## 4 Wind Field Characteristics
### Coherence Function  
Spatial correlation is one of the basic characteristics of fluctuating wind. It is very important when the effect of fluctuating wind on structures is considered. Coherence function is often used to describe the correlation degree in wind engineering, and the formula is:  
\[\text{coh}(z_{1},z_{2},n)=\frac{|S_{z_{1}z_{2}}(n)|}{\sqrt{S_{z_{1}}(n)S_{z_{2 }}(n)}} \tag{11}\]  
Figure 14: Comparison of PSD of reduced turbulent fluctuation of the thunderstorm at 16 m (**a**) and 64 m (**b**) and empirical models. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**h (m)** & **14:20** & **14:25** & **14:27** & **14:28** & **14:29** & **14:30** & **14:35** \\ \hline
8 & 1.22 & 1.  
where \(z_{1}\) and \(z_{2}\) are two values of height, \(n\) is the frequency, \(S_{z_{1}z_{2}}(n)\) is the co-spectral density function between \(z_{1}\) and \(z_{2}\), and \(S_{z_{1}}(n)\) and \(S_{z_{2}}(n)\) are the spectral density function at \(z_{1}\) and \(z_{2}\). As noted earlier, the thunderstorm event was observed by nine ultrasonic anemometers located at different heights in the same location. Figure 15 shows the coherence function curves of the reduced turbulent fluctuation, detected by different pairs of anemometers installed at the 325 m meteorological tower in Beijing. The three higher anemometers (140 m, 200 m, and 280 m) are not taken into account, because they are far away from the lower anemometers. As shown in the figure, in general, the coherence between two wind speed records is related to the distances between different anemometers. The coherence related to shorter distances is larger than that of longer distances. However, the relationship between distance and correlation does not show a monotonic trend, which confirms that the distance between two points is not the only influencing factor. The coherence decreases exponentially with the increase of frequency, and this is similar to the feature of synoptic winds. At present, the coherence function of thunderstorm wind is mainly based on the estimation of the wind-speed time history of low-altitude thunderstorm wind. Because of this, it is difficult to analyze the response of high-rise structures, for instance, ultra-high- voltage transmission towers and so on, under thunderstorm wind. Therefore, the establishment of the empirical coherence function model of thunderstorm wind with respect to height is necessary. The spacing, frequency, and average wind speed are important influencing factors of the coherence function of the longitudinal fluctuating wind with respect to height. For simplicity, the following empirical exponential function model is usually adopted in wind engineering, which can be used to describe the coherence function between wind speeds at two heights [33]:  
\[\text{coh}(z_{1},z_{2},n)=exp(-K(z_{1},z_{2})n|z_{1}-z_{2}|) \tag{12}\]  
where \(K\) is a parameter changing with \(z_{1}\) and \(z_{2}\). \(K(z_{1},z_{2})\) is associated with an average wind speed, and it is defined as follows:  
\[K(z_{1},z_{2})=2k(z_{1},z_{2})/[\overline{u}(z_{1})+\overline{u}(z_{2})] \tag{13}\]  
where \(\overline{u}(z_{1})\) and \(\overline{u}(z_{2})\) are the maximum values of slowly-varying mean wind speed at \(z_{1}\) and \(z_{2}\), while \(k(z_{1},z_{2})\) is a parameter depending on \(\Delta z=|z_{1}-z_{2}|\). According to the least squares method, \(K\) is related to different separation distances and this can be obtained as shown in Figure 16. [PERSON] et al. [33] gave the empirical formula:  
\[k(\Delta z)=-k_{1}\times\ln(\Delta z)+k_{0} \tag{14}\]  
Figure 15: Coherence function with frequency obtained by the field measurement. In this thunderstorm event, it is estimated \(k_{1}=5.20\) and \(k_{0}=26.65\) in the empirical Equation (14). Figure 16 shows the coherence function fitting result (black curve) with a slight deviation based on the model.