## 4 Discussion
### Advective Ambipolar Self-Coagulation
#### 4.2.1 Comet Structure for Advective Type  
(a). [PERSON]'s criterion of sheath  
In Equation (30), the energy conservation equation of ions in an assumption of neglecting elastic collisions is given. Herein, \(M\) is the mass of ions and \(\Phi(x)\) is the potential of sheath. \(u_{s}\) is the velocity of ions at the beginning of considered region, i.e., \(x=0\), which is taken the interface of neutral plasma and sheath. \[\frac{1}{2}Mu^{2}(x)=\frac{1}{2}Mu_{s}^{2}-e\Phi(x). \tag{30}\]  
Utilizing another assumption that neglects the inelastic collision in the sheath (i.e., without the ionization source), the continuity equation of ions evolves into Equation (31). \[n_{i}(x)u(x)=n_{is}u_{s}. \tag{31}\]  
Herein, \(n_{i}(x)\) is the ion density of arbitrary position and \(n_{is}\) is the constant density at \(x=0\), respectively. Correlating the Equations (30) and (31) and eliminating the arbitrary velocity of ions, \(u(x)\), an expression is achieved for the ion density in Equation (32). \[n_{i}(x)=n_{is}\bigg{(}1-\frac{2e\Phi}{Mu_{s}^{2}}\bigg{)}^{-1/2}. \tag{32}\]In Equation (33), the Boltzmann's relation of electron density is presented. Herein, \(n_{e}(x)\) is the arbitrary electron density and \(n_{es}\) is the constant density at \(x=0\), respectively. \(T_{e}\) is the electrons temperature. \[n_{e}(x)=n_{es}e^{\Phi(x)/T_{e}}. \tag{33}\]  
In Equation (34), the relation between \(n_{is}\) and \(n_{es}\) is given. As seen, they are equal since the sheath is always connected to bulk plasma. In Equation (35), the Poisson's equation is shown. And in Equation (36), the ion density and electron density of the Poisson's equation are replaced by the Equations (32) and (33), respectively. Herein, a new parameter, \(\xi_{sd}\), is introduced and the term \(e\xi_{s}\) represents the kinetic energy of ions at \(x=0\), as illustrated in Equation (37). \[n_{es}=n_{is}=n_{s}, \tag{34}\]  
\[\frac{d^{2}\Phi}{dx^{2}}=\frac{e}{\epsilon_{0}}(n_{e}-n_{l}), \tag{35}\]  
\[\frac{d^{2}\Phi}{dx^{2}}=\frac{en_{s}}{\epsilon_{0}}\left[\exp\frac{\Phi}{T_{ e}}-\left(1-\frac{\Phi}{\xi_{s}}\right)^{-1/2}\right], \tag{36}\]  
\[e\xi_{s}=\frac{1}{2}Mu_{s}^{2}. \tag{37}\]  
In Equation (38) the reformed [PERSON]'s equation is to be integrated (herein a particular mathematical skill of integration is used), and in Equation (39) the integrated result is calculated by utilizing the conditions of integrating limits that are expressed in Equation (40). In Equation (39), the [PERSON]'s expansion is applied to the exponential function and square root function, and then an inequality is obtained in Equation (41) at the first-order approximations of these [PERSON]'s expansions, given by the fact that the term on the left side of Equation (39), \(\frac{1}{2}\left(\frac{d\Phi}{dx}\right)^{2}\), should be larger than zero. This inequality is reformed in Equation (42) and the [PERSON]'s creetron of electropositive plasma for the red sheath of cations is obtained. \[\int_{0}^{\Phi}\frac{d\Phi}{dx}\frac{d}{dx}\left(\frac{d\Phi}{dx}\right)dx= \frac{en_{s}}{\epsilon_{0}}\int_{0}^{\Phi}\frac{d\[PERSON]}{dx}\left[\exp\frac{ \[PERSON]}{T_{e}}-\left(1-\frac{\Phi}{\xi_{s}}\right)^{-1/2}\right]dx, \tag{38}\]  
\[\frac{1}{2}\left(\frac{d\Phi}{dx}\right)^{2}=\frac{en_{s}}{\epsilon_{0}}\left[ T_{e}\exp\frac{\Phi}{T_{e}}-T_{e}+2\xi_{s}\left(1-\frac{\[PERSON]}{\xi_{s}} \right)^{1/2}-2\xi_{s}\right], \tag{39}\]  
\[n_{i}(x)u(x)=n_{is}u_{s}. \tag{31}\]0  
\[n_{i}(x)u(x)=n_{is}u_{s}. \tag{31}\]1  
\[n_{i}(x)u(x)=n_{is}u_{s}. \tag{31}\]2  
(b). Early-stage dynamics of simulated O\({}^{-}\) species  
In Figures 14 and 15, the temporal evolution of the O\({}^{-}\) density profile in the two early stages of simulation, i.e., before the self-coagulation happens, is shown. It is seen that upon increasing the simulating time, the anions are kept to be pushed inward by the ambipolar potential due to the inefficient creation source of them (i.e., low electronegativity).