Out of the 195 sonic anemometers, this study relies on the instruments on towers 20/trSE_04 and 25/trSE_09. Both of these towers were 100 m high with sonic anemometers at 10, 20, 30, 40, 60, 80, and 100 m levels on booms pointing to ∼135∘ (∼155∘) for tower 20/trSE_04 (25/trSE_09). All sonic anemometers on these two masts were Gill WM Pro sonic anemometers, sampling the three-dimensional wind vector at a rate of 20 Hz. The two sonic anemometers at 80 and 100 m at the 100 m meteorological mast 25/trSE_09 were within the height limits of the lidar scans and are therefore used for intercomparison of turbulence measurements. The ground levels of towers 20/trSE_04 and 25/trSE_09 with respect to ridge height (i.e. wind turbine base height) are -10 and -178 m, respectively.

The University of Colorado Boulder’s tethered lifting system (TLS), a specially designed tethersonde system, enables unique in situ high-rate measurements of wind speed, wind direction, and temperature. From these high-rate measurements, the TKE dissipation rate can be estimated . TLS capabilities for observing detailed wind speed, temperature, and dissipation rate profiles have been demonstrated in several field campaigns , including measurements of dissipation rate in wind turbine wakes . used data from the TLS to assess small-scale and large-scale turbulence intermittency in flat terrain, while used the system to explore microscale turbulence in the stable boundary layer.

The Perdigão TLS instrument packages were similar to those of . Fast wind speed measurements at 1 kHz were from 1.25 mm length, 5 µm diameter, tungsten wires. Other measurements included 1 kHz cold-wire anemometer temperature measurements (Auspex Scientific, custom-made device), 100 Hz thermistors (Honeywell 111-103EAJ-H01), solid-state measurements for temperature (Analog Devices Inc TMP36) and relative humidity (Honeywell HIH-4000), a 100 Hz Pitot tube (Dwyer Instruments, model 166-6), and a pressure sensor (Honeywell DC001NDC4) for velocity and pressure measurements, as well as GPS and compass measurements. GPS measurements of latitude, longitude, and altitude were sampled every 5 s. While the TLS can be deployed in either a profiling or a hovering mode, most Perdigão measurements consisted of profiles. The ascent and descent of the TLS was controlled by a custom-made winch system with an average ascent and descent rate of 0.3 m s-1. The payloads were lifted using a 16 m3 Helikite system (Allsopp) with a lightweight but strong tether (1120 kg Dyneema line, 2.5 mm diameter).

Here we focus on the flow in the center of the valley and in a cross section through the location of the wind turbine. For this purpose, three Leosphere Windcube 200S lidars of the German Aerospace Center (DLR) were deployed at the locations indicated in Fig. . All of the systems performed RHI scans as indicated by the dashed lines in Fig. . Lidar RHI#1 and RHI#2 were aligned with the wind turbine along the primary wind direction. RHI#2, in the valley, probed the flow with a high elevation angle so that the line-of-sight (LOS) measurements included a significant contribution from the vertical wind component. RHI#1, on the northeast ridge, probed the valley flow at a low elevation angle, thus measuring primarily contributions of the horizontal wind components. Synthesizing data from these two lidars allows for coplanar wind speed retrievals as described in . RHI#3 provided additional information about the out-of-plane flow and wind turbine (WT) wake position. A combination of the three lidars also allowed multi-Doppler measurements of the wind turbine wake for a range of wind directions far from the main wind direction . The parameters of the RHI scans and lidar specifications are given in Table . The horizontal distance from the masts, and thus the sonic anemometers, to the RHI plane of lidars RHI#1 and RHI#2 is 150 m.

As part of the Collaborative Lower Atmospheric Mobile Profiling System CLAMPS; , the University of Oklahoma (OU) deployed a Halo Photonics Stream Line scanning lidar at the so-called Lower Orange site in the Vale do Cobrão, approximately 100 m from the cross section through the WT. The scanning scenarios for this lidar during the campaign comprised a regular sequence of velocity–azimuth display (VAD) scans (2 min), RHI along-valley (2 min) and cross-valley scans (2 min), and vertical stare measurements (9 min). In this study, the vertical stare measurements are used to derive turbulence dissipation rate and the results from the VAD scan is used for wind speed information. Table  gives an overview of the CLAMPS lidar parameters for the vertical stare measurements that are relevant for the turbulence retrieval (see Sect. ). The horizontal distance of CLAMPS to the sonic anemometers on tower 25/trSE_09 is 250 m.

From the valley, radiosondes were launched regularly every 6 h, aiming to reach 20 km at 00:00, 06:00, 12:00, and 18:00 UTC. The radiosondes provide vertical profiles of pressure, temperature, and humidity as well as wind speed and wind direction.

TKE dissipation rate from the sonic anemometers on the meteorological towers, εs, is calculated from the 2nd-order structure function of the horizontal velocity (Eq. ). εs is calculated every 30 s, and the fit to the Kolmogorov model is done using a temporal separation between τ1=0.1 s and τ2=2 s see also.

Estimates of ε obtained by the TLS are retrieved using the inertial dissipation technique . For each 1 s of data, a Hamming window was applied and the streamwise velocity spectra as a function of frequency was computed. The spectra were then smoothed and the mean structure function parameter Cu2 was computed over the frequency band 5 to 10 Hz. The dissipation rate εt was then computed using the Corrsin relation: 9εt=0.52Cu23/2.

In the case of a profiling lidar (or a scanning lidar used in a vertical stare mode as the CLAMPS lidar), TKE dissipation rate can be derived from the variance σv2 of the line-of-sight (LOS) velocities (which in this case equals vertical velocity) following the approach described in and further refined and validated in . Before processing, the LOS data are filtered for good carrier-to-noise ratio (CNR) with a threshold of -23 dB. By assuming locally homogeneous and isotropic turbulence, the turbulence spectrum (Eq. ) derived from the measured LOS velocity can be integrated within the inertial subrange: 10σv2=∫κκ1Sv(κ)dκ=-32αε2/3κ1-2/3-κ-2/3.

For the CLAMPS lidar's vertical scans, which measured only the vertical component of velocity, the sample length N to use for this integration is chosen by fitting the experimental spectra to the model spectrum described in and following the approach described in and . Dissipation rate εv can then be derived as 11εv=2π23α3/2σv2-σe2LN2/3-L12/33/2, where L1=Ut, with U being the horizontal wind speed; t is the dwell time; LN=NL1; and α=0.55. Since this method is based on measurements in the inertial subrange, the interruption of vertical stare measurements by VAD and RHI scans, as described in Sect. , does not compromise the retrieval. The horizontal wind speed U is retrieved from a sine-wave fitting from the velocity–azimuth display (VAD) scans, which were performed every 15 min. σe2 accounts for the instrumental noise which affects the measured variance, and it is defined as in , using the technical parameters in Table . When the instrumental noise is too large, the inertial subrange is difficult to detect in the lidar observations, and the dissipation retrievals are undermined. For this reason, for each spectral fit, we calculate the deviation between the measured lidar spectrum Sv^ and the spectral model S over the n spectral frequencies used, and we quantify the error in the fit as 12ES=1n∑i=1n|Sv^,i-Si|Si. Retrievals of εv are discarded when ES>10. This threshold was chosen as it reliably removes noise-dominated spectra and provides the best agreement with the retrievals from other instruments as shown in Sect. .

Retrieving turbulence parameters from an RHI scan cannot be done with the same method as for vertical scans. Since the duration of a single scan is usually on the order of tens of seconds and in this experiment even 1 min, it is not possible to derive turbulence from the variance in LOS measurements only. The sampling time is too long to resolve the relevant scales in the inertial subrange in weak turbulence conditions. Here we propose an algorithm to retrieve eddy dissipation rate and integral length scales from RHI scans following the principal idea by . We introduce a modification of the Doppler spectrum width (DSW) method which uses variance in LOS velocities σv2 and the turbulent broadening of the Doppler spectrum σt2. In , the RHI scans are used to calculate vertical profiles of ε by binning data points from the RHI scan into height bins over the whole scan area. The complex flow over the Perdigão double ridges compromises this approach. In the modification of the method, we divide the area covered by the RHI scan into square subareas with a defined side length (here sa=20 m). Within these subareas, the LOS variance is calculated as a space and time average over a half-hour period: 13σ^v2=1N∑i=0Nv^r,i-v‾r2, where N is the number of single LOS measurements within the time and space bin and v‾r is the mean of all measurements in the bin. Variables with a hat (overscore) denote measured variables. The number of measurement points in the subareas varies with distance from the lidar. Close to the lidar, there are as many as 15 points, while far from the lidar it is reduced to 3–5 points. Most of the large-scale variance is captured through the temporal average of 30 min, and sensitivity tests showed that an increase or decrease in measurement points in the subareas does not change the results significantly. The half-hour averaging period has been chosen as a common averaging time for turbulence measurements in the ABL. Longer periods could be affected by the mesoscale changes in the flow field, and shorter periods reduce the number of single RHI scans, which increases the uncertainty in variance measurements. Large cells and coherent structures are important contributions to turbulent mixing. More specific studies related to these phenomena will be necessary in the future to understand their impact on turbulence measurements with lidars and the implications for averaging periods.

The turbulent broadening of the Doppler spectrum is defined as 14σt2=σ^sw2-σ02-σ^s2-E with σ^sw2 being the measured spectral width, σ02 being the spectral width at constant wind speed in the sensing volume, σ^s2 being the measured spectral broadening caused by shear, and E being the random error. According to , we set the noise threshold for derivation of parameters from the Doppler spectrum, i.e., nth=1.01. This value is much smaller than for the lidar used in due to the high number of accumulations by the fiber-based system used in this study. We then assume E to be negligible.

To maximize the signal-to-noise ratio of the spectra and thus estimate more reliable spectral widths at low signal strength, all the spectra within a time and space bin are interpolated in the Fourier domain, aligned according to their maxima, and accumulated. The spectral width of the accumulated spectra is used as σ^sw2.

The contribution from shear σ^s2 is calculated according to for each LOS measurement and averaged over the time and space bin: 15σ^s2=12π1N∑i=0Nv^r,ir+ΔR-v^r,ir-ΔRΔz2ΔR2, where N is the number of LOS measurements in the time and space bin, v^r,i(r) is the measured radial velocity of the range gate at location r, ΔR is the distance between two range gates, and Δz is the physical resolution of the lidar measurement.

The spectral width at zero wind speed, σ02, as well as Δz can be theoretically derived from the lidar parameters Tw (time window) and σp (pulse width) through a model of the Doppler lidar echo signal as described in . The echo signal models assume a specific pulse shape and require knowledge of the lidar parameters Tw and σp. These vary for different systems and are only given as estimations by the lidar manufacturer. Here we will consider σ0 and Δz as unknown parameters that need to be tuned within physically reasonable limits to achieve good agreement with reference instruments. As an initial guess for σ0, the mean of all observed spectral widths can be used. For Δz, the initial guess is the physical resolution as provided by the manufacturer for the used lidar settings (in this case 50 m). It has to be noted that since the calibration of these parameters will also account for inaccuracies in the assumptions made for the theoretical turbulence model, the estimated parameters are not necessarily the real physical lidar parameters.

A model for the volume averaging of the lidar measurement and basic turbulence theory as described above is used to derive the equations for the retrieval of turbulence from RHI scans. Assuming that the lidar pulses have Gaussian shapes, a window function for LOS measurements of wind speed can be defined as 16Qs(z)=1Δze-πz2/Δz2, so the wind speed measured by the lidar is the convolution of the actual wind speed with the window function, 17v^(r)=∫-∞∞dzQs(z)v(r+z).

The transfer function of the low-pass spatial filter of the lidar, derived from the window function, is 18Hp(κ)=∫-∞∞dzQs(z)e-2πjκz2.

The total variance in the LOS velocity σv2 is the sum of measured variance σ^v2, turbulent broadening of the spectra σt2, and an error term accounting for instrumental noise σe2 (see also Sect. ): 19σv2=σ^v2+σt2+σe2.

The variances are the integral of the power spectra multiplied by the respective filter function: 20σ^v2=2∫0∞dκSv(κ)Hp(κ),21σt2=2∫0∞dκSv(κ)1-Hp(κ).

A generic spectrum, subdivided into areas of lidar-measured variances, appears in Fig. . It illustrates the energy that is measured by the lidar LOS measurements by the solid line and the energy that is measured through the turbulent broadening of the spectra by the dashed line. The corresponding variances are the integral of the hatched areas.

Substituting Sv for the von Kármán model (Eq. ), we obtain 22σt2=4σv2Lv∫0∞dκ1-Hp(κ)1+8.42Lvκ25/6. In the inertial subrange of turbulence, the van Kármán model can be simplified to 23Sv,i(κ)=2σv2Lv8.42Lvκ-5/3, and in combination with Eq. () it can be solved for σv2: 24σv2=Ck1.573ε2/3Lv2/3. Substituting σv2 in Eq. () with Eq. () results in 25σt2=2.54Ckε2/3Lv5/3∫0∞dκ1-Hp(κ)1+8.42Lvκ25/6.

With the simplified equation for a Gaussian-shaped filter function Hp, 26Hp(κ)=exp⁡-2πΔzκ2, and substitution of κ for ξ=2πΔzκ, the equation can be rewritten as a function of Δz and Lv: 27σt2=0.2485Ckε2/3Δz2/3∫0∞dξ1-exp⁡-ξ2/(2π)ξ2+0.746Δz/Lv25/6.

Substituting ε in Eq. () with the solution for ε from Eq. (), the equation for σt2/σv2 as a function of Δz and integral length scale Lv, as they appear in , can be formulated: 28σt2/σv2=FwLv,Δz,29FwLv,Δz=(1.972)2/3Ck-1Lv-2/3GwΔz,Lv, 30GwΔz,Lv=0.2485CkΔz2/3∫0∞dξ1-exp⁡-ξ2/(2π)ξ2+0.746Δz/Lv25/6. The only unknown is Lv. A downhill simplex algorithm is used to minimize Eq. () for Lv 31arg minLv-exp⁡-FwLv,Δz-σt2σv22.

Minimization of Eq. () is computationally expensive. To accelerate the data processing, a power function can be defined which approximates the relationship between Lv and σt2σv-2: 32L^v=c1σt2σv2c2+c3. The coefficients c1, c2, and c3 are determined by a curve fit over the range of Lv=3 to 1000 m to Eq. (). The coefficients are specific for each lidar, since they depend on Δz, and will be determined in Sect. . The fitting curve and residuals as obtained through minimization of Eq. () appear in Fig. . It shows that the error, Lv-L^v, that is made with the power-law approximation is in the range of ±1 m. For simplicity, we will use the variable name Lv for integral length scales calculated with Eq. () in the following.

With Lv and measured variance σv2 in Eq. (), TKE dissipation rate for the RHI measurements εr can finally be calculated as 33εr=1.972Ck3/2σv3Lv.

For optimal accuracy of the dissipation rate retrieval, the two unknowns, σ0 and Δz, need to be calibrated according to reference instruments. In this study, the sonic anemometer at 100 m on tower 25/trSE_09 is the closest in situ observation to the RHI scans in the valley. It is used for the calibration by minimizing the root mean square error between this measurement and the respective RHI scan. The resulting parameters differ slightly for lidars RHI#1 and RHI#2 and appear in Table  along with the coefficients for the power-law fit of Lv. The difference between Δz and σ0 can be partially attributed to instrumental variability but will also incorporate other sources of error in the turbulence model and data retrieval.

To estimate the uncertainty in the retrievals of εs, we apply the law of combination of errors, which describes how random errors propagate through a series of calculations . For a function g=g(xi), with xi being the independent and uncorrelated variables, the law of combination of errors states that, for small errors (i.e. if we ignore 2nd order and higher terms), the variance in the function g, approximated by the sample variance σg2, is given by 34σg2=∂g∂xi2σxi2, where σxi2 values are the sample variances in the xi values. By applying this method to Eq. (), the fractional standard deviation in the ε estimate is the following : 35σε,s=32σIIε, where I is the sample mean of τ-2/3D(τ) and σI2 is its sample variance.

Similarly, uncertainties are calculated for the TLS measurements εt. However, since the determination of dissipation rate is done in the frequency domain, I in this case is the sample mean of κ5/3S(κ).

The uncertainty in the εv retrievals from the profiling lidars can be estimated from the uncertainty in the LOS velocity variance by also applying the law of combination of errors (Eq. ) to Eq. (): 36σε,v=∂εv∂σvσσ,v=2π23α3/2σv2-σe2LN2/3-L12/31/23σvLN2/3-L12/3σσ,v 37=εv3σvσv2-σe2σσ,v, where σσ,v is the uncertainty in the sample variance. This value is not known but is considered to be on the same order of magnitude as the instrument noise and thus set to the value of σe.

The uncertainty in εr can be calculated by Gaussian uncertainty propagation through Eq. () if the uncertainties of the estimation of the integral length scale σL,v2 and the uncertainty in the measurement of the wind speed variance σσ,v are known: 38σε,r=5.916σv2Ck3/2Lvσσ,v2+-1.972σv3Ck3/2Lv2σL,v2. The uncertainty in the variance in radial velocities σσ,v can be determined from the uncertainty in the turbulent broadening estimation σσ,t and the uncertainty in measured LOS velocities σσ^,v: 39σσ,v=σσ,t2+σσ^,v2. The uncertainty in the measurement of the integral length scale Lv cannot be determined directly from Eq. (). A propagation of uncertainties is not possible here, because the function is not differentiable. The approximated function Eq. () can, however, be differentiated with respect to σv and σt and can thus be used to propagate uncertainties of the measured values to Lv so that 40σL,v=2c1c2σtσv2c2σσ,v2+-2c1c2σt2σv3c2σσ,t2, with c1, c2, and c3 given from Table . The uncertainties that are found through this approach are then fed into Eq. () in order to calculate the uncertainty in εr.

As can be seen from Eq. (), the uncertainty in the retrieval of εr depends strongly on the combination of σv and σt. A two-dimensional map visualizing the relative error σε,rεr-1100% appears in Fig. . The contour lines show that uncertainties grow very large for dissipation rates smaller than 10-3 m2 s-3. Uncertainties are also large for values in excess of 10-1 m2 s-3 if σv2 is small at the same time. The input uncertainty σσ,v depends on the CNR and is assumed to be on the order of the corresponding instrumental noise σe. The values in Fig.  are calculated with input uncertainties σσ,v=σσ,t=0.05 m s-1, which correspond to a CNR value of approximately -12 dB, which is common for the signal strength during the Perdigão campaign for the DLR lidars.