Lidar Estimates of the Anisotropy of Wind Turbulence in a Stable Atmospheric Boundary Layer

In this paper, a method is proposed to estimate wind turbulence parameters using measurements recorded by a conically scanning coherent Doppler lidar with two different elevation angles. This methodology helps determine the anisotropy of the spatial correlation of wind velocity turbulent fluctuations. The proposed method was tested in a field experiment with a Stream Line lidar (Halo Photonics, Brockamin, Worcester, United Kingdom) under stable temperature stratification conditions in the atmospheric boundary layer. The results show that the studied anisotropy coefficient in a stable boundary layer may be up to three or larger.


Introduction
One of the consequences of density stratification in the atmosphere or ocean is the formation of disk-shaped density inhomogeneities due to the collapse of internal waves arising in stratified fluids [1]. In a stably stratified atmosphere, this leads to the appearance of anisotropic discoid turbulent temperature inhomogeneities. Radio occultation methods are widely used to study anisotropic turbulence in free atmosphere [2][3][4]. In accordance with previously published results [2,3], the spatial spectrum of the temperature turbulence in a stable atmosphere can be represented as the sum of an isotropic and anisotropic component, which are statistically independent. The results presented in [4] show that anisotropic turbulence begins to play the predominant role at heights above four to five kilometers, where the anisotropy coefficient (ratio of the horizontal scale to the vertical scale of the correlation of air temperature fluctuations) can exceed 30.
The anisotropy of turbulent fluctuations of temperature and wind velocity manifests in the atmospheric boundary layer (ABL) as well. To date, a number of questions regarding the anisotropy of wind turbulence remain unexplored. In particular, in the scientific literature, no data have been published on the anisotropy of wind turbulence in a stable ABL in the presence of low-level jets (LLJs). Typically, LLJs with maximum wind speed of 15-25 m/s occur at night, when stratification of the air temperature is stable, at a height of 200-700 m above ground level. The mechanism of formation of the nocturnal LLJ was described in detail by Blackadar [5]. Accounting for wind turbulence anisotropy is important in the development of ABL mathematical models used for various practical applications, such as weather forecast, wind energy, air transport safety, diffusion of atmospheric impurities, etc.
In the ABL, air flow is always turbulent. Therefore, components of the wind vector V = {V z ,V x ,V y } (where V z is the vertical component, and V x and V y are horizontal components) are random functions at time t and radius vector r = {z, x, y} in a Cartesian coordinate system centered at a point on the Earth's surface (z is the vertical coordinate, x and y are the horizontal coordinates). Let the direction of the average horizontal wind be parallel to the axis x, and the average value of the vertical component be zero. In this case, V x is the longitudinal component and V y is the transverse (on the horizontal plane) component of the wind velocity vector. Average values < V z > and < V y > are equal to zero. Hereinafter, the angular brackets denote ensemble averaging. Denote the fluctuations of the vertical, longitudinal, and transversal components of the wind as w = V z , u = V x -< V x >, and v = V y , respectively.
A method was proposed for estimation of the turbulent energy dissipation rate ε and the radial velocity variance σ 2 r with measurements by a conically scanning PCDL at elevation angle φ [19,20]. The estimate for σ 2 r found by the method described [19,20], is a result of averaging the variances of the radial velocities measured with the lidar at all azimuth angles θ, which are set during the scan. With the assumption that the turbulence structure is described by the von Kàrmàn model [26,27], the integral scale of longitudinal correlation of the radial velocity fluctuations L V can be calculated from the lidar estimates ε and σ 2 r by the following equation [19,20]: The integral scale L V is, just as σ 2 r , the result of averaging over all the azimuth angles θ. Similar to σ 2 r , the scale L V at a fixed height depends on the elevation angle φ employed in the conical scanning due to the turbulence anisotropy, even at horizontally statistically homogeneous wind velocity fluctuations. If the scanning is conducted at the angle φ = tan −1 (1/

√
2) ≈ 35.3 • , then the kinetic energy of turbulence E = (1/2)(σ 2 w + σ 2 u + σ 2 v ) can be calculated from the lidar estimates σ 2 r as E = (3/2)σ 2 r [6,19,20]. In this paper, a method is proposed to estimate wind turbulence parameters, including the integral spatial scales of turbulence based on Equation (1), from lidar measurements employing conical scanning at different elevation angles. This method also allows us to determine the anisotropy of the spatial correlation of wind velocity fluctuations. The method was tested in an experiment with a Stream Line lidar (Halo Photonics, Brockamin, Worcester, United Kingdom) under stable temperature stratification conditions in the atmospheric boundary layer.

Lidar Method to Determine Parameters of Anisotropic Turbulence
The instantaneous radial velocity V r , which is the projection of the wind velocity vector onto the optical axis of the probing beam, at point r = RS(θ, φ), can be represented in the form [14]: where R is the distance from the lidar to the point r and S(θ, φ) = {sin φ, cos φ cos θ, cos φ sin θ} is the unit vector along the optical axis. Difference V r ' = V r -< V r > represents turbulent fluctuations in the radial velocity. Due to statistical inhomogeneity and anisotropy, the radial velocity variance σ 2 r = < (V r ) 2 > is a function of both the parameters, characterizing wind turbulence, and of the distance R and the angles θ and φ. We assume that the wind field is statistically homogeneous in the horizontal plane. Then, the radial velocity variance σ 2 r (R, ϕ) = (2π) −1 2π 0 dθ σ 2 r (R, θ, ϕ) averaged over all azimuth angles θ (in the range from 0 to 2π) can be represented in the form [6,20]: where h = Rsinφ = z is the height and σ 2 H = (σ 2 u + σ 2 v )/2 is the wind velocity variance defined as an average of variances of the longitudinal and transversal components of the wind velocity vector in the horizontal plane. Experiments [28] showed that the variance σ 2 w differs much more from the variances σ 2 u and σ 2 v than variances σ 2 v and σ 2 u differ from each other. Often, the variances σ 2 v and σ 2 u coincide. Therefore, for an experimental study of the anisotropy of wind turbulence, measuring only two variances: σ 2 w and σ 2 H = (σ 2 u + σ 2 v )/2 may suffice. According to Equation (3), the variance σ 2 r , besides the height h, depends on the elevation angle φ due to the anisotropy of turbulence (σ 2 w (h) σ 2 H (h)). To obtain estimates of the variances σ 2 w and σ 2 H , simultaneous lidar measurements at two different elevation angles are necessary. Given two different angles φ = φ 1 and φ = φ 2 (φ 1 φ 2 ) in Equation (3), we obtain a system of equations for the height profiles of the variances σ 2 w (h) and σ 2 H (h): To retrieve the vertical profile of the turbulent kinetic energy E(h) from lidar data, we take φ 1 = 35.3 • in Equation (4) [6]. The angle φ 2 is taken as 60 • . Then, the solution of the system of Equation (4) has the following form: where φ 1 = 35.3 • and φ 2 = 60 • . The turbulent energy dissipation rate ε is a characteristic of locally isotropic wind turbulence within the inertial subrange. In the case of statistical homogeneity of the wind field in the horizontal plane, the dissipation rate is a function of height h and does not depend on horizontal coordinates x and y. After substitution of the values of the variance σ 2 r (h/ sin ϕ i , ϕ i ) and the dissipation rate ε(h) obtained from lidar measurements at different elevation angles φ i (i = 1, 2) into Equation (1), the integral scales L V (h, φ i ) at the angles φ i are calculated. The estimation method of σ 2 r and ε from the data measured by the conically scanning PCDL is described in [20]. Employing the von Kàrmàn model in Equation (1) yields the vertical profiles of longitudinal correlation scales of fluctuations of the vertical L w (h) and horizontal L H (h) wind velocity, if the variance σ 2 r in Equation (1) is replaced with σ 2 w (h) and σ 2 H (h), respectively. Because σ 2 H = (σ 2 u + σ 2 v )/2, we assume that the value of L H is between values of L u and L v . On obtaining scales L V1 = L V (h, φ 1 ) and L V2 = L V (h, φ 2 ) from lidar measurements at the scanning elevation angles φ 1 = 35.3 • and φ 2 = 60 • , and employing Equations (1), (5), and (6), the scales L w and L H can be respectively calculated as: To apply the described method to estimate the variances and correlation scales of fluctuations of vertical and horizontal wind velocity, measurements should be recorded simultaneously by two lidars scanning at different elevation angles. However, considering that the lidar data has to be averaged over a long period of time to obtain a statistically significant estimate of the variance of radial velocity, this method can be applied using a single lidar.
Therefore, as shown in Figure 1, the lidar probing beam conducts conical scanning about the vertical axis alternatively at the elevation angles of 35.3 • (odd scan numbers) and 60 • (even scan numbers).
Remote Sens. 2019, ХХ, x FOR PEER REVIEW 4 of 14 To apply the described method to estimate the variances and correlation scales of fluctuations of vertical and horizontal wind velocity, measurements should be recorded simultaneously by two lidars scanning at different elevation angles. However, considering that the lidar data has to be averaged over a long period of time to obtain a statistically significant estimate of the variance of radial velocity, this method can be applied using a single lidar.
Therefore, as shown in Figure 1, the lidar probing beam conducts conical scanning about the vertical axis alternatively at the elevation angles of 35.3° (odd scan numbers) and 60° (even scan numbers). Before the measurements were taken, elevation angle was set to ϕ = 35.3°. Then, the scanning starts with constant angular velocity ωs = dθ/dt. During the scanning, the azimuth angle θ of the probing direction varies from 0° to 360°. After a full turn, the scanning stops and the elevation angle of 35.3° changes to 60°, which takes approximately 1 s. Then, the scanning starts in the opposite direction. The azimuth angle θ changes from 360° to 0°. At θ = 0°, the elevation angle of 60° again changes to 35.3° and the procedure repeats.

Experiment
The experiment was conducted on 6-24 June 2018 at the Basic Experimental Observatory (BEO) of the Institute of Atmospheric Optics SB RAS (IAO) in Tomsk, Russia (56.475448° N, 85.048115° E), employing a Stream Line PCDL (Halo Photonics, Brockamin, Worcester, United Kingdom). The experimental geometry is shown in Figure 1, which shows conical scanning with alternating elevation angle (35.3° and 60°). The duration of every scan was Tscan = 60 s. Adding the time δt ≈ 1 s needed to change the elevation angle from 35.3° to 60° and vice versa, the duration of one measurement cycle Tcircl = 2(Tscan + δt) was a little bit longer than 2 min. For accumulation of raw lidar data, Na = 7500 laser shots were used. The pulse repetition frequency was fp = 15 kHz. Thus, the duration of measurements for every azimuth angle ∆t = Na/fp = 0.5 s. The azimuth resolution was ∆θ = 360°/M = 3°, where M = Tscan/∆t = 120 was the number of rays for one conical scan.
As a result of the measurements, we obtained arrays of estimates of the signal-to-noise ratio SNR (Rk, θm; n) and the radial velocity VL (Rk, θm; n). Here, the SNR is the ratio of average heterodyne signal power to the average detector noise power in a 50 MHz bandwidth. Estimates of SNR and VL are functions of the parameters Rk, θm, and n, where Rk = R0 + k∆R is the distance from the lidar to the center of the sensing volume, k = 0,1,2,…,K−1; ∆R = 18 m is the range gate length; θm = m∆θ is the dependence of the azimuth angle on the ray number m = 0,1,2,…,M−1 at the elevation angle ϕ = 35.3°; θm = 360°−m∆θ is the azimuth angle at ϕ = 60° ; n = 1,3,5,7,… is the conical scan number at the elevation angle ϕ = 35.3°; and n = 2,4,6,8,… is the conical scan number at the elevation angle ϕ = 60°. Each of the arrays SNR (Rk, θm; n) and VL (Rk, θm; n) is divided into two sub-arrays: SNRi (Rk, θm; n) and VLi (Rk, θm; n) with odd (i = 1) and even (i = 2) numbers n. The subscript i = 1 corresponds to the elevation angle ϕ = 35.3°, whereas i = 2 denotes ϕ = 60°. Before the measurements were taken, elevation angle was set to φ = 35.3 • . Then, the scanning starts with constant angular velocity ω s = dθ/dt. During the scanning, the azimuth angle θ of the probing direction varies from 0 • to 360 • . After a full turn, the scanning stops and the elevation angle of 35.3 • changes to 60 • , which takes approximately 1 s. Then, the scanning starts in the opposite direction. The azimuth angle θ changes from 360 • to 0 • . At θ = 0 • , the elevation angle of 60 • again changes to 35.3 • and the procedure repeats.

Experiment
The experiment was conducted on 6-24 June 2018 at the Basic Experimental Observatory (BEO) of the Institute of Atmospheric Optics SB RAS (IAO) in Tomsk, Russia (56.475448 • N, 85.048115 • E), employing a Stream Line PCDL (Halo Photonics, Brockamin, Worcester, United Kingdom). The experimental geometry is shown in Figure 1, which shows conical scanning with alternating elevation angle (35.3 • and 60 • ). The duration of every scan was T scan = 60 s. Adding the time δt ≈ 1 s needed to change the elevation angle from 35.3 • to 60 • and vice versa, the duration of one measurement cycle T circl = 2(T scan + δt) was a little bit longer than 2 min. For accumulation of raw lidar data, N a = 7500 laser shots were used. The pulse repetition frequency was f p = 15 kHz. Thus, the duration of measurements for every azimuth angle ∆t = N a /f p = 0.5 s. The azimuth resolution was ∆θ = 360 • /M = 3 • , where M = T scan /∆t = 120 was the number of rays for one conical scan.
As a result of the measurements, we obtained arrays of estimates of the signal-to-noise ratio SNR (R k , θ m ; n) and the radial velocity V L (R k , θ m ; n). Here, the SNR is the ratio of average heterodyne signal power to the average detector noise power in a 50 MHz bandwidth. Estimates of SNR and V L are functions of the parameters R k , θ m , and n, where R k = R 0 + k∆R is the distance from the lidar to the center of the sensing volume, k = 0,1,2, . . . Each of the arrays SNR (R k , θ m ; n) and V L (R k , θ m ; n) is divided into two sub-arrays: SNR i (R k , θ m ; n) and V Li (R k , θ m ; n) with odd (i = 1) and even (i = 2) numbers n. The subscript i = 1 corresponds to the elevation angle φ = 35.3 • , whereas i = 2 denotes φ = 60 • .
To increase the accuracy of SNR measurement, SNR estimates were averaged over all azimuth angles for every distance R k and scan number n, where V zi is the vertical component and V xi and V yi are the horizontal components, were determined from the arrays of lidar estimates of the radial velocity V Li (R k , θ m ; n) using the direct sine-wave fitting method [14,29]. The wind speed U i and the wind direction angle θ Vi were calculated from the horizontal components of the velocity vector as U i = |V xi + jV yi | and θ Vi = arg{V xi + jV yi }, respectively, where j is an imaginary unit.

Results of the Measurements
To retrieve information about turbulence from lidar data, the probability of bad (false) estimates of radial velocity has to be nearly zero [30]. That is, the SNR during the measurements should be high. For accumulation number N a = 7500, the SNR should exceed -16 dB. Therefore, to study the anisotropy of wind turbulence in a stable ABL, we had to select data corresponding to the stable temperature stratification and SNR > -16 dB. Lidar data obtained in the 12-hour period from 20:00 on 23 July to 08:00 on 24 July met this requirement. Using the data for the average temperature of air T(h, t) measured by sonic anemometers during this time interval at heights h 1 = 3 m and h 1 = 42 m, we calculated the potential temperature derivative dT p /dh by the equation: where y a = 0.0098 • K/m is the adiabatic gradient of dry air temperature. Under the condition dT p /dh > 0, the temperature stratification is stable; if dT p /dh = 0, the stratification is neutral; and when dT p /dh < 0, the stratification is unstable. Calculations show that beginning with 20:00 on 23 July to 08:00 on 24 July, the value of dT p /dh varied in the range of 0.005 to 0.02 • K/m. Therefore, the temperature stratification of the surface atmospheric layer was stable in this period.
The estimates of the radial velocity with subscript k obtained from lidar measurements at different values of the elevation angles, φ 1 = 35.3 • and φ 2 = 60 • , correspond to different heights, h ki = h 0i + k∆h i , where h 0i = R 0 sin φ i is the initial height and ∆h i = ∆R sin φ i is the vertical step. By sorting the obtained data in heights, two two-dimensional (2D) composite plots corresponding to the elevation angles of 35.3 • and 60 • were sampled for each of the following parameters: the signal-to-noise ratio SNR i (h ki , t n ) , wind speed U i (h ki, t n ), wind direction angle θ Vi (h ki, t n ), and the vertical wind speed V zi (h ki, t n ). Here, t n = t 0 + nT circl / 2 is the current time and t 0 is the measurement starting time. The obtained distributions are shown in Figure 2, which shows that the SNR 1 (h, t n ) for the elevation angle of 35.3 • becomes much smaller than SNR 2 (h, t n ) for the elevation angle of 60 • starting from heights at 400-500 m. This is caused by the longer distance to the probed volume at φ = 35.3 • than at φ = 60 • and, possibly, by the horizontal inhomogeneity of the aerosol backscatter coefficient. In the zones where SNR i (h ki , t n ) > -16 dB, differences in the height-time distributions of the wind speeds U 1 (h , t n ) and U 2 (h , t n ) and the wind direction angles θ V1 (h , t n ) and θ V2 (h , t n ) are not as significant.  (Figure 2g-h). This agrees with the results reported by Lolli et al. [31]. Due to large-scale turbulent inhomogeneities of the wind flow and fast mesoscale processes, a significant difference was observed in estimates of vertical wind velocity V z1 (h , t n ) and V z2 (h , t n ).
To obtain estimates for the wind turbulence parameters, we used the measured data obtained at SNR i (h ki , t n ) > −16dB , when the probability of bad estimates of the radial velocity [32] was close to zero. According to the data in Figure 2a-b, this requirement was fulfilled for heights up to 600 m. The turbulent energy dissipation rates ε 1 (h k1, t n ) and ε 2 (h k2, t n ), radial velocity variances σ 2 r1 (h k1 , t n ) and σ 2 r2 (h k2 , t n ), and integral scales of turbulence L V1 (h k1, t n ) and L V2 (h k2, t n ) were estimated from the arrays of fluctuations of lidar estimates of the radial velocity V Li ' using the method described thoroughly by Smalikho and Banakh [20]. Here,     Figure 3 depicts the obtained spatiotemporal distributions of turbulence parameters σ 2 ri (h ki , t n ), ε i (h ki, t n ), and L Vi (h ki, t n ). In every distribution, the black curve shows the temporal profile of the height of maximal wind speed assessed from the data in Figure 2c. This curve is a boundary between the top and bottom parts of the LLJ and can be considered as its center. The wind turbulence is very weak in the top part of the LLJ; in particular, the turbulent energy dissipation rate does not exceed 10 -4 m 2 /s 3 , and the variance of radial velocity σ 2 r2 (h k2 , t n ) at the elevation angle of 60 • is below 0.08 m 2 /s 2 . Assuming horizontal statistical homogeneity of the wind, the kinetic energy of turbulence was calculated as We calculated the errors of estimation of the kinetic energy E and the dissipation rate ε using the technique described in [30]. According to our calculations, the relative error of the kinetic energy estimates was 7%-10% in the bottom part of the LLJ, whereas the accuracy of the dissipation rate estimates was 6%-8%. In this experiment, the minimum height for retrieving vertical profiles of velocity from lidar measurements with acceptable accuracy was 100 m. The maximum height at which measurements were recorded with sonic anemometers was 42 m (a sonic anemometer was installed on top of the mast, located 150 m from the lidar). In Figure 4, in addition to lidar data, we depict the results of measurements with the sonic anemometer at a height of 42 m. Sonic anemometer measurement data show that, despite the conditions of stable temperature stratification, in the surface layer of the atmosphere (layer thickness of about 100 m in height above the ground), wind turbulence was quite strong. Assuming horizontal statistical homogeneity of the wind, the kinetic energy of turbulence was calculated as E(h k1 , t n ) = (3/2)σ 2 r1 (h k1 , t n ) from the data for the variance of radial velocity σ 2 r1 (h k1 , t n ) obtained from measurements at the elevation angle of 35.3 • . Combining calculated E(h k1, t n ) and data of Figure 2c,e, and Figure 3a,c, we drew vertical profiles of the wind speed and direction, kinetic energy of turbulence, and its dissipation rate in Figure 4. The turbulence was strong on 23 July in the bottom part of the LLJ at heights 200-400 m at wind speeds exceeding 15 m/s. The kinetic energy E ranged from 1.0 to 0.5 m 2 /s 2 , and the dissipation rate varied within 7 10 -3 -10 -3 m 2 /s 3 . We calculated the errors of estimation of the kinetic energy E and the dissipation rate ε using the technique described in [30]. According to our calculations, the relative error of the kinetic energy estimates was 7%-10% in the bottom part of the LLJ, whereas the accuracy of the dissipation rate estimates was 6%-8%.
In this experiment, the minimum height for retrieving vertical profiles of velocity from lidar measurements with acceptable accuracy was 100 m. The maximum height at which measurements were recorded with sonic anemometers was 42 m (a sonic anemometer was installed on top of the mast, located 150 m from the lidar). In Figure 4, in addition to lidar data, we depict the results of measurements with the sonic anemometer at a height of 42 m. Sonic anemometer measurement data show that, despite the conditions of stable temperature stratification, in the surface layer of the atmosphere (layer thickness of about 100 m in height above the ground), wind turbulence was quite strong. The turbulent energy dissipation rate is a characteristic of the locally isotropic field of wind velocities. Assuming that the wind velocity field is statistically homogeneous in the horizontal plane, the lidar estimates of the dissipation rates ε1(h, tn) and ε2(h, tn) obtained for the same height h, but for different elevation angles (35.5° and 60°), should fully coincide on average. To verify this, we used the data in Figures 3c-d and their vertical interpolation with a step of 10 m, starting from 100 m and ending at a height of 500 m. Figure 5 depicts the results of the comparison of the lidar estimates of the dissipation rate ε1(h, tn) and ε2(h, tn) in four vertical layers: (1) 100-200, (2) 200-300, (3) 300-400, and (4) 400-500 m. The analysis of these results showed that, on average, ε1 is 20% less than ε2 in the first layer, ε1 and ε2 practically coincide in the second layer, and ε1 exceeds ε2 1.5 and 2 times in the third and fourth layers, respectively. The reason for the small discrepancy in dissipation rate estimates in the lowest layers is unclear. On average, at heights from 170 m to 320 m, the absolute value of the discrepancy between of the dissipation rate estimates [2(ε1 -ε2)/(ε1 + ε2)]×100% does not exceed 10%. This indicates the horizontal statistical homogeneity of the wind velocity field at these heights. The significant discrepancy of the dissipation rate estimates from lidar measurements at the elevation angles of 35.3° and 60° in the upper layers may originate from horizontal mesoscale inhomogeneities in the wind field caused, for example, by internal atmospheric waves being stronger against the weakly turbulent background. The circles indicate the results of measurements with a sonic anemometer at a height of 42 m. The dotted lines "stitch" the corresponding data, which is measured using a sonic anemometer and lidar.
The turbulent energy dissipation rate is a characteristic of the locally isotropic field of wind velocities. Assuming that the wind velocity field is statistically homogeneous in the horizontal plane, the lidar estimates of the dissipation rates ε 1 (h , t n ) and ε 2 (h , t n ) obtained for the same height h, but for different elevation angles (35.5 • and 60 • ), should fully coincide on average. To verify this, we used the data in Figure 3c-d and their vertical interpolation with a step of 10 m, starting from 100 m and ending at a height of 500 m. Figure 5 depicts the results of the comparison of the lidar estimates of the dissipation rate ε 1 (h , t n ) and ε 2 (h , t n ) in four vertical layers: (1) 100-200, (2) 200-300, (3) 300-400, and (4) 400-500 m. The analysis of these results showed that, on average, ε 1 is 20% less than ε 2 in the first layer, ε 1 and ε 2 practically coincide in the second layer, and ε 1 exceeds ε 2 1.5 and 2 times in the third and fourth layers, respectively. The reason for the small discrepancy in dissipation rate estimates in the lowest layers is unclear. On average, at heights from 170 m to 320 m, the absolute value of the discrepancy between of the dissipation rate estimates [2(ε 1 -ε 2 )/(ε 1 + ε 2 )]×100% does not exceed 10%. This indicates the horizontal statistical homogeneity of the wind velocity field at these heights. The significant discrepancy of the dissipation rate estimates from lidar measurements at the elevation angles of 35.3 • and 60 • in the upper layers may originate from horizontal mesoscale inhomogeneities in the wind field caused, for example, by internal atmospheric waves being stronger against the weakly turbulent background.  Accessible experimental data (see, for example, [28]) indicate that in the atmospheric boundary layer, the variance of horizontal wind velocity σ 2 H (h) exceeds the variance of the vertical component σ 2 w (h), and the correlation scale of the fluctuations of the horizontal velocity L u (h) is larger than the correlation scale of fluctuations of the vertical velocity L W (h) due to anisotropy of wind turbulence. Since σ 2 H > σ 2 w , according to Equation (4), we expect that the variance σ 2 r (h/ sin ϕ 1 , ϕ 1 ) exceeds σ 2 r (h/ sin ϕ 2 , ϕ 2 ) in our experiment at the elevation angles φ 1 = 35.3 • < φ 2 = 60 • . Figure 6 shows the vertical profiles of the variances σ 2 r1 (h k1 ) = σ 2 r (R k , ϕ 1 ) and σ 2 r2 (h k2 ) = σ 2 r (R k , ϕ 2 ), and integral correlation scales L V1 (h k1 ) = L V (R k , φ 1 ) and L V2 (h k2 ) = L V (R k , φ 2 ) of the radial velocity fluctuations, as retrieved from the lidar measurements at elevation angles φ = φ 1 = 35.3 • and φ = φ 2 = 60 • , respectively. At the same height h, the variance satisfies σ 2 r1 > σ 2 r2 (Figure 6a,c,e). The integral scales L Vi were calculated, according to Equation (1), as: From the analysis of the results of estimation of the turbulent energy dissipation rate εi shown in Figure 5, ε1 ≈ ε2 at heights of 100-320 m and ε1 > ε2 at heights of 320-500 m. This means that LV1 obtained by Equation (9) in the 320-500 m layer is underestimated, whereas the LV2 in this layer is overestimated. In spite of that, the inequality LV1(h) > LV2(h) holds at all heights, due to the anisotropy of turbulence (Figures 6b, d, and f).
The data in Figures 3a-b show that the ratio of the radial velocity variances  From the analysis of the results of estimation of the turbulent energy dissipation rate ε i shown in Figure 5, ε 1 ≈ ε 2 at heights of 100-320 m and ε 1 > ε 2 at heights of 320-500 m. This means that L V1 obtained by Equation (9) in the 320-500 m layer is underestimated, whereas the L V2 in this layer is overestimated. In spite of that, the inequality L V1 (h) > L V2 (h) holds at all heights, due to the anisotropy of turbulence (Figure 6b,d,f).
The data in Figure 3a-b show that the ratio of the radial velocity variances µ = σ 2 r1 /σ 2 r2 averaged employing all estimates in the 100-300 m layer is equal to 1.4. Thus, on average, the ratio of the integral scales is L V1 /L V2 = µ 3/2 = 1.7. Approximately the same result L V1 /L V2 ≈ 1.8 was obtained from direct averaging of the ratios of the integral scales in this layer based on the data in Figure 3e-f. Thus, due to the anisotropy of turbulence, the variance of the radial velocity at the elevation angle of 35.3 • is 1.4 times larger than that at 60 • , whereas the integral scale of turbulence at the elevation angle of 35.3 • exceeds the integral scale at the elevation angle of 60 • by 1.7 times.
Assuming that the wind field is horizontally statistically homogeneous in the 100-300 m layer, applying σ 2 ri = σ 2 w sin 2 ϕ i + σ 2 H cos 2 ϕ i (Equation (3) [20]), where φ 1 = 35.3 • and φ 2 = 60 • , we find: for the ratio of the average between the variances of the longitudinal and transversal components of the wind vector σ 2 H = (σ 2 u + σ 2 v )/2 to the variance of vertical component σ 2 w . Substituting µ = 1.4 into Equation (10) yields that the variance σ 2 H is 2.26 times larger than the variance σ 2 w . On the assumption that the spatial structure of the wind turbulence is described by the von Kàrmàn model (Equation (1) The obtained estimates of the anisotropy coefficient of wind turbulence parameters do not contradict the known experimental data [28].
Using the data (three components of the wind vector) measured with a sonic anemometer at a height of 42 m (Figure 4), we obtained the variances σ 2 H (t) and σ 2 w (t)versus time t for the period under consideration from 20:00 on 23 July to 08:00 on 24 July 2018. Each variance estimate was derived from 30-minute measurement data. The anisotropy coefficient σ 2 H /σ 2 w averaged over all the measurement period was approximately 2, which is only 13% less than the lidar estimate (v σ = 2.26) for heights of 100-300 m.
As follows from Figures 3e-f and 6b,d,f, the lidar estimate of the integral scale L Vi increases with height and takes overstated values at heights above 450 m at the central and upper parts of the LLJ.
In Figure 3e-f, the zones where the estimates of the integral scale of turbulence L Vi (h ki ,t n ) exceed 200 m are shown in white. To find the reason for this error, by analogy with Banakh and Smalikho. [25], we studied fluctuations in the lidar estimates of the radial velocity V Li (R k , θ m ; n) . As a result, we found that inhomogeneities in the wind flow with scales comparable to the base radius of the scan cone occur above 300 m. This effect is mostly pronounced at the central part of the LLJ and especially in the zones where the wind direction changes strongly with height (Figures 2e-f and 4b). It is possible that mesoscale processes occur here, like internal atmospheric waves arising under stable temperature stratification conditions and a strong change in the wind velocity with height.
For a more accurate estimation of the scales L H and L w in the upper part of the LLJ, we used the following approach. Consider the 100-300 m layer, where the contribution of mesoscale processes to wind variations is negligibly small in comparison with wind turbulence, and the conditions of horizontally statistically homogeneous turbulence are fulfilled (ε 1 ≈ ε 2 , Figure 5). In this layer, as depicted in Figure 6b,d,f, on average, the integral scales L V1 and L V2 increase linearly with height and have realistic values. Thus, at a height 100 m, on average, L V1 = 45 m and L V2 = 26 m, whereas at 300 m L V1 = 93 m and L V2 = 51 m. Assuming that in the 300-500 m layer, the characteristic of the change in the integral scale L Vi with height does not considerably differ from that observed in the lower layer, we linearly extrapolated the scales in the 100-300 m layer to higher layers. As a result, we found that at h = 500 m, the integral scales are L V1 = 136 m and L V2 = 80 m. These values of the turbulent scales L Vi for the central part of the LLJ do not contradict the published experimental data [25]. According to Equations (7) and (8)

Conclusions
In this paper, we proposed a method for the determination of wind turbulence parameters from PCDL measurements employing conical scanning at two different elevation angles. This method assesses the anisotropy of the spatial correlation of the wind velocity turbulent fluctuations. The method was successfully tested in an atmospheric experiment with a Stream Line lidar (Halo Photonics, Brockamin, Worcester, United Kingdom) under conditions of stable temperature stratification of the atmospheric boundary layer.
We found that the variance in radial velocity measured at an elevation angle of 35.3 • always exceeds the variance in radial velocity measured at an elevation angle of 60 • . The averaged ratio of these variances is 1.4. Assuming that the turbulence spatial spectrum is described by the von Kàrmàn model, we obtained that the ratio of the integral scales of fluctuations of the radial velocity measured with scanning at angles of 35.3 • and 60 • is approximately equal to 1.7. It follows from the obtained estimates of turbulence anisotropy that the variance of the horizontal velocity exceeds the variance of the vertical wind velocity 2.26 times. Correspondingly, the ratio of the integral correlation scale of fluctuations of the horizontal wind velocity to that of the vertical velocity is approximately 3.4. At the central part of the LLJ observed during the experiment, at heights of 400-500 m, the integral scales of fluctuations of the horizontal and vertical wind velocity components are, on average, 183 and 54 m, respectively.