## 3 Interaction between Plasma Sheath and HPM  
### The EM Properties of Plasma Sheath  
The fluid parameters are obtained through the CFD simulation calculation, which can be converted and calculated to produce the electron density \(n_{e}\) and the neutral particle density \(n_{m}\). The formulas are  
\[n_{e}=6.02\times 10^{23}\frac{\rho Y_{g}}{M_{g}}, \tag{1}\]  
\[n_{m}=6.02\times 10^{23}\sum_{s=1}^{5}\frac{\rho Y_{s}}{M_{s}}, \tag{2}\]  
where \(\rho\) is the plasma density, \(M_{s}\) is the relative atomic mass of component \(s\) (N\({}_{2}\), O\({}_{2}\), N, O, NO, NO\({}^{+}\), and e\({}^{-}\)), and \(Y_{s}\) is the mass fraction of component \(s\). After obtaining the neutral particle density \(n_{m}\), the electron density \(n_{e}\), and the electron temperature \(T_{e}\), the EM parameters of plasma [33] can be established: plasma frequency \(\omega_{p}\) and the collision frequency \(v_{c}\)[18]. \[\omega_{p}=\sqrt{\frac{e^{2}n_{e}}{\epsilon_{0}m_{e}}}, \tag{3}\]  
\[v_{c}=6.3\times 10^{-15}n_{m}\sqrt{\frac{T_{e}}{300}}, \tag{4}\]  
where \(e\) is electron charge, \(\epsilon_{0}\) is the dielectric constant of vacuum, and \(m_{e}\) is the electron mass. In EM simulation calculations, the computational EM algorithm uses ADE-FDTD [23; 24; 25]. On the one hand, ADE-FDTD can be used to observe the propagation of EM waves in plasma over a wide frequency range. The polarization current, on the other hand, can be utilized to finish the conversion from plasma absorption energy to temperature increase when using ADE-FDTD to compute the polarization current of intermediate variables in the propagation process of EM waves. Figure 3: The distribution characteristics of the peak electron number density of RAM-C II along the blunt cone streamline at an altitude of 71 km. In the ADE-FDTD algorithm, the polarization current is described as  
\[\mathbf{J}^{n+1}=\frac{2-v_{c}\bigtriangleup t}{2+v_{c}\bigtriangleup t}\mathbf{ J}^{n}+\frac{\varepsilon_{0}\omega_{p}^{2}\bigtriangleup t}{2+v_{c}\bigtriangleup t }(\mathbf{E}^{n+1}+\mathbf{E}^{n}). \tag{5}\]  
It is known that the relationship between the polarization current and the electric field is  
\[\mathbf{J}=\frac{\varepsilon_{0}\omega_{p}^{2}}{j\omega+v_{c}}\mathbf{E}= \frac{(v_{c}-j\omega)\varepsilon_{0}\omega_{p}^{2}}{v_{c}^{2}+\omega^{2}} \mathbf{E}. \tag{6}\]  
The ADE-FDTD algorithm must satisfy a certain relationship between both temporal and spatial accuracy in terms of numerical stability; otherwise, it will make the calculated values unstable. This is manifested by an infinite increase in the value of the calculated field quantity as the number of calculation steps increases. This increase is not due to the accumulation of errors but to the disruption of the propagation relations of the EM waves. The following conditions must be satisfied for computational space and time for non-uniform media compositions:  
\[\Delta s(\Delta x,\Delta y,\Delta z)\leq\frac{\lambda}{12}, \tag{7}\]  
\[\Delta t\leq\frac{\sqrt{\mu\varepsilon}}{\sqrt{\left(\frac{1}{\Delta x}\right) ^{2}+\left(\frac{1}{\Delta y}\right)^{2}+\left(\frac{1}{\Delta z}\right)^{2}}}. \tag{8}\]  
Depending on the size of the CFD computational fluid domain, the FDTD region was set at 2.5 m, 1.8 m, and 1.8 m in the X, Y, and \(Z\) directions, respectively. The FDTD mesh size is set to \(\Delta x=\Delta y=\Delta z=1.5\times 10^{-3}\) m, which is 1/20 th of the wavelength of a 10 GHz frequency EM wave traveling in free space. The overall FDTD mesh quantity is 1670 \(\times\) 1200 \(\times\) 1200. According to the CFL stability condition, the minimum time step of FDTD is set as \(\Delta t_{FDTD}=2.5\times 10^{-12}\) s.  
In the process of EM field simulation and flow field simulation, the data of CFD mesh nodes and FDTD mesh nodes can be known by calculating N-S equations and Maxwell equations. According to the comparison between the fluid and the EM mesh, we can know that computational fluid mesh is much larger than computational EM mesh, so each computational fluid mesh contains many computational EM meshes.