## 5 Discharge Characteristics and Energy
### Energy Transfer Process  
The characteristics of the energy are analyzed by separating each parts of the energy following the energy transfer path, as shown in Figure 1. \(Q_{A_{-}ui}\) is the sum of the energies of the electrons \(Q_{electrons}\) and the ions \(Q_{ions}\) gain from the electric field, and \(Q_{electrons}\) is dissipated to the electron-energy change \(\Delta En_{e}\) and the energies of the electron impact endothermic reactions \(Q_{Ee}\), \(Q_{Ei}\), and \(Q_{Ve}\). First, because \(Q_{Ee}\), \(Q_{Ei}\), \(Q_{Ve}\), and \(Q_{ions}\) are determined by the number densities of the corresponding species, the total number densities of \(N_{2}(A^{3}\Sigma)\), \(N_{2}(B^{3}\Pi)\), \(N_{2}(C^{3}\Pi)\), \(N_{2}(a^{{}^{\prime}1}\Sigma)\), the ions \(\mathrm{N}_{2}^{+}\), \(\mathrm{O}_{2}^{+}\), and the vibrational excited species \(N_{2}\) (\(v=1\)-\(8\)) are shown in Figures 14-16, respectively. Figure 12: Comparison of \(n_{e}\), the gradient of \(n_{e}\), and the voltage at the point \(x\) = 0.09 mm (Point “A”) at the edge of the sheath in R99. Figure 13: Comparison of the total \(n_{e}\) in the whole domain \(n_{etot}\), the gradient of \(dn_{etot}/dt\), and the voltage in R99. As the number densities are adopted from the 7 th pulse, the initial number densities of the species are not same to that in the initial values. The number densities of the nitrogen excited molecules increase mainly in the two discharges during the pulse, and the increments in the two discharges are nearly the same. The total number density of \(N_{2}(A^{3}\Sigma)\) increases up to \(4.9\times 10^{16}\) m\({}^{-2}\), which is about 4 times larger than the maximum density of \(N_{2}(B^{3}\Pi)\) and 6 times larger than the maximum densities of \(N_{2}(C^{3}\Pi)\) and \(N_{2}(a^{{}^{\prime}1}\Sigma)\). After the pulse, \(N_{2}(C^{3}\Pi)\) and \(N_{2}(a^{{}^{\prime}1}\Sigma)\) decay rapidly with nearly the same rate, however, \(N_{2}(A^{3}\Sigma)\) and \(N_{2}(B^{3}\Pi)\) decay more slowly. The number density of \(N_{2}(A^{3}\Sigma)\) decreases in about \(10^{4}\) ns, which correlates with the time scale of the energy release of the quench of the nitrogen excited molecules, as shown in Figure 21a. The number densities of N\({}_{2}^{+}\) and O\({}_{2}^{+}\) also increase in the two discharges, and the increments in the two discharges are almost the same. The maximum density of O\({}_{2}^{+}\) is about 2 times larger than that of N\({}_{2}^{+}\). The time scales of the decay of N\({}_{2}^{+}\) and O\({}_{2}^{+}\) are about 500 ns and \(8.7\times 10^{5}\) ns, which also correlates to the time scale of the change of the chemical energy stored in the ions, as shown in Figure 20. The number densities of \(N_{2}\) (\(v=1\)-8) also increase mainly in the two discharges. The number densities of N\({}_{2}^{+}\) and O\({}_{2}^{+}\) are about 500 ns and \(8.7\times 10^{5}\) ns, which are about 100 discharges during the pulse, which is the same as that of the excited species and the ions. The number densities of the vibrational excited nitrogen decrease with the increase of the vibrational quantum number, and the number density of \(N_{2}\left(v=1\right)\) is about two orders larger than that of \(N_{2}\left(v=8\right)\) after the pulse, which are \(1.1\times 10^{17}\) m\({}^{-2}\) and \(1.3\times 10^{15}\) m\({}^{-2}\), respectively. Because the VT relaxation rates are low, the decay of \(N_{2}\left(v=1\text{--}8\right)\) starts at a time scale of around 1 ms, which is close to the time scale 0.5 ms in the experiments conducted by [PERSON] and the co-workers [44]. The time scale correlates with the time scale of the energy release of \(N_{2}\left(v=1\text{--}8\right)\), as shown in Figure 21a, which corresponds to the slow gas heating.