Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence

applied sciences

Article Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence

Steven Fiorino 1,* , Santasri Bose-Pillai 1 and Kevin Keefer 1,2

1 Department of Engineering Physics, Air Force Institute of Technology, Wright Patterson Air Force Base, 2950 Hobson Way, Dayton, OH 45433, USA; santasri.bose-pillai@afit.edu (S.B.-P.); kevin.keefer.ctr@afit.edu (K.K.) 2 Applied Research Solutions, 51 Plum Street, Suite 240, Beavercreek, OH 45440, USA * Correspondence: steven.fiorino@afit.edu

Abstract: Optical turbulence, as determined by the widely accepted practice of profiling the tempera- 2 ture structure constant, CT, via the measurement of ambient atmospheric temperature gradients, can be found to differ quite significantly when characterizing such gradients via thermal-couple differential temperature sensors as compared to doing so with acoustic probes such as those commonly used in sonic anemometry. Similar inconsistencies are observed when comparing optical turbulence 2 strength derived via CT as compared to those through direct optical or imaging measurements of small fluctuations of the index of refraction of air (i.e., scintillation). These irregularities are especially apparent in stable atmospheric layers and during diurnal quiescent periods. Our research demonstrates that when care is taken to properly remove large-scale index of refraction gradients, the sonic 2 anemometer-derived velocity structure constant, Cv, coupled with the similarly derived turbulence- driven index of refraction and vertical wind shear gradients, provides a refractive index structure 2 constant, Cn, that can more closely match the optical turbulence strengths inferred by more direct means such as scintillometers or differential image motion techniques. The research also illustrates 2 2 Citation: Fiorino, S.; Bose-Pillai, S.; the utility and robustness of quantifying Cn from CT at a point using a single sonic anemometer 2 Keefer, K. Re-Visiting Acoustic and establishes a clear set of equations to calculate volumetric Cn data using instrumentation that Sounding to Advance the measures wind velocities with more spatial/temporal fidelity than temperature. Measurement of Optical Turbulence. Appl. Sci. 2021, 11, 7658. https:// Keywords: optical turbulence; index of refraction structure constant; temperature structure constant; doi.org/10.3390/app11167658 velocity structure constant; sonic anemometry; sound detection and ranging; scintillometers; time- lapse imaging Academic Editors: Mikhail Vorontsov and Steve Hammel

Received: 19 July 2021 1. Introduction Accepted: 16 August 2021 2 Published: 20 August 2021 The index of the refraction structure constant, Cn, is commonly used to quantify optical turbulence strength, which, in turn, can degrade both active and passive system

Publisher’s Note: MDPI stays neutral performance, whether that of a laser central to a free space optical communication uplink, with regard to jurisdictional claims in an inter-continental relay mirror high energy laser power-beaming architecture to trans- published maps and institutional affil- port energy to remote sites, or an upward-looking telescope pointed at a distant space iations. object. Though valuable and ever more capable, a path-averaged and even a path-resolved 2 measurement of Cn is available via optical means such as scintillometry [1] and time-lapse imagery [2,3], respectively. Even so, continuous, high frequency, very localized, point measurement of causative temperature, moisture, and wind gradients near the source or image plane of one’s active/passive system can be difficult using optical means because one is Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. observing a net optical effect. Such localized point measurements, however, are especially This article is an open access article beneficial to engineers and scientists who develop and apply compensation and correction distributed under the terms and techniques and associated modalities, as well as those individuals endeavoring to develop, conditions of the Creative Commons verify, and validate optical turbulence and effects models through carefully instrumented Attribution (CC BY) license (https:// field tests and post-test forensics analysis. Similarly, these high-fidelity flux measurements creativecommons.org/licenses/by/ are of interest to mission planners and system operators seeking to update/nudge an 4.0/). otherwise long range, days-to-weeks advance forecast of system performance to support

Appl. Sci. 2021, 11, 7658. https://doi.org/10.3390/app11167658 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 7658 2 of 18

up-to-the-minute mission briefs [4] and current-mission system settings. The latter advance forecasts are built on a combination of increasingly sophisticated laser propagation and numerical weather prediction models. By emphasizing the high frequency measurement of the time of flight of an acoustic pulse transmitted between two probes, 3D sonic anemometers were originally conceived, and subsequently extensively applied, to assess temperature gradients and, thus, the temperature structure constant. However, limiting one’s analysis to temperature gradients can lead to a sub-optimal assessment of optical turbulence strength during, for example, diurnal quiescent periods. During these periods, temperature gradients in all directions diminish to near zero—the evening quiescent period ends because the ground (due to radia- tive losses to space) cools the air near it and creates a vertical temperature gradient, and as the night goes along, horizontal gradients are established because of different cooling rates of different surfaces. The morning quiescent period breaks down because of the opposite effect of the sun heating the ground. At night (in stable conditions), waves along the stable layers—caused by wind and large object movement—create a spatial and temporal index of refraction fluctuations that create “optical turbulence” effects on laser/optical propagation. During the day, the sun heating the ground creates convection that sup- ports more statistically characterize-able advecting turbulent eddies (e.g., Kolmogorov turbulence) that produce a spatial and temporal index of refraction fluctuations that, in turn, effect electromagnetic—especially optical and near infrared—wave propagation in the classical “optical turbulence” way. While temperature gradients are, in general, the largest contributor to the index of refraction gradients, humidity gradients and pressure 2 2 perturbations also contribute. Calculating Cn from CT, in some cases, makes quiescent periods seem far more marked than they often are as measured by an optical instrument such as a scintillometer. Arrays of sonic anemometers measure the speed of sound in air in many directions, thus, they can conceivably capture gradients in the temperature, humidity (since this affects air density and the speed of sound), and the velocity. The velocity fluctuations are created by the small turbulent eddies or waves on the stratified layers that have small circulations (e.g., vortices) associated with them, which, in turn, produce pressure/density gradients that establish an index of refraction gradients with very little temperature change. These humidity and pressure/density variations often 2 continue to exist during quiescent periods and in adiabatic layers, keeping actual Cn levels 2 significantly higher than those characterized through a consideration of CT only. This suggests the rationale for employing arrays of sonic anemometers as control research and test instrumentation to capture more fully the physics that create optical turbulence effects. This study demonstrates the implementation of 3D sonic anemometers arrayed in a way to do what each sonic anemometer was originally designed to do best: acoustically interrogate eddies and sense temperature, moisture, and wind velocity (and hence pressure) gradients, which, in combination, lead to index of refraction gradients. Nosov et al. [5] 2 2 2 have similarly demonstrated a calibrated method to obtain CT, Cv, and Cn from a small array of ultrasonic anemometers. This paper extends the relationships shown in [5] with ev- 2 2 idence demonstrating optical turbulence strength, Cn, can be derived from a parameter, Cv, which is more directly tied to the turbulent eddy distribution independent of non-adiabatic temperature gradients that many techniques tend to exploit to their advantage, but that can disappear when turbulence does not. Shikhovtsev et al. [6] have noted the potential importance of quantifying the non-temperature gradient component that partially drives 2 turbulent eddy production with their seasonal study of Cn strength near Lake Baykal that peaked in winter when winds are strongest. While underlining the benefits of employing local sonic anemometer arrays, this paper also seeks to set forth a solid set of relationships of equal relevance when one is using acoustic sounding—whether sonic anemometry or other familiar techniques such as Sound Detection and Ranging (SODAR)—to evaluate and characterize optical turbulence. A validation of these relationships will primarily be accom- plished through a comparative evaluation of the results obtained through path-resolved optical means. Materials and Methods for the present work are described in Section2. This Appl. Sci. 2021, 11, 7658 3 of 18

section derives, or when of common usage simply references, the key physical expressions enabling an investigation of optical turbulence strengths and effects using acoustically observed micro-meteorological parameters. Section2 also provides a description of the sonic anemometer arrays, supporting instrumentation, and the general field experiment set-up. It is in this section that the equivalency of the sonic and virtual temperature is demonstrated (they have previously been assumed different but approximately equal), and a new method to remove the non-turbulent layer vertical temperature change that is independent of atmospheric stability is introduced. Furthermore, a summary description of a path-resolved optical turbulence profiler used for comparative reference is included here as well. Representative optical turbulence point measurements, as collected from sonic anemometers installed on a static tower, will be profiled, analyzed, and compar- atively evaluated with the reference path-resolved optical turbulence measurements in Section3. Section4 will conclude by summarizing the analysis and discussion of the preceding section, the implications of the findings, and future research directions.

2. Materials and Methods 2.1. Structure Parameters In their widely acknowledged development of atmospheric electromagnetic wave propagation amidst fluctuations associated with the temperature, moisture, and pressure, Kolmogorov and then Tatarskii advanced the concept of structure functions. In combination, this led to an analytical basis for quantifying optical turbulence, the correlation of the associated random localized small-scale fluctuations of the former physical parameters, the spatial/temporal characteristics of turbulent eddies in the atmosphere, and, subsequently, the effects on the index of refraction [7,8]. In those instances where the turbulent flow can be characterized as Kolmogorov—isotropic, homogenous, and an associated energy spectrum falling within the inertial sub-range where energy is neither created nor dissi- pated by the large and small eddies—the structure function is described according to the Kolmogorov–Obhukov two-third power law. For the sake of illustration, to account for temperature fluctuations, the structure function, DT, is expressed as follows [7]:

D 2E 2 2/3 DT(r) = [T(r2) − T(r1)] = CT|r2 − r1| (1)

2 where CT is the structure parameter for temperature, T, and the brackets, hi, signify an ensemble average. The underlying analytical theory presumed temperature measurements at two points separated by the distance r. In practice though, such micrometeorological parameters are generally measured at a single point as a function of time, t. In order to correlate these temporally varying point measurements to spatial variability in temperature, Taylor’s frozen turbulence hypothesis generally is invoked. Taylor’s hypothesis postulates the shapes of the turbulent eddies is frozen in time and move perpendicularly past a probe or observer at the convection velocity, hV⊥i, which for Taylor’s frozen turbulence hypothesis is taken as equivalent to the local mean wind velocity. Subsequently, Equation (1) can be re-written as follows:

2 2 −2/3 CT = h[T(t + ∆t) − T(t)]i (hV⊥i∆t) (2)

Kaimal and Finnigan [9] (pp. 61–62), however, counter the tenets of this hypothesis through references to ﬁeld studies that demonstrated larger eddies moved faster than their smaller counterparts. Naturally, such a variable ﬂow and consequent non-uniform momentum, heat, and mass transfer throughout the atmosphere, especially in the mixed atmospheric boundary layer, lead one to discount assumptions of frozen eddy shapes and sizes. Nonetheless, this reference argued as convincingly that such assumptions have been validated through a large number of observations and associated time-averaged statistics. Additionally, many measurements, such as those recorded with sonic anemometers, are conducted at extremely high frequencies, wherein eddies are not likely to change over the course of the measurement period. Appl. Sci. 2021, 11, 7658 4 of 18

Specific to optical turbulence and its effects on electromagnetic wave propagation, the causal index of refraction fluctuations are characterized by the structure parameter for the index of refraction, which can be expressed in a form similar to Equation (2) but with regard to each of the physical scalars influencing the atmospheric index of refraction— temperature, moisture, and pressure. In large part, though, scientists and engineers characterizing atmospheric and optical turbulence effects in the visible and near infrared 2 2 are satisfied with the following straightforward expression tying Cn to CT:

P 2 C2 = 79.2 × 10−6 C2 (3) n T2 T

where P is pressure (hPa) and T is the ambient air temperature (K). This expression was originally derived from first principles and presented by Tatarskii [8] (p. 102) but was also arrived at through empirical measurement of fine scale temperature fluctuations in the course of optical experiments [10]. Several key assumptions underlie the application of Equation (3) [11]. These include statistically homogeneous and isotropic Kolmogorov turbulence; moisture fluctuations are of small consequence in both the optical (visible) and near infra-red portions of the spectrum, which generally is acknowledged as reasonable except over water surfaces [12]; and pressure fluctuations that are presumed to rapidly dissipate as compared to the temperature fluctuations, and generally thought to be inconsequential. Interestingly then, though it is widely known that the original theory underpinning optical turbulence and its effects originated with regard to velocity fluctuations (pulsations) [7], the velocity structure constant is not explicitly represented in Equation (3). Indeed, very 2 satisfactory results have been obtained by simply calculating CT using two high-speed, fine-wire thermometers to record temperature fluctuations as a function of time. With comparable accuracy, 3D sonic anemometers have also been used extensively to record 2 temperature fluctuations [13]. Yet, for instances of neutral stratification, CT will trend towards zero, and one would presume optical turbulence strengths would trend likewise. However, observations of such tendencies, as might be anticipated during the diurnal quiescent periods after sunrise or before sunset or on windy or cloudy days, have not been so absolute [14]. Tatarskii’s original development attempted to quantitatively characterize optical turbulence even for these instances of neutral stratification by accounting for the vertical gradients in wind velocity, pressure, and moisture, as well as temperature—which led to fluctuations in the index of refraction and its structure constant [8]. As will be discussed shortly, sonic anemometers are inherently designed to measure speed of sound fluctuations caused by moisture, temperature, and velocity fluctuations and, hence, pressure fluctuations. Turbulent eddies, each with characteristic moisture, temperature, and pressure, cross the sonic anemometer’s acoustic propagation path at a velocity linked to that of the localized mean wind velocity as well as fluctuations in the flow’s velocity. In fact, Bernoulli’s principle suggests the latter fluctuations can lead to pressure fluctuations. These velocity fluctuations can be characterized according to a derived structure parameter for velocity. As earlier, assuming Kolmogorov turbulence and Taylor’s frozen turbulence hypothe- 2 sis, the velocity structure parameter, Cv, which characterizes the eddy and turbulent flow’s velocity fluctuations, can be written as follows [11]:

2 2 −2/3 Cv = [V⊥(t + ∆t) − V⊥(t)] (V⊥∆t) , (4)

where V⊥ represents the measurement of the velocity fluctuations transverse to the field instrumentation’s characteristic observation path(s). Returning to Kolmogorov’s original 2 derivation, where Cv was shown to describe the turbulence flow’s total energy, an explicit 2 2 expression showing the link between Cn and Cv is derived in AppendixA and is as follows:

2 2 2 Cn = 1.4Cv[∇hni/(∂hV(z)i∂z)] , (5) Appl. Sci. 2021, 11, 7658 5 of 18

where ∇hni is the gradient of the ensemble average of the fluctuations of the index of refraction, and ∂hV(z)i∂z is the ensemble average of the vertical gradient of the wind velocity fluctuations. As will be made clear shortly, this result accounts for fluctuations in temperature as well as moisture, by way of ∇n. This expression brings forward the centrality of the turbulent flow’s velocity gradients and fluctuations, which anchored Obhukov, Kolmogorov, Tatarskii, and others’ optical turbulence theory. The expression in Equation (5) also leads one to consider an alternative point measurement approach wherein one exploits sonic anemometery, positioning a primary and secondary sonic anemometer vertically one above the other, to derive the requisite gradients between two layers, while 2 simultaneously evaluating Cv at the primary point of interest.

2.2. Sonic Anemometry and Optical Turbulence Point Measurement For 3D sonic anemometers, the characteristic observation paths include three axial piezo-electric transducers pairs, each transducer probe emitting ultra-short, very high frequency, acoustic signals in unison with its partner, and then recording the time delay difference between the transmitted and received acoustic signals. In sum, six acoustic signals will have travelled through an air parcel moving across the ﬁxed transducer separation paths with a velocity governed by that parcel’s characteristic horizontal (“u” and “v”) and vertical “w” wind velocity. The time delay difference will also be inﬂuenced by the speed of sound, which for ambient air can be expressed by adapting the ideal gas law to account for water vapor via the parcel’s virtual temperature (the temperature at which a dry air parcel has the same density as its moist counterpart held at the same pressure), as follows:

1/2 1/2 1/2 c = (γdP/ρ) = (γdRdTv) = (γdRdTv) , (6)

where c is the velocity (m/s) of sound in ambient air, P is the absolute air pressure (Pa), 3 ρ = P/RdTv is an expression for the density kg/m of a moist air parcel, Tv is the parcel’s virtual temperature (K), and γd (1.4) and Rd (287 J/kg·K) are the standard dry air speciﬁc heat ratio and gas constant, respectively. In practice, meteorologists deﬁne and approximate Tv as follows: εe h e i T = T[1 + 0.61q] = T 1 + 0.61 ≈ T 1 + 0.61ε ≈ T[1 + 0.38e/P], (7) v P − 0.378e P

where q is the speciﬁc humidity (the mass of water vapor per unit mass of the moist air parcel sampled), ε is the ratio of the molecular masses of water vapor and dry air (i.e., 0.622), and e is the vapor pressure (Pa). Thus,

1/2 c ≈ (γdRdT[1 + 0.38e/P]) (8)

Alternatively, it is common practice among those using sonic anemometers to express c as follows [15,16]:

1/2 1/2 c = (γdRdT[1 + 0.51q]) ≈ (γdRdT[1 + 0.32e/P]) , (9)

where the sonic temperature, Ts, is often referred to as the sonic virtual temperature, and defined as follows: Ts = T[1 + 0.32e/P] (10) Through investigation, Kaimal and Gaynor [16] identified the source for the difference between the sonic virtual temperature (Equation (10)) and meteorological virtual temperature (Equation (7)). The distinction originated in the leading coefficient to the ratio of e/P, where that for Equation (7) is effectively (1 − Mw/Ma) and that for Equation (10) is derived from (γw/γa − Mw/Ma), in which γw/γa is the ratio of the specific heat ratios of water vapor and dry air (1.33/1.4) and Mw/Ma is the ratio of the molecular masses of water vapor and dry air (0.622) [17]. This effectively adjusts the 0.61 coefficient in Equation (7) Appl. Sci. 2021, 11, 7658 6 of 18

to 0.51 in Equation (9). However, this explanation appears to ignore that q ≈ εe/P is itself an approximation (the left side of Equation (7) captures the definition of specific humidity) that slightly underestimates the value of q, especially in very moist air. This suggests that the 0.32 coefficient that appears in Equations (9) and (10) should be larger, and perhaps closer to the 0.38 value that appears in Equations (7) and (8). Furthermore, Fleagle and Businger [18] show that the effect of water vapor content on the dry air values of the specific heat at constant pressure (cp) and constant volume (cv) can be quantified as follows: c = (1 + 0.84q)c p pa (11) cv = (1 + 0.93q)cva, −1 −1 −1 −1 where cpa = 1005 J kg K and cva = 718 J kg K are the dry air values of the specific heats and q is, again, the specific humidity. When the air is 100% water vapor, q = 1 and −1 −1 Equation (11) provides the specific heat values for pure water vapor, cpw = 1847 J kg K , −1 −1 cvw = 1386 J kg K , and γw/γa = 1.33/1.4. However, air in the free atmosphere rarely exceeds about 2% water vapor content by mass; with q = 0.02—which is equivalent to a near sea level dew point temperature of ~300 K—γw/γa = 1.39/1.4 ≈ 1. Thus, the leading coefficient to the ratio of e/P is effectively only (1 − Mw/Ma). Accordingly, it had become common practice to assume that the sonic virtual temperature very closely approximates the meteorological virtual temperature and can for all intents and purposes be treated as equivalent; the arguments herein remove any concern about the equivalency of these two parameters. The virtual temperature simplifies the treatment of moisture contributions to many local atmospheric processes including buoyancy and index of refraction gradients, which will be discussed further below. In sum, the sonic virtual temperature, derived through processing of time-delayed acoustic wave propagation, provides ready-access to the meteorological atmospheric virtual temperature. The virtual temperature, in turn, provides a means to quantify the local index of refraction’s dependence on both temperature and moisture. 2 Returning to Equation (5), while the results for Cv and ∂hV(z)i∂z are regularly re- trieved directly from the sonic anemometer outputs, those for the gradient of the index of refraction, ∇n, require consideration of the contributing factors. Beginning with Tatarskii, the potential temperature (really the potential virtual temperature)—a conserved quantity— naturally is referenced quite frequently when attempting to infer or evaluate the latter gradient [8,19,20]. With the advent of new optical techniques to deduce the index of refraction fluctuations, some have chosen Tv as an alternative to using a conservative quantity to assist with quantifying these fluctuations through optical measurements [21]. By opting to take the alternative path, separately obtaining accurate water vapor fluctuation measurements is not necessary since the moisture quantification is contained in the virtual temperature. Proceeding in this manner, one begins by selecting an expression for air’s index of refraction. Given its lineage, an attractive option is that developed by Edlen, which culminated in what has been labelled as modified-Edlen or Edlen66 [22]. Amongst other enhancements, Edlen66 accounted for the refractivity of water vapor. Consistent with the development’s underlying principle, the air’s refractive index is linked to its density according to the following: n − 1 = aρ, (12) where, according to Edlen66 with air density normalized to 1 kg-m−3 [22], h i n − 1 = 10−6 83.4212 + 24060.3/ 130 − 1/λ2 + 159.97/ 38.9 − 1/λ2 (13)

On combining Equations (12) and (13), one can solve for a, which is found to be 0.227 × 10−3 m3-kg−1 for visible light (λ = 0.550 µm) for a standard atmosphere at sea level where the air density is 1.225 kg-m−3. Consequently, for an air parcel displaced Appl. Sci. 2021, 11, 7658 7 of 18

from level z1 to level z2, the gradient of the displaced parcel’s index of refraction can be approximated as follows:

∂a ∂ρ ∂ρ ∇n → = ρ + a ≈ a , (14) z1 z2 ∂z ∂z ∂z where the ﬁrst term in Equation (14) is, by Edlen’s premise, zero: a is normalized to air density. As introduced earlier, a modiﬁed form of the ideal gas law can be written as ρ = P/RdTv to account for moisture, which results in the following: ∂ρ a −P ∂Tv ∂P ∇nz1→z2 ≈ a ≈ + , (15) ∂z RdTv Tv ∂z ∂z Recalling that ∂P/∂z = −ρg, where g is the acceleration due to gravity, one can re-write Equation (15) as follows: −aP ∂Tv g −aρ ∂Tv g −(n − 1) ∂Tv g ∇n → ≈ + ≈ + ≈ + , (16) z1 z2 2 ∂z R ∂z R ∂z R RdTv d Tv d Tv d

As developed earlier in this paper, Tv is effectively the sonic temperature recorded using the sonic anemometer. It is not a conserved quantity as is the potential virtual temperature. In order to discern the dynamic range of the fluctuations in the index of refraction gradient, as is implied in the derivation associated with Equation (5), converting to a potential virtual temperature is warranted, but not necessary if one removes the large-scale virtual temperature lapse rate with height. However, to ensure consistency and preserve the underlying nature of the causal source of the gradients (i.e., turbulent mixing), the gradient of the index of refraction can be separated into a large-scale index component— an ensemble average—and an eddy component that represents the small-scale turbulent fluctuations as follows: ∇n = ∇n + ∇n0 (17) By doing so, the dominant large-scale index of refraction gradients, otherwise con- cealing high frequency (small-scale) fluctuations in the index of refraction gradient that 2 determine Cn, can be removed to arrive at the following:

−(n − 1) ∂T g −(n − 1) ∂T g ∇n0 ≈ v + − v + z1→z2 (18) Tv ∂z Rd Tv ∂z Rd

By assuming the mean layer virtual temperature at any one time, Tv is equivalent to

the ensemble average mean layer virtual temperature, Tv , Equation (18) can be simpliﬁed to the following: −(n − 1) ∂T − ∂hT i ∇ 0 ≈ v v n z1→z2 (19) Tv ∂z The ensemble average of the right side of Equation (19) provides ∇hn0i, which is then 2 combined with the other terms in Equation (5) to arrive at a Cn proﬁle.

2.3. Path Resolved Optical Turbulence Proﬁler for Comparative Reference The Delayed Tilt Anisoplanatism (DELTA) instrument shown in Figure1 is a turbulence proﬁling system developed by MZA [23]. The DELTA system uses an 8-inch aperture to image a distant target onto a science camera. Images are captured at 100 frames per second. The system measures the differential motion, or tilts between pairs of feature points on the target as a function of their angular separation. The differential tilt variance 2 between a pair of feature points is a path-weighted integral of Cn. Figure2 shows the target board used for DELTA measurements in the present work and weighting functions for differential tilt variances, corresponding to four different feature point separations on the target board. These weighting functions drop to zero at both ends of the path. Sub-aperture Appl.Appl. Sci. Sci. 20 202121, 11, 11, x, FORx FOR PEER PEER REVIEW REVIEW 8 8of of18 18

Appl. Sci. 2021, 11, 7658 8 of 18 forfor differential differential tilt tilt variances, variances, corresponding corresponding to to four four different different feature feature point point separations separations on on thethe target target board. board. These These w weightingeighting functions functions drop drop to to zero zero at at both both ends ends of of the the path. path. Sub Sub- - apertureaperture separations separations have have weighting weighting functions functions with with peaks peaks closer closer to to the the source source end end of of the the separations have weighting functions with peaks closer to the source end of the path. path.path. Weighting Weighting functions functions for for large large separations separations have have peaks peaks closer closer to to the the aperture aperture end end of of Weighting functions for large separations have peaks closer to the aperture end of the path. thethe path. path. The The DE DELTALTA processing processing software software uses uses the the measured measured differential differential tilt tilt variances variances for for The DELTA processing software uses the measured differential tilt variances for various variousvarious separations, separations, along along with with the the corresponding corresponding weighting weighting functions functions to to generate generate turbu- turbu- separations, along with the corresponding weighting functions to generate turbulence lencelence profiles profiles along along the the path. path. Turbulence Turbulence values values are are typically typically reported reported in in 10 10 bins bins distrib- distrib- profiles along the path. Turbulence values are typically reported in 10 bins distributed uteduted along along the the path. path. Frames Frames captured captured during during a athree three second second window window are are used used to to generate generate aalong turbulence the path. profile. Frames Turbulence captured parameters during a such three as second Rytov number, window Fried’s are used coherence to generate di- aa turbulenceturbulence profile.profile. Turbulence Turbulence parameters parameters such such as as Rytov Rytov number, number, Fried’s Fried’s coherence coherence diameter, Scintillation index as well as crosswind speeds are calculated from this profile. diameter,ameter, Scintillation Scintillation index index as as well well as as crosswind crosswind speeds speeds are are calculated from this profile.profile. The maximum operational range for a DELTA system is approximately two kilometers. TheThe maximummaximum operationaloperational range range for for a a DELTA DELTA system system is is approximately approximately two two kilometers. kilometers.

Figure 1. DELTA turbulence profiling system.

FigureFigure 1.1.DELTA DELTA turbulence turbulence proﬁling profiling system. system.

Weighting Functions Weighting Weighting Functions Weighting

Figure 2. Differential tilt variances. (a) DELTA target board; (b) Weighting functions (dimensionless) FigureFigure 2. 2.Differential Differential tilt tilt variances. variances. (a ()a DELTA) DELTA target target board; board; (b ()b Weighting) Weighting functions functions (dimension- (dimension- for 4 different feature point separations. z = 0 at the DELTA location. less)less) for for 4 different4 different feature feature point point separations. separations. z =z 0= at0 at the the DELTA DELTA location. location. When turbulence is strong, the feature points on the target become blurry and can When turbulence is strong, the feature points on the target become blurry and can no no longerWhen be turbulence tracked as is points. strong, While the feature trying points to still on track the feature target become points, DELTAblurry and measures can no longer be tracked as points. While trying to still track feature points, DELTA measures smallerlonger be differential tracked as tilt points. variances While due trying to the to blurring. still track This feature leads topoints, an underestimation DELTA measures of smallersmaller2 differential differential tilt tilt variances variances due due to to the the blurring. blurring. This This leads leads to to an an underestimation underestimation of of Cn values during periods of strong turbulence. Since DELTA measures differential motion Cn2 values2 during periods of strong turbulence. Since DELTA measures differential mo- betweenCn values feature during points, periods the of whole strong image turbulence. shift, due Since to changes DELTA in measures the large-scale differential refractive motion between feature points, the whole image shift, due to changes in the large-scale re- indextion between gradient, feature does not points, affect the the whole measurements. image shift, due to changes in the large-scale re- fractivefractive index index gradient gradient, does, does not not affect affect the the measurements. measurements. 2.4. Field Test Description: Propagation Path and Instrumentation Bed-Down 2.4. Field Test Description: Propagation Path and Instrumentation Bed-Down 2.4. FieldSection Test3 belowDescription: presents Propagation optical turbulence Path and profiles Instrumentation derived by Bed in situ-Down micro-meteorological point measurements recorded using two time-synchronized sonic anemometers, one designated as primary and the other secondary, installed at arbitrary heights, z1 and z2, vertically Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 18

Appl. Sci. 2021, 11, 7658 Section 3 below presents optical turbulence profiles derived by in situ micro-meteor-9 of 18 ological point measurements recorded using two time-synchronized sonic anemometers, one designated as primary and the other secondary, installed at arbitrary heights, 푧1 and 푧2, vertically one above the other. Being vertically stacked, the associated sonic anemom- one above the other. Being vertically stacked, the associated sonic anemometer stations eter stations are hereafter referred to as stacked sonic anemometer masts. Subsequently, a are hereafter referred to as stacked sonic anemometer masts. Subsequently, a diurnal, diurnal, temporally varying optical turbulence profile at a primary point and height of temporally varying optical turbulence profile at a primary point and height of interest interest is derived using Equation (5) above. Namely, the profile is arrived at by combin- is derived using Equation (5) above. Namely, the profile is arrived at by combining the ing the wind velocity and calculated small scale fluctuations of the index of refraction wind velocity and calculated small scale fluctuations of the index of refraction gradients gradients between the secondary and primary level with the velocity structure constant between the secondary and primary level with the velocity structure constant as derived as derived using the primary instrument’s measurements. As the sonic anemometers are using the primary instrument’s measurements. As the sonic anemometers are installed installed one directly above the other to ensure there is no lateral offset, the gradients one directly above the other to ensure there is no lateral offset, the gradients essentially essentially are with respect to the vertical direction. As reference, the primary sonic ane- are with respect to the vertical direction. As reference, the primary sonic anemometer’s recordedmometer’s temperature recorded temperature fluctuations fluctuations are used to calculate are used theto calculate temperature the temperature structure function struc- attur thee function primary at point the primary of interest, point and of hence interest, profile and opticalhence profile turbulence optical using turbulence the standard using approachthe standard through approach the application through the of application Equation (3).of Equation Given the (3 10). Given Hz sampling the 10 Hz rate sampling of the sonicrate of anemometers, the sonic anemometers, a 90-second a 9 ensemble0-second ensemble averaging averaging time was time largely was utilized largely u fortilized the 2 presentedfor the presented results results because becauseC2 profiles 퐶푛 profiles obtained obtained via Equation via Equation (5) and (5) those and those obtained obtained via 2n via optical methods and2 퐶푇 converged best in terms of variability and magnitude. It is optical methods and CT converged best in terms of variability and magnitude. It is also closealso toclose the to 100-second the 100-s ensembleecond ensemble averaging averaging time suggested time suggested in [5]. The in selection [5]. The ofselection optimum of ensembleoptimum averagingensemble times,averaging especially times, when especially pursuing when the pursuing technique the presented technique here, presented would likelyhere, would benefit likely from furtherbenefit investigation.from further investigation. TheThe fieldfield measurementsmeasurements werewere conductedconducted atat WrightWright PattersonPatterson AFB,AFB, OhioOhio duringduring thethe summersummer ofof 20202020 andand thethe heightheight ofof thethe COVID-19COVID-19 pandemic.pandemic. AtmosphericAtmospheric micrometeoro-micrometeoro- logicallogical as as well well as as soil soil measurements measurements were were recorded recorded at at multiple multiple points points along along a 1.9-kilometera 1.9-kilome- propagationter propagation path. path. While While the the slant slant path path traversed traversed many many types types of of soil/canopy soil/canopy cover cover and and topographicaltopographical features,features, equipmentequipment andand personnelpersonnel availabilityavailability limitedlimited thethe deploymentdeployment ofof stackedstacked sonicsonic anemometeranemometer mastsmasts toto twotwo locationslocations alongalong thethe path.path. AnAn aerialaerial viewview ofof thethe slantslant path path and and surrounding surrounding area area is shown is shown in Figure in Figure3 along 3 with along the with locations the locations of the stacked of the sonicstacked anemometer sonic anemometer mast stations. mast stations.

FigureFigure 3.3.Propagation Propagation Path.Path. Left:Left: thethe 30-meter30-meter mastmast andand locationslocations ofof thethe sonic sonic anemometers. anemometers. Upper Upper right:right: aerialaerial viewview ofof thethe opticallyoptically sampledsampled 1.9-kilometer1.9-kilometer turbulence turbulence path path and and location location of of the the 30-meter 30-meter sonic anemometer mast station as well as the DELTA system (Telescope/Camera Receiver and Tar- sonic anemometer mast station as well as the DELTA system (Telescope/Camera Receiver and Target get Board). The 6-meter sonic anemometer mast station was co-located with the DELTA target Board). The 6-meter sonic anemometer mast station was co-located with the DELTA target board. board. Lower right: Schematic (looking south) showing the propagation path topographical eleva- Lowertion change. right: Elevations Schematic are (looking shownsouth) in meters showing above themean propagation sea level (MSL). path topographical elevation change. Elevations are shown in meters above mean sea level (MSL).

A portion of the propagation path was situated alongside the quasi-active runway- taxiway used to ﬂy in aircraft for display at the National Museum of the United States Appl. Sci. 2021, 11, 7658 10 of 18

Air Force. The topography, a portion of which is level and open, can lead to strong wind shears, and indeed one of the Dayton area tornados of May 2019 had passed through this area. Sonic anemometry continues to benefit from a strong vendor base, which makes high-quality instrumentation and a wide variety of probe array configurations available, many optimized according to one’s application (e.g., optical imaging and communication, near-surface fundamental eddy covariance flux assessments, or canopy wind effects and simulations). Others are designed to operate well in extreme ambient conditions (i.e., some probes are configured for especially high wind velocity), and/or to minimize flow asymmetries and interference associated with towers and other structures on which the sonic anemometers are mounted. For this research, a variety of probe configurations designed and built by Applied Technologies Inc. (ATI) were considered and fielded, amongst which is ATI’s non-orthogonal A-probe. This probe’s specifications include the ability to preserve sonic temperature and wind velocity precision and accuracy even in conditions of very high wind speeds. Ultimately, the best correlated, multi-instrument dataset for the study herein occurred during a 2-day period of benign weather: 14–15 July 2020. Conditions during this period were near 30-year climatological means. On 14 July, the sky condition was clear to partly cloudy, temperatures varied from 17 to 30 ◦C, with light east to northeast winds, and a period of no wind from ~0200UTC to 1200UTC where a significant temperature inversion near the surface occurred. On 15 July, the sky condition was nearly clear all day, temperatures varied from 19 to 32 ◦C, with light south to southwest winds, and periods of calm from ~0200UTC to 1200UTC. Dew point temperatures remained between 15 to 18 ◦C for the entire 2-day period; this means relative humidity varied from ~30% in the afternoons to ~90% in the mornings just before sunrise.

2.5. Data Analysis: Summary Description The relevant structure functions were calculated in the spatial domain using the recorded measurements in their raw form as opposed to implementing an intermediate step converting the data to the frequency domain. Additionally, given the topographical non-homogeneities and acknowledging the challenges associated with precision leveling and alignment of a single sonic anemometer, much less a coupled pair, stream-wise rotation was applied to the sonic anemometer velocity measurements [24].

3. Results and Discussion 2 The top set of plots in Figure4 shows calculated Cn profiles for 14 July 2020 that originated from Ts measurements recorded using the ATI sonic anemometers mounted at 6.4, 10.2, and 24.5 m above the surface. The profiles are a direct result of calculation 2 of the structure function CT according to Equation (2) and then through application of Equation (3). These profiles show typical quiescent periods at approximately an hour after sunrise (1130UTC) and an hour prior to sunset (2330UTC). Generally, the highest 2 anemometer recorded the lower Cn values, while the lowest anemometer recorded the highest values. Of note are the periods where this general characteristic is reversed (e.g., ~0500 to ~0730UTC); these are periods of marked atmospheric stability. The bottom set of plots shown in Figure4 illustrate the importance of removing the large-scale index of refraction and only using the portion of the refractive gradient that is driven by turbulence (i.e., the small-scale fluctuations in the index of refraction gradient 2 2 ∇n’, per Equation (19)). The 24.5-meter Cn from the CT plot in the bottom portion of Figure4 is the same as the 24.5-meter plot shown in the top portion of the figure, except that 2 2 a 90-second ensemble averaging time is used instead of 180 s; this Cn from the CT profile was 2 2 used as a reference to verify the suitability of the Cn from the Cv calculation. Subsequently 2 2 profiled in the bottom portion of Figure4 are Cn from Cv using alternate forms of the 2 temperature gradient when deriving the vertical gradient of the index of refraction. The Cn 2 from Cv using Tv for the vertical temperature gradient (∂Tv/∂z) plot deviates significantly 2 2 from the Cn from CT baseline plot during quiescent periods, stable periods, and the late Appl. Sci. 2021, 11, 7658 11 of 18

afternoon—virtually throughout the entire period. This is expected, since ∂Tv/∂z contains the temperature decrease with height caused by air expanding adiabatically—this is the primary component of the large-scale index of refraction that is always present in the atmosphere to some degree and is not due to turbulence and the index of refraction structure function. A common practice to remove this adiabatic expansion cooling effect is to convert the temperature or virtual temperature to potential temperature or potential virtual temperature as conducted in the AppendixA to arrive at Equation (5). This was 2 2 conducted for the Cn from Cv using the potential virtual temperature, θv, for the vertical temperature gradient (∂θv/∂z)—and hence the index of refraction gradient—plot in the 2 2 lower part of Figure4. While this brought the Cn from Cv using the ∂θv/∂z plot much 2 2 more in line with the Cn from the CT baseline plot, there remained a signiﬁcant deviation during the stable period from ~0500UTC to ~0730UTC when θv increased with height. 2 2 Using ∂θv/∂z for the lapse with height should have corrected the Cn from the Cv plot to the baseline at all times. However, it did not because an accurate conversion from Tv to θv requires the instantaneous pressure value associated with each sonic Tv measurement at each level utilized in the ∂θv/∂z calculation. Since the sonic anemometers did not provide instantaneous pressure measurements with each Tv value, the pressure was calculated at the level of each sonic anemometer from the pressure at the level of the DELTA target board weather station assuming hydrostatic balance. This produced a dry adiabatic lapse of temperature and introduced large-scale index of refraction ∂θv/∂z errors at times when the atmosphere was markedly stable (not adiabatic). Thus, this research presents the methodology encapsulated in Equations (17)–(19) whereby the large-scale Tv vertical gradient, and, in turn, the large-scale index of refraction gradient, is removed without the need for instantaneous pressure at each sonic Tv measurement. The impact is quite 2 2 apparent. As can be seen in the bottom part of Figure4, the Cn from Cv with the large scale Appl. Sci. 2021, 11, x FOR PEER REVIEWT C2 C2 11 of 18 v vertical gradient removed plot follows the n from the T baseline plot throughout the entire day.

2n2 FigureFigure 4.4. Top:Top: proﬁlesprofiles ofofC Cn at at multiple multiple points points above above the the surface surface as as extracted extracted from from calculations calculations ofof 2 the temperature structure constant, 퐶2 푇, using in situ point measurement of sonic temperature, 푇푠; the temperature structure constant, CT, using in situ point measurement of sonic temperature, Ts; a a three-minute ensemble averaging time is used for each of the plots. Bottom: Cn2 from CT2 compared three-minute ensemble averaging time is used for each of the plots. Bottom: C2 from C2 compared 2 2 n T to C2n from C2v and the gradient of index of refraction in which the operative vertical temperature to Cn from Cv and the gradient of index of refraction in which the operative vertical temperature gradient is either θv, Tv, or Tv but with the large scale Tv vertical gradient removed per Equation (19). gradient is either θ , T , or T but with the large scale T vertical gradient removed per Equation (19). All plots are for 24.5v mv and vuse a 90-second ensemblev averaging time. All plots are for 24.5 m and use a 90-second ensemble averaging time. The bottom set of plots shown in Figure 4 illustrate the importance of removing the large-scale index of refraction and only using the portion of the refractive gradient that is driven by turbulence (i.e., the small-scale fluctuations in the index of refraction gradient ∇n’, per Equation (19)). The 24.5-meter Cn2 from the CT2 plot in the bottom portion of Figure 4 is the same as the 24.5-meter plot shown in the top portion of the figure, except that a 90-second ensemble averaging time is used instead of 180 s; this Cn2 from the CT2 profile was used as a reference to verify the suitability of the Cn2 from the Cv2 calculation. Subse- quently profiled in the bottom portion of Figure 4 are Cn2 from Cv2 using alternate forms of the temperature gradient when deriving the vertical gradient of the index of refraction. The Cn2 from Cv2 using Tv for the vertical temperature gradient (∂Tv/∂z) plot deviates significantly from the Cn2 from CT2 baseline plot during quiescent periods, stable periods, and the late afternoon—virtually throughout the entire period. This is expected, since ∂Tv/∂z contains the temperature decrease with height caused by air expanding adiabatically— this is the primary component of the large-scale index of refraction that is always present in the atmosphere to some degree and is not due to turbulence and the index of refraction structure function. A common practice to remove this adiabatic expansion cooling effect is to convert the temperature or virtual temperature to potential temperature or potential virtual temperature as conducted in the Appendix A to arrive at Equation (5). This was conducted for the Cn2 from Cv2 using the potential virtual temperature, θv, for the vertical temperature gradient (∂θv/∂z)—and hence the index of refraction gradient—plot in the lower part of Figure 4. While this brought the Cn2 from Cv2 using the ∂θv/∂z plot much more in line with the Cn2 from the CT2 baseline plot, there remained a significant deviation during the stable period from ~0500UTC to ~0730UTC when θv increased with height. Using ∂θv/∂z for the lapse with height should have corrected the Cn2 from the Cv2 plot to the baseline at all times. However, it did not because an accurate conversion from Tv to θv requires the instantaneous pressure value associated with each sonic Tv measurement at each level utilized in the ∂θv/∂z calculation. Since the sonic anemometers did not provide instantaneous pressure measurements with each Tv value, the pressure was calculated at the level Appl.Appl. Sci.Sci. 20202121,, 1111,, xx FORFOR PEERPEER REVIEWREVIEW 1212 ofof 1818

ofof each each sonic sonic anemometer anemometer from from the the pressure pressure at at the the level level of of the the DELTA DELTA target target board board weatherweather stationstation assumingassuming hydrostatichydrostatic balancebalance.. ThisThis produceproducedd aa drydry adiabaticadiabatic lapselapse ofof tem-tem- peratureperature andand introducedintroduced largelarge--scscaleale indexindex ofof refractionrefraction ∂θ∂θvv/∂z/∂z errorserrors atat timestimes whenwhen thethe atmosphereatmosphere waswas markedlymarkedly stablestable (not(not adiabatic).adiabatic). ThusThus,, thisthis researchresearch presentspresents thethe method-method- ologyology encapsulatedencapsulated inin EquationEquationss ((1717))––((1919)) wherebywhereby thethe largelarge--scalescale TTvv verticalvertical gradient,gradient, andand,, inin turnturn,, thethe largelarge--scalescale indexindex ofof refractionrefraction gradient,gradient, isis removedremoved withoutwithout thethe needneed forfor instantaneousinstantaneous pressurepressure atat eacheach sonicsonic TTvv measurement.measurement. TheThe impactimpact isis quitequite apparent.apparent. AsAs Appl. Sci. 2021, 11, 7658 2 2 12 of 18 cancan bebe seenseen inin thethe bottombottom partpart ofof FigureFigure 4,4, thethe CCnn2 fromfrom CCvv2 withwith thethe largelarge scalescale TTvv verticverticalal 2 2 gradientgradient removedremoved plotplot followsfollows thethe CCnn2 fromfrom thethe CCTT2 baselinebaseline plotplot throughoutthroughout thethe entireentire day.day. 퐶22 FigureFigureFigure5 5 5shows showsshows the thethe calculated calculatedcalculated C퐶푛2푛 profilesprofilesprofiles for forfor 14 1414 JulyJuly 20202020 at at 24.5 24.5 mm aboveabove thethe the surfacesurface surface n 22 predicatedpredicated onon aa calculationcalculation ofof thethe velocityvelocity structurestructure constant,constant, 퐶퐶푣2,, atat thatthat heightheight asas wellwell asas predicated on a calculation of the velocity structure constant, C푣v, at that height as well as thethethe ensemble ensembleensemble averageaverageaverage ofof thethe smallsmall scale scale fluctuations fluctuationsfluctuations ofof of thethe the indexindex index ofof of refractionrefraction refraction gradient,gradient, gradient, ∇〈푛′′〉 ∇∇〈n푛0,〉 and,, andand the thethe ensemble ensembleensemble average averageaverage of ofof the thethe gradient gradientgradient of ofof the thethe wind windwind velocity velocityvelocity in inin the thethe vertical verticalvertical direction direc-direc- 휕〈푽(푧)〉 ⁄ 휕푧 ∂tiontionhV( z휕)〈i푽∂z(푧.) As〉 ⁄ described휕푧.. AsAs describeddescribed in Section inin 2SectionSection, the latter 22,, thethe gradients latterlatter gradientsgradients are determined areare determineddetermined via measurements viavia meas-meas- recordedurementsurements with recordedrecorded a primary withwith andaa primaryprimary secondary andand secondary sonicsecondary anemometer, sonicsonic anemometer,anemometer, in this case inin the thisthis primary casecase thethe was pri-pri- at 24.5marymary m waswas above atat 24.524.5 the groundmm aboveabove and thethe the groundground secondary andand thethe at secondarysecondary 10.2 m above atat 10.210.2 the mm ground. aboveabove thethe The ground.ground. calculations TheThe werecalculationscalculations implemented w wereere implemented afterimplemented converting afterafter the convertingconverting primary and thethe secondary primary primary andand sonic secondary secondary anemometer sonic sonic velocity ane-ane- measurementsmometermometer velocity velocity to the measurements measurements mean wind streamlined to to the the mean mean coordinate wind wind streamlined streamlined system through coordinate coordinate triple rotationsystem system aboutthroughthrough the tripletriple sonic rotationrotation anemometer’s aboutabout thethe u, sonicsonic v, and anemometer’sanemometer’s w coordinate u,u, v, systemv, andand ww summarily coordinatecoordinate referencedsystemsystem sum-sum- in 퐶22 Sectionmarilymarily 2.5referencedreferenced above. Figure inin SectSect5 ionionalso 22 includes.5.5 above.above. the FigureFigureC2 profile 55 alsoalso asincludesincludes derived thethe from 퐶푛푛 Cprofileprofile2 . The asas correlation derivedderived 퐶22 n T isfromfrom quite 퐶푇 good.푇.. TheThe correlationcorrelation isis quitequite good.good.

22 FigureFigureFigure 5. 5.5. Proﬁles ProfilesProfiles of ofof C CC2nn at atat 24.5 24.524.5 m mm above aboveabove thethethe surfacesurfacesurface usingusing in in situ situ sonic sonic anemometeranemometer measurementsmeasurements measurements 22 n ′′ derivedderived fromfrom ((a.)a.) 퐶퐶푣2푣 atat 24.524.5 m,m, asas wellwell asas ∇∇푛푛0 (large(large scalescale indexindex ofof refractionrefraction gradientgradient removed),removed), derived from (a.) Cv at 24.5 m, as well as ∇n (large scale index of refraction gradient removed), andand 휕휕푽푽⁄⁄휕푧휕푧 derivedderived onon sonicsonic anemometeranemometer measurementsmeasurements atat bothboth 24.524.5 mm andand 10.210.2 m;m; andand ((b.)b.) derivedderived and ∂V/∂2z derived on sonic anemometer measurements at both 24.5 m and 10.2 m; and (b.) derived fromfrom CCTT2 calculatedcalculated usingusing thethe inin situsitu pointpoint measurementmeasurement ofof thethe sonicsonic temperature,temperature, 푇푇푠,, atat 24.524.5 m.m. AA 2 푠 from9090--ssecondecondCT calculated ensembleensemble using averagingaveraging the in timetime situ isis point usedused measurement forfor thethe plots.plots. of the sonic temperature, Ts, at 24.5 m. A 90-second ensemble averaging time is used for the plots. InInIn turn, turn,turn, FiguresFiguresFigures6 66 and andand7 77show showshow the thethe effect effecteffect of ofof the thethe streamwise streamwisestreamwise rotationrotationrotation ofofof thethe windwind vector vector,vector,, whereinwhereinwherein the thethe sonic sonicsonic anemometer anemometeranemometer datadatadata isis treatedtreated inin aa streamlinedstreamlined coordinate coordinate system.system.

Appl. Sci. 2021, 11, x FOR PEER REVIEWFigure 6. Profiles of C2n22 at 24.5 m above the surface after applying streamwise (SW) triple rotation13 of 18 FigureFigure 6.6. ProﬁlesProfiles ofof CCnn at at 24.524.5 mm aboveabove thethe surface after applying streamwise streamwise (SW) (SW) triple triple rotation rotation schemeschemescheme about aboutabout the thethe sonic sonicsonic anemometer’s anemometer’sanemometer’s “u”, “u”, “v”, “v”, and and “w”“w” coordinatecoordinate system.system. system.

Figure 7. Profiles of C2n2 at 10.2 m above the surface after applying streamwise (SW) triple rotation Figure 7. Proﬁles of Cn at 10.2 m above the surface after applying streamwise (SW) triple rotation schemescheme about about thethe sonicsonic anemometer’sanemometer’s “u”, “v”, and “w” coordinate system. system.

The impact of streamwise rotation is especially significant for the measurements nearer the surface (i.e., at 10.2 m), which is to be expected given the increased proximity to topographical inhomogeneities at the lower height. The profiles of Cn2 at both 24.5 and 10.2 m for July 14 are overlaid in Figure 8. As is the case in Figures 6 and 7 above, each profile at the height of interest is obtained by combining the wind velocity and calculated small scale fluctuations of the index of refraction gradients with the velocity structure constant, Cv, using the primary sonic anemometer’s (whether at 24.5 or 10.2 m) point measurements. There is no discernible difference in the magnitude of the Cn2 at each level in Figure 8, as is evident in the top image of Figure 4; this is likely due to the turbulent refractive index gradient in the 10.2 to 24.5 m layer being the primary calculation component for the Cn2 at both levels.

휕〈푽〉 Figure 8. Profiles of 퐶2 derived from 퐶2, ∇〈푛′〉, at heights 24.5 and 10.2 m above the surface. 푛 푉 휕푧

In Figure 9, the DELTA Cn2 profile overlays those of the sonic anemometers quite well through the day, even at the sunset/sunrise quiescent periods, though there is a notable separation at approximately 1700UTC to 2400UTC. The DELTA’s lower afternoon Cn2 values brought its mean Cn2 value for the day to less than half the mean value found using both sonic anemometer methods. Examining the DELTA camera images shown in Figure 10 below seems to suggest this separation occurred as the target board’s feature points became more difficult to discern and track, specifically from approximately 1700UTC through 2100UTC, when the target board became more backlit as the sun transited to the western portion of the sky. Additionally, the 14/0006UTC image in Figure 10 shows that the target board illumination that was used during the field test appears to overwhelm the near sunset lighting conditions causing such a high contrast that only large feature separation distances are discernible. This could have had the effect of limiting the number of weighting functions such that the DELTA would provide more of an integrated path Cn2 value weighted slightly toward the camera end of the path—which was 30 m high on a 60-meter tower. This increases the likelihood that the DELTA and 30-meter mast sonic anemometer instruments were sampling significantly different volumes of air. The 14/1200UTC image in Figure 10 shows the target board clearly displayed many different Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 18

Appl. Sci. 2021, 11, 7658 13 of 18 Figure 7. Profiles of Cn2 at 10.2 m above the surface after applying streamwise (SW) triple rotation scheme about the sonic anemometer’s “u”, “v”, and “w” coordinate system.

TheThe impact impact of streamwise rotation is especially significant significant for for the the measurements measurements nearernearer thethe surfacesurface (i.e.,(i.e.,at at 10.210.2 m), m), which which is is to to be be expected expected given given the the increased increased proximity proximity to topographicalto topographical inhomogeneities inhomogeneities at at the the lower lower height. height. The profiles of C2n2 at both 24.5 and 10.2 m for July 14 are overlaid in Figure 8. As is The profiles of Cn at both 24.5 and 10.2 m for July 14 are overlaid in Figure8. As is the casethe case in Figures in Figures6 and 67 a above,nd 7 above, each profileeach profile at the at height the height of interest of interest is obtained is obtained by combining by com- thebining wind the velocity wind velocity and calculated and calculated small scale small fluctuations scale fluctuations of the index of the of index refraction of refraction gradients v withgradients the velocity with the structure velocity constant, structureC constant,v, using the C , primary using the sonic primary anemometer’s sonic anemometer’s (whether at 24.5(wheth or 10.2er at m)24.5 point or 10.2 measurements. m) point measurements. There is no There discernible is no discernible difference indifference the magnitude in the magnitude2 of the Cn2 at each level in Figure 8, as is evident in the top image of Figure 4; of the Cn at each level in Figure8, as is evident in the top image of Figure4; this is likely duethis tois likely the turbulent due to the refractive turbulent index refractive gradient index in thegradient 10.2 to in 24.5 the m10.2 layer to 24.5 being m layer the primary being the primary calculation component2 for the Cn2 at both levels. calculation component for the Cn at both levels.

휕〈푽〉 Figure 8. Profiles of 2퐶2 derived from 2퐶2∇, ∇h〈푛0′i〉, ∂hV iat heights 24.5 and 10.2 m above the surface. Figure 8. Proﬁles of Cn 푛derived from CV,푉 n , 휕푧∂z at heights 24.5 and 10.2 m above the surface.

2 InIn FigureFigure9 9,, thethe DELTADELTA Cn2 profileprofile overlays those of of the the sonic sonic anemometers anemometers quite quite well well throughthrough thethe day,day, eveneven atat thethe sunset/sunrisesunset/sunrise quiescent quiescent periods, periods, though though there there is is a anotable notable 2 separationseparation atat approximatelyapproximately 1700UTC1700UTC toto 2400UTC.2400UTC. The The DELTA’sDELTA’s lower lower afternoon afternoonC Cn nvalues2 val- 2 broughtues brought its mean its meanCn value Cn2 value for the forday the today less to thanless than half thehalf meanthe mean value value found found using using both sonicboth sonic anemometer anemometer methods. methods. Examining Examining the the DELTA DELTA camera camera images images shown shown in in Figure Figure 10 below10 below seems seems to suggest to suggest this separationthis separation occurred occurred as the as target the target board’s board’s feature feature points becamepoints morebecame difficult more todifficult discern to and discern track, and specifically track, specifically from approximately from approximately 1700UTC 1700UTC through 2100UTC,through 2100UTC, when the when target the board target became board morebecame backlit more as backlit the sun as the transited sun transited to the western to the portionwestern of portion the sky. of Additionally, the sky. Additionally the 14/0006UTC, the 14/0006UTC image in image Figure in 10 Figure shows 10 that shows the targetthat boardthe target illumination board illumination that was used that during was used the field during test appearsthe fieldto test overwhelm appears to the overwhelm near sunset lightingthe near conditions sunset lighting causing conditions such a high causing contrast such that a onlyhigh largecontrast feature that separationonly large distances feature areseparation discernible. distances This could are discernible. have had the This effect could of have limiting had the the number effect of of limiting weighting the functionsnumber 2 suchof weighting that the DELTAfunctions would such provide that the more DELTA of an would integrated provide path moreCn value of an weightedintegrated slightly path towardCn2 value the weighted camera slightly end of toward the path—which the camera was end 30of the m highpath— onwhich a 60-meter was 30 tower. m high This on increasesa 60-meter the tower. likelihood This increases that the DELTA the likelihood and 30-meter that the mast DELTA sonic and anemometer 30-meter instrumentsmast sonic wereanemometer sampling instruments significantly were different sampling volumes significantly of air. The 14/1200UTCdifferent volumes image of in air.Figure The 10 shows14/1200UTC the target image board in Figure clearly 10 displayed shows the many target different board clearly feature displayed separations many such different that the full number of weighting functions—including one peaking near the 600-meter distance from the DELTA receiver to the 30-meter mast—should have been available for a true profiling capability. Additionally, it is evident in Figure 10 that the target board position shifted significantly in the images throughout the day, this had little consequence on the system’s ability to quantify and profile turbulence along the path since multiple feature separation distances were always visible within the camera’s processing “mask” field 2 of view. Thus, the DELTA Cn profile measurement method is not susceptible to large- 2 scale effects on the Cn quantification since DELTA only uses the measured differential tilt variances for the various feature separations rather than feature positions within the field of view. Appl.Appl. Sci.Sci. 20202121,, 1111,, xx FORFOR PEERPEER REVIEWREVIEW 1414 ofof 1818

featurefeature separationsseparations suchsuch thatthat thethe fullfull numbernumber ofof weightingweighting functionsfunctions——includingincluding oneone peak-peak- inging nearnear thethe 606000--mmetereter distancedistance fromfrom thethe DELTADELTA receiverreceiver toto thethe 3030--mmetereter mastmast——shouldshould havehave bebeenen availableavailable forfor aa truetrue profilingprofiling capability.capability. Additionally,Additionally, itit iiss evidentevident inin FigureFigure 1010 thatthat thethe targettarget boardboard positionposition shiftedshifted significantlysignificantly inin thethe imagesimages throughoutthroughout thethe day,day, thisthis hadhad littlelittle consequenceconsequence onon thethe system’ssystem’s abilityability toto quantifyquantify andand profileprofile turbulenceturbulence alongalong thethe pathpath sincesince multiplemultiple featurefeature separationseparation distancesdistances werewere alwaysalways visiblevisible withinwithin thethe camera’scamera’s processing “mask” field of view. Thus, the DELTA Cn2 profile measurement method is not processing “mask” field of view. Thus, the DELTA Cn2 profile measurement method is not susceptible to large-scale effects on the Cn2 quantification since DELTA only uses the meas- Appl. Sci. 2021, 11, 7658 susceptible to large-scale effects on the Cn2 quantification since DELTA only uses the meas-14 of 18 uredured differentialdifferential tilttilt variancesvariances forfor thethe variousvarious featurefeature separationsseparations ratherrather thanthan featurefeature po-po- sitionssitions withinwithin thethe fieldfield ofof viewview..

2 FigureFigureFigure 9.9.9. ProﬁlesProfilesProfiles of of C퐶퐶푛22 atatat 24.524.5 aboveabove the the surface surface using using inin situsitu sonicsonic anemometeranemometer anemometer measurementsmeasurements measurements n푛 휕〈푽〉 2 ′ 휕〈푽∂〉hVi −14 −14 (derived from 퐶2푉2, ∇〈푛 ′〉,0 , mean = 1.3 × 10 −14, std−14 dev = 3.6 × 10 −14) as well−14 as the DELTA system (derived(derived fromfrom C퐶V푉,, ∇∇〈푛hn〉,i휕푧, ∂, zmean, mean = 1.3 = 1.3× 10× 10, std dev, std = dev3.6 × = 10 3.6 )× as10 well )as as the well DELTA as the system DELTA −15 휕푧 −15 2 (mean = 4.8 × 10 −15, std dev− 15= 3.4 × 10 −15). For comparison,−15 a profile of 퐶푛2 at 24.5 m as derived2 from system(mean = (mean 4.8 × 10 = 4.8, std× 10dev =, 3.4 std × dev 10 =). 3.4For× comparison,10 ). For a comparison,profile of 퐶푛 aat proﬁle 24.5 m of asC derivedn at 24.5 from m as 2 −15 −14 퐶푇2 (mean = 9.8 2× 10 −15, std dev = 1.9− 15× 10 −14) on 14 July 20 at−14 Wright-Patterson AFB. derived퐶푇 (mean from = 9.8CT ×(mean 10 , std = 9.8 dev× =10 1.9 ×, 10 std dev) on =14 1.9 Jul×y 2010 at Wright) on 14-Patterson July 20 at AFB. Wright-Patterson AFB.

Figure 10. Representative, individual image frames recorded at multiple times on 14 July using the FigureFigure 10.10.Representative, Representative, individualindividual image frames recorded recorded at at multiple multiple times times on on 14 14 July July using using the the DELTA eight-inch aperture camera at 100 Hz frame rate. As noted earlier, the system measures the DELTADELTA eight-incheight-inch aperture camera camera at at 100 100 Hz Hz frame frame rate. rate. As Asnoted noted earlier, earlier, the system the system measures measures the differentialdifferential motion,motion, oror tiltstilts betweenbetween pairspairs ofof featurefeature poipointsnts onon thethe targettarget asas aa functionfunction ofof theirtheir angularangular the differential motion, or tilts between pairs of feature points on the target as a function of their separation.separation. angular separation. 2 Figure 11 shows Cn22 profiles for 15 July 2020 at 24.5 m above the surface using 퐶 2 2or FigureFigure 1111 showsshows Cn2 proﬁlesprofiles forfor 15 July 2020 at at 24.5 24.5 m m above above the the surface surface using using 퐶C푇 oror n 2 푇T Appl. Sci. 2021, 11, x FOR PEER REVIEWalternatively via direct calculation of the structure function 퐶2푣2 as was carried out for15 theof 18 alternativelyalternatively viavia direct direct calculation calculation of of the the structure structure function functionC v퐶푣as as was was carried carried out out for for the the 14 July1414 JulyJuly 2020 20202020 proﬁle. profile.profile.

2 FigureFigure 11.11. ProﬁlesProfiles ofof C퐶2푛 atat 24.5 24.5 above above the the surface surface using using in in situ situ sonic sonic anemometer anemometermeasurements measurements n 휕〈푽〉 (derived from 퐶22, ∇〈푛′〉0, ∂h,V meani = 1.2 × 10−14, std−14 dev = 1.6 × 10−14) as −well14 as the DELTA system (derived from CV푉, h∇n i,휕푧 ∂z , mean = 1.2 × 10 , std dev = 1.6 × 10 ) as well as the DELTA system(mean = (mean 4.6 × =10 4.6−15, ×std10 dev−15 ,= std 3.0 dev × 10 =−15 3.0) on× 1510 −Jul15y) on2021; 15 JulyDELTA 2021; measurements DELTA measurements were not wereavailable not 2 2 −14 after 1200UTC. For comparison, a profile of 퐶푛 at 24.52 m, as derived from 퐶푇 (mean2 = 1.0 × 10 , available after 1200UTC. For comparison, a proﬁle of Cn at 24.5 m, as derived from CT (mean = 1.0 × −14 10std− 14dev, std = 1.1 dev × =10 1.1)× on10 15−14 Jul) ony 20 15 at July Wright 20 at-Patterson Wright-Patterson AFB. AFB.

The Cn2 plots using CT2 and those obtained using Cv2 at 24.5 m match quite well, but do not show pronounced quiescent dips near sunset (15/0000UTC to 15/0130UTC) and sunrise (15/1130UTC) to 15/1230UTC). The DELTA plot, however, does, as it did for 14 July, show significant minima of Cn2 just prior to sunset on 15 July (UTC, 14 July local time) and just after sunrise on 15 July. The DELTA’s lower quiescent period Cn2 values again brought its mean Cn2 value for the day to less than half the mean value found using both sonic anemometer methods. Figure 12 below shows DELTA camera images at two points in time, from which differential tilt variances are derived to arrive at the Cn2 profile shown in Figure 11 (DELTA images and data were not available after 12UTC on 15 July). The image at 15/0006UTC in Figure 12 looks nearly identical to the one at 14/0006UTC in Figure 10. This suggests that the limited feature separation distances resulted in the lower DELTA Cn2 value being influenced more by the more quiescent air near the receiver camera that was ~30 m higher than the air volume the sonic anemometers were sampling at 10–25 m above the ground at the 30-meter mast (see Figure 3). However, the 15/1122UTC image in Figure 12 also looks identical to its counterpart in Figure 10—this indicates that even with the multiple feature separation distances available, the DELTA weighting functions cannot be expected to provide exactly comparable data to that measured at a point along the path where the weighting function may peak. Additionally, the turbulence in- duced random motion of the high contrast features on the target board against relatively darker background areas in the morning results in some spurious banding effects in the images. This probably caused issues with tracking and error in estimating Cn2.

Figure 12. DELTA camera images as recorded at the times shown on 15 July 2021. Spurious banding effects are evident on the left edges of the leftmost white target board patches in the 1122UTC image.

4. Conclusions Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 18

2 Figure 11. Profiles of 퐶푛 at 24.5 above the surface using in situ sonic anemometer measurements 휕〈푽〉 (derived from 퐶2, ∇〈푛′〉, , mean = 1.2 × 10−14, std dev = 1.6 × 10−14) as well as the DELTA system 푉 휕푧 (mean = 4.6 × 10−15, std dev = 3.0 × 10−15) on 15 July 2021; DELTA measurements were not available after 1200UTC. For comparison, a profile of 퐶2 at 24.5 m, as derived from 퐶2 (mean = 1.0 × 10−14, Appl. Sci. 2021, 11, 7658 푛 푇 15 of 18 std dev = 1.1 × 10−14) on 15 July 20 at Wright-Patterson AFB.

The Cn2 plots using CT2 and those obtained using Cv2 at 24.5 m match quite well, but do not show2 pronounced2 quiescent dips near sunset 2(15/0000UTC to 15/0130UTC) and The Cn plots using CT and those obtained using Cv at 24.5 m match quite well, but dosunrise not show (15/1130UTC) pronounced to quiescent15/1230UTC) dips. The near DELTA sunset (15/0000UTCplot, however, to does, 15/0130UTC) as it did for and 14 sunriseJuly, show (15/1130UTC) significant tominima 15/1230UTC). of Cn2 just The prior DELTA to sunset plot, however,on 15 July does, (UTC, as it14 did July for local 14 2 2 July,time) show and significant just after sunrise minima on of C15n justJuly. prior The toDELTA’s sunset on lower 15 July quiescent (UTC, 14 period July local Cn values time) 2 2 andagain just brought after sunrise its mean on C 15n value July. Thefor the DELTA’s day to less lower tha quiescentn half the periodmean valueCn values found again using 2 broughtboth sonic its meananemometerCn value methods. for the day Figure to less 12 thanbelow half shows the meanDELTA value camera found images using at both two sonicpoints anemometer in time, from methods. which differential Figure 12 below tilt variances shows DELTA are derived camera to images arrive at twothe C pointsn2 profile in 2 time,shown from in Figure which differential11 (DELTA tiltimages variances and data are derived were not to available arrive at theafterC n12UTCprofile on shown 15 July). in FigureThe image 11 (DELTA at 15/0006UTC images and in Fig dataure were 12 looks not available nearly identical after 12UTC to the on one 15 July).at 14/0006UTC The image in atFigure 15/0006UTC 10. This insuggest Figures that12 looks the limited nearly feature identical separation to the one distances at 14/0006UTC resulted in in Figure the lower 10. ThisDELTA suggests Cn2 value that being the limited influenced feature more separation by the more distances quiescent resulted air near in the the lower receiver DELTA cam- 2 Ceran value that beingwas ~30 influenced m higher more than bythe the air morevolume quiescent the sonic air anemometers near the receiver were camera sampling that at was10–25 ~30 m m above higher the than ground the airat the volume 30-m theeter sonic mast anemometers(see Figure 3). were However, sampling the 15/1122UTC at 10–25 m aboveimage the in Figure ground 12 at also the 30-meterlooks identical mast (seeto its Figure counterpart3). However, in Figure the 10 15/1122UTC—this indicates image that ineven Figure with 12 the also multiple looks identicalfeature separation to its counterpart distances in available, Figure 10 the—this DELTA indicates weighting that even func- withtions the cannot multiple be expected featureseparation to providedistances exactly comparable available,the data DELTA to that weighting measured functions at a point cannotalong the be expectedpath where to provide the weighting exactly function comparable may data peak. to Additionally, that measured the at aturbulence point along in- theduced path random where themotion weighting of the high function contrast may features peak. Additionally, on the target the board turbulence against inducedrelatively randomdarker background motion of the areas high in contrast the morning features results on the in target some boardspurious against banding relatively effects darker in the background areas in the morning results in some spurious banding effects in2 the images. images. This probably caused issues with tracking and error in estimating Cn . 2 This probably caused issues with tracking and error in estimating Cn.

FigureFigure 12. 12.DELTA DELTA camera camera images images as as recorded recorded at at the the times times shown shown on on 15 15 July July 2021. 2021. Spurious Spurious banding banding effects are evident on the left edges of the leftmost white target board patches in the 1122UTC image. effects are evident on the left edges of the leftmost white target board patches in the 1122UTC image.

4.4. Conclusions Conclusions 2 In this paper, we revisit in detail an alternative approach to extracting Cn, which capitalizes on another structure constant rarely harvested from sonic anemometers, the 2 velocity structure constant, Cv. As noted earlier, the 3D sonic anemometer provides a unique opportunity to simultaneously sample all of the interrelated physical parameters, which bear on the index of refraction fluctuations. To underscore this advantage, a stepwise 2 2 summary of key physical relationships leading to an expression tying Cv to Cn, is presented. 2 Beyond the traditional processing of wind velocity data to calculate Cv, the expression provides the occasion to draw on a key parameter of sonic anemometry, the sonic temperature and its equivalency to virtual temperature, to infer inherent air density and moisture fluctuations and, consequently, index of refraction gradients. The paper includes details on the field experiments and data analysis used to corroborate the feasibility and suitability 2 2 of extracting representative Cn in this alternative manner using Cv. While the study did 2 2 illustrate the ease and accuracy of obtaining Cn via CT from a single anemometer, it is 2 2 anticipated that the set of Cn from Cv equations (Equations (4), (5), and (19)) that do not require instantaneous sonic pressure measurements, could serve to amplify the strengths of volumetric instrumentation that measure wind velocities with more spatial/temporal fidelity than temperature, such as SODAR and Doppler LiDAR/LaDAR (Light Detection Appl. Sci. 2021, 11, 7658 16 of 18

and Ranging, Laser Detection, and Ranging). Furthermore, the AppendixA herein pro- 2 vides a unique derivation of how the velocity structure function, Cv, relates to the refractive 2 index structure function, Cn, (Equation (5)) and eliminates the need to ignore the water vapor contribution to the index of refraction by combining temperature and moisture into one parameter, the virtual temperature.

Author Contributions: Conceptualization, S.F. and K.K.; methodology, S.F. and K.K.; software, K.K.; validation, S.F., K.K. and S.B.-P.; formal analysis, S.F., S.B.-P. and K.K.; investigation, K.K.; resources, K.K.; data curation, K.K.; writing—original draft preparation, K.K., S.F. and S.B.-P.; writing—review and editing, S.F., K.K. and S.B.-P.; visualization, K.K., S.F. and S.B.-P.; supervision, S.F.; project administration, S.F.; funding acquisition, S.F. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the Directed Energy Joint Transition Office (DE-JTO) and the US Office of Naval Research (ONR). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors thank the Directed Energy Professional Society (DEPS) for spon- sorship and the US Air Force Research Laboratory (AFRL) for access to its 30-meter mast as well as AFRL support personnel who assisted with mast construction. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A According to Tatarskii [25], the structure constant of a conservative passive additive υ in a turbulent ﬂow takes the following form:

1/3 " 2 # 2 2 K 2 Cυ = a (∇hυi) (A1) (∂hVi/∂z)2

where a is a numerical constant, K is the coefﬁcient of turbulent diffusion, hVi is the mean wind velocity. Potential temperature θ and speciﬁc humidity q being conservative passive additives, their structure constants can be expressed by Equation (A1). Since the refractive index n can be expressed as a function of θ and q, the structure constant for refractive index can be expressed in terms of the structure constants of θ and q as well [26]: 2 2 2 2 2 Cn = (∂n/∂θ) Cθ + (∂n/∂q) Cq + 2(∂n/∂θ)(∂n/∂q)Cθq (A2) In most cases of optical propagation, the dry air term dominates, and the other terms can be ignored. However, using the potential virtual temperature θv instead of the potential 2 temperature, the effects of humidity can be included in the analysis. Hence, Cn can be approximated as follows:

2 2 2 Cn ≈ (∂n/∂θv) Cθv 1/3 (A3) 2 2 K2 2 ≈ a (∂n/∂θv) (∇hθvi) (∂hVi/∂z)2

Since the mean gradient of θv is primarily in the vertical direction,

∂n/∂z ∇hθvi = ∂θv/∂z = (A4) ∂n/∂θv Appl. Sci. 2021, 11, 7658 17 of 18

.. The energy dissipation rate ε is related to the coefﬁcient of turbulent diffusion [25],

2 .. ∂hVi ε = K (A5) ∂z

2 Additionally, the velocity structure constant Cv can be expressed as follows [26]:

2 ..2/3 Cv = 2ε (A6)

where the constant has been determined experimentally. Using Equations. (A4), (A5), and (A6) in Equation (A3),

a2 ∂n/∂z 2 C2 = C2 (A7) n 2 v ∂hVi/∂z

Since the mean gradient of the refractive index is primarily in the vertical direction,∇hni ≈ ∂n/∂z. Hence, Equation A7 can be written as follows:

a2 ∇hni 2 C2 = C2 (A8) n 2 v ∂hVi/∂z

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