Turbulence measurements hold paramount significance in atmospheric sciences and renewable energy applications. These measurements are used to understand and quantify aerosols, heat, moisture, and momentum in the atmospheric boundary layer . In the troposphere, turbulence governs energy and momentum exchange between the Earth's surface and the air above it. Turbulence measurements are also used to monitor and understand air quality, pollutant dispersion, and heat dynamics in urban meteorology. Atmospheric turbulence is usually measured with in situ instruments, such as sonic anemometers, that can be installed on a meteorological mast. These instruments measure turbulence at a fixed point in space. Erecting a meteorological mast for turbulence measurements may not always be feasible, especially in complex terrain, deep offshore seas, or in conditions where in situ measurements are not possible, e.g., aircraft wake vortices. In such cases, remote sensing devices such as Doppler wind lidars can be used to estimate atmospheric turbulence. Commercial scanning lidars also provide the capability to get multi-point statistics and vertical profiles, thus presenting a more detailed view of the atmosphere.

Estimating turbulence from a Doppler wind lidar is not a novel concept. There are two key challenges associated with it, as delineated by : turbulence contamination and probe-volume averaging. The former arises from reconstructing wind velocity components from lidar line-of-sight (LOS) velocities, while the latter is inherent to the measuring principle of Doppler lidars, which do not record point measurements; rather, they measure within a probe volume, acting as a turbulence filter. A number of methods and scanning strategies have been proposed to operate Doppler lidars as standalone instruments for measuring atmospheric turbulence. used a short-pulse Doppler wind lidar with a range gate resolution of 150 m to measure Reynolds stresses and turbulence kinetic energy (TKE) profiles in the atmosphere. Their method relies on the conical scanning of the atmosphere at a fixed elevation angle and deducing Reynolds stress values directly from the individual beam LOS variances. This approach avoids cross-contamination among the wind field components; however, it does not address turbulence filtering due to probe volume averaging . This method has been further improved and optimized for a Doppler beam swing (DBS) scanning pattern, where only five azimuth measurements at a fixed elevation angle and one vertical measurement are obtained to deduce the full Reynolds stress tensor .

A significant amount of research efforts has been made to address the turbulence filtering or the attenuation of small-scale turbulence due to probe-volume averaging. and utilized the LOS velocity structure function (D(r), where r is the separation distance) for a pulsed lidar to estimate the turbulence energy dissipation rate ε. They removed the effect of spatial averaging using theoretical corrections, such as modeling the lidar range weighting as a Gaussian function. In this approach, the measured second-order structure function is fitted to a theoretical D(r) model that accounts for the probe-volume averaging of the lidar LOS velocities. obtained ε from the longitudinal structure function of LOS velocities measured by a 2 µm lidar with a range gate resolution of 30 m and compared it with the ε estimates obtained from the Doppler spectrum width (DSW) method, confirming the results with numerical simulations. Their results showed a high correlation between the two methods and a reduction in bias as ε increased. and utilized the azimuthal structure function of LOS velocities obtained from the conically scanning vertical profiling lidars to estimate ε and TKE. Both studies account for the probe volume averaging by defining a transverse filter function. However, the latter introduced an additional advection filter to account for turbulence advection within the scanning plane. The ε estimates were then compared with the sonic anemometers. Correcting for advection reduced the bias; however, the random error, as indicated by the observed scatter, remained relatively high.

A major drawback of these methods is that the presented D(r) models are also dependent on the outer scale of turbulence Lo. To obtain Lo, it was often assumed that the large-scale turbulence is isotropic and can be modeled by an isotropic spectral model, such as the von Kármán spectral model. However, the von Kármán spectral model applies only to special cases of isotropic, homogeneous turbulence. In reality, the atmospheric turbulence is highly anisotropic . In order to find ε from such methods, the modeled D(r) was fitted to the measurements in such a way that the best fitting takes place for both variables: ε and Lo . The best fit obtained using this method may not provide the best ε value, since Lo is also being fitted to minimize the error function. Any anisotropy in the large-scale turbulence would distort ε estimates. Furthermore, it unnecessarily complicates the fitting procedure as more than one variable is involved.

Turbulence energy dissipation rate can also be obtained through the one-dimensional LOS velocity spectrum, provided that the inertial sub-range is detected . However, this method requires invoking Taylor's hypothesis of frozen turbulence and a suitable choice of Lo. The effect of turbulence filtering due to probe volume averaging was also modeled to obtain the corrected velocity spectrum, thereby improving the prediction of ε . In summary, there are two primary ways of estimating ε from the Doppler lidar measurements reported in the scientific literature: the structure function method and the velocity spectrum method. Both these methods usually require knowledge about the outer scale of turbulence, Lo, and may invoke the assumption of frozen turbulence.

In this article, we present an improved second-order longitudinal structure function model for lidar LOS velocities that also accounts for probe volume averaging in a pulsed lidar. The presented structure function model improves on the previously described methods in the following two ways: (i) the lidar system used in this study provides LOS velocity estimates at closely spaced range gates, with a range-gate spacing of 3 m along the beam. Such a small range-gate spacing enables evaluation of the structure function D(r) at small separations, thereby improving the identification of the inertial subrange. and (ii) the use of longitudinal D(r) means that there is no need to assume Taylor's frozen turbulence hypothesis, as all the LOS measurements along the lidar beam have a known constant separation distance r between the range gates. This ensures that the structure function model only depends on ε in the inertial subrange. The model is then used to estimate the turbulence energy dissipation rate ε in the atmospheric boundary layer, and the values are validated against sonic anemometer measurements at three heights of 103, 175, and 241 m above ground level at a test site in Denmark.

The second-order longitudinal structure function model for lidar LOS velocities is presented in detail in Sect. 2. In Sect. 3, the test site and information about the lidar and the validation data (ultrasonic anemometers) are provided. Section 4 presents the complete workflow for estimating ε from ultrasonic anemometer and lidar data. Section 5 describes the validation of lidar-derived ε estimates against sonic anemometer data and the relevant discussion. Section 6 concludes the article by describing the salient features of the model and summarizing the study's significant results.

The second-order velocity structure function is the variance of the difference in velocity between two points x and x+r, where r is the separation vector between them. The second-order structure function can be defined as: 1D(r)=n⋅v(x)-n⋅v(x+r)2, where n=r/|r|. Since the component n⋅v of the velocity vector is aligned with n, this is called a longitudinal structure function. Kolmogorov hypothesized that in the inertial subrange, the turbulence is locally isotropic and D(r) is only a function of the turbulent energy dissipation rate ε, the separation distance r, and the universal Kolmogorov constant CK≈2. Hence, for |r|≪Lo, where Lo is the integral length scale or the size of the largest eddies, 2D(r)=CK(εr)2/3. The pulsed Doppler lidar measures backscattered light over a finite spatial volume, so-called probe volume, the extent of which is governed by the pulse duration. The return signal, therefore, contains contributions from the aerosol particles distributed throughout the probe volume. The center of the probe volume is typically assigned as the nominal range gate position and determined from the pulse time-of-flight . Volume measurement is inherently different from point measurements used in reference instruments, such as sonic or cup anemometers.

The lidar used in this study measures LOS velocities at several range gates along the beam. Lidar manufacturers use various methods to estimate the representative LOS velocity from the Doppler spectrum. Following , if we assume that the effect of probe volume on turbulence attenuation can be modeled by the centroid method of determining the dominant frequency in the Doppler spectrum, then the line-of-sight velocity of the lidar beam can be defined as: 3vr(x)=∫-∞∞φ(s)n⋅v(sn+x)ds, where n is a unit vector along the beam direction, x is the center of the lidar measuring volume at the point of interest, and s is the distance along the beam from the point x. φ(s) is the weighting function normalized to unit integral, and a Gaussian shape is assumed here: 4φ(s)=1σ2πe-s22σ2, where σ is the standard deviation of the Gaussian distribution. Though defined Eq. () for a continuous-wave lidar system, the same can be assumed for a pulsed lidar system. evaluated the accuracy of the centroid frequency estimator method with a coherent pulsed lidar system and found it to perform well in calculating both the first- and second-order moments of the Doppler spectrum. The longitudinal structure function along the beam then becomes: 5Dlidar(r)=∫-∞∞v(x+s)-v(x+r+s)φ(s)ds2, where r is the distance between two range gates along the beam and v=n⋅v is the velocity component along the beam direction as a function of distance along the beam. It is assumed that r and σ are small as compared to the turbulence length scale Lo and the flow is homogeneous. This can be solved for the inertial subrange as (see Appendix for a detailed derivation): 6Dlidar(r)=CKπ22/3σ2/3ε2/3Γ56×1F1-13,12,-r24σ2-1, where Γ is the Gamma function, and 1F1 is the confluent hypergeometric function. The structure function model described in Eq. () only considers the spatial averaging and ignores any effect of temporal averaging on the lidar LOS velocities. By including the effect of the time Δt over which the lidar's Doppler signal is averaged, we can derive a modified second-order longitudinal structure function model containing both spatial and temporal averaging effects: 7Dlidar(r)=αε2/3Γ432π1/2Γ116∫02π∫0∞e-(σκcos⁡(β-θ))2×1-cos⁡(rκcos⁡(β-θ))sinc2ΔtUκcos⁡β21-811cos⁡2(β-θ)κ5/3dκdβ, where U is the mean wind speed, α is the spectral Kolmogorov constant, θ is the beam-to-wind angle measured in the plane containing both beam and wind unit vectors, and κ and β are the polar coordinates in k1-k2 plane (k1 and k2 are the along-wind and cross-wind wavenumbers). Full details on how this expression is derived are given in Appendix A.

Here we note that the lidar structure function also depends on the standard deviation of the Gaussian weighting function σ. The main assumption for accurately estimating ε is that σ should be less than Lo. If atmospheric turbulence is less intense and its length scale is smaller than the lidar's sampling volume, then turbulence fluctuations will be filtered out. The choice of σ depends on the lidar instrument and can be obtained by fitting the data for a range of σ values and then selecting a value that provides the least amount of bias against true values. Figure a describes how Dlidar (Eq. ) changes as a function of r for three different σ values. The change in local slope of the structure function is also a function of r, as seen in Fig. b. Given the measured structure function from a high-resolution lidar beam, Eq. () or () can be fitted using the least squares method or other optimization techniques to obtain the turbulence energy dissipation rate ε.

In order to estimate the turbulent energy dissipation rate ε, the primary challenge is the detection of the inertial subrange. This can be done in several ways: either through the power spectral density of the wind components or by using the structure function method. For sonic anemometers, the straightforward way is to plot a wavenumber spectrum of the u, v, or w wind component by assuming Taylor's frozen turbulence hypothesis. In the inertial subrange, the one-point, two-sided velocity spectra in terms of wavenumber k1 (where k1=2πf/U‾; f is the frequency and U‾ is the mean wind speed) are given by: 9Fu(k1)=955αε2/3k1-5/3, for the along-wind spectrum and 10Fv(k1)=Fw(k1)=1255αε2/3k1-5/3, for the cross-wind and vertical components, where α is the spectral Kolmogorov constant with a value of ∼ 1.52 . To detect the inertial subrange, a compensated spectrum k15/3⋅Fu(k1) is plotted against k1. The region where d(k15/3⋅Fu(k1))/dk1≈0 represents the inertial subrange . Here, we only used wavenumber in the range 5×10-2 m⁻¹ < k1 < 100 m⁻¹ to detect the inertial subrange and allowed a slope deviation of only 10 %. This interval was selected after inspecting the spectra across the dataset's wind-speed range, where the inertial-subrange plateau was consistently observed within these wavenumber bounds. After detecting the range, εsonic is calculated using: 11εsonic=559αk15/3⋅Fu(k1)‾3/2, where ⋅‾ represents the mean value in the inertial subrange.

For the εlidar calculation from the BEAM 6x lidar, the structure function approach is adopted, meaning that we fit Eq. (), where ε is the only unknown, to the measured lidar data. The benefit of such an approach is that since lidar beams measure LOS velocities at range gates physically separated by a distance at the same time, there is no need to assume Taylor's frozen turbulence hypothesis. The distance between the range gates is constant, i.e., 3 m. Furthermore, assuming that the sampling volume σ<Lo, it would allow resolving the ε without Lo dependence. The following steps are taken to derive εlidar from a single 10 min period from lidar LOS data:

For a selected height h, range gates within h ± 14 m are selected to get the measured longitudinal structure function from LOS velocities. For each beam, a mean structure function is computed from all scans over the 10 min period. Before the structure function is calculated, the mean line-of-sight velocity is subtracted at all heights. This is necessary to avoid an extra term in the structure function originating from the mean wind profile.

The atmospheric conditions observed at the Østerild test site during the lidar measurement period were predominantly neutral, with about 55 % of the data falling into one of three categories: nns, n, or nnu. Stable atmospheric conditions (s, vs) occurred about 30 % of the measurement time, while the unstable conditions (u, vu) were present for about only 15 % of the time. With this in mind, let's view the Dlidar(r) model fitting error, described in Eq. (), as a function of the atmospheric stability in Fig.  at the three measurement heights of 103, 175, and 241 m. The neutral conditions (nns, n, nnu) produced the lowest fitting errors at all three heights. The unstable conditions (u and vu) showed slightly higher errors when fitting the structure function model over the measured data. The highest model-fitting errors were observed under very stable (vs) atmospheric conditions, as these conditions have the least atmospheric turbulence, making it difficult to identify the inertial subrange. We also observed an increase in fitting errors with height, with the median and maximum error values highest at 241 m across all stability conditions. This is caused by a combination of a weaker lidar return signal with increasing height and weaker turbulence fluctuations. The resulting lower signal-to-noise ratio makes it harder to detect the inertial subrange and fitting of ε.

The comparison of turbulence energy dissipation rates obtained from lidar and sonic anemometers at the three heights of 103, 175, and 241 m is displayed in Figs.  and for the strict and relaxed error criteria, respectively. The data shown here represent all seven stability classes and are therefore not subdivided. Each figure is further divided into two panels: (I) only spatial averaging, where the εlidar values are obtained by fitting Eq. () over the measured structure function data, and (II) spatial + temporal averaging, where Eq. () is fitted over the data. In both panels, the number of available 10 min time periods (N) that qualify for all the requirements, i.e., wind speed, wind direction, and whether the fitting error is within the defined limit, is highest at 103 m. The sample number subsequently decreases at higher heights due to increased model-fitting error and greater difficulty in detecting the inertial subrange.

In Fig. , the scatter between the data points at all heights is minimal for both panels which can be observed by nearly identical values of ρ, i.e., 0.96–0.97 at 106 m, 0.95 at 175 m, and 0.91 at 241 m. These large values of ρ imply a strong linear association between the model-predicted εlidar and measured εsonic values. However, ρ alone does not describe the full picture. Looking at the OLS regression lines for both the spatial-averaging and spatial + temporal-averaging panels, one can clearly see the utility of including temporal averaging in the fitting process. The slope of the OLS regression line (a) represents the rate of change of εlidar with εsonic while the intercept (b) reveals any offset. Including temporal averaging in the fitting process significantly improves predictions of εlidar values: a 5-percentage-point increase in a at 103 m and a 3-point increase at 175 m and 241 m. Similarly, the intercept b approaches zero with temporal averaging. The results show that the model without temporal averaging (in Eq. ) tends to overestimate εlidar when turbulence is low and to underestimate it under strong turbulence.

In Fig. , similar trends are observed for the relaxed criterion (≤ 30 % model fitting error). For the relaxed criterion, however, the number of available 10 min periods are increased significantly: from 90 % of the available data to almost 100 % at 103 m, from 60 % of the available data to 85 % at 175 m, and from 40 % of the available data to 70 % at 241 m. This increase is marked by a very slight decrease in ρ values: 0.93–0.94 at 103 m, 0.91–0.92 at 175 m, and 0.89 at 241 m. This implies that relaxing the error criterion significantly increases the amount of available data without severely sacrificing the strong linear association between lidar and sonic ε. Although the number of 10 min periods where lidar overestimates ε increases, they are still relatively smaller in number, as indicated by the color in the density plots (see Fig. ). The OLS regression parameters a and b also improve under the relaxed fitting criterion when temporal averaging effects are included. 4 percentage points increase in a at 103 m, 2 points increase at 175 m, and 3 points increase at 241 m. The intercept value also moves closer to zero in the plots with spatial + temporal averaging. These results highlight the importance of including both kinds of probe volume-averaging effects, i.e., spatial and temporal, when calculating the second-order structure functions. Further plots will show only the results from fitting the spatial + temporal model (Eq. ) to the data.

The bias analysis of the lidar-derived turbulence energy dissipation rates is presented in Figs.  and . Histograms of the log⁡10(εlidar/εsonic) are illustrated in Fig.  for the three heights while the bias ranges of εlidar relative to εsonic are shown in Fig. . These plots are shown for both relaxed and strict error criteria. From the histogram plots, one can see that most of the data is centered at log⁡10(εlidar/εsonic)=0. For the strict criterion, there is no tail in the distribution on either side of the center. But for the relaxed criterion, a small tail on the right side, i.e. log⁡10(εlidar/εsonic)>0 is observed, which shows those 10 min periods where the lidar overestimates ε. Combined, these data points represent less than 5 % of the total data analyzed.

Another interesting way to look at the bias is to categorize the data in bias ranges, as seen in Fig. . For all three heights, more than 50 % of the lidar obtained εlidar values are within ±20 % range of the εsonic. Similarly, > 90 % and > 75 % of εlidar values obtained are inside the ±40 % range of the εsonic for the strict and relaxed error criteria, respectively. These results clearly show the high accuracy of the structure function method in determining the turbulence energy dissipation rate from a lidar with a small range-gate spacing.

The utility of the modeled structure function containing the lidar filtering effect in estimating ε is clearly observed in the above shown comparison. The largest source of random error in this method arises from the detection of the inertial subrange. It was challenging to detect an inertial subrange during very stable atmospheric conditions and during low wind speeds (< 5 m s⁻¹) due to the lidar's filtering of small-scale turbulence structures. Since we used only the high-level lidar data, i.e., LOS velocities, to estimate ε, any fluctuations smaller than the lidar probe volume length are filtered out. One way to overcome this is to use low-level lidar data, e.g., Doppler spectrum width, or to minimize the range gate spacing of the lidar beams. A comparison between the DSW and the structure function methods is not performed here but is left for future investigation.

The choice of the Kolmogorov constant CK also impacts the estimated ε values. CK is reported to be 2.1 ± 0.1 in the literature . For the fixed measured structure function, the turbulence energy dissipation rate scales as ε∝CK-3/2, thus using CK=2.1 instead of CK=2 would lower the ε estimates by approximately 7 %. This is true for the absolute comparison between ε estimates obtained from a single instrument/method. However, in the relative comparison between ε values obtained from lidar and sonic anemometer, the choice of CK does not matter. The sonic-based dissipation rates are inferred using the spectral Kolmogorov constant α=1.52, which comes from CK=2 (a value of CK=2.1 results in α=1.6). Therefore, although the choice of CK does introduce uncertainty in the absolute magnitude of the obtained ε values, it does not affect the relative lidar-sonic comparison. It should be noted that two different methods are used to estimate ε from sonic anemometer and lidar measurements: the spectral and structure function methods, respectively. However, both estimates are based on the same fundamental assumptions of local isotropy in the inertial subrange and Kolmogorov scaling. Some slight systematic differences may arise between the two methods owing to differences in the fitting procedure and measurement filtering.

Unlike in older systems, instrument noise in modern lidar systems, such as the one used in this study, can be neglected . Since the magnitude of ε spans the order of multiple decades (10⁻⁴ to 10⁻¹), the instrument noise can possibly pollute the measurements, especially at low ε values. However, the random errors from the instrument noise are reduced by using a conservative SNR threshold. A lower threshold would allow more data points at the cost of increasing the random error. Furthermore, the statistical uncertainty associated with the lidar-measured ε is reduced by taking the mean structure function across all six beams over the 10 min period and then fitting the model to that averaged measured structure function.

The method presented in this article is not specific to the BEAM 6x lidar. In principle, it could be applied to any other pulsed Doppler lidar if the range-gate spacing is sufficiently smaller over the region of interest. The advantage of the BEAM 6x configuration here is that 3 m range gate spacing is available throughout the atmospheric column, allowing the method to be applicable over an extended range. The excellent correlation between ε obtained from the lidar and sonic anemometer indicates that ε can be computed directly from the inertial subrange, provided that σ<Lo. This also requires that the inertial subrange is detected with a high certainty, where the high range-gate resolution of the Beam 6x lidar proves especially useful. Large model fitting errors observed in the very stable atmospheric conditions (see Fig. ) may indicate that the inertial subrange may not be detected properly or that the turbulence length scales are relatively small, i.e., Lo<σ. A simple possible solution would be to include a correction parameter for very stable conditions in the structure function model, or to use a lidar with even smaller range-gate spacing. Recently introduced Halo Photonics by Lumibird BEAM 6x Wind Power lidar with a range gate resolution of 1.5 m along the beam can be used to resolve small-scale turbulence even further, but has not been tested yet. For wind energy applications, turbulence intensity (TI) is a more practical parameter to describe turbulence instead of ε. An ongoing study aims to accurately estimate the TI parameter using the methods described in this paper. The aim is to correct the LOS variance estimates for each beam once we know ε, and then use LOS variances to compute the Reynolds stress tensor using a method described by and .

A second-order longitudinal structure function model for lidar LOS velocities is presented in this article. The model also includes a Gaussian filter to account for probe volume (both spatial and temporal averaging effects) within a pulsed lidar beam. The model is used to measure ε at three heights: 103, 175, and 241 m, using a six-beam pulsed lidar. The lidar-obtained ε values were compared with those from ultrasonic anemometers.

The second-order structure function model presented here can be used to estimate ε using a standalone lidar with high range-gate resolution (i.e., small spacing between range gates). The lidar used here has a range-gate spacing of 3 m and can effectively detect the inertial subrange. We compared almost 8 months of lidar-measured ε with the corresponding sonic-measured ε and found an excellent correlation between the two datasets. We observed Pearson correlation coefficient (ρ) values of more than 0.9 across all three heights, significantly higher than in similar previous studies and methods . The observed bias was also very small, i.e., more than 50 % of all the lidar-measured ε values lie within ±20 % of the sonic-measured values.

It was observed that the main source of random errors in lidar-measured ε values is the improper detection of the inertial subrange. Since the structure function method relies heavily on detecting the inertial subrange, relaxing the error criterion during model fitting can significantly increase random errors in the estimated ε. It is therefore advisable to be cautious when using this method under low wind-speed conditions (< 5 m s⁻¹), where reduced atmospheric turbulence may lead to contamination by instrument noise. Under very stable atmospheric conditions, detecting the inertial subrange becomes even more challenging due to smaller turbulence length scales, which are filtered out by the lidar's probe volume-averaging process. Prospective studies may include quantifying the random error arising from instrumental noise and comparing it with that of other similar methods, such as the Doppler spectrum width method, to estimate ε. Additionally, the effect of lidar motion on ε measurements in a floating configuration, which we expect to be null, would also be an interesting investigation.