The accuracy of the method of azimuth structure function for estimation of the dissipation rate of the kinetic energy of turbulence from an array of radial velocities measured by low-energy micropulse coherent Doppler lidars with conical scanning by a probing beam around the vertical axis has been studied numerically. The applicability of the method in dependence on the turbulence intensity and the signal-to-noise ratio has been determined. The method of azimuth structure function was applied for estimation of the turbulent energy dissipation rate from radial velocities measured by the lidar in the experiments on the coast of Lake Baikal. Two dimensional time–height patterns of the wind turbulence energy dissipation rate were obtained. Part of them were obtained in presence of the atmospheric internal waves (AIWs) and low-level jet streams. It is observed that the wind turbulence in the area occupied by jet streams is very weak. In the process of dissipation of AIWs the wind turbulence strength increases.
© 2017 Optical Society of America
The rate of dissipation of the kinetic energy of turbulence is one of the key parameters of the turbulent wind field. In the inertial range of spatial scales of inhomogeneities of the velocity field, the cascade process of turbulent energy transfer from large-scale inhomogeneities to small-scale ones is fully determined by this parameter and obeys the Kolmogorov 2/3 law . The dissipation rate characterizes the intensity of turbulent processes in the atmosphere, and it is important to know how this parameter changes in space and time. Many papers are devoted to development of methods for remote and, in particular, lidar measurement of the dissipation rate (see monograph  and review  and references therein). There are also more recent publications concerning this issue [4–7]. Thus, in  it was proposed to estimate the dissipation rate within the inertial range of turbulence in the atmosphere from the transverse (azimuth) structure function of the radial velocity measured by a pulsed coherent Doppler wind lidar with conical scanning by a probing beam around the vertical axis. It was shown in  that estimates of the dissipation rate obtained by this method from data of the Halo Photonics Stream Line lidar and values of the dissipation rate measured simultaneously by a sonic anemometer at the same height are in good agreement. However, in the general case, the error of estimation of the dissipation rate by this method from wind data obtained with micropulse lidars of the class of Stream Line lidar under various atmospheric conditions was not analyzed yet.
The point is that fiber micropulse coherent Doppler lidars (CDL) similar Stream Line lidar are characterized by the very low signal-to-noise ratio at the background concentration of atmospheric aerosol. In , a method was developed for estimation of the wind speed and direction with acceptable accuracy from data measured by such lidars which can contain up to 50% of “bad”  estimates of the radial velocity. This method uses the procedure of filtering of good estimates of the radial velocity. There are no similar filtering procedures, which could be used in estimation of wind turbulence parameters. Therefore, for determination of turbulence parameters, only slightly noised data, in which the probability of bad estimate of the radial velocity is close to zero, can be used. As a result, the height limit for reconstruction of profiles of turbulent parameters from lidar data is much lower than the maximal height of reconstruction of wind speed and direction profiles, and it is necessary to know previously which data can be used for estimation of turbulence parameters.
Below, in the next part of this paper, we analyze the accuracy of the method of azimuth structure function (MASF) in dependence on the atmospheric conditions: concentration of aerosol (signal-to-noise ratio) and turbulence intensity (dissipation rate). In third section of the paper, we present the results of lidar studies of wind turbulence in the atmospheric boundary layer in the shore zone of Lake Baikal with this method.
2. Theoretical foundation
2.1 Basic equations
During the measurement of the radial velocity by a lidar with the use of conical scanning by the probing beam around the vertical axis, the beam elevation angle is fixed, and the azimuth angle with time varies from 0° to 360° with the angular scanning rate . The raw data of measurements by the Stream Line lidar are arrays of estimates of correlation functions of the complex signal , where ; ; = 50 MHz is the receiver bandwidth; is the distance from the lidar to the center of the sensing volume; is the range step, which is usually taken equal to 18 or 30 m; ; is the azimuth angle; ; is the resolution in the azimuth angle, and is the scan number. For every correlation function , the accumulation of data with the same number of probing pulses is used. In the case of the Stream Line lidar, the pulse repetition frequency is = 15 kHz. If = 1500 (the minimal possible value for this lidar), the time for measurement of the correlation function of the complex lidar signal is = 0.1 s. Then, of the scanning rate = 10°/s, the resolution in the azimuth angle is = 1°, the number of rays for one full scan is = 360, and the duration of one scan is = 36 s.
For estimating turbulence parameters from wind data, it is important to conduct measurements with the high resolution in the azimuth angle. At the same time, the tangential velocity of motion of the sensing volume should exceed significantly the average wind velocity at the height of measurement.
The estimates of the correlation functions are used to calculate the Doppler spectra , and the positions of their maximum are then used to obtain the array of estimates of the radial velocities . In addition, the array of estimates of the signal-to-noise ratio is determined from the array as , where is the noise power. The further procedure of processing of lidar data in order to obtain estimates of the dissipation rate at heights consists in the following.
For every height , the vector of average wind velocity is determined from the array of estimates of radial velocities with the aid of sine-wave fitting , and fluctuations of the radial velocity are calculated as
In Eq. (2), and (here is measured in radians). The condition allows the azimuth structure function to be replaced with the transverse  structure function of the radial velocity , where is transverse (with respect to the probing beam optical axis) size of the sensing volume. The estimate of the turbulence energy dissipation rate can be found from the difference of lidar estimates of the azimuth structure function , 1) if does not exceed the upper boundary of the inertial range of turbulence, 2) the probability of bad estimate  of the radial velocity is zero, and 3) can be represented in the form Eq. (3), is the radial velocity averaged over the sensing volume, and is the random instrumental error having properties of white noise with zero mean and the variance .
The probability of bad estimate of the radial velocity from lidar data depends on atmospheric conditions, mostly, on the concentration of aerosol particles, which manifests itself from the signal-to-noise ratio. Thus, to determine the conditions of applicability of MASF for estimation of the dissipation rate, it is necessary to find the minimal value of the signal-to-noise ratio , at which the probability of bad estimate in the array of estimates obtained from measurements micropulse lidar is close to zero (does not exceed an arbitrarily small preset value). Toward this end, we should know the dependence . This dependence can be found on the basis of the theory developed in Ref , or using numerical simulation of random realizations of lidar estimates of radial velocity.
Using the results of numerical simulation of lidar signals at different values of signal-to-noise ratio (description of the simulation algorithm is given in [2,9]), we found the dependence for the Stream Line lidar with the following parameters: laser wavelength = 1.5 μm, probing pulse duration = 170 ns, and the width of the time window used to estimate the Doppler spectra = 120 ns. In simulation we set that the noise component of the lidar signal is a white noise with Gaussian probability density function (PDF) and neglected by homogeneity of the radial velocity within the sensing volume. It is valid in the case of weak wind turbulence and absence of a strong wind shear. For spectral accumulation we used = 1500 laser shots. For each SNR value, we obtained 106 statistically independent radial velocity estimates . From the obtained array of simulated radial velocity estimates, we calculated the PDF of the radial velocity estimate . Then we determined the probability (or fraction) of a bad estimate of the radial velocity by minimizing the mean square difference between the PDF obtained in the numerical experiment and the model PDF described by Eq. (34) in Ref .
From the obtained dependence at = 1500 it follows that in the case of = 360 and = 20 the probability is less than , if 0.03 (−15.2 dB). We also obtained dependencies for other values of accumulation number . Analysis of the obtained results has shown that the probability of a bad estimate of the radial velocity is a function of the product of and , at least for 1000. Therefore, if for = 1500 ( = 0.1 s) the condition ( = 360 and = 20) is fulfilled, when 0.03 (−15.2 dB), then for = 15000 ( = 1 s) this condition is realized at (−20.2 dB).
Thus, obtained relations determine the measurement conditions at which the azimuth structure function of the radial velocity can be calculated from the Stream Line lidar array of radial velocity estimates without using a special data filtration procedure.
2.2 Azimuth structure function of the radial velocity
The condition , where is the outer scale of the inertial range, allows the turbulence to be considered as isotropic in the calculation of the azimuth structure function of the radial velocity . Then, assuming horizontal statistical homogeneity and stationarity of the turbulent wind field, with allowance for Eq. (3), for we obtainEq. (4) at = 0, we used method Monte Carlo based on the algorithm of computer simulation of random realizations of two-dimensional distributions of the radial velocity [2,13].
Figure 1 shows an example of simulation of random realizations of the dependence on the azimuth angle for the radial velocity (blue curve) and the lidar estimate of the radial velocity from positions of maxima of the Doppler spectra (red curve). In simulation we set the average wind speed 10 m/s, the elevation angle = 60°, the distance to the center of the sensing volume = 344 m (height 300 m), resolution in the azimuth angle = 1° ( = 3 m), the turbulence energy dissipation rate = 10−2 m2/s3, and the integral scale of correlation of wind velocity fluctuations = 250 m. Since = 0, according to Eq. (3) the lidar estimate of the radial velocity is . One can see that the red curve in Fig. 1 is smoother in comparison with the blue curve, that is, the averaging over the sensing volume takes place. In the case of the Stream Line lidar, the longitudinal dimension of the sensing volume is = 30 m.
According to Eq. (4), the difference structure function = is independent of the instrumental error of the estimate of radial velocity and is a difference structure function averaged over sensing volume , where . Figure 2 demonstrates the effect of averaging over the sensing volume on the difference structure function.
Figure 2 shows the calculated by Monte Carlo method difference structure functions , where is the structure function of the radial velocity (without averaging over the sensing volume), and , and , where the subscripts “” and “” indicate that the radial velocity was estimated, respectively, from the position of the spectral maximum and from the centroid of the spectral distribution. It can be seen that, due to the averaging of the radial velocity over the sensing volume, and are less than in the considered range of variation. Therefore, when estimating the dissipation rate from the lidar data, one should necessarily take into account the averaging over the sensing volume.
2.3 Relative error of lidar estimate of the dissipation rate
The relative error of lidar estimate of the dissipation rate was calculated for = 1500, = 20, = 360, = 3 m, and = 7 ( = 21 m) in dependence the signal-to-noise ratio and magnitude of the dissipation rate .
For the ratio of the estimate of dissipation rate to the given (true) value , the following is valid: , where . We assume that . Then the error can be represented in the formEqs. (2) and (3) with allowance for the inequality and the following properties of the random instrumental error of estimation of the radial velocity: , (the probability density function of is Gaussian one ), and ( is the Kronecker delta) in Eq. (5), after carrying out rather cumbersome analytical calculations, we derived the following equation for :
The value of was calculated from the model data of a pulsed lidar with the use of numerical simulation of random realizations and estimation of the radial velocity from the positions of maxima of the Doppler spectra. At = 20, = 360, = 3 m, and = 7, we have obtained = 0.04. This value is independent of the dissipation rate . The model difference transverse structure function averaged over sensing volume in accordance with the theory  is represented in a formEqs. (9)-(11) is the Kolmogorov constant, , , , and is a gamma function.
To calculate the instrumental error of estimation of the radial velocity as a function of , we used the data of numerical simulation of the lidar signal based on the approach [2,8,9,13]. The simulation was performed under assumption of homogenous wind without taking into account the turbulent fluctuations of wind velocity. It follows from the obtained numerical results that the dependence for the Stream Line lidar parameters ( = 170 ns, = 120 ns and the sampling interval of the complex lidar signal 20 ns) can be approximated well by the simple formula:
Rough estimates demonstrate that for such small values of the pulse duration and the time window, the turbulent broadening of the Doppler spectrum is negligible as compared to the instrumental broadening, at least in the case of weak and moderate wind turbulence (< 0.001 m2/s3). Thus, at least in the range of the turbulent intensity determined by the condition < 0.001 m2/s3, Eq. (12) can be used for estimation of .
Figure 3 shows the calculated dependence of the relative error in the lidar estimate of dissipation rate of the kinetic energy of turbulence on the signal-to-noise ratio at the different intensity of wind turbulence. It can be seen that the dissipation rate can be estimated with the acceptable accuracy from measurements by the Stream Line lidar even at the very weak turbulence, when = 10−6 m2/s3, if is no less than 0.24.
The data of numerical simulation were used to determine, by Eqs. (6), (7), and (12) the values of and , at which . With allowance for = 0.04, the relative error in this case should not exceed 16%. Figure 4 shows the dependence of the signal-to-noise ratio on the turbulence energy dissipation rate, at which the relative error = 16%.
In Fig. 4, the grey color indicate an area of and values, at which the error of lidar estimate of dissipation rate exceed 16%. The dashed line in the figure shows the level = 0.03, below which the estimation of the dissipation rate is impossible at any value of because of the presence of bad estimates in the measured array of radial velocities. Figure 4 allows us to judge the representativeness of estimation of the dissipation rate from the measured signal-to-noise ratio .
Wind turbulence in the boundary layer of atmosphere was studied on the coast of the Lake Baikal in summer campaigns of 2015 and 2016 with the use of a Stream Line lidar. The lidar was mounted 340 m away from the coast of the lake, on the territory of the Baikal Astrophysical Observatory of Institute of Solar-Terrestrial Physics, Siberian Branch, Russian Academy of Sciences (51°50′47.17”N, 104°53′31.21”E); the lidar altitude above the lake level was 180 m. Using the initial data of these measurements (arrays of radial velocities and signal-to-noise ratios) we retrieved the time–height patterns of the turbulence kinetic energy dissipation rate and calculated the relative error of assessment of the dissipation rate from the lidar data.
Stream Line lidar measurements were carried out under the following parameters: elevation angle = 60°; number of pulses used for data accumulation = 3000; angular speed of probing beam scanning = 10°/s; measurement time per scanning 36 s; resolution in the azimuth angle = 2°, and the number of rays per scanning = 180.
To retrieve the vertical profiles of the wind speed U, the wind direction angle , and the vertical component of the wind velocity , the method of filtered sine wave fitting was used [2,8]. As shown in  for = 3000, the minimal signal to noise ratio is 0.005 (–23 dB), under which the wind speed estimation error does not exceed 0.1 m/s.
To assess the turbulence energy dissipation rate , we used the method of transverse (azimuth) structure function of the radial velocity measured within the turbulence inertial interval by the scanning lidar. The time–height patterns of the turbulence energy dissipation rate were calculated with a height step of 26 m and a time step of 36 s based on the array of radial velocities measured with the use of data received for = 25 scannings (measurement time is 15 min) for each vertical profile of . For each height h, the SNR values from the array of estimates were averaged over all the azimuth angles and over = 25 scannings. Then the relative error of the dissipation rate was calculated by Eqs. (6) and (7). The error of the radial velocity estimate , which enters into Eq. (7), was determined from the experimental data by two methods: 1) calculation of by Eq. (12), using measured SNR and 2) estimation of , using Eq. (22) given in . The second method is applicable for arbitrary intensity of the wind turbulence.
The analysis of the lidar data processing results revealed several AIW events occurred in the presence of jet streams. The measurements on August 23, 2015, showed an AIW with a period of 9 min and longitudinal and vertical components of the velocity of about 1 m/s and 0.3 m/s, respectively, which was propagating through the region of experiment for almost 6 h in the presence of two jet streams. The initial data measured by the lidar from 11:00 to 17:00 on August 23, 2015, satisfied the condition 0.05 up to 500 m. We limited to this altitude in the retrieval of the vertical profile of the dissipation rate, while the profiles of the wind velocity components were retrieved up to height of about 900 m.
Figure 5 shows the time–height patterns of the wind speed , vertical component of the wind velocity , , , , and . The SNR is below a threshold of 0.005 in the black regions. The values for the error in estimating the radial velocity presented in Fig. 5 were obtained by the second method (using Eq. (22) in ), since in many cases the dissipation rate exceeded 10−3 m2/s3. These values were used to determine . SNR varied from 0.05 to 0.2 in the 100–500 m layer during that measurement. The radial velocity error does not exceed 0.16 m/s. The data on the wind speed in Fig. 5 clearly show the presence of two jet streams with speed maxima at altitudes of about 250 m and 750 m, respectively (hereinafter, the altitudes are given relative to the lidar location point). The directions of these jet streams are almost perpendicular to each other (the bottom jet stream was directed from north to south through mountains, and the top jet stream was directed from east to west, i.e., from the lake side) . Near-harmonic oscillations of and are clearly seen in Fig. 5 from 14:20 to 15:00. The data on the turbulence energy dissipation rate in Fig. 5 show that takes values from a wide range, attaining a maximum of about 0.006 m2/s3 before the arising of a jet stream. Relatively strong wind turbulence is also observed under the jet stream. The turbulence is quite weak inside the jet stream, including during an AIW event.
Figures 6 and 7 show the vertical profiles of , , , and measured with an interval of 1 h (the data are taken from Fig. 5).The turbulence energy dissipation rate obviously can be retrieved from lidar measurements with an error of no larger than 30% at m2/s3. The SNR should be no less than 0.075 in this case. The turbulence spatial structure should include an inertial interval with the upper boundary of no less than a half of the longitudinal size of the sensing volume ( = 30 m for the Stream Line lidar).
In Figs. 6 and 7, the dashed curves show the results for and calculated by Eq. (12) and Eq. (6) (the first method). Using all the data for given in Fig. 5 (the second method) and the data for found by the first method, we calculated values and , where and angular brackets mean spatial-temporal averaging over all obtained data for and . As a result, we found that = −0.07 and = 0.13. It means that the errors of radial velocity estimate (and, correspondingly, the errors of the turbulent energy dissipation rate estimate) calculated by these two methods differ insignificantly.
In August of 2016, in contrast to 2015, when the concentration of aerosol particles was increased due to forest fires, the measurements were conducted at the rather weak lidar echo signal. Long rains washing out the aerosol from the atmosphere were often falling. Therefore, we succeeded in conducting adequate measurements in the absence of rain only for three days. To increase the signal-to-noise ratio at the height of localization of the low-level jet flow, we often had to refocus the probing beam to the closer distance = 300 m. The elevation angle , as in 2015, was taken equal to 60°, the scanning rate was 5°/s (duration of one conical scan of 72 s), and = 3000 ( = 1°, = 360). Despite unfavorable weather conditions, we still succeeded in revealing one case of appearance of an atmospheric internal wave from the lidar data.
Figure 8 depicts the two-dimensional distributions of the wind speed , wind direction angle , and vertical component of the wind vector obtained from raw data of the Stream Line lidar as measured on August 6 of 2016. During the measurements, slightly overcast weather was observed approximately to 01:00 Local Time. Nevertheless, a jet flow was observed, in which the height of maximum of the wind velocity decreased from 350 m to 200 m with time. After 5:30 LT, the jet flow practically disappeared, and wind oscillations took place, which especially clearly seen in the figures for the wind direction angle and the vertical component. We believe that these oscillations are associated with appearance of an atmospheric internal wave early in the morning, when the stratification of air temperature is most stable.
Since the measurements were conducted under conditions of weak echo signal, it appeared impossible to obtain two-dimensional distributions of the dissipation rate . Only in the thin layer of 250–350 m, owing to the focusing of the probing beam to 300 m, the signal-to-noise ratio met the requirement of very low probability of bad estimate of the radial velocity. To obtain estimates of the dissipation rate in this layer, we used the array of radial velocities measured by the lidar for 25 scans (measurement duration of 30 min).
Figure 9 shows the dynamics of the wind speed, wind direction angle, vertical component of the wind velocity vector, turbulence energy dissipation rate, and relative error of lidar estimate of the dissipation rate at a height of 300 m (480 m above the surface of Lake Baikal). Every point in Fig. 9(a), Fig. 9(b), Fig. 9(c) corresponds to the measurement for 72 s (one full scan). The error of estimation of the dissipation rate was determined as described above in section 2.3. At this height till approximately 07:30 LT, the wind direction was from the shore to the lake. Then, for the relatively short time, the wind direction alternated by approximately 90°.
As can be seen from Fig. 9, since 01:30 LT for five hours at a height of 300 m the vertical component of wind Fig. 9(c) took only negative values (downward flow). Starting from 04:40 LT, the time profile of began to oscillate. The period of these oscillations and their amplitude decreased, on average, with time. According to Fig. 9(c), since 05:00 till 07:30 LT the amplitude of oscillations of the vertical velocity was approximately 0.6 m/s. At the same time, the amplitude of oscillations of the horizontal velocity Fig. 9(a) was, at least, two times smaller. This behavior of the internal wave differs significantly from the results of the experiment in 2015 , when we observed nearly harmonic oscillations of the three components of the wind velocity vector with the amplitudes, on average, of 1 m/s for the horizontal speed and 0.3 m/s for the vertical one.
We succeeded in obtaining estimates for the turbulence energy dissipation rate with an acceptable accuracy only for time interval, whose total duration is about 30% of the total measurement time (see Fig. 9(e), Fig. 9(f)). The data of Fig. 9 demonstrate that the turbulence inside the jet flow is very weak. It is very similar that the inertial range of turbulence is completely absent, and the flow in the jet is nearly laminar, low frequency wind variations in the jet are mostly caused by mesoscale processes in the atmosphere. We can also suppose that observed increase in the turbulence intensity after 07:30 LT is caused by decay of the internal wave and transfer of its energy to turbulization of the wind flow.
The paper has proposed the method for calculation of the relative error in lidar estimate of the dissipation rate of the kinetic energy of turbulence. The accuracy of the method of azimuth structure function for estimation of the dissipation rate from the array of radial velocities measured by micropulse coherent Doppler lidars at the conical scanning by the probing beam around the vertical axis has been analyzed numerically. It was shown that the validity of this method depends on the magnitude of the dissipation rate measured and the signal to noise ratio. Obtained in the paper quantitative results allow one to determine beforehand the representativeness of lidar estimation of the turbulence energy dissipation rate from the measured signal to noise ratios. It is especially important when using the low energy micropulse CDL such as Halo Photonics commercial lidar system Stream Line.
The method of azimuth structure function was applied for estimation of the turbulent energy dissipation rate from radial velocities measured by the scanning micropulse CDL Stream Line in the boundary layer of atmosphere on the coast of Lake Baikal. Two dimensional time–height patterns of the wind turbulence energy dissipation rate were obtained and accuracy of lidar estimate of the dissipation rate was determined. Few events of the internal atmospheric waves were registered during the experimental campaigns. All the registered AIWs were observed in the presence of low level jet streams. It is observed that the wind turbulence in the area occupied by jet streams is very weak. Possibly, it is absent at all, and wind velocity variations in the area of jet streams are caused by mesoscale atmospheric processes. In the process of dissipation of AIWs the wind turbulence strength increases. These results are important for understanding the turbulence - wave interaction processes in the atmosphere and evaluation of wind energy variations [7,16].
Russian Science Foundation (No. 14-17-00386-П).
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