# Unfrozen Skewed Turbulence for Wind Loading on Structures
## 4 Results
### Coherence Estimates  
The co-coherence and quad-coherence estimates obtained using the record collected from 2:00 a.m. to 5:00 a.m. on 8 December 2017 are displayed in Figure 5. They are superposed to the fitted functions \(\gamma_{ii}\) (Equation (32)) and \(\rho_{ii}\) (Equation (33)). The vertical root-coherence was studied using the sensors located on the same side of the mast and their horizontal separation was thus zero. Similarly, the horizontal root-coherence was studied using only the sensors at the top of the mast, which were both at 49 m above ground. It should be noted that only frequencies up to 1 Hz were considered for the least-square fit of Equations (32) and (33) since the co-coherence is close to zero at higher frequencies. Table 4 displays the decay coefficients obtained by least-square fit of Equation (32). For lateral separations, the coefficients are slightly smaller than observed by [PERSON] and [PERSON] [9] on the Sotra bridge, located north of the Bjornafjord. The parameter \(c_{yl}^{w}\) is, however, relatively close to the value estimated in the Lysefjord [36]. For vertical separations, the decay coefficients are similar to those estimated above the sea [83], suggesting that the flow incorporates combined characteristics from both offshore and hilly environments. Overall, the identified decay coefficients are within the range of values found by  
Figure 4: Estimated (scatter) and fitted (solid lines) one-point auto- and cross-power spectral estimates at \(z=49\) m above ground from 2:00 a.m. to 5:00 a.m. on 8 December 2017. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Coefficients** & \(a_{u}\) & \(a_{v}\) & \(a_{w}\) & \(a_{uvv}\) & \(b_{uw}\) \\ \hline Values & 118 & 24 & 3.6 & 12 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Spectral parameters from Equations (1)–(3) and (7) estimated using wind data recorded from 2:00 a.m. to 5:00 a.m. on 8 December 2017 on MW1. [PERSON] et al. [84] from multiple masts in three other Norwegian fjords. The along-wind decay coefficients \(c_{x1}^{j}\) (\(j=u,v,w\)) are equal to 1, which is also within the range of values derived from the data recorded in the Bjernafjord with long-range Doppler Wind lidar instruments [22]. A value \(c_{x1}^{j}\leq 1\) suggests that [PERSON]'s hypothesis of frozen turbulence is suitable, at least for the largest eddies. In reality, as soon as \(c_{x1}^{j}>0\), there exists a range of eddies that cannot be considered frozen any longer and this range increases with the along-wind separation. Finally, it should be noted that the possible dependency of the Davenport decay coefficients on the separation distance (e.g., [58; 59]), cannot be explored in the present study because the separation distances are less than 20 m.  
Figure 5 shows that the quad-coherence of the \(u\) and \(w\) component is nearly zero for vertical separations, contrary to the \(v\) component. This result supports thus the recommendation provided by ESDU [19]. The solid line used to approximate \(\rho_{vv}(f,d_{z})\) is computed using the eddy slope value described in Equation (30), which shows that the low-frequency range is only partly portrayed by the empirical model described in [PERSON] et al. [59]. On the other hand, the eddy slope has no visible influence on \(\gamma_{vv}(f,d_{z})\). A more in-depth investigation is required to establish the relevance of the eddy slope when modelling the dynamic wind loading. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Component \(i\)** & \(c_{x1}^{j}\) & \(c_{y1}^{j}\) & \(c_{y2}^{j}\) & \(c_{z1}^{j}\) & \(c_{z2}^{j}\) \\ \hline \(u\) & 1 & 8 & 0.01 & 11 & 0.03 \\ \(v\) & 1 & 4 & 0.01 & 9 & 0.30 \\ \(w\) & 1 & 5 & 0.36 & 4 & 0.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Decay coefficients estimated on 8 December 2017 from 2:00 to 5:00 on MW1. These coefficients are estimated by least-square fit of Equation (13).