## 2 Numerical Model
### Explosion Shock Wave Model of Plasma  
When corona discharge occurs, high-energy electrons will collide with neutral molecules along the direction of the electric field, and new electron avalanches will be produced continuously; thus, the plasma composed of ions continues to develop forward. At the initial stage, the shock wave will propagate outward from the plasma channel, and its pressure waveform is similar to the explosion shock wave, so the shock wave theory can be used to simulate the explosion process of plasma. Based on the method of equivalent explosion energy, the plasma in the discharge channel is replaced by explosives with the same energy, and the plasma energy in the gap can be calculated using the _U-I_ formula:  
\[E_{p}=E_{T}=m_{T}W_{T}=\int_{0}^{\tau}P(t)dt=U\int_{0}^{\tau}I(t)dt \tag{9}\]  
where \(E_{p}\) is the plasma energy; \(E_{T}\) is the explosion energy of TNT; \(m_{T}\) is the mass of TNT; \(W_{T}\) is the explosive heat value; \(\tau\) is the duration time of a single pulse discharge; \(P(t)\) is the power of the plasma channel; \(U\) is the voltage applied to the electrode; and \(I(t)\) is the current of corona discharge. This paper adopts the dynamic simulation software AUTODYN (version 11.0) to simulate the propagation process of the plasma shock wave. Its built-in Euler with 2D multi-material algorithm can simulate the shock wave generated by explosions accurately. The shock wave pressure generated by TNT with the same energy of plasma is described by the JWL state equation [26], and its specific form is  
\[P=A(1-\frac{\omega}{R_{1}V})e^{-R_{1}V}+B(1-\frac{\omega}{R_{2}V})e^{-R_{2}V}+ \frac{\omega E_{0}}{V} \tag{10}\]  
where \(P\) is the explosion pressure, \(V\) is the relative volume specific volume, \(E_{0}\) is the initial internal energy per unit volume of the explosive; \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) are the material parameters. TNT density \(\rho\) = 1520 kg/m\({}^{3}\), and the initial velocity of explosion \(D\) = 6930 m/s, initial pressure \(P_{0}\)=2.1 \(\times\) 10\({}^{4}\) MPa. The JWL state equation parameters of TNT are shown in Table 2.  
The applied equation of state of ideal gas for the air substance and its equation are as follows  
\[P=(\gamma-1)\frac{\rho}{\rho_{0}}e+P_{shift} \tag{11}\]  
where \(P\) is the gas pressure, \(\rho_{0}\) is the initial density of air, \(\rho\) is the density of air after compression or expansion, \(e\) is the internal air energy, \(\gamma\) is the adiabatic exponent, and \(P_{shift}\) represents the original gas pressure. In an ideal gas, \(e\) is only a function of temperature, and its form can be written as  
\[e=c_{v}T \tag{12}\]  
where \(c_{v}\) is the specific heat, and \(T\) is the air temperature. The air parameters are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \(A\) (MPa) & \(B\) (MPa) & \(R_{1}\) & \(R_{2}\) & \(\omega\) & \(E_{0}\) (J/m\({}^{3}\)) & \(V\) \\ \hline \(3.74\times 10^{5}\) & \(3.747\times 10^{3}\) & \(4.15\) & \(0.9\) & \(0.35\) & \(3\times 10^{9}\) & \(1\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the JWL state equation for explosives. By substituting the discharge current which is solved by Equation (8) into Equation (9), we can derive the explosive mass with the same energy as the plasma. The shape of the explosive is set as a sphere, and the 2D simulation model is established, which is shown in Figure 3. The air domain area of the model is 20 \(\times\) 12 cm\({}^{2}\), the explosive is placed at the midpoint of the x-axis, and the coverage area is calculated based on TNT density. The detonation point is located at the sphere center of the explosive. The mesh partition adopts the Euler method, with 2000 horizontal nodes and 1200 vertical nodes. The left, right, and upper boundaries of the air are set as outflow boundaries to simulate the infinite air domain, while the lower boundary is the default axisymmetric boundary.