurring turbulence. The objective of the experiment presented herein was to measure the effects of optical turbulence in clear ocean water on the propagation of laser beams and to quantitatively characterize these effects on the optical channel. To the best of our knowledge the index of refraction structure constant, , quantifying the strength of optical turbulence which is widely used in atmospheric optics, has not been determined in the ocean.
The effect of optical turbulence on the propagation of laser beams in the atmosphere has been studied in great detail. These investigations depend heavily on the particular shape of the refractive index power spectral density for the smallest scales in the turbulent structure.
The smallest structures expected in the turbulent velocity field are on the order of the Kolmogorov microscale (). At the Kolmogorov microscale, the kinetic energy is rapidly dissipated due to the viscosity of the medium,18–20],
The shape of the spectrum at these high wave numbers, where diffusion of heat and salinity smooths out the index of refraction variability, influences the beam distortion. Due to the high Prandtl number of temperature () and the even higher Schmidt number of salinity ()  as compared to for air, the spectral shape of the index of refraction variation is different for the ocean than for the atmosphere.
Figure 1 shows the atmospheric power spectrum according to Kolmogorov and Hill, as well as the spectral shape in the ocean for different contributions from temperature and salinity according to Nikishov [8,21]. The parameter indicates the contribution of temperature and salinity to the index of refraction spectrum according to Ref. , where corresponds to pure salinity contribution while indicates a temperature-dominated spectrum. For single diffusive media, i.e., only temperature or salinity gradients exist, Nikishov’s spectra reduce to the Hill spectrum identified in Fig. 1 by a single Prandtl number. The difference of the spectral shape becomes very apparent if the well-known scaling is divided out. Clearly, the atmospheric spectrum is not a good proxy for the spectral shape of the ocean in the dissipation range; therefore, one does not expect theories developed for the atmosphere to be applicable for the ocean.
In the theoretical analysis of beam-wander effects, one finds a low-pass filtering effect which blocks the contribution of scales smaller than the beam diameter. Also, scales much larger than the beam diameter do not contribute to beam wander [3,22]. Using the beam width of 3.5 mm and the estimated minimum outer scale of 1 m, the effect of the finite outer scale can be estimated via Eq. (3) to be less than 15%,4). Consequently, these models are insensitive to the specific shape of the index of refraction spectrum at the outer scale and the dissipation regime, and hence only scales within the inertial sub-range are relevant. Within the inertial sub-range, the spectral shape of the turbulence has the slope independent of the specifics of the turbulent generation and dissipation. For these reasons, models describing the beam wander in the atmosphere should be applicable to beam wander in the ocean. The standard deviation (std) of the beam wander for a focused Gaussian beam is given by  23]. Here, is the propagation distance, is the index of refraction structure constant, and is the beam radius at the projector. Equations (3) and (4) are applicable for weak turbulence characterized by and, in the case of Gaussian beam, the additional requirement: . Here, is the Rytov variance and is the dimensionless beam parameter describing the geometry of the beam . During the deployment of the telescoping rigid underwater sensor structure (TRUSS), was between 0.0007 and 0.23 while ranged between 0.002 and 0.58. This indicates that the experimental conditions are within the weak fluctuation regime.
The index of refraction structure constant is the local strength of optical turbulence defined by
Similarly, the temperature structure parameter is defined by9) can be expressed by [24,25]
The temperature structure constant can be derived from the time series of a moving fast response thermistor. By converting time into distance via the speed of the probe and applying the frozen turbulence hypothesis, is found from the temperature profile in the ocean. With these tools, the index of refraction structure constant is accessible from the temperature profile by fitting Eq. (7) to the temperature structure constant [Eq. (6)] and using the proportionality between and [Eq. (9)].
3. EXPERIMENTAL SETUP
A laser beam propagating through an optically active turbulent region suffers random changes to the phase front, which causes distortions to the beam shape as well as beam deflections. In order to measure beam deflection, it is necessary that the relative position and orientation of the transmitter and receiver be known at all times during the experiment. To this end, a space truss was constructed which consisted of 5 m sections that, when joined, form the TRUSS. The structure was designed to be mechanically stiff in order to ensure static measurement geometry and, at the same time, be open to water flow so as to not interfere with the natural turbulence. A single space truss section, constructed from aluminum pipes, has an equilateral triangle cross section with sides of 1.38 m (Fig. 2). Up to three sections can be connected to allow for a 15 m long separation between transmitter and receiver. To deploy the TRUSS, individual sections are placed horizontally inside a deployment cradle and pushed out from the stern of the ship as new sections are added. When all desired sections of the TRUSS are assembled, the deployment cradle is pivoted into a vertical position [see transition between Figs. 2(a) and 2(b)]. In the vertical position, the TRUSS is lowered via a steel cable to the desired depth. At the desired depth, the TRUSS was held for the duration of the measurements in order to minimize the effect of turbulence production due to vertical motion of the platform. Wave motion, which also would introduce a periodic and undesirable up and down motion of the measurement apparatus, was minimal during the time of the measurements. Unfortunately, the significant wave height was not recorded and therefore a quantities value cannot be given. A detailed explanation of the deployment procedures can be found in . Transmitter and receiver housings were manufactured at Harbor Branch Oceanographic Institute and pressure-tested to 150 m. Each housing is held in place by six turnbuckles, which allow for angular adjustment (See Fig. 2) and keep the obstruction of the water flow to a minimum. Communication with the instruments is via Ethernet and power is provided from the deck of the ship by means of a 100 m cable.
Bogucki and Domaradzki have simulated the contribution of light scattering due to boundary flow over an optical phase front sensor and found that the effect is negligible compared to the volume effect of optical turbulence between transmitter and receiver . This result has been obtained for geometry where the boundary layer effects occupy a much greater fraction of the measurement volume than in the setup used in the work described herein. We therefore assume that additional turbulence due to turbulent kinetic energy production around the transmitter and receiver housing, as well as from the TRUSS structure, does not significantly alter the measured results in terms of order of magnitude. This assumption seems to be validated by the fact that the estimates of by the vertical microstructure profiler (VMP) have the same order of magnitude and are not subject to turbulence creation of the instrument. However, for a more precise measurement of optical turbulence in the ocean, these effects would have to be quantified. This is a subject of future research.
The transmitter projects a grid of beams toward a inch active area custom-built high-speed position-sensitive light detector or beam profiler. The beams are formed by a diffractive optical element (DOE; HOLOEYE Photonics AG Part#: DE-R 241) illuminated by a CW 532 nm, 5 mW DPSS laser manufactured by Laser Quantum Limited. The laser is expanded in order to fill the entirety of the DOE and then focused onto the beam profiler by a set of lenses. The large-area beam profiler consists of a pixel CCD monochrome high-speed camera (Phantom Miro eX4) equipped with an 8 mm objective (Thorlabs Part# MVL8M1), which records an intensity image from the back side of the ground glass plate (B&H Photo & Electronics Corp., Part#: 211221). Due to the spreading of the beam grid (0.17° between neighboring beams) and the limited size of the detector, only about 20 individual beams fall onto the ground glass plate, where they can be recorded by the camera. Each laser spot is resolved by about 10 CCD pixels across the beam diameter. At full resolution, the camera’s internal 1 Gbit of high-speed RAM holds 2178 frames. A schematic of the experimental setup as well as a typical image of the grid of laser beams captured by the camera is shown in Fig. 3. During the Bahamas optical turbulence exercise (BOTEX), two sections of the TRUSS were joined to form an optical path of 8.75 m between the exit window of the receiver and the entrance window of the transmitter.
4. DATA ACQUISITION
On July 10th, 2011 conditions at the Tongue of the Ocean site prevailed suitable for the deployment of the TRUSS. Measurements were taken during a day cast (11 AM–12:30 PM) and a night cast (1 AM–3:30 AM the next day) at coordinates 25° and 25° , respectively. The range of TKED and the fine structure of the temperature profile were determined from several VMP casts taken in between the day and night TRUSS deployments. The locations, just off the Coast of New Providence Island, and times of the various measurements are shown in Fig. 4. Due to the different operational principles of the sensors used to measure TKED and temperature, namely a shear probe and a fast-response thermistor, the fall velocity of the VMP had to be adjusted in order for each sensor to perform. This means that the temperature fine structure is extracted from slow VMP casts with a fall speed of and the TKED is determined from fast VMP casts with fall speeds of . The TKED was found to be between and . It is evident from the VMP data, shown in Fig. 5(c), that strong turbulence was present near the ocean surface, probably related to near-surface flow dynamics such as surface waves, wind-driven flow, tidal currents, and/or convective mixing due to diurnal heat flux variability. Beneath the surface zone, between 20 and 55 m of depth, there was a relative calm constant temperature zone (28.7°C) with low TKED. The TKED increased again at the top of the thermocline layer, i.e., between 55 and 80 m. Below 80 m, another calm and constant-temperature (26.4°C) region followed at depths from 80 to 105 m before the turbulence strength increased again. Figure 5(a) presents the temperature profile of three slow VMP casts as well as the temperature profiles as measured by the thermistor on the TRUSS. The location of the first strong temperature gradient in the thermocline and the magnitude of the temperature difference of about 2.5°C remained the same over the duration of the various measurements.
5. RESULTS AND DISCUSSION
In order to extract the temperature structure function from the VMP temperature profile, the temperature gradient due to the thermocline has to be removed. The thermocline temperature gradient is not part of the turbulence and would bias the temperature structure function toward higher values. Removal of the gradient or “detrending” is accomplished by subtracting the moving average over 20 cm from the temperature profile. From the temperature structure function obtained in this manner, the proportionality constant between the square of the temperature difference and the separation distance to the power of 2/3 is determined. This can be done by a simple curve fit where the only fitting parameter is . The temperature structure function is fitted to Eq. (7) over regions spanning . Fitting over a range between 1 and 20 cm excludes the inner scale () as well as outer scale region () of the structure function, and also ensures that the turbulence can be treated as locally isotropic. During the deployment of the TRUSS, was as measured from the temperature and salinity difference at 60 and 100 m. With this, [Eq. (9)] can be evaluated using an empirical relation for () and () to be . Finally, is calculated from via Eq. (9).
Figures 5(b)–5(c) show the detrended VMP temperature profiles, i.e., the high spatial frequency temperature variation [Fig. 5(b)] and the index of refraction structure constant obtained from the temperature variation [Fig. 5(c)]. It is clear from the data that, while the vertical position and extent of the thermocline shows some variability over time, the highest detrended temperature variability and correlate well with the largest gradient of the thermocline, where presumably the mixing of cold and warm water is the strongest. These are also the regions where the TKED is largest [Fig. 5(d)] and where we will later find the strongest optical effects.
For 28°C water, the Batchelor scale is on the order of 2 mm for the calm regions and on the order of 0.2 mm for the most turbulent region as calculated from minimum and maximum TKED dissipation measured by the VMP. The Batchelor scale for salinity is about a factor of 10 smaller than the temperature scale. However, the conditions during BOTEX were such that the contribution of salinity to the index of refraction was compared to the contribution of temperature. This can be seen by comparing the contributions to the total differential in Eq. (10). Taylor’s frozen turbulence hypothesis [3,29], in combination with both the average vessel drift velocity of during the measurements and the Batchelor scale, leads to the shortest expected beam distortion time scale of 5 and 0.5 ms for the calmest and most turbulent regions, respectively. The time scales are calculated from the transition time of the smallest structures over their own dimension. However, the smallest structures have dimensions smaller than the diameter of the beam and therefore only lead to beam distortion, sometimes called beam breathing . Beam wander, on the other hand, is caused by turbulent eddies that are larger than or at least comparable in size to the diameter of the beam. At the transmitter, the beam diameter is 7 mm. Any deflection close to the transmitter will have the greatest effect since any angular deflection is amplified by the subsequent beam propagation. If we therefore assume that beam deflection originates from turbulent eddies no smaller the , we calculate the shortest time scales for the beam deflection to be on the order of 15 ms (). Figures 6(a) and 6(c) show beam deflection along the vertical axis of the CCD camera, as determined by a tracking algorithm. The tracking algorithm uses the first-order Fourier components of a CCD section containing only a single bright spot to track the center position of a dot after it is initially identified manually in the first frame of the series . The output of the tracking algorithm is a matrix containing the 2178 (one set for each frame) coordinates along the vertical and horizontal dimension of the CCD for each dot on the ground glass plate. Images of the moving dots were recorded at 300 fps (frames per second) rather than at the full speed of the camera in order to increase the observation time before the 1 GBit camera memory was filled. Approximately 15 distinct light dots could be traced over the entirety of the recording time of 7 s, for the measurements at the different depths to which the TRUSS was lowered. In Fig. 6, individual dot traces are shown as gray lines, while the ensemble average of all dots is indicated by red lines. The inserts in the top left corner of Figures 6(a)–6(d) show the trace of a single light dot. The sampling interval is expressed as a circle for each sample point. The spectral density generated by a fast Fourier transform of the displacement data (shown in the lower left inserts) drops to zero at frequencies greater than , justifying the assumption that a sample rate of 300 fps is adequate to capture the motion of the dots. The sampling interval of 300 fps also is well within the 67 Hz frequency component expected from the 15 ms transition time of a turbulent cell through the beam. In all cases, the Nyquist criterion is satisfied.
Shown in Figs. 6(a) and 6(c) is raw data as extracted by the tracking routine for sampling depths of 70 m and 80 m, respectively. The highly periodic deflection of the dots in the upper left panel is likely due to a vibrational mode of the TRUSS excited by strumming of the cable as it moves through the water. This periodic deflection is common to all dots and exhibits a sharp spike in the Fourier transform at around 12 Hz, as shown in the lower insert to Fig. 6(a). The pronounced periodic deflection of the dots at 70 m is an extreme case. A more common situation is depicted in Fig. 6(c), where no periodic motion is visible in the data and the spike in the Fourier transform is absent. Motion that is highly periodic and common to all light dots can be removed by subtracting the average deflection of all dots from the position for a single dot for each frame, and hence the ensemble average deflection of all dots is zero [red line in Figs. 6(b) and 6(d)]. Comparison of the Fourier transforms (lower inserts in the panels of Fig. 6) shows that most of the deflection occurs independently of the other dots, as evident from the unchanged shape and amplitude of the spectrum after removal of the common motion, i.e., the relative motion of the beams carries most of the turbulence signature. This is somewhat at odds with the idea that turbulent cells closest to the transmitter have the largest effect on the beam wander since turbulent eddies in the vicinity of the transmitter cross all beams simultaneously and should therefore introduce motion common to all beams.
Removing this common motion from the ensemble average should reduce the measured beam wander. However, this is not what we found. As is evident in Figs. 6(c) and 6(d), removal of the common motion does not reduce the overall beam wander nor does it change the amplitude of the Fourier transform. One possible explanation is that, as the beam width diminishes along the propagation path (focused beam), turbulent eddies too small to deflect the beam at the transmitter become relevant. This would result in a weighting function similar to path-weighting function for scintillations ; since the majority of beam deflection would then accrue somewhat “downstream” from the transmitter where the beams are separated from each other, the deflection would be uncorrelated. However, we are not aware of any such study. Recently a geometrically similar method, but propagating in the reverse direction, has been put forward by Gladysz et al. . Here, light emerges from an array of spatially separated LED and propagates to a common aperture in the form of a camera lens. We believe that, due to the principle of channel reciprocity , a very similar analysis of our data as proposed by Gladysz et al. is possible. However, due to the limited number of laser spots observed during our experiment, the sophisticated analysis technique proposed by Gladysz et al. was not attempted. An analysis of the beam wander frequency spectrum and a method to extract the mean flow orientation of the turbulence from our data can be found in a separate publication .
Since our goal is to determine from the std of the beam’s center motion, adequate sampling of the beam wander and reasonable Gaussian distribution has to be ensured. Histograms of the beam center deflection from its mean position are presented in Fig. 7 for two depths of relatively calm and relatively turbulent conditions. The histograms are fitted with Gaussian envelope functions, and their standard deviation as well as their correlation to a Gaussian shape is determined. The fitting to a Gaussian is generally very good, with a correlation value of better than 0.998 for all data sets recorded between 5 and 85 m. The std varies from 0.014 to 0.185 mm for a depth of 48.6 and 75.1 m, respectively, which are the extreme cases.
Figure 8 summarizes the results during the descent of the TRUSS night cast. Shown in the figure is the thermal profile as measured by the temperature probe mounted on the TRUSS. Also shown is the std along orthogonal directions on the CCD plotted at the respective depth (green squares). From the std, is calculated using Eq. (4) for a beam of , propagating over a length (black circles). The range of calculated from three separate casts of the VMP is indicated by the gray shaded area for comparison. The same data, together with the measurements of the day cast, are represented in a different fashion in Fig. 9. Here , derived from the TRUSS measurement, is plotted versus attained from the VMP cast temperature profiles at the same depth. The error bars indicate the variability in the measurement of from the three VMP casts. As expected, there is a large variability in the data due to the unsteady nature of the turbulence and the non-coinciding measurement times and places. Nevertheless, the measurements correlate reasonably well and the overall trend and magnitude is clear. We would like to direct the attention of the reader to the temperature variation measured by the slow thermistors mounted on the TRUSS at a depth between 20 and 30 m in Fig. 8. This variation is absent from the temperature profile during the VMP casts [Fig. 5(a)] and can explain the outliers in the night cast at 30.9 and 22.6 m in Fig. 9.
Optical turbulence is one of the most important factors limiting optical systems in the atmosphere. The same phenomena has the potential to also disrupt optical communications and imaging in the ocean. Nevertheless, relatively little is known about the strength of optical turbulence in the marine environment. By focusing on the center motion of a laser beam, i.e., beam wander, we were able to use a model developed for the atmosphere to optically determine in the ocean. These models only rely on the large-scale structures found in the inertial sub-range of the turbulence and are insensitive to the particular shape of the index of refraction spectra in the vicinity of the viscous dissipation range. The values were then compared to determined from the temperature microstructure measured from a separate platform. The strong correlations between the temperature gradient measured from the two platforms, namely from the VMP and from the TRUSS, indicate reasonably stable turbulent conditions over the time and space over which the measurements took place. This allowed the comparison of the optically measured values of with values determined from the temperature data. Due to the nature of turbulence, the data shows significant variability, but the overall trend and magnitude of is the same for both measurement principles. The apparent trend of the optical measurement to underestimate at regions of strong temperature gradient compared to the values derived from the temperature structure could be attributed to outer scales smaller than what we estimated. However, the underestimation is within the uncertainty of the measurement and might simply be due to insufficient statistics, i.e., due to the limited amount of data during the changing conditions.
The optical method has the advantage of data being averaged over the propagating length of several meters and measurement time of many seconds, while the VMP only is measuring at a single point and relies on its motion to sample the profile. This is of particular importance for an unsteady quantity like turbulence, where measuring over many realizations is necessary in order to arrive at a meaningful statistical average. We found that optical turbulence in the ocean can be quite strong and will significantly affect the resolution of imaging systems over path length of . This is well within the range over which optical schemes can function, and therefore is an important factor for their operation. We also would like to point out that TRUSS deployments during BOTEX in the Gulf Stream indicated much stronger optical turbulence. However, for reasons outside the scope of this paper, these values could not be quantified further and are therefore only to be taken as an indicator of possibly much higher values of in parts of the ocean. Future surveys will have to be conducted to establish a range of for different geographical locations and seasons.
Office of Naval Research (ONR) (62782N); Florida Atlantic University (FAU)-Harbor Branch SAVE OUR SEAS Florida specialty license plate program.
The authors would like to acknowledge Dr. W. T. Rhodes for insightful suggestions and help editing the present paper.
1. H. T. Yura, “Small-angle scattering of light by ocean water,” Appl. Opt. 10, 114–118 (1971). [CrossRef]
2. H. Horvath, “On the applicability of the Koschmieder visibility formula,” Atmos. Environ. 5, 177–184 (1971). [CrossRef]
3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005), p. xxiii
4. F. R. Dalgleish, A. K. Vuorenkoski, and B. Ouyang, “Extended-range undersea laser imaging: current research status and a glimpse at future technologies,” Mar. Technol. Soc. J. 47, 128–147 (2013). [CrossRef]
5. J. S. Jaffe, “Underwater optical imaging: the past, the present, and the prospects,” IEEE J. Ocean. Eng. 40, 683–700 (2015). [CrossRef]
6. W. Hou, “A simple underwater imaging model,” Opt. Lett. 34, 2688–2690 (2009). [CrossRef]
7. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012). [CrossRef]
8. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000). [CrossRef]
9. N. H. Farwell and O. Korotkova, “Multiple phase-screen simulation of oceanic beam propagation,” Proc. SPIE 9224, 922416 (2014). [CrossRef]
10. R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68, 1067–1072 (1978). [CrossRef]
11. A. V. Kanaev, W. Hou, S. R. Restaino, S. Matt, and S. Gladysz, “Restoration of images degraded by underwater turbulence using structure tensor oriented image quality (STOIQ) metric,” Opt. Express 23, 17077–17090 (2015). [CrossRef]
12. S. Matt, W. Hou, S. Woods, W. Goode, E. Jarosz, and A. D. Weidemann, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods Oceanogr. 11, 39–58 (2014). [CrossRef]
13. W. Hou, E. Jarosz, S. Woods, W. Goode, and A. D. Weidemann, “Impacts of underwater turbulence on acoustical and optical signals and their linkage,” Opt. Express 21, 4367–4375 (2013). [CrossRef]
14. W. Hou, S. Woods, E. Jarosz, W. Goode, and A. D. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51, 2678–2686 (2012). [CrossRef]
15. D. J. Bogucki, J. A. Domaradzki, C. Anderson, H. W. Wijesekera, J. R. V. Zaneveld, and C. Moore, “Optical measurement of rates of dissipation of temperature variance due to oceanic turbulence,” Opt. Express 15, 7224–7230 (2007). [CrossRef]
16. S. Q. Duntley, “Light in the Sea*,” J. Opt. Soc. Am. 53, 214–233 (1963). [CrossRef]
17. F. Dalgleish, W. L. Hou, A. Vuorenkoski, G. Nootz, and B. Ouyang, “In situ laser sensing of mixed layer turbulence,” Proc. SPIE 8724, 87240D (2013). [CrossRef]
18. G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid, 1: general discussion and the case of small conductivity,” J. Fluid Mech. 5, 113–133 (1959). [CrossRef]
19. G. K. Batchelor, I. D. Howells, and A. A. Townsend, “Small-scale variation of convected quantities like temperature in turbulent fluid, 2: the case of large conductivity,” J. Fluid Mech. 5, 134–139 (1959). [CrossRef]
20. S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2005), p. xviii.
21. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978). [CrossRef]
22. D. H. Tofsted, “Outer-scale effects on beam-wander and angle-of-arrival variances,” Appl. Opt. 31, 5865–5870 (1992). [CrossRef]
23. L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006). [CrossRef]
24. L. Washburn, T. F. Duda, and D. C. Jacobs, “Interpreting conductivity microstructure: estimating the temperature variance dissipation rate,” J. Atmos. Ocean. Technol. 13, 1166–1188 (1996). [CrossRef]
25. J. D. Nash and J. N. Moum, “Estimating salinity variance dissipation rate from conductivity microstructure measurements,” J. Atmos. Ocean. Tech. 16, 263–274 (1999). [CrossRef]
26. W. Hou, E. Jarosz, F. R. Dalgleish, G. Nootz, S. Woods, A. D. Weidemann, W. Goode, A. Vuorenkoski, B. Metzger, and B. Ramos, “Bahamas optical turbulence exercise (BOTEX): preliminary results,” Proc. SPIE 8372, 837206 (2012). [CrossRef]
27. D. J. Bogucki and J. A. Domaradzki, “Numerical study of light scattering by a boundary-layer flow,” Appl. Opt. 44, 5286–5291 (2005). [CrossRef]
28. X. H. Quan and E. S. Fry, “Empirical-equation for the index of refraction of seawater,” Appl. Opt. 34, 3477–3480 (1995). [CrossRef]
29. G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London Ser. A 164, 0476–0490 (1938). [CrossRef]
30. F. Rainer, “Find centroid of a monochrome laserspot within a dark background,” https://www.mathworks.com/matlabcentral/fileexchange/6490-find-centroid-of-a-monochrome-laserspot-within-a-dark-background.
31. R. S. Lawrence, G. R. Ochs, and S. F. Clifford, “Use of scintillations to measure average wind across a light-beam,” Appl. Opt. 11, 239–243 (1972). [CrossRef]
32. S. Gladysz, M. Segel, C. Eisele, R. Barros, and E. Sucher, “Estimation of turbulence strength, anisotropy, outer scale and spectral slope from an LED array,” Proc. SPIE 9614, 961402 (2015). [CrossRef]
33. D. L. Fried and H. T. Yura, “Telescope-performance reciprocity for propagation in a turbulent medium,” J. Opt. Soc. Am. 62, 600–602 (1972). [CrossRef]
34. G. Nootz, W. Hou, F. R. Dalgleish, and W. T. Rhodes, “Determination of flow orientation of an optically active turbulent field by means of a single beam,” Opt. Lett. 38, 2185–2187 (2013). [CrossRef]