## 5 Influence of Urban Thermal Effect on Thunderstorm Processes and Lightning Activity
Assuming the incoming flow direction of the thunderstorm system is the \(x\)-axis (from northwest to southeast), the Boussinesq approximation equation can be expressed as follows:  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\omega\frac{ \partial u}{\partial z}=-\frac{\partial\pi}{\partial x}+k\frac{\partial^{2}u} {\partial z^{2}} \tag{1}\]  
\[\frac{\partial\theta}{\partial t}+u\frac{\partial\theta}{\partial x}+\omega \frac{\partial\theta}{\partial z}=k\frac{\partial^{2}\theta}{\partial z^{2}} \tag{2}\]  
\[\frac{\partial\pi}{\partial z}=\lambda\theta \tag{3}\]  
\[\frac{\partial u}{\partial x}+\omega\frac{\partial\omega}{\partial z}=0 \tag{4}\]  
Taking the partial derivatives of z for Equations (1) and (3), respectively, we can obtain:  
\[\frac{\partial}{\partial t}(\frac{\partial u}{\partial z})=-\frac{\partial}{ \partial z}(u\frac{\partial u}{\partial x}+\omega\frac{\partial u}{\partial z })-\frac{\partial}{\partial x}(\frac{\partial\pi}{\partial z})+k\frac{ \partial^{3}u}{\partial z^{3}} \tag{5}\]  
\[\frac{\partial}{\partial x}(\frac{\partial\pi}{\partial z})=\lambda\frac{ \partial\theta}{\partial x} \tag{6}\]  
By omitting the influence of \(\frac{\partial}{\partial z}(u\frac{\partial u}{\partial x}+\omega\frac{ \partial u}{\partial z})\) and \(\frac{\partial^{3}u}{\partial z^{3}}\), it can be concluded from Equations (5) and (6) that:  
\[\frac{\partial}{\partial t}(\frac{\partial u}{\partial z})\cong-\lambda\frac {\partial\theta}{\partial x} \tag{7}\]  
Figure 7: Schematic diagrams of urban-related thermodynamic disturbances associated with an afternoon thunderstorm over the Great Bay Area in South China [12]. It is evident from the provided equation that when a thunderstorm system passes through a built-up area, the horizontal temperature gradient between the cold pool outflow and the urban underlying surface exerts a force that enhances the vertical shear of the wind speed. A strong horizontal wind speed vertical shear can significantly enhance vertical airflow movements [90]. When the upwelling airflow enters the city, it is propelled forward in the upper atmosphere due to the force of environmental winds, leading to a tilt in the main body of the thunderstorm. This ensures the continuous upwelling of the airflow [91]. Consequently, the enhanced vertical wind shear resulting from the thermal differences on the underlying surface plays a crucial role in maintaining and intensifying the thunderstorm system. The distinct physical differences in the properties between urban underlying surfaces and suburban areas result in a faster temperature rise in built-up areas. The phenomenon of surface wind fields converging toward urban areas can impact the vertical structural characteristics of the entire boundary layer flow field, facilitating an elevation in the boundary layer height and an increase in the vertical mixing height [15; 92]. This maintains a strong upward movement in the thunderstorm center. However, some scholars have suggested that the impact of the UHI effect on thunderstorms is not significant [21; 22], and that it may actually accelerate raindrop evaporation within clouds and alter water vapor exchange between the ground surface and the atmosphere, potentially inhibiting urban thunderstorm activities [93; 94]. Overall, the UHI effect has a certain enhancing effect on thunderstorm processes and lightning activity.