## 3 Results
### Case Study of the Spectral Signature of a Turbine Wake  
In order to look at frequency signatures of the above averaged time series, we calculated spectral energy densities of the three wind components as well as for the raw TKE estimate. The LiDAR spectra are derived from processed DBS raw data, as described in Section 2. By comparing LiDAR spectra to spectra from sonic anemometers, we found that Windcube v1 measurements can qualitatively represent turbulence spectra of the atmospheric boundary layer up to the frequency of 0.125 Hz. Spectral energy densities of TKE are well represented in free-stream and wake conditions, while the agreement in the spectra of the horizontal wind components is dependent on the wind direction and beam locations; spectra of the vertical component are underestimated (Figures 2 and 3). To highlight the difference between upstream and downstream spectra, we analyzed a four-hour period from 19:00 to 23:00 UTC on 4 March 2014. During the selected period, south-westerly winds from around 230\({}^{\circ}\) dominated with an average horizontal wind speed of around 5 m\(\cdot\)s\({}^{-1}\) at hub height. The measured wind speeds are below the rated wind speed of the Nordex research turbines, meaning that the blades are blocking most of the flow without an increased pitch angle. Typically, for winter night-time conditions, the atmosphere was stably stratified, prohibiting vertical mixing and allowing longer life times of disturbances. Under these conditions, the Windcube wls67 and the met-mast are upstream of the research turbine WT6, and the Windcubes wls37 and wls65 are located downstream inside the wake. Figure 8: Energy spectra of four hours on 4 March 2014 in free stream (blue and black) and wake conditions (green and red) of the along-wind (**a**), cross-wind (**b**) and vertical component (**c**) u, v, w and TKE (**d**). Black colors show data from the sonic anemometer and blue; red and green colors show Windcube wls67, Windcubes wls37 and wls65 data at 80 m (hub height). The dashed line indicates the \(-2/3\) Kolmogorov slope. When comparing the turbulence spectra of the free-stream to the wake conditions, a clear maximum at around 0.02 Hz becomes visible in the spectral energy densities of the along-wind component u at hub height (Figure 8). The maximum in u goes along with an increased level of turbulence in the other wind components, as well as in the TKE estimate. Although the vertical component is not accurately represented in the four-beam Windcube v1 data, the relative differences between the two Windcube instruments can be seen as a qualitative indication of variances in the vertical wind speed. These differences in vertical wind speed fluctuations seem also to influence the variance in TKE. The level of variances in TKE is decreasing for the wls65, located further downstream. Above blade tip height (at 160 m), the distinct maximum in the spectra disappears, but the level of variance is still elevated. In contrast to the observed wind spectra (Figure 9a-c), the spectra of TKE (Figure 9d), which represents the sum of squared variances of the three wind components, show a still visible wake signature. At the presented measurement height, the spectral energy density of TKE is highest at 3.2 D downstream. Due to the measurement height above the blade tip and the horizontal wake expansion, the LiDAR beams of the Windcube located closer to the turbine are outside of the rotor swept area, while those of the Windcube further downstream are inside the expanded area. Therefore, the variance of TKE is highest further downstream. This effect is obliterated for the averaged profiles presented above, as the rotor swept area moves in and out of the LiDAR beam geometry for the selected wind direction sector. Figure 9: Energy spectra of four hours on 4 March 2014 in free stream (blue and black) and wake conditions (green and red) of the along-wind (**a**), cross-wind (**b**) and vertical component (**c**) u, v, w and TKE (**d**).