# Departure from Flux-Gradient Relation in the Planetary Boundary Layer
## 3 Results
### Momentum Fluxes and Lidar's Turbulence Attenuation  
For the analyzed westerly winds, both the observed and NEWA-WRF vertical profiles of momentum flux (Figures 3b and 4b) show \(\langle v^{\prime}w^{\prime}\rangle\) values close to zero and \(\langle u^{\prime}w^{\prime}\rangle\) values decreasing in magnitude almost linearly with height. However, at both sites, there is an important mismatch between the lidar observations and NEWA-WRF for \(\langle u^{\prime}w^{\prime}\rangle\). This could be caused either by the computations involving Equation (6) or turbulence attenuation due to the lidar's spatial averaging (or both). For the analyzed period, the observed and NEWA-WRF vertical profiles of velocity gradients are rather similar (not shown). Therefore, if the bias comes mainly from the NEWA-WRF momentum fluxes, then our estimate of \(K_{m}\) using Equation (6) could be flawed. However, we know that the lidar-derived momentum fluxes have an inherent bias due to spatial averaging along the lidar beam [4]. To gain insights into the origin of this bias, we estimate the lidar's turbulence attenuation under neutral conditions using the turbulence spectral model of [PERSON] [23] (hereafter, the [PERSON] model), with a turbulent eddy-lifetime parameter \(\Gamma=3.9\), as recommended by  
Figure 5: Mean profiles of wind turning bias at FINO3 (**a**) and Hamburg (**b**), given by the difference between the lidar and NEWA-WRF wind directions (\(\Delta\theta\)), under unstable (blue) and stable (green) conditions. The mean PBL height under stable conditions, based on NEWA-WRF, is plotted as a horizontal green line. the IEC standard [24]. \(\Gamma\) describes the degree of anisotropy of turbulence; \(\Gamma=0\) corresponds to fully isotropic turbulence. Figure 6 shows the ratio of the filtered to the unfiltered momentum flux \(\langle u^{\prime}w^{\prime}\rangle_{f}/\langle u^{\prime}w^{\prime}\rangle_{u}\) as a function of the ratio of the lidar's probe volume (i.e., the FWHM) to the turbulence length scale (\(L_{MM}\)). \(L_{MM}\) is another parameter of the [PERSON] model. The black full line in Figure 6 shows the prediction of the ratio \(\langle u^{\prime}w^{\prime}\rangle_{f}/\langle u^{\prime}w^{\prime}\rangle_{u}\) using the [PERSON] model. For a detailed description of the calculations, refer to [PERSON] et al. [4]. For the WLS70 used in this study, \(\text{FWHM}=75\) m, so we only need to know the value of \(L_{MM}\) to compute the degree of filtering. We can estimate the turbulence length scale using the approximation by [PERSON] [25],  
\[L_{MM}\approx\frac{\sigma_{U_{h}}}{\partial U_{h}/\partial z}, \tag{7}\]  
where \(\sigma_{U_{h}}\) is the standard deviation of the horizontal velocity and \(\partial U_{h}/\partial z\) is the horizontal wind shear, which is computed here using the polynomial of \(U_{h}\) suggested by [PERSON] [26]. We account for the observations at all vertical levels of the masts. The length scale at 100 m is computed as a linear interpolation between the closest measurements at each mast. The dashed lines in Figure 6 show that the turbulence length scale estimated at FINO3 is larger than that at Hamburg, which results in a larger attenuation of the momentum flux at Hamburg (74.4%) compared to that at FINO3 (83.4%). As turbulence length scales depend on atmospheric stability [27], we expect that the dominant unstable conditions at FINO3 result in larger turbulence length scales compared to those at Hamburg where neutral conditions are more often observed. By compensating the lidar-observed \(\langle u^{\prime}w^{\prime}\rangle\) values at 100 m with the results in Figure 6, we find a better agreement between the lidar- and NEWA-WRF derived values. However, its worth noticing that the NEWA-WRF momentum fluxes are parameterized from Equation (6) so the simulated momentum fluxes do not necessarily have the same magnitude as the observations.