and show an expression for the spectra measured by a lidar beam Fvr(k1)=ninj∬ϕ^(k⋅n)2Φij(k)dk2dk3, where ϕ^ is the Fourier transform of the lidar weighting function that considers the probe volume. For a CW lidar, this is typically approximated by ϕ^(k1)=exp⁡(-|k|zR), where zR is the Rayleigh length that can be estimated as zR=λdf2πrb2, where λ is the laser wavelength and rb the beam radius at the output lens. For the pulsed lidar

ϕ^(k1)=sinc2kzR/2. In Eq. () zR is not the Rayleigh length as in Eq. (), but rather half the length of a rectangular pulse . Despite this discrepancy, we use the same symbol because zR is the parameter that characterizes both weighting functions. Notice that Fvr is not a function of df/L because the turbulence is assumed homogeneous.

Examples of radial velocity spectra of the CW and pulsed lidars calculated from Eq. () with a half opening angle of φ=15∘, which are compared with the “ideal” sonic u spectrum, are shown in Figs.  and , respectively (within the range of wave numbers that we are interested in, sonic anemometers resolve the u spectrum well). As explained in , the negative correlation between the vertical and horizontal velocities causes the variance of the upward (and forward) pointing beam to generally be the highest of all beams, while the variance is generally the lowest for the downward beam. The difference between the downward and upward pointing beam spectra is smaller than the differences between u, v, and w spectra and deteriorates with increasing zR/L. This ratio indicates the amount of filtering of eddies due to the probe volume. We can also see that for the pulsed lidar the radial spectra of the top beams (1 and 2) are above the sonic u spectrum for zR/L=0.25, which is due to contributions from different components of the spectral tensor. Similar mechanisms can result in a middle beam radial velocity spectrum above the top beam one, particularly for zR/L≥1.

Figure 's left panel shows the behavior of the ratio of the lidar beam radial velocity variance, σvr2, to the variance of the u component, σu2, for a number of zR/L values and for both types of lidars based on the Mann model with Γ=3. As expected from the results in Figs.  and , the ratio increases with decreasing zR/L and for the 0/middle beam of the pulsed/CW lidars, σvr2/σu2=1 at zR/L=0 as no averaging due to probe volume occurs. Furthermore, both lidars' top beams variances can be higher than σu2 for zR/L≈0. Another way to study the contributions of the different velocity components to σvr2 is shown in Fig.  right panel. There we illustrate the ratio of the variance of the other two components, σv2 and σw2, as well as σu2 to σvr2 as a function of the beam azimuthal position for zR/L=0. With φ=15∘ and such turbulence characteristics, we can only measure a portion of σv2 and σw2, and σvr2≈σu2 at θ≈11∘/169∘ (also if a middle beam is used, no matter the turbulence characteristics). For the same turbulence characteristics as those used in Fig. , if we use a lidar with φ=60∘, σvr2<σu2 for all azimuthal positions, whereas σvr2≈σv2 at θ≈200∘/340∘ and σvr2≈σw2 at θ≈237∘/303∘ (not shown). It is also observed that for the same zR/L value, the averaging by the CW lidar has a stronger effect on the variance than the pulsed lidar.

The strategy is to calculate theoretical spectra (in the form of a LUT) that include both the effect of pointing the lidar in the direction ni and of averaging. Then, the measured spectra are fitted to the LUT to get the turbulence parameters. One can expect this procedure to be unsuccessful for zR>L, i.e., if the lidar is averaging out most eddies as shown in Figs.  and .

The computational burden of creating a lidar-based LUT using Eq. () is larger than in the standard case, i.e., using Eq. (), because Fvr is not only a function of the two parameters k1L and Γ but also of zR/L, φ, and θ. Furthermore, lidar beam misalignment can be an issue. Therefore, we need to add an extra dimension to the LUT because such misalignment has a large effect on the lidar radial velocity spectrum.

Figure  illustrates the effect of misalignment (β=-2∘) on the pulsed lidar radial velocity spectra for a set of Mann parameters. The effect of the relatively small misalignment is noticeable; the spectrum of the beams that become more parallel to the wind is clearly above that of those that become less parallel at the same height. For this particular pulsed lidar configuration, misalignment can result in a similar spectrum for the beams 0 (middle) and 1/2 (depending on the sign of the misalignment).

The Nørrekær Enge wind farm is located in the Himmerland region in northern Jutland, Denmark, ≈ 300–400 m south-east of the waters of Limfjorden (see Fig. ). It comprises 13 Siemens 2.3 MW-93 wind turbines with hub height of 81.8 m and a rotor diameter D of 92.6 m. They are aligned on a row at a direction 73.9∘ with the north. The distance between the turbines is 487 m (5.2 D). A meteorological mast was located at 101.2∘, at a distance of 232 m (2.5 D) from turbine number 4 (from left to right on the row). The wind farm is located over flat terrain and the surface is characterized by a mix between croplands and grasslands, and the fjord to the north. At ≈ 2 km southwest of turbine 4, the terrain is no longer flat.

The measurements here analyzed correspond to the period 27 October 2015 to 7 January 2016. There are three types of measurements: supervisory control and data acquisition (SCADA) on turbine 4, FL nacelle-lidar measurements from systems mounted on the nacelle of turbine 4, and meteorological mast observations. Both lidars were pre-tilted down ≈ 0.30∘ so that their axes pointed at hub height, at a position 2.5 D from the turbine for maximum power–performance operating conditions, based on aeroelastic simulations of the tower bending (A. Vignaroli, personal communication, 2016).

We analyze the time series of all data and their statistics in 10 min periods. The total number of 10 min periods available for analysis is 9586. The next steps are followed in the analysis:

We use the 10 min vane measurements to concentrate the analysis on a wake-free sector covering the mast location (88.85–238.85∘) that takes into account the obliquity of the wind farm row and a 15∘ wake expansion (see Fig. ). A total of 5825 10 min measurements are available for analysis where both lidars are also working (based on a 10 min status signal of both lidars) and turbine 4 is grid-connected and available.

Furthermore, each 10 min time series has been post-processed. For the Avent data, we linearly detrend each radial velocity time series for each beam before applying a despiking filter, where values above and below 3 standard deviations from the mean are filtered out. The missing values are then filled in using linear interpolation. The top left panel of Fig.  shows an example of a 10 min time series of the Avent beams' radial velocity. The solid lines show the final interpolated time series and the markers show original radial velocities before post-processing.

For the ZephIR data, we construct time series of radial velocities at azimuthal positions similar to those of the Avent. Since the azimuthal positions of the ZephIR change from revolution to revolution, we extract radial velocities within azimuthal position bins of 7.2∘ on a fixed frame of reference. Three of such bins, 43, 6, and 31, are “aligned” with the Avent beams 1, 2, and 4, respectively. The time series per bin is then threshold-filtered with a minimum radial velocity of 2 m s-1, and detrended and despiked as with the Avent data. The top right panel of Fig.  shows the time series per bin; we include four more bins (0, 12, 18, and 37) than those aligned with the Avent beams and their positions can be inferred by color coding using the bottom left panel of Fig.  which shows the radial velocities in a polar plot. In the top panels in Fig.  the effect of despiking is noticeable (the filtered time series are shown in solid lines and the original are shown in markers), and in the bottom left panel of Fig. , all the radial velocities estimated from the Doppler spectrum within the 10 min period at the 120 m range by the ZephIR are shown. Since the lidars were mounted behind the rotor, the rotating blades sometimes interfered with the beam and the estimated radial velocity became the projection of the radial velocity of the blade onto the beam direction; the result is the figure of eight close to zero radial velocity shown in the bottom left panel of Fig. . In this latter plot, we also include the radial velocities of the three Avent beams that are aligned with the ZephIR bin positions. At these three positions, both lidars show good agreement; a comparison of all 10 min mean radial velocities estimated by the Avent and ZephIR for one of these “aligned” positions, beam 2 and bin 6, respectively, is shown in the bottom right panel of Fig. . The top right panel of Fig.  also shows that it is possible to get more than one radial velocity value within the same azimuthal bin (sometimes up to three values). Finally, the ZephIR's time series are “completed” using linear interpolation.

For each 10 min period, the 1 Hz sonic and cup anemometer data are detrended and despiked as with the lidar data, and mean and turbulence statistics are computed. The sonic-anemometer wind-speed components are rotated so that u is aligned with the mean wind. We estimate the friction velocity, from the sonic wind speed and temperature fluctuations, as u*=u′w′‾2+v′w′‾21/4, and the Obukhov length estimated as LO=-u*3κ(g/T‾)w′Θv′‾, where κ is the von Kármán constant (≈ 0.4), g the gravitational acceleration, T a reference temperature, and Θv the virtual potential temperature. Spectra of all lidar radial velocities, sonic-anemometer wind speed components and cup-anemometer horizontal wind velocity are computed for each 10 min period. All 10 min turbulence statistics and spectra from the sonic anemometer are also computed on a 5 and a 18 s basis, mimicking the lidar sampling frequencies.

When compared to the measurements from the 80 m cup anemometer, the sonic-anemometer mean horizontal wind speeds are 2.6 % lower (see Fig.  left panel). This bias is higher than 0.6%, which is the estimation that results from assuming, between the two instruments' heights, the logarithmic wind profile U=u*κln⁡zz0, where z0 is the roughness length, ≈ 0.012 m, which is a typical value of these surface conditions . When looking at variances, the bias is 12 % (Fig. , right panel) if we use the u component or the combined u and v components for the estimation of the sonic-anemometer variance. The latter means that for this site and at this height, the v variance has a low contribution to the horizontal velocity variance (which is what a cup anemometer does theoretically measure) and so we could assume the cup-anemometer variance to give a good estimate of the u variance. On the other hand, the bias between both instruments' variances cannot be explained simply; a 4 % bias is expected, assuming the 2 % bias of the mean wind speed.

The behavior of the sonic-derived velocity spectra does not correspond well with the notion of turbulence local isotropy within the inertial subrange, where we expect the same spectral density for the v and w components and the u component is 25 % lower . Figure  shows that within the inertial subrange the ensemble-average sonic u spectrum (of all 10 min observed spectra) is indeed ≈ 25 % lower than the v spectrum but so is the w spectrum. Possible explanations for this are path-averaging errors and transducer shadowing mainly attenuating the w spectrum measured by the CSAT3 . Figure  also illustrates the fit to the three auto-spectra and cross-spectrum using the Mann model (see Sect. ), which shows the expected behavior within the inertial subrange (k1⪆0.03 m-1 for this case). The fit is performed on the ensemble-average spectra that have been logarithmically-averaged on the basis of the wavenumber (we will use such logarithmically-averaged spectra when fitting Mann parameters). These “average” Mann parameters (Γ=3.00, αε2/3=0.14 m4/3 s-2, and L=35.38 m) are similar to those observed at a site with similar surface and turbulence characteristics , but it should be noticed that these are the average of spectra for a number of atmospheric and turbulence conditions and that the Mann-model fitting procedure is normally performed over specific wind speed, turbulence, or atmospheric-stability ranges.

Due to the uncertainty on the sonic-derived statistics, we will use the cup-anemometer variance as a proxy for σu2. However, we will use the sonic-based Mann parameters for comparison with the lidar-based Mann parameters (and for estimations of σv,w2 and u′w′‾) because it is the only reference we have for three-dimensional turbulence measurements.

Although the variances of a velocity time series sampled over a 10 min period at a frequency fs of 0.2 or 0.06 Hz are not statistically different from those estimated from 1 or 10 Hz records, aliasing and noise might appear both in the sonic-anemometer and the lidar radial velocity spectra. The left panel of Fig.  shows the Avent radial velocity spectrum that has been ensemble-averaged from all the 10 min observed spectra for each of the beams. We conjecture that the increase in the spectral densities at high frequencies is due to noise. The right panel of Fig.  shows the effect of a noise filter, which is based on the method by , on the ensemble-average Avent radial velocity spectra.

The noise filter seems to recover the shape of the Avent radial velocity spectra. However, when tested on the 18 s sonic ensemble-average u spectrum (not shown), the filter highly distorts the shape and the peak of the spectrum. Therefore, we focus the spectra analysis on the measurements performed at fs≥0.2 Hz, i.e., we exclude the ZephIR radial velocity spectra for the analysis. For the results presented hereafter, the noise filter is only applied to the Avent radial velocity spectra.

Figure  also shows that for these ensemble-averages, the spectral density of beam 2 is the highest, followed by that of beams 0 and 1, and then that of beam 4. This behavior might be due to three reasons: Excessive rolling of the Avent, so that beam 2 points higher than beam 1; that the turbulence characteristics at the position of beam 2 are rather different than those at the position of beam 1; or that there is yaw misalignment so that beam 2 points closer to the direction of the mean wind compared to beam 1 (see Fig. ). Both ZephIR and Avent have tilt and roll signals, and for the 10 min samples analyzed here the maximum absolute 10 min mean tilt and roll are only 0.56 and 0.31∘, respectively. Also, the very flat terrain characteristics should not have such an impact on the ensemble-average spectrum of two beams that point at the same height, like beams 1 and 2 in this particular case. So, the most plausible explanation is that beams 2 and 3 are more aligned with the mean wind than beams 1 and 4.

In the right panel of Fig. , we can see that the spectrum of beam 2 is slightly higher than that of the 80 m cup anemometer and higher than that of the sonic anemometer (up to f≈0.04 and 0.07  Hz, respectively). Such behavior is expected for low zR/L values (see Fig.  top left panel) or under lidar misalignment conditions (see Fig. ). We can also see that the cup-anemometer spectrum is higher than that of the sonic anemometer, as expected from the variance results in the right panel in Fig. .

For both lidars we need to reconstruct the horizontal wind speed at the specific range of the lidars, which can later be used for spectral analysis and for filtered along-wind variance estimates. We use a simplified version of the linear-gradient model of , vr(θ)=-cos⁡φu+Rddudzsin⁡θ+vcos⁡θsin⁡φ, where Rd is the radius of the disc formed by the scanning pattern at the given range, to estimate u, v, and the vertical gradient of the along-wind component, du/dz. In Eq. (), we ignore w and other vertical and horizontal gradients of the wind components because their contribution is small. For both lidars, the beams selected in Sect.  are used for the reconstruction, which can be done on the time-series basis or the 10 min averages. Figure  shows the results of the lidar-based reconstruction on all 10 min means compared to the 80 m cup anemometer and between the lidars; for both lidars we show the horizontal wind-speed magnitude but when using the mean wind speed we obtain the same results. The same results, regarding the linear regression and R2 (not shown), as those given in the left panel of Fig.  are found when comparing the radial velocity of beam 0 with the 80 m cup-anemometer wind speed on a 10 min basis.

The Doppler-spectrum analysis is performed over all the 2273 10 min periods using the ZephIR data (the Doppler spectrum information is not available for the Avent). While each of the 10 min radial velocity time series per bin position is thresholded and despiked (see Sect. ), we extract the normalized Doppler radial velocity spectrum for each of the samples within that 10 min and bin position. We then sum all the normalized Doppler spectra within the 10 min period and the resulting Doppler spectrum is normalized to unit area before we estimate the variance in two ways: by computing the second moment from the spectrum and by fitting a normal distribution to the spectrum to extract its variance . Figure  illustrates examples of ensemble-average Doppler spectra for different 10 min periods for the positions of bins 0 and 31, where we intentionally show 10 min radial velocity distributions with high and low mean values, and high and low variances, including double-peak distributions (there are only a few of them). These few distributions give us an idea of the variety of turbulence characteristics of the dataset. Distributions with high and low radial velocities generally show high and low variances, respectively, as expected. Particularly in the examples, there is a 10 min period with very low variance for both bin positions with clear larger radial velocities for bin 0 compared to those for bin 31, indicating very high wind shear, which is normally associated with atmospheric stable conditions. This is an early morning 10 min period in late October, in which the sonic-derived LO value is 1.82 m, corresponding to extremely stable conditions.

Figure  shows the overall turbulence characteristics of the site based on cup- and sonic-anemometer observations, using the 2273 10 min concurrent data. In the left frame, we illustrate the behavior of the turbulence intensity, σU/U, with wind speed, using the 80 m cup-anemometer measurements; wind speeds are in the range ≈ 5–23 m s-1 with low σU/U values within the low wind-speed range and σU/U increasing with wind speed. In the right frame, we illustrate the behavior of the dimensionless wind shear, ϕm=κz/u*∂U/∂z, with dimensionless atmospheric stability, z/LO; we use the cup-anemometer wind-speed measurements at 78 and 56 m to estimate ∂U/∂z (≈ΔU/Δz) and the sonic-derived u* to compute LO and ϕm. Figure. , in the right panel, shows that the atmosphere during the analyzed period is mostly stable (z/LO>0) and that, as expected, ϕm increases with increasing z/LO. Such atmospheric conditions explain the low σU/U values for low wind speeds. In the left panel of Fig. , we include a prediction of σU/U, using Eq. () with σU=2.5u* and zo=0.012 m, which fairly agrees with the data for high wind speeds only, as expected. In Fig.  (right panel), we include, for comparison only, the prediction ϕm=1+4.7z/LO from surface-layer theory that is offset with the data because ϕm, and so z/LO, is estimated at a mean height of z=67 m with only two wind-speed observations that were 22 m apart, whereas the turbulence estimates are from the sonic anemometer at 76 m.

Based on the observed turbulence characteristics and knowing that we need to average a number of 10 min spectra to be able to robustly extract the Mann parameters , we classify the concurrent data into 10 classes as illustrated in Table , ensuring that there are a close to 100 10 min samples per class as a minimum. From the atmospheric-stability classes, we can see that the data comprise mainly stable conditions, with stability 1 being the only close-to-neutral class (〈z/LO〉=0.0625, with z=67 m). The more stable the atmospheric conditions the lower the wind speed and the friction velocity, as expected. Most of the data range within the stability 2 class (〈z/LO〉=0.1489), i.e., most of the observations are nearly stable. From the wind-speed classes, we observe most of the data within a high-speed range (11–13 m s-1) and, similarly to the stability classification, the lower the wind speed the more stable the atmosphere (except for the speed 1 and 2 classes), and so the lower the friction velocity. Interestingly, for the speed 1 and 2 classes z/LÕ=0.5084 and 0.7196, respectively (where ̃ indicates the median value), which are higher values than the mean dimensionless stability of the most stable class (stability 5). We use the median for the speed classes since the LO values highly fluctuate within those speed ranges.

Based on the ZephIR configuration (φ=15.05∘), we are able to predict all variances' ratios σvr2/σu,v,w2, using the Mann model with a given Γ parameter for the unfiltered lidar radial velocity variances, i.e., using Eq. () with zR/L=0. This is a procedure similar to the one we use for the results in Fig.  (right panel). Figure  shows a comparison of the ZephIR “unfiltered” radial velocity variances (for bins 0 and 31) with the cup-anemometer variances for all the 2273 10 min data, together with the Mann-model prediction, using Γ=3. We present variance estimations that are computed from the normal distribution fit to the average normalized Doppler spectrum, instead of those calculating the second moment from the spectrum, since the latter method is more sensitive to “spurious” data that appear far from the area where most radial velocities are concentrated. This is particularly seen for the lower bins (18 and 31) and might be due to non-filtered blade-obstructed data, noise, or sudden jumps in the radial velocity within the 10 min period.

As expected, based on the results in Fig. , the top (bin 0) and a lower beam (bin 31) show a higher and lower variance, respectively, than that of the “u” component (in quotation marks because we use the cup-anemometer measurements). The Mann-model-based results slightly underpredict the ratio σvr2/σu2 for these two beams compared to the raw data. Reducing the value of Γ or accounting for misalignment improves the predictions; e.g., with Γ=2.5 and β=0∘ the Mann-model results predict biases of 11 and -13 % for bins 0 and 31, respectively (not shown). It is important to mention that the original (filtered) radial velocity variances for these two bins are 13 and 31 % lower than the cup-anemometer measurements (not shown) with slightly higher R2 values, 0.785 and 0.798, respectively.

Furthermore, we can also estimate σu,v,w2 and u′w′‾ for each 10 min period through a least-squares fit of Eq. () that does not depend on the Mann parameters but assumes homogeneous turbulence within the scanned volume, using the unfiltered radial velocity variances. Figure  (left panel) shows the estimate of σu2 based on the unfiltered radial velocity variances of all bins without accounting for misalignment compared to σcup2. The lidar-variance estimate is only 2 % larger than the cup-anemometer value and the R2 value is higher than that of any other comparison between cup-anemometer and lidar beam radial velocity variances (filtered or not).

In Fig.  (right panel) we show a similar comparison to the plot in the left panel but for the “filtered” u variance, which was computed by reconstructing the u and v components, as described in Sect. , using the ZephIR measurements on the seven bins, but from the 18 s radial velocity measurements. The comparison with the filtered values shows poor agreement with a 50 % underestimation of the variance by the ZephIR. However, reconstructed u velocities from the 18 s radial velocities and averaged within 10 min periods compare well with the reconstructed values from the 10 min means; the mean bias is 0 % and R2=0.999 (not shown).

We also compare the lidar-derived σv,w2 and u′w′‾ values with the sonic-anemometer estimates; the biases are very high and R2 values are very low (not shown). This is not surprising given the weight of the σv,w2 and u′w′‾ terms in Eq. () when using low φ values. With this lidar configuration, the reconstruction of the v component from, for example, Eq. (14) is not sound either; the yaw misalignment based on both the Avent and ZephIR reconstructed u and v components shows poor agreement when compared to the difference between the wind-vane and the turbine-yaw 10 min signals.

We can also estimate σu2 through a least-squares fit of Eq. () but using the unfiltered radial velocity variances of the horizontal bins (12 and 37) only and the comparison with σcup2 shows similar results (bias of 3 % and R2=0.842). This indicates, firstly, the small but positive effect of adding the top and lower beams' variances, and secondly, that the contributions of other velocity components are not that significant for the estimation of σu2 with the actual lidar configuration. Accounting for misalignment does not improve the variance comparison (the bias increases from 2 to 7 %).

We also classify the 10 min 80 m cup anemometer variances and Avent radial velocity spectra into the classes given in Table , ensemble-average the spectra within each class, and compute the variance of each ensemble-average spectrum. The comparison of such variances, for each Avent beam, is illustrated in Fig.  (raw). We also show a similar comparison but for the noise-filtered Avent radial velocity ensemble-average spectra. Furthermore, we include the prediction σvr2/σu2 based on the Avent lidar configuration using the Mann model with fixed Mann parameters (same as those found in Sect.  using the ensemble-average sonic-anemometer velocity spectra).

When the noise filter is applied, the ratio σvr2/σcup2 is well predicted by the Mann model. The largest difference is observed for beam 4 but this is because the noise filter highly reduces the variance for one particular class only. For beams 1 and 2, the Mann model predicts the same σvr2/σcup2 value as here we do not take into account lidar misalignment.

The ensemble-average sonic and Avent radial velocity spectra are used separately to extract two independent sets of Mann parameters for each of the atmospheric stability classes in Table  by fitting the sonic- and lidar-based LUTs computed through the use of Eqs. () and (). Figure  shows the results of the two stability classes farthest apart (stabilities 1 and 5).

For the stability 1 class, the Mann model agrees well with the sonic velocity spectra and for stability 5 the differences between the model and the sonic-anemometer observations are larger, as expected, since the Mann model was developed for near-neutral atmospheric conditions. Both the sonic observations and the Mann model show the spectral peaks to move to higher wave numbers with increasing stability because the size of the turbulence eddies decreases with stability in agreement with the study of . The lidar radial velocity spectra also show similar features to the sonic-based spectra: higher normalized spectral densities for the most stable compared to the close to neutral class and spectral peaks that move to higher wave numbers with increasing stability. The former feature might be due to the way we normalize the spectra: we make use of the 76 m u* value instead of one close to the ground where surface-layer scaling is more valid, particularly for stable conditions. The agreement of the Mann-model-based spectra also deteriorates with stability but the lidar-based LUT seems to follow the behavior of the radial velocity spectra for these two classes fairly well.

In Fig. , we show the results of the Mann parameters extracted from the ensemble-average sonic and lidar radial velocity spectra for all atmospheric stability classes. There is a slight decrease in Γ with stability (based on the sonic-anemometer data) and the lidar-based value closely follows the sonic-based one, with best agreement at the highest stability range. The sonic-based αε2/3–parameter slightly decreases with stability and, for the near-neutral stability class, the lidar-based value is close to the sonic-based one. A similar feature is found for the L parameter; both types of data show a very close value for near-neutral conditions and the sonic-based value slightly decreases with stability as expected. The increasing differences between the sonic- and the lidar-based αε2/3 and L parameters with stability are interconnected. We cannot expect to measure eddies below the size of the lidar probe volume, which in this case means that we are not able to accurately estimate the length scale when L⪅zR. This occurs already at the stability 3 class. These two Mann parameters are, in practical terms, scaling factors in the velocity spectra as seen from Eq. (), and so an underestimation of L generally leads to an overestimation of αε2/3 when fitting the lidar-based LUT.

We also have to notice that when using this type of lidar configuration, we are extracting turbulence information from the radial velocity spectra of beams, whose spectral densities are rather close (since all beams measure a close to u spectrum), whereas in the case of the sonic-anemometer observations we use three auto-spectra and a cross-spectrum that are relatively far apart in terms of spectral densities. This issue is discussed further in Sect. .

We now perform a similar procedure as that in Sect.  but for each of the wind-speed classes in Table , and the results of the two wind-speed classes most far apart (speeds 1 and 5) are shown in Fig. . For the speed 1 class, the Mann model does not agree with the sonic velocity spectra as well as it does when compared to the speed 5 class, as expected, since the atmospheric conditions are closer to neutral for the latter class. Both the sonic-anemometer observations and the Mann model show spectral peaks that move to lower wave numbers with increasing wind speed because of the combined effect of stability and wind speed; the larger the turbulent eddies, the higher the wind speed and the lower the stability.

The lidar radial velocity spectra also show similar features to the sonic-based spectra; lower normalized spectral densities for the high-wind compared to the low-wind class and spectral peaks that move to lower wave numbers with increasing wind speed. The agreement of the Mann-model-based spectra deteriorates with decreasing wind speed, but the lidar-based LUT also seems to follow the behavior of the radial velocity spectra for these two classes fairly well (similarly as it does when comparing spectra for the range of stability classes).

In Fig. , we show the results of the Mann parameters but for the wind-speed classes. Based on the sonic-anemometer data, Γ, is rather constant with wind speed, a behavior already observed by for the same height and the lidar-based value agrees well with the sonic-based one for all wind-speed classes, particularly the two low wind-speed classes. Similar to the results from the atmospheric-stability classes, the differences between the sonic- and the lidar-based αε2/3 and L parameters are larger than those for Γ, but for these wind-speed classes the αε2/3 parameter does not differ largely under the classes where L differs the most, i.e., speed classes 1 and 2, where the average conditions are very stable. Turbulence characteristics under these two classes are similar and L is higher within speed 1 compared to the speed 2 class. The highest differences in the estimations of L are also found for those classes in which L⪅zR (speed classes 1–3).