## 5 Discharge Characteristics and Energy
### Discharge Characteristics  
In this section, the discharge characteristics of the NS-DBD actuator in the high voltage pulse and post-discharge periods are investigated. The energy is investigated by analyzing the magnitudes, the time scales, the species number densities, the discharge energy, and gas heating. The spatial-temporal distributions of the number densities of the electrons \(n_{e}\) and \(N_{2}^{+}\) ion \(n_{N_{2}^{+}}\) are shown in Figure 9a,b, respectively. Before the discharge breakdown at a time of about 23 ns, the electric field in the whole domain is low and nearly distributes uniformly, as shown in Figure 10. As a result, the ionization process is weak in this phase, and the low densities of the electrons and ions are shown. After the breakdown, the number densities of the electrons and the ions increase as a result of the electron avalanche. \(E/N\), \(n_{e}\), and \(N_{2}^{+}\) at time 25 ns after the breakdown are shown in Figure 11. A sheath forms because of the charge accumulation on the dielectric surface of the negatively biased driven electrode. The sheath is featured by a strong electric field and the violation of quasi-neutrality. The sheath thickness is characterized by the length of the high-electric field area according to the Bohm criterion [43]. The sheath edge is considered at the location where the electric field magnitude is decreased by 99%. The thickness of the sheath is 0.179 mm and the maximum \(E/N\) in the sheath is about 7290 Td at time 25 ns. In the sheath, an electric field builds up until an equilibrium of electron and ion fluxes towards the surface is established. Because the drift and diffusion coefficients of electrons are much higher than that of the ions, \(n_{e}\) is lower than \(n_{N_{2}^{+}}\) in the sheath. \(n_{e}\) and \  
at time 25 ns during the pulse in the discharge gap are shown in Figure 11. The sheath has a length of about 0.1 mm, where \(E/N\) is larger than 2000 Td. The sheath is nearly free of electrons. \(n_{e}\) and \(n_{N_{2}^{+}}\) display high values in the edge of the sheath, which reach magnitudes of \(6.5\times 10^{19}\) m\({}^{-3}\) and \(4\times 10^{19}\) m\({}^{-3}\), respectively. According to the research of [PERSON] et al. [7], the error of the two-term approximation of the Boltzmann equation becomes substantial above \(E/N=3000\) Td. However, \(n_{e}\) is very low in the sheath. Therefore, the present model for ionization at the edge of the sheath is accurate enough to study the discharge characteristics. To study the characteristics of \(n_{e}\) locally and globally, \(n_{e}\) at the point \(x=0.09\) mm (Point \"A\") at the edge of the sheath and the total \(n_{e}\) in the whole domain \(n_{etot}\) are shown in Figures 12 and 13, respectively. Because the computational model is one dimensional, the number densities in the other two orthogonal dimensional directions are homogeneous. The total number density \(n_{etot}\) per unit area is calculated by integrating the number density \(n_{e}\) in the computational domain, the unit of \(n_{etot}\) is consequently m\({}^{-2}\). The gradient of \(n_{e}\), \(n_{etot}\), and the voltage are also shown in the figures. \(n_{e}\) and \(n_{etot}\) both increase rapidly in the fall phase in the negative pulse and the ascending phase in the  
Figure 9: The spatial-temporal distributions of the number densities of the electrons (**a**) and \(N_{2}^{+}\) (**b**) in \(R99\). positive pulse, corresponding to two discharges. In the rise phase of the negative pulse, \(n_{e}\) and \(n_{etot}\) increase slowly, while \(n_{e}\) and \(n_{etot}\) start to decrease from the decay phase of the pulse. The time of the maximum \(dn_{e}/dt\) and \(dn_{etot}/dt\) are about 24 ns and 23 ns, when the breakdown happens. Therefore, the breakdown voltages are predicted as 11.9 kV and 10.3 kV. The breakdown voltage predicted by \(n_{e}\) locally is in a closer agreement with the experimental estimation 13 kV [27].