# The Spatial Structure of Passively Simulated Atmospheric Boundary Layer Turbulence
## 4 Results and Discussion
### Vertical Spatial Structure  
Due to the complex vertical spatial structure of the natural shear flow, isotropic turbulence theory is no longer applicable in the vertical direction. However, an in-depth study of the vertical spatial structure of atmospheric turbulence is an important direction for the refined analysis of high-rise buildings' wind resistance performance. Because the vertical spatial structure of atmospheric turbulence is so complicated, the theoretical derivation is still difficult to solve. The primary research method is still to collect a large amount of data via field measurements of natural atmospheres or wind tunnel tests, followed by fitting to obtain an approximate empirical model. This study obtained the spatial correlation of 55 vertical spacing combinations in each wind field by real-time measurement of fluctuating wind time history data at different heights. According to the definition of Equation (1), the cross-correlation coefficients under different spacing combinations were obtained, and the variation trend of the correlation coefficient with the dimensionless spacing \(\Delta z/H\) is shown in Figure 11, where \(H\) is the height of the highest collection point. The relationship between the vertical correlations of each velocity component is \(R_{u}>R_{v}>R_{w}\), which is quite different from the horizontal lateral correlation. The vertical correlation coefficients of the three types of simulated turbulent wind fields are the same with the change in the distance, and the values are relatively close. This demonstrates that the type of ground roughness has little effect on the vertical correlation of the turbulence field, which is a key feature of simulating atmospheric turbulence. The partial spacing combinations were used as examples in the frequency domain to demonstrate the influence of vertical distance on the coherence of each turbulence component, as shown in Figure 12.  
The results show that the vertical coherence function is not only related to wind field factors such as the turbulence intensity, length scale, and wave number but also the average height of the measurement point combination, which is mainly reflected by the average wind speed at two heights, namely, \(U=(U_{z_{1}}+U_{z_{2}})/2\). The overall distribution trend of the vertical coherence function with the wave number is similar to the lateral coherence. As the distance and wave number increase, the coherence function gradually decreases, and the attenuation rate increases continuously. Figure 11: The correlation coefficients of different turbulent fluctuation components: (**a**) BL1 flow; (**b**) BL3 flow. Following extensive comparative analysis, it was discovered that there is a special relationship between the vertical coherence function and parameter \(\Delta z/U\). In Figure 12, two coherence curves with approximate coefficients \(\Delta z/U\) can be found, in which the distribution characteristics are almost the same. Therefore, it may be concluded that if the values of \(\Delta z/U\) are close, the two coherences can be regarded as similar. For any vertical spacing, the vertical coherence is basically inversely proportional to the size \(\Delta z/U\). That is, if the value \(\Delta z/U\) is smaller, the correlation of the two points will be stronger and the decay rate of coherences will be relatively small. As a result, despite the fact that the spatial structure of turbulence in the vertical direction is complex and has many influencing factors, the coherences of other arbitrary heights can be roughly and quickly deduced based on the ratio \(\Delta z/U\) after measuring a few wind velocity data, which is useful in engineering practice. Considering that atmospheric turbulence has a complex vertical spatial structure, which is affected by many factors, it is difficult to use theoretical isotropic turbulence models to describe it completely.