Figure  presents the integral length scales of the velocity components as a function of height. The IEC 61400-1 values (Eq. ) are compared with estimates obtained from a least-square fit of the Kaimal spectrum (Eq. ) to measured spectra under neutral conditions, as well as from an exponential fit to the auto-correlation function according to Eqs. ()–().

At heights z<60m, the length scales estimated using both methods generally fall below the IEC reference values. The only exception is the vertical length scale obtained from the fitted Kaimal spectrum, which slightly exceeds the standard value. At z>100m, the Kaimal-based estimates are consistently larger than the IEC reference. In contrast, the length scales derived from the auto-correlation functions tend to underestimate the horizontal components while providing closer agreement for Lw.

The IEC 61400-1 standard proposes linearly increasing length scales up to a level of 60m, remaining constant above this height. The estimates obtained from the auto-correlation functions mostly support this trend, showing only minor increases in Li from 55 to 125m. The length scales derived by fitting the Kaimal spectrum, however, appear to increase linearly up to a height of 110m and even stronger at the highest level, which is based on SRWS measurements. A similar overestimation of lidar-based length scales was reported by for the horizontal velocity components obtained from dual-lidar measurements and partially attributed to spatial averaging effects.

The dependence of the integral length scales on atmospheric stability and wind direction is illustrated in Fig. . Under neutral conditions, the integral length scales are within the ranges of 100 to 250m for Lu, 30 to 100m for Lv, and 10 to 50m for Lw. From neutral to slightly unstable conditions, the length scales increase, consistent with enhanced turbulence generation due to buoyancy. Unexpectedly, this trend does not persist in the highly convective regime, which may be related to limitations of Taylor's hypothesis of frozen turbulence. The estimation of integral length scales from single-point temporal auto-correlation functions assumes that turbulent structures are advected past the measurement location without undergoing significant evolution. Under unstable stratification, buoyancy-driven turbulence may accelerate the temporal evolution of turbulent eddies, causing the measured auto-correlation functions to decay more rapidly. Consequently, the slightly smaller integral length scales estimated under highly unstable conditions may not necessarily reflect smaller turbulent structures but rather indicate departures from Taylor’s hypothesis of frozen turbulence. From neutral to slightly stable conditions, all length scales decrease substantially as buoyancy transitions from enhancing to suppressing turbulence. Under highly stable conditions, however, a slight increase in Lu and Lv is observed, which can be attributed to large-scale mesoscale motions increasingly captured within the 30 min averaging period. This effect is limited to the horizontal components, supporting the common assumption of quasi-two-dimensional mesoscale structures.

Figure  shows the ratio of the along-wind to the vertical length scales. Under unstable conditions, the ratio remains relatively constant between values of 2 and 6 but increases approximately by a factor of 2 towards neutral and even further under stable stratification. This trend reflects the deformation of turbulent eddies due to negative buoyancy and reduced vertical mixing under stable conditions, with the effect being most pronounced at lower measurement heights.

The variation in length scales with wind direction (Fig. d–f) is less systematic and generally weaker than the dependence on atmospheric stability. Nevertheless, a consistent increase in Li is observed for wind directions ϕ‾>180°, corresponding to flow over both flat terrain and from the estuary. Smaller length scales occur for ϕ‾<180°, likely due to enhanced small-scale turbulence generated by the upstream built area.

The usable frequency range of the spectral estimates is assessed in Fig. . Figure a shows the ensemble-averaged horizontal wind speed spectra from the sonic and cup anemometers under neutral conditions. Residual high-frequency noise is present even after despiking the time series. When normalizing the frequencies by zU-1, this noise may bias the estimates closer to the turbulence peak. To avoid this, the spectra are truncated prior to normalization and ensemble averaging. For the cup anemometers, a cut-off frequency of 0.3Hz, corresponding to 0.6fN, is applied. The sonic spectra are truncated at 2Hz for z={55,110}m and at 6Hz for the instrument at z=25m, which is from a different manufacturer.

A high level of agreement between sonic and cup spectra is observed up to approximately 0.07Hz. At higher frequencies within the inertial subrange, the sonic spectra continue to follow the expected fS∝f-2/3 scaling, whereas the cup spectra exhibit pronounced high-frequency attenuation. This behaviour is attributed to spatial averaging associated with the cup geometry and temporal filtering due to the rotor inertia.

The normalized horizontal wind speed spectra from the sonic and cup anemometers are shown in Fig.  for five stability classes, plotted as a function of the reduced frequency. The number of samples N is indicated for each case in the legend. Under unstable conditions, the combined influence of shear- and buoyancy-driven turbulence results in a pronounced spectral plateau. As stability approaches neutral conditions, the plateau narrows due to the diminishing buoyancy contribution. Under stable stratification, the turbulence peak shifts towards higher frequencies as negative buoyancy suppresses the micro-scale turbulence and reduces the size of the coherent structures. Simultaneously, low-frequency contributions emerge, likely related to mesoscale motions.

Aside from the high-frequency attenuation, the spectra from sonic and cup anemometers show good agreement at corresponding heights. The largest discrepancies occur within the spectral gap under highly stable conditions, where the sonic spectra indicate slightly higher spectral energy.

Figure b and c compare along-wind and vertical spectra under neutral conditions obtained from the sonic anemometer at a height of 110m and the SRWS at the bow-tie centre (z=125m). The raw lidar time series were found to be relatively noisy, leading to artificial spectral energy above approximately 0.06Hz. After despiking, the usable frequency range extends to about 0.1Hz. Additional smoothing is achieved by extracting the median of five neighbouring values within a grid cell rather than a single sample, which, at the bow-tie centre, corresponds to a trajectory length of approximately 4.7m.

Similar to the cup anemometer results, the SRWS spectra display high-frequency attenuation, primarily due to the lidars' finite probe volume, starting around 0.04Hz for both velocity components shown in Fig. . While the along-wind spectrum is only affected in the inertial subrange, the spectral peak of the vertical component cannot entirely be resolved by the SRWS.

The resulting velocity spectra are presented in Fig.  for the three sonic anemometers and the SRWS recordings in the bow-tie centre. Under unstable conditions, the along- and cross-wind spectra exhibit a distinct spectral plateau. The plateau is more pronounced for the cross-wind component and, unlike for the along-wind velocity, does not persist under neutral conditions. Instead, a spectral gap becomes apparent in the neutral fSv estimate, consistent with offshore observations by and . In further agreement with , the spectral gap is located at reduced frequencies of approximately 5×10-3 to 2×10-2 under slightly stable conditions and 2×10-2 to 5×10-2 under highly stable conditions. The spectral gap reaches deeper for the cross-wind spectra than for the along-wind component. Its depth and position vary with height, with higher elevations exhibiting a shallower gap shifted towards higher frequencies, consistent with observations by at the coastal Høvsøre site.

For the vertical velocity component, the spectral plateau is less distinct but still observable under convective conditions. Under neutral conditions, the w spectra follow the basic fSi(n)∝f1 and fSi(n)∝f-2/3 scaling, as neither a plateau nor a low-frequency contribution is present. For the 30 min averaging period considered here, a clearly defined spectral gap is not observed under stable conditions, although indications are present at reduced frequencies below 2×10-2, consistent with and .

The SRWS spectra appear less smooth due to the smaller sample size. However, they seem to capture the main spectral features, including the plateau, the turbulence peak, and the spectral gap, for most cases. Furthermore, the depth of the spectral gap follows the height-dependent trend observed across the sonic measurements. For the along-wind component, deviations from the sonic results are limited to the inertial subrange, where attenuation in the SRWS spectra leads to an underestimation of spectral energy. The spatial averaging effect is more severe for the v and w components, where both the magnitude and position of the turbulence peak are distorted. In the cross-wind spectra, this is the case under stable conditions only, whereas the vertical turbulence peaks are affected across all stability regimes.

In addition to the measurements, Fig.  includes least-square fits of the empirical turbulence models (Eqs. –) together with the IEC parametrization of the Kaimal spectrum for neutral conditions. For the horizontal components, the Kaimal spectrum matches the peak and inertial subrange but underestimates the low-frequency energy due to the presence of the spectral plateau in u and the mesoscale contributions in v. For the vertical component, the overall spectral shape is captured reasonably well, although the peak frequency is shifted by approximately n≈0.5. The peak estimated around a reduced frequency of around 5×10-1 is, however, consistent with the observations of .

The pointed–blunt model (Eq. ) provides a good representation of the spectral plateau through a double-peak, resulting in accurate fits for unstable and neutral conditions. The pointed model with mesoscale contribution (Eq. ) used under stable conditions captures the spectral gap well, except in Fig. j, where the particularly narrow gap at 25 and 55m cannot be reproduced accurately. The coefficients derived from fitting the turbulence models to the observations are provided in Appendix  as functions of atmospheric stability.

The low-frequency contribution captured within the 30 min averaging window is generally well approximated by the Si(n)∝f-5/3 scaling. Under highly stable conditions, the u and v spectra suggest the presence of a secondary peak at frequencies below the spectral gap, which is not represented by the turbulence model. Similar observations were reported by and near the Brunt–Väisälä frequency and attributed to wave-like motions in stable stratification. A more detailed investigation of this feature would require longer averaging periods to resolve lower-frequency contributions.

The auto-spectral estimates presented in Figs. – are ensemble averages over all wind directions that are not affected by turbine or mast wake interference. Prior to ensemble averaging, each spectrum is normalized by its corresponding variance to reduce the influence of directional differences in turbulence intensity. However, as shown in Fig. d–f, the integral length scales vary with wind direction, indicating that the spectral shape is also expected to differ based on the upstream terrain. This effect is examined in Fig. , which presents normalized spectra under neutral conditions for four wind-direction sectors with contrasting terrain characteristics. As the upstream roughness decreases from Fig. a (residential area) to Fig. d (river estuary), the low-frequency energy increases, causing the along-wind turbulence peak around n=0.1 to gradually broaden into a spectral plateau. This behaviour is consistent with the corresponding increase in the integral length scales discussed previously. A similar increase in low-frequency energy is observed in the cross-wind spectra, whereas the vertical velocity spectra are comparatively insensitive to the upstream terrain. Overall, the sonic anemometer and SRWS spectra exhibit similar trends across the different sectors. Some estimates, however, such as the sonic spectrum at z=110m in Fig. c, show increased uncertainty due to the smaller number of available samples.

Vertical coherence of the horizontal wind speed is evaluated along the meteorological mast using both sonic and cup anemometers. The ensemble-averaged results, shown in Fig. , are expressed as a function of the normalized wavenumber kdz. The coherence estimates are consistent across all separations and stability classes, indicating that they are largely unaffected by attenuation in the inertial subrange of the cup anemometer spectra. This can partially be explained by the fact that attenuation occurs at f>0.07Hz, whereas the coherence of the horizontal wind speed decays at normalized wavenumbers below 1.5. Contributions from frequencies above 0.07Hz can only be reflected at such low values of kdz at relatively high mean wind speeds of u¯>8.8ms-1 for dz=30m, u¯>17.9ms-1 for dz=61m, and u¯>26.7ms-1 for dz=91m. In addition, spatial averaging effects may partially cancel when the cross-spectrum between two points is normalized by the corresponding auto-spectra, as discussed by for coherence estimates derived from dual-lidar measurements.

In the present study, a direct comparison between sonic- and lidar-based coherence estimates is challenging due to differences in measurement height. For instance, the sonic estimate for dz=30m corresponds to an average height of z¯=40m, whereas the lowest SRWS estimate is available at z¯=70m. The closest correspondence is found between the sonic separation of dz=55m at z¯=82.5m and the SRWS estimate between the first to seventh grid cell along the vertical scan line (Fig. b), yielding dz=60m at z¯=85m.

The corresponding coherence estimates are compared in Fig. . The horizontal velocity components exhibit a high agreement across all stability classes. Significant deviations are observed only for the along-wind component under highly unstable conditions (Fig. a), where the limited number of samples increases the uncertainty in the SRWS estimate.

For the vertical velocity component, however, the SRWS predicts systematically lower coherence, approximately by a factor of 2 compared to the sonic estimates. Part of this discrepancy may be attributed to the slight difference in separation distance (5m), as γwz is typically more sensitive to dz than the horizontal components. The remaining difference may reflect either an underestimation by the SRWS or an overestimation by the sonic measurements. The sonic-based coherence aligns well with results reported by for dz=40m at the FINO1 offshore platform. This agreement may suggest that the sonic anemometers' γwz in Fig.  is overestimated, as a 60m separation onshore would be expected to exhibit lower coherence than a 40m spacing offshore. As reported by , a reduction in the lidar-based coherence estimates due to spectral attenuation related to volume-averaging is unlikely because the cross-spectra are normalized by the correspondingly attenuated auto-spectra. Any residual influence is expected to be negligible for normalized wavenumbers below 1, as the attenuation primarily affects frequencies above 0.04Hz. For a separation of 60m, these frequencies correspond to kdz<1 only for mean wind speeds exceeding 15ms-1, which are rarely encountered in the present dataset.

The velocity reconstruction and coherence estimation based on the SRWS scans may, however, introduce additional decorrelation effects that are specific to the w component. Due to its greater spatial variability within the finite measurement volumes, the vertical velocity is more sensitive than the horizontal components to departures from the assumptions underlying the velocity reconstruction. Furthermore, the correction for the time lag between the vertically separated extraction points relies on Taylor's hypothesis of frozen turbulence and is thus expected to be less accurate for the vertical velocity component because of the more rapid temporal evolution of vertical motions. Both effects may contribute to reducing the correlation between the SRWS-derived vertical velocity time series at different heights. A conclusive assessment would require additional γwz estimates at comparable heights and separations.

Despite the differences between instruments, the coherence of the vertical turbulence exhibits consistent trends with atmospheric stability. From unstable to stable conditions, coherence decreases, reflecting the reduction in turbulent length scales. For the horizontal components, this is expressed through a steeper decay and a moderate reduction in coherence at k=0. For the vertical component, the reduction is more pronounced, leading to substantially lower intercepts with the y axis under stable conditions.

Figure  presents the SRWS-based coherence estimates for separations between 10 and 160m along the vertical line of the scan. To obtain meaningful averages as a function of stability and separation distance, the ensembles are restricted to estimates obtained at comparable measurement heights of 110m≤z¯≤140m. The fitted curves correspond to Eq. (). The model provides an accurate representation of the observations while allowing for γiz(k=0)<1.

According to the Davenport model (Eq. ), coherence scales with fdzu¯-1, implying that all estimates should collapse into a single curve when plotted against kdz. This is not observed, suggesting a stronger dependency on dz. Additionally, coherence is affected by the measurement height, which is not accounted for by the Davenport model. The height dependence of the vertical coherence is examined in Fig.  for a fixed separation distance of dz=60m and neutral conditions. A reduction in coherence with decreasing height is observed for all wind components while being most pronounced for the vertical velocity. The coherence of the along-wind turbulence appears to be least sensitive to z¯ and exhibits a noticeable dependence only at normalized wavenumbers kdz>0.25.

To account for the combined influence of measurement height and separation distance, the model coefficients A and C in Eq. () are fitted for a wide range of combinations across the bow-tie pattern. The resulting coefficients are shown in Fig.  as a function of the normalized separation dzz¯-1. For small values of dzz¯-1, the scaling coefficient A approaches 1, corresponding to identity coherence at k=0. With increasing separation or decreasing height, A decreases approximately exponentially. The exponential formulation in Eq. () provides an accurate fit to Aw. Greater scatter is observed for the horizontal components, particularly for dzz¯-1>0.5.

established a linear relationship between the decay coefficient C and dzz¯-1 based on propeller anemometer measurements within 20m over open rural terrain. In the present results, a better representation is achieved using the two-parameter exponential function given in Eq. ().

Combining the derived parameterizations of A and C with the basic coherence model yields the three-parameter model given in Eq. (). The corresponding model coefficients are summarized in Appendix  as a function of atmospheric stability.

The lateral coherence derived from the SRWS scans is shown in Fig. . The reduction in turbulent length scales from unstable to stable conditions is reflected in the lateral coherence. However, the associated reduction in coherence is less pronounced than for vertical separations. This behaviour is consistent with the integral length scales (Fig. ), which indicate that turbulent structures are more highly compressed in the vertical than in the lateral direction. As a result, laterally separated points are less sensitive to stability effects than vertically separated ones. Unlike for the vertical direction, the lateral coherence of the cross-wind component γvy appears to scale relatively well with the normalized wavenumber kdy.

To obtain meaningful ensemble averages, the lateral coherence estimates shown in Fig.  are restricted to a measurement height of 125m, corresponding to estimates obtained along the horizontal segment of the bow-tie pattern. For lateral separations between 10 and 90m, one additional estimate is available both above and below this height from the diagonal segments connecting to the vertical line, as illustrated in Fig. . For dy=60m, these estimates are located at z¯=70m and z¯=192m and are compared to the z¯=125m estimate in Fig. . Although only three measurement heights are available, and the estimates at the lowest and highest elevations are associated with relatively large uncertainty, a reduction in lateral coherence with decreasing measurement height is evident. Consistent with the vertical coherence results (Fig. ), the height sensitivity is most pronounced for γwy. At z¯=70m, the vertical velocity is quasi-incoherent across the dy=60m separation.

A direct comparison between vertical and lateral coherence is provided in Fig.  in terms of the ratios of the fitted scaling and decay coefficients A and C. To ensure comparability, only estimates at similar heights within the range of z¯=120 to 130m are included. The analysis is further restricted to separations up to 40m, as the representative height z¯ becomes less meaningful for larger vertical spacings.

While no clear trend is observed for Ciz/Ciy, the ratio of the scaling coefficient A decreases slightly under stable conditions, suggesting that the vertical coherence reduces more rapidly than the lateral coherence. This observation is consistent with the flattening of turbulent structures caused by the suppression of vertical mixing under stable stratification.

The combination of slightly smaller decay coefficients and larger scaling coefficients for lateral separations indicates that lateral coherence generally exceeds vertical coherence.