Study On The Distribution Law Of Lightning In Mesoscopic Terrain Based On Conformal Transformation

As one of the ten natural disasters,lightning disaster has always been a serious problem for people and property damage.It is self-evident that it is necessary to do well in lightning protection.The research on lightning activity such as lightning distribution and ground lightning density is the theoretical basis of lightning protection.However,there are some differences in the distribution of thunderdrop in different terrain.There are more thunderstorms in high altitude areas such as plateau and high mountain,while in low altitude areas such as plain,there are less thunderstorms,even in different areas of the same terrain,such as half mountainside,peak or foot.Due to the difference of the lightning drop distribution in different terrain areas,the research on the macro topography with a scale between the large-scale and the micro terrain considering only the pole tower or viaduct is of great significance for the differential lightning protection.This paper mainly focuses on the following aspects:In view of the problem that the finite difference method is widely used in the calculation of space electric field in the lightning development process,the calculation speed is very slow,and it can not meet the research demand of lightning falling distribution.The method of calculating space electric field by conformal transformation method and mirror image method is proposed.The method can speed up the simulation speed greatly,and provide the premise for the subsequent research of the large amount of simulation data.The basic principle of conformal transformation and its applicability in calculating space electric field are described.The advantages and disadvantages of the finite difference method and conformal transformation method are compared in the calculation of space electric field in the process of lightning development.Based on the theory of complex function,the conformal transformation function of the basic mountain shape is studied,the expression of the basic mountain shape is obtained,and the spatial mapping relation before and after the conformal transformation is derived.According to the spatial mapping relationship,the change of terrain on spatial grid and the influence on space electric field are analyzed.The space potential formula under the terrain boundary is obtained by combining with the mirror image method.The comparison between the conformal transformation algorithm and the finite element simulation results proves that the algorithm is correct,flexible and fast in solving the calculation of space electric field in the process of lightning development.Taking the height,half height and slope as the terrain parameters,the mathematical relations between the terrain parameters and the conformal transformation parameters are derived,and the influence of the conformal transformation parameters on the shape of the mountain is analyzed.Based on the theory of conformal transformation,the conformal transformation function of the basic mountain shape is extended and verified to meet the characteristics of the actual terrain in asymmetrical shape.The extended conformal transformation function is analyzed,and three mathematical expressions of asymmetric terrain are derived.According to the theory of complex function,the mathematical relations of spatial mapping under different terrain conditions are derived.The change of space grid under different terrain boundary is analyzed.The results show that it conforms to the law of conformal transformation,and proves the correctness of the extended conformal transformation function.Based on the fractal development theory of lightning forerunner,a mesoscopic terrain lightning leading fractal development model based on conformal transformation method is established.The basic mountain shape and the extended three mountain shapes are simulated by fixed pilot starting position and random pilot starting position respectively,and the final lightning point position is counted.The difference of the falling thunder distribution at different terrain boundary is analyzed from the point of probability theory and mathematical statistics,and the influence law of different terrain boundary shape and lightning lead starting position on the lightning drop distribution is analyzed.