# The Spatial Structure of Passively Simulated Atmospheric Boundary Layer Turbulence
## 4 Results and Discussion
### Vertical Spatial Structure
Because the vertical coherence of fluctuating wind is a fundamental equation necessary for calculating the wind-induced response of high-rise buildings, the empirical coherence function model is typically obtained by fitting a large amount of measured data. The modified [PERSON] model (Equation (11)), the ESDU 86010 simplified model (Equations (12) and (13)), and [PERSON]'s model (Equation (14)) are the most commonly used coherence function models. Based on extensive research on the spatial structure of atmospheric turbulence, it has been discovered that existing models have some shortcomings in describing the vertical spatial structure of the natural shear flow. Therefore, this study attempted to make some amendments to those existing coherence function models. In structural wind engineering, the vertical coherence function of the along-wind and lateral components is the focus of attention. In order to conduct a comparative analysis, we referred to the above correction method to improve the theoretical models of [PERSON] and [PERSON] and then obtained the undetermined parameters by fitting the experimental data. The vertical coherence model expression of the along-wind component is  
\[Coh_{u}(k_{1},\Delta y)=\frac{2^{1+\zeta/2}}{\Gamma(-\zeta/2)}\cdot\left[\eta_ {x}^{-\zeta/2}K_{\zeta/2}(\eta_{\chi})-\frac{\eta_{\chi}{}^{1-\zeta/2}K_{1+ \zeta/2}(\eta_{\chi})}{2}\right] \tag{19}\]  
\[Coh_{u}(k_{1},\Delta y)=\theta_{\chi}K_{1}(\theta_{\chi})-\frac{\theta_{\chi} {}^{2}}{2}K_{0}(\theta_{\chi}) \tag{20}\]  
Figure 12: Variation of the coherence function with \(k_{1}\) for various vertical separations: (**a**) along\(-\)wind velocity component; (**b**) lateral velocity component; (**c**) vertical velocity component. where the correction coefficients \(\eta_{\chi}\) and \(\theta_{\chi}\) are as follows:  
\[ \begin{array}{l}\eta_{\chi}=\frac{\Delta y\delta}{\pi_{1}L_{u}^{x}}\cdot\sqrt{1 +\left[(2\pi/\beta)\cdot(a_{1}k_{1}L_{u}^{x})\right]^{b_{1}}}\\ \theta_{\chi}=\frac{\Delta y}{a_{2}L_{u}^{x}}\cdot\sqrt{1+\left(2\pi\right)^{ 2}\cdot\left[(k_{1}U)^{\chi}(\frac{a_{1}L_{u}^{x}}{U})\right]^{2}}\end{array}  \tag{21}\]  
The vertical coherence correction model expression of the lateral component is  
\[Coh_{v}(k_{1},\Delta y)=\frac{2^{1+\zeta/2}}{\Gamma(-\zeta/2)}\cdot\left[\eta _{\chi}{}^{-\zeta/2}K_{\zeta/2}(\eta_{\chi})+\frac{{v_{\chi}}^{2}{\eta_{\chi} }^{1-\zeta/2}K_{1+\zeta/2}(\eta_{\chi})}{B_{\chi}}\right] \tag{22}\]  
\[Coh_{v}(k_{1},\Delta y)=\frac{2^{1+\zeta/2}}{\Gamma(-\zeta/2)}\cdot\left[\eta _{\chi}{}^{-\zeta/2}K_{\zeta/2}(\eta_{\chi})+\frac{{v_{\chi}}^{2}{\eta_{\chi} }^{1-\zeta/2}K_{1+\zeta/2}(\eta_{\chi})}{B_{\chi}}\right] \tag{23}\]  
where \(B_{\chi}\), \(B_{D}\), and \(v_{\chi}\) are correction terms:  
\[ \begin{array}{l}B_{K}=1+\frac{8}{3}\cdot(2\pi k_{1}\alpha_{1}L_{u}^{x}/\beta )^{b_{1}}\\ B_{D}=1+3\cdot(2\pi k_{1}\alpha_{2}L_{u}^{x})^{b_{2}}\\ v_{\chi}=2\pi a_{1}L_{u}^{x}k_{1}\text{ or }v_{\chi}=2\pi a_{2}L_{u}^{x}k_{1} \end{array}  \tag{24}\]  
At the same time, the model is verified with reference to an empirical model of the vertical coherence function proposed by [PERSON] [26], and its expression is as follows:  
\[Coh_{u}(k_{1},\Delta z)=\exp(-C_{1}k_{1}\Delta z\cdot\sqrt{1+\left(\frac{A_{1 }}{f}\right)^{2}}) \tag{25}\]  
\[Coh_{v}(k_{1},\Delta z)=A_{2}\cdot(1-\frac{C_{2}}{k_{1}\Delta z\cdot\sqrt{1+ \left(A_{1}/f\right)^{2}}})\exp(-C_{1}k_{1}\Delta z\cdot\sqrt{1+\left(A_{1}/f \right)^{2}}) \tag{26}\]  
where \(A_{1}\), \(A_{2}\), \(C_{1}\), and \(C_{2}\) are undetermined parameters. Taking the selected vertical spacing combinations as an example, Figure 13 shows the coherence function fitting of the along-wind component in the BL1 wind field. According to the fitting results, it can be found that several types of coherence function models can roughly reflect the vertical coherence function of the along-wind component.