# The Spatial Structure of Passively Simulated Atmospheric Boundary Layer Turbulence
## 2 Theoretical Considerations
\begin{table}
\begin{tabular}{l l} \hline \(u_{i}(x,t)\) & Time history of wind velocity \\ \(R_{ij}(r,\tau)\) & Correlation function \\ \(R_{uu}\), \(R_{v}\), \(R_{w}\) & Correlation coefficients \\ \(r\) & Distance at lateral and vertical directions \\ \(x\), \(y\), \(z\) & Corresponding structural axis \\ \(\Delta y\), \(\Delta z\) & Span-wise distance of two directions \\ \(u\), \(v\), \(w\) & Velocity components of turbulence \\ \(S_{ij}()\), \(S_{i}()\), \(P_{ij}()\), \(Q_{ij}()\) & Spectrum \\ \(a\), \(n\) & Shape and length scale factor \\ \(L\), \(L_{uu}\), \(L_{up}\), \(L_{up}\) & Integral length scale \\ \(l_{u}\), \(l_{v}\), \(l_{w}\) & Turbulence intensity \\ \(\sigma_{u}\), \(\sigma_{ii}\) & Root mean square values \\ \(U\), \(U_{2}\), \(U_{0}\) & Average wind velocity \\ \(k_{1}\), \(k\), & Wave number \\ \(f\) & Frequency \\ \(Coh_{u}\) ( ), \(Coh_{i}()\) & Coherence function \\ \(l_{0}\), \(l_{1}\) & Bessel function of the first kind \\ \(\Gamma\) & Gamma function \\ \(K_{\zeta}\), \(K_{1-\zeta}\), \(K_{\gamma}\), \(K_{1-\gamma}\), \(K_{0}\), \(K_{1}\) & Bessel function of the second kind \\ \(\eta\), \(\zeta\), \(K_{1}\), \(\gamma\), \(\theta\), \(\phi\), \(\upsilon\), \(\psi\), \(\theta\), \(c\), \(\delta\) & Coefficients of the theoretical coherence model \\ \(c_{uu}^{j}\), \(c_{ij}^{j}\), \(c_{w}^{j}\) & Decay factor of coherence model \\ \(\eta_{X}\), \(\theta_{X}\), \(B_{x}\), \(B_{D}\) & Correction parameters in empirical coherence models \\ \(a_{1}\), \(b_{1}\), \(a_{2}\), \(b_{2}\), \(A_{1}\), \(A_{2}\), \(C_{1}\), \(C_{2}\) & Fitted await parameters in empirical coherence models \\ \hline \end{tabular}
\end{table} 
Table 1: List of symbols. a coherence function \(Coh_{u}(k_{1},\Delta y)\) based on spectral tensor analysis, which was extended by [PERSON] and [PERSON] [18] for horizontal separation:  
\[Coh_{u}(k_{1},\Delta y)=\frac{S_{u}(k_{1},\Delta y)}{S_{u}(k_{1})}=1-\Delta y^{2 }\cdot\int_{0}^{\infty}\frac{kS_{u}(k)}{S_{u}(k_{1})}\cdot G(\Delta y\cdot\sqrt {k^{2}-k_{1}^{2}})dk \tag{5}\]  
where \(G(\psi)=J_{0}(\psi)/2+J(\psi)/\psi\), and \(\psi=\Delta y\cdot\sqrt{k^{2}-k_{1}^{2}}\). By substituting the wind fluctuation spectrum into Equations (4) and (5), the theoretical coherence model can be obtained, and the following expression will be obtained by substituting the von [PERSON] spectrum model:  
\[Coh_{u}(k_{1},\Delta y)=\frac{2^{1+1/\zeta}}{\Gamma(-\zeta/2)}\cdot\left[\eta ^{-\zeta/2}K_{\zeta/2}(\eta)-\frac{\eta^{1-\zeta/2}}{2}K_{1+\zeta/2}(\eta)\right] \tag{6}\]  
\[Coh_{i}(k_{1},\Delta y)=\frac{2^{1+\zeta/2}}{\Gamma(-\zeta/2)}\cdot\left[\eta ^{-\zeta/2}K_{\zeta/2}(\eta)+\xi\cdot\frac{v^{2}\eta^{1-\zeta/2}K_{1+\zeta/2} (\eta)}{1+(32\pi k_{1}L_{u}^{x}/3\beta)}\right] \tag{7}\]  
where \(\eta=\frac{\Delta y}{L}\cdot\sqrt{1+(v/\beta)^{2}}\), \(\beta=\sqrt{\pi}\cdot\Gamma(\frac{\zeta}{2})/\Gamma(\frac{\zeta+1}{2})\), \(v=2\pi Lk_{1}\), and \(\zeta\) is a constant related to the fluctuating velocity component, taking \(1\) for the lateral component and \(-1\) for the vertical component. If substituted into the Dryden spectrum model, the following theoretical expression can be obtained:  
\[Coh_{u}(k_{1},\Delta y)=\theta K_{1}(\theta)-\frac{\theta^{2}}{2}K_{0}(\theta) \tag{8}\]  
\[Coh_{i}(k_{1},\Delta y)=\theta K_{1}(\theta)+\zeta\cdot\frac{v^{2}\theta^{2}} {1+3(2\pi k_{1}L)^{2}}K_{0}(\theta) \tag{9}\]  
where \(\theta=\frac{\Delta y}{L}\cdot\sqrt{1+\left(2\pi k_{1}L\right)^{2}}\). The derivation and the calculation process of the theoretical model are complex, making it unsuitable for engineering practice applications. Based on the normalized von [PERSON] spectrum model, [PERSON] [32] proposed the following normalized cross-spectral density:  
\[P_{ij}(\kappa_{1},r)=\frac{1}{\Gamma(\gamma)}\cdot\left[(\frac{\kappa_{1}r}{2 })^{\gamma}K_{\gamma}(\kappa_{1}r)-(\frac{\kappa_{1}r}{2})^{\gamma+1}K_{1- \gamma}(\kappa_{1}r)\right] \tag{10}\]  
where \(\gamma\) reflects high-frequency effects, and \(\kappa_{1}\) is the corrected wave number, \(\kappa_{1}=\sqrt{\left(2\pi k_{1}\right)^{2}+1/L^{2}}\).