# Departure from Flux-Gradient Relation in the Planetary Boundary Layer
## 4 Discussion and Conclusions  
We extend the work from [PERSON] et al. [11] and present novel lidar observations that show a clear misalignment between the vertical gradient of the mean wind vector and the stress vector, \(\beta\), up to 500 m for both offshore and onshore conditions. The measurements were performed with a long-range Doppler wind lidar on the FINO3 offshore platform and close to the Hamburg weather mast. Such a misalignment is assumed to be zero by the PBL schemes that apply a flux-gradient relation and that are normally utilized within current numerical weather models and for wind resource assessment, e.g., the WRF model. The observed \(\langle\beta\rangle\)-values increase from about \(-5^{\circ}\) at 100 m to about \(-20^{\circ}\) at 500 m at both offshore and onshore sites. During stably stratified conditions, where larger vertical wind gradients are present, \(\beta\) is still negative but above \(-10^{\circ}\) and approaches zero at the PBL top. The \(\beta\) value has a larger magnitude for unstable compared to stable conditions at all heights for both sites, and this magnitude increases with height onshore. Hence, the basic assumption of the flux-gradient relation, which we utilize to derive the eddy fluxes from the NEWA-WRF outputs holds better under stable compared to unstable atmospheric conditions, as expected, due to the larger coherent structures in the latter. Our results show a bias between the \(\langle u^{\prime}w^{\prime}\rangle\) vertical profiles derived from radial velocity measurements of a pulsed profiling lidar with those derived from the NEWA-WRF outputs. We assess the filtering effect inherent to the lidar's probe volume averaging by combining the Mann model and estimations of the length scale for both sites. The angle between the stress vector relative to the mean wind vector (i.e., \(\alpha\)) derived from the NEWA-WRF outputs agrees well with the lidar-derived values, both offshore and onshore. Figure 8: Vertical profiles of turning angles computed from the idealized neutral WRF-LES over (**a**) water and (**b**) land. This agreement is likely caused by a balance between the wind direction and momentum fluxes' biases. We observe an underestimation of the wind-turning by the NEWA-WRF output at the offshore site and an opposite turning at the onshore site. We speculate that this is mainly due to a wrong surface roughness assignment (or wrong land-use description) at around Hamburg tower in NEWA-WRF. It is worth noting that the heterogeneous condition at the onshore site is far from the PBL scheme assumptions used to derive the flux-gradient relation. This challenge can be partly overcome by the use of three-dimensional PBL schemes [31]. The wind direction bias is larger (in magnitude) under stable atmospheric conditions for both sites. At the offshore site, the wind direction bias is small (\(\approx\)1.5\({}^{\circ}\)) at 100 m and increases with height, whereas the bias decreases (in magnitude) with height at the onshore site. [PERSON] et al. [2] argued that such wind turning biases in mesoscale models are the product of artificial enhancement of the mixing by turbulence closures, which, in turn, modifies the Ekman pumping and has a direct impact on, e.g., the ability to simulate cloud formation mechanisms. The idealized WRF-LESs, both onshore and offshore, also show \(\beta\) values close to 0\({}^{\circ}\) within the bulk of the PBL, further demonstrating first, the need to investigate the conditions at which this misalignment occurs and, second, the need to better represent the PBL with numerical models. Note that the simulations of [PERSON] et al. [7] and [PERSON] and [PERSON] [9] used a different LES framework than our WRF-LES. A real-time WRF-LES, like that performed by [PERSON] et al. [32], might be useful for further investigating, e.g., whether or not forcing conditions affect the behavior of \(\beta\).