## 2 Models and Numerical Methods
### Air Chemistry  
\(N_{2}/O_{2}\) mixture is assumed as the initial neutral air, and the molar fraction of \(N_{2}\) and \(O_{2}\) is 4:1. There are 46 species and 395 chemical reactions [3] and the chemical reactions are very complex even for this simplified air. Kinetic analysis conducted by [PERSON] et al. [37] reveals that the electrons impact dissociation of \(O_{2}\) molecules, the quenching of electronic excited \(N_{2}\) molecules by \(O_{2}\) molecules and the quenching of excited \(O\) atoms by \(N_{2}\) molecules mainly contribute to the fast gas heating when the reduced electric field \(E/N\leq 200\) Td. The electron impact dissociation of \(N_{2}\) molecules and the processes of the charged species are the major processes when \(E/N>400\) Td.  
A reduced chemistry model is considered by containing the main neutral and charged species involved processes. The reduced kinetic model consists of 31 species and 99 reactions, which is denoted as \(R99\) hereafter. The species and the reactions are shown in Table 1 in Appendix A. The abbreviations of the types of reactions are shown in Table 2. The details of the reaction are shown in Table 3.  
As the fractions of the contribution of the dissociation reactions are smaller than the corresponding quench and charged species processes on the gas heating in low and high electric field zones [37]. A further reduced kinetic model is considered based on \(R99\) with all the dissociation processes and the associated species excluded. The reduced kinetic reaction model consists of 28 species and 85 reactions and is denoted as \(R85\) hereafter.  
Based on the sensitivity analysis conducted by [PERSON] et al. [18], the electronic excited state molecules \(N_{2}(A^{3}\Sigma),N_{2}(B^{3}\Pi),N_{2}(C^{3}\Pi),N_{2}(a^{1}\Sigma)\) account for a large portion of total input power. Number densities of \(N_{2}^{+},O_{2}^{+}\) are larger than those of \(N_{4}^{+},O_{4}^{+},O_{2}^{-}\) in non-equilibrium plasma. A much further reduced air kinetic reaction model is developed by including the main electronic excited states species and charged species while excluding the \(O_{2}^{-}\). The model is the same as the one used by [PERSON] et al. [7], which consists 23 species and 50 reactions. It is denoted as \(R50\). The details of \(R50\) can be referred from [7].  
Vibrational excited nitrogen contributes to the gas heating through the vibration-translation (VT) relaxations at a large timescale. However, it contributes no more than 10% of the total energy released at the relatively high reduced field (\(E/N>200\) Td) [38]. Therefore, a kinetic model is further reduced based on \(R50\) by excluding all the vibrational excited nitrogen species \(N_{2}\) (\(v=1\)-\(8\)). It is denoted as \(R42\). The details of the reactions are also be referred from [7] but excluding all the 8 vibrational excitation reactions.  
Another simple kinetic model is considered by averaging the \(N_{2}\) and \(O_{2}\) species as an \"Air\" molecule, which incorporates 4-species and 4-reaction mechanism [29; 30; 31], as shown in Table 1. It is denoted as \(R4\) hereafter. Details of the reactions are shown in Table 4. The model is the same as the one used in the previous works [20; 39; 40].