## 2 Description of the Model  
A fluid model is applied in this paper to investigate the effect of the gap width on temporal nonlinear behaviors in CO\({}_{2}\) DBDs under Martian conditions. In this model, the continuity equations coupled with the drift-diffusion approximation describe the dynamic behaviors of each neutral and charged species, the Poisson equation is used to obtain the electric field in the discharge region, and the electron energy balance equation is also incorporated to calculate the electron temperature [39; 40; 41]. \[\frac{\partial N_{\text{e,a,i,n}}}{\partial t}+\frac{\partial T_{\text{e,a,i,n} }}{\partial x}=R_{\text{e,a,i,n}} \tag{1}\]  
\[\Gamma_{\text{e,a,i,n}}=\mp\mu_{\text{e,a,i}}N_{\text{e,a,i}}E-D_{\text{e,a,i, n}}\frac{\partial N_{\text{e,a,i,n}}}{\partial x} \tag{2}\]  
\[\varepsilon_{0}\frac{\partial E}{\partial x}=e_{0}\left(\sum_{i}N_{i}-\sum_{ a}N_{\text{e,a}}\right) \tag{3}\]  
\[\frac{\partial}{\partial t}\left(\frac{3}{2}k_{\text{b}}N_{\text{e}}T_{\text{ e}}\right)=-\frac{\partial Q_{\text{e}}}{\partial x}-e_{0}\Gamma_{\text{e}}E-\sum_{j}k_{j}N_{ \text{e}}N_{j}H_{j}-3k_{\text{b}}N_{\text{e}}\frac{m_{\text{e}}}{m_{\text{g}} }\left(T_{\text{e}}-T_{\text{g}}\right)\overline{v_{\text{e}}} \tag{4}\]  
where the indices e, a, i, and n are electrons, negative ions, positive ions, and neutral particles, respectively. \(N\) and \(\Gamma\) are the density and flux of species, respectively. \(R\) is the source and lose items of species. \(\mu\) and \(D\) are the mobility and diffusion coefficients, respectively, and only the diffusion item is considered for neutral particles. \(\varepsilon_{0}\) is the vacuum permittivity and \(e_{0}\) is the elementary charge. \(E\) denotes the electric field in the discharge region. \(k_{b}\) is the Boltzmann constant. \(Q_{\text{e}}\) is the electron thermal flux. \(k_{j}\) and \(H_{j}\) are the reaction rate and threshold energy of the inelastic impact reaction \(j\), respectively. \(m_{\text{e}}\), \(m_{\text{g}}\), \(T_{\text{e}}\), and \(T_{\text{g}}\) are the electron mass, gas species mass, electron temperature, and gas temperature, respectively. \(\overline{v_{\text{e}}}\) denotes the electron momentum transfer collision frequency with the background gas, which is obtained from Ref. [42]. The total current density \(I_{0}(t)\) is given as  
\[I_{0}=\frac{\varepsilon_{0}}{d_{\text{g}}+2d_{\text{s}}/\varepsilon_{\text{ r}}}\frac{\text{d}V_{\text{a}}}{\text{d}t}+\frac{1}{d_{\text{g}}+2d_{\text{s}}/ \varepsilon_{\text{r}}}\int_{0}^{d_{\text{g}}}j_{\text{g}}\text{d}x \tag{5}\]  
where \(d_{\text{g}}\) denotes the gap width. \(d_{\text{s}}\) and \(\varepsilon_{\text{r}}\) represent the the thickness and relative permittivity of the dielectric barriers, respectively. \(j_{\text{g}}\) is the conduction current density, which is determined by \(j_{\text{g}}=e_{0}\left(\sum\limits_{i}T_{i}-\sum\limits_{a}\Gamma_{\text{e,a }}\right)\). In this paper, the applied voltage \(V_{\text{a}}\) with the power-on phase (\(P_{\text{on}}\)) and power-off phase (\(P_{\text{off}}\)) is described by a segmented function, which is given by  
\[V_{\text{a}}= \begin{cases}V_{0}\text{sin}(2\pi ft)&t\in P_{\text{on}}\\ 0&t\in P_{\text{off}}\end{cases}  \tag{6}\]  
where \(V_{0}\) is the amplitude of the sinusoidal voltage, and \(f\) is the driving frequency. The waveform of applied voltage is given in Figure 1, where a positive half-cycle sinusoidal voltage with a voltage amplitude of 1500 V and a driving frequency of 50 kHz is applied during the power-on phase, and no negative voltage is applied. In addition, the applied voltage is set to zero during the power-off phase.