# Departure from Flux-Gradient Relation in the Planetary Boundary Layer
## 3 Results
### Momentum Fluxes and Lidar's Turbulence Attenuation
Furthermore, the attenuation estimated in Figure 6 is subject to uncertainties from the spectral turbulence model, the FWHM value, and the \(L_{MM}\) estimation (refer to Appendix A for the vertical profiles of estimated \(L_{MM}\) at each site). Figure 6: Ratio of the filtered to unfiltered momentum flux as a function of the ratio of the lidar’s probe length to the turbulence length scale (\(L_{MM}\)). The dotted lines represent estimated values for onshore and offshore sites at 100 m.  
The Angles of the Stress Vector with Mean Wind Vector (\(\alpha\)) and with Mean Wind Shear Vector (\(\beta\))  
Figures 3c and 4c show the vertical profiles of \(\alpha\) and \(\beta\) for offshore and onshore conditions, respectively. As expected from the LESs of [PERSON] et al. [7], \(\langle\alpha\rangle\) is close to zero near the surface and increases with height. Furthermore, the results using the NEWA-WRF output agree well with those of the lidar at both sites. However, note that a good agreement on \(\langle\alpha\rangle\) could occur due to a combination of biases in both wind direction and momentum fluxes. The most noticeable discrepancy between the lidar measurements and the NEWA-WRF output is on \(\langle\beta\rangle\), since we use flux-gradient relation to derive the eddy fluxes with the NEWA-WRF output. This theory assumes this misalignment is zero. The misalignment \(\langle\beta\rangle\) for all heights is on average close to \(-18^{\circ}\) offshore and \(-12^{\circ}\) onshore. If we reduce the wind sector to \(\pm 30^{\circ}\), the mean \(\langle\beta\rangle\) for all heights is \(-17^{\circ}\) offshore and \(-9^{\circ}\) onshore, i.e., the result \(\langle\beta\rangle<0\) is not sensitive to the wind sector at either of the sites. Here, the non-linear behavior of \(\langle\beta\rangle\) with height, with a kink at 250 m, is attributed to faulty lidar measurements during both campaigns, as pointed out in Section 2.1. The \(\beta\) profiles and, hence, the validity of the flux-gradient relation are sensitive to the vertical gradients of the mean wind. Therefore, we expect that PBL schemes using the assumption of a flux-gradient relation (such as MYNN2 and other local PBL schemes) show a better performance when the vertical mean wind gradients are more significant, i.e., under stable conditions [28]. Figure 7 shows the vertical profiles of \(\beta\) as a function of atmospheric stability for both sites. To avoid the spikes seen in Figures 3 and 4, the value at the third measurement height is the result of a linear interpolation. The vertical profiles of \(\langle\beta\rangle\), derived from the lidar observations, show that the misalignment can be as small as \(\approx\)\(-7^{\circ}\) on average for offshore conditions and close to \(0^{\circ}\) onshore, under stable conditions. Additionally, under stable conditions \(\langle\beta\rangle\) approaches zero close to the PBL top. Although fluctuating, the \(\langle\beta\rangle\) values under unstable conditions are always larger (in module) than those under stable conditions at both sites, as expected.