## 2 Numerical Model  
### Hydrodynamic Model of Corona Discharge  
Corona discharge often occurs near the tip electrode with a small curvature radius. Therefore, we adopt the typical pin-plate electrode structure as the research object. Figure 1 shows the schematic diagram of a pin-plate discharge model, including a pin electrode with high voltage and a plate electrode with zero potential, and the corona discharge channel is located between the two electrodes. The single corona discharge process in a short gap is usually completed in nanoseconds. The charge mass ratio of positive and negative ions is four to five orders of magnitude smaller than that of electrons. Compared with electrons, the motion speed of positive and negative ions is very small; therefore, the diffusion process of positive and negative ions can usually be ignored. In addition, corona discharge belongs to low-temperature plasma discharge, so it can be assumed that the temperature rise process of ions is adiabatic, and the temperature of neutral molecules does not change. Ignoring the effect of the magnetic field, the hydrodynamic equations which describe corona discharge include the continuity equation of electrons, positive ions and negative ions and the Poisson equation, which describes the potential distribution. These equations are expressed as  
\[ \begin{array}{rcl}\frac{\partial N_{e}}{\partial t}&=&-\frac{1}{r}\frac{ \partial}{\partial r}(r\mu_{e}N_{e}E_{r})-\frac{\partial}{\partial z}(\mu_{e} N_{e}E_{z})+\frac{1}{r}\frac{\partial}{\partial z}(rD\frac{\partial N_{e}}{ \partial r})+\frac{\partial}{\partial z}(D\frac{\partial N_{e}}{\partial z}) \\ &&+S_{ph}+(\alpha-\eta)N_{e}\mu_{e}|\@vec{E}|-k_{ep}N_{e}N_{p}\end{array}  \tag{1}\]  
\[ \begin{array}{rcl}\frac{\partial N_{p}}{\partial t}&=&-\frac{1}{r}\frac{ \partial}{\partial r}(r\mu_{p}N_{p}E_{r})-\frac{\partial}{\partial z}(\mu_{p} N_{p}E_{z})\\ &&+S_{ph}+\alpha N_{e}\mu_{e}|\@vec{E}|-k_{ep}N_{e}N_{p}-k_{pn}N_{p}N_{n}\end{array}  \tag{2}\]  
\[ \begin{array}{rcl}\frac{\partial N_{n}}{\partial t}=-\frac{1}{r}\frac{ \partial}{\partial r}(r\mu_{n}N_{n}E_{r})-\frac{\partial}{\partial z}(\mu_{n} N_{n}E_{z})+\eta N_{e}\mu_{e}|\@vec{E}|-k_{pn}N_{p}N_{n}\end{array}  \tag{3}\]  
\[ \begin{array}{rcl}\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial \varphi}{\partial r})+\frac{\partial}{\partial z}(\frac{\partial\varphi}{ \partial z})=-\frac{e}{\varepsilon}(N_{p}-N_{n}-N_{e})\end{array}  \tag{4}\]  
where \(t\) is the simulation time; \(N_{e}\), \(N_{p}\), and \(N_{n}\) are the number densities of electrons, positive ions and negative ions, respectively; \(\mu_{e}\), \(\mu_{p}\), and \(\mu_{n}\) are the mobility of electrons, positive ions and negative ions, respectively; \(D\) is the electron diffusion coefficient; \(\alpha\), \(\eta\), \(k_{ep}\), and  
Figure 1: Schematic diagram of pin–plate discharge model. Where \(g\) represents the normal direction of the pin electrode surface. denote the ionization, attachment, electron-ion, and ion-ion recombination coefficient; \(E\) is the electric field; \(S_{ph}\) is the photoionization term; \(\varepsilon\) is the permittivity of air; and \(e\) is the electron charge. The data for the swarm parameters including \(\mu_{e}\), \(\mu_{p}\), \(\mu_{n}\), \(D\), \(\alpha\), \(\eta\), \(k_{cp}\), and \(k_{pn}\) as a function of the local reduced electric field expressed in \(\left|\,E\,\right|/N\) were taken from [19], where \(N\) is the air neutral density. By solving Equations (1)-(4), the space-time evolution and distribution law of the electric field and charged ions during discharge can be calculated and analyzed. The continuity equations of charged particles are coupled with the Poisson equation, and the solution of ions density is crucial to ensure the stability of the result. Photoionization adopts the three-exponential fitting Helmholtz model proposed by [PERSON] et al. [20], and the photoionization \(S_{ph}\) is expressed by the weighted sum of three components.