## 4 Discussion
### Self-Coagulation Theory  
The steady-state continuity equation of anions that consists of free diffusion flux and negative source term (represented by the recombination) is expressed in Equation (22). \[-D_{-}\
abla^{2}n_{-}=-n_{-}n_{+}k_{rec}=-n_{-}v_{rec}. \tag{22}\]  
Slightly reforming the Equation (22) and meanwhile introducing a parameter, \(k_{-}\), a quasi-Helmholtz equation is constructed in Equation (23). \[ \begin{split}\
abla^{2}n_{-}-n_{-}\frac{v_{rec}}{D_{-}}=\
abla^ {2}n_{-}-n_{-}k_{-}^{2}=0,\\ \
abla^{2}n_{-}-n_{-}k_{-}^{2}=0.\end{split}  \tag{23}\]  
Herein, \(k_{-}^{2}=\frac{v_{rec}}{D_{-}}\). For simplicity, in Equation (24), all the subscripts of quantities are omitted. \[\
abla^{2}n-nk^{2}=0. \tag{24}\]  
Figure 13: Simulated two-dimensional profile of net chemical source, S\({}^{+}\), in (**a**) full range and (**b**) numerically truncated range. Refer to the maxima of legends of panels (**a**,**b**). The discharge conditions are the same as in Figure 4.  
In Equation (25), this quasi-Helmholtz equation is reformed by the method of separation of variables in the cylindrical coordinate system, at the assumption of azimuthal symmetry. \[ \begin{array}{l}\frac{1}{\rho}\frac{\partial}{\partial p}\left(\rho\frac{ \partial m}{\partial\rho}\right)+\frac{\partial^{2}n}{\partial z^{2}}-k^{2}n=0, \\ n(\rho,\,z)=R(\rho)Z(z),\\ Z^{\prime\prime}+v^{2}Z=0,\\ \frac{d^{2}R}{d\rho^{2}}+\frac{1}{\rho}\frac{dR}{d\rho}-\left(k^{2}+v^{2} \right)R=0.\end{array}  \tag{25}\]  
Herein, \(v^{2}\) represents the eigenvalues. By means of utilizing the homogeneous boundary conditions of the axial and ordinary differential equation, \(Z^{\prime\prime}+v^{2}Z=0\), the eigenvalues of \(v^{2}\) and the eigen functions, \(Z_{m}(z)\), are acquired in Equation (26). \[ \begin{array}{l}{\
u_{m}}^{2}=m^{2}\pi^{2}/l^{2},\\ Z_{m}=\sin(m\pi z/l),\\ Z=\sum\limits_{m=0}^{\infty}c_{m}Z_{m}=\sum\limits_{m=0}^{\infty}c_{m}\sin(m \pi z/l).\end{array}  \tag{26}\]  
As noticed, the radial and ordinary differential equation, \(\frac{d^{2}R}{d\rho^{2}}+\frac{1}{\rho}\frac{dR}{d\rho}-\left(k^{2}+{\
u_{m}}^ {2}\right)R=0\), is one zero-order imaginary Bessel equation, because of the property of negative source. Considering that the density value is limited at the axial center, the imaginary Bessel function, not the Hankel function, is adopted. Then, we obtain the expression of \(R(\rho)\) as follows in Equation (27). \[R(\rho)=d_{m}I_{0}(\sqrt{k^{2}+{\
u_{m}}^{2}}\rho)=d_{m}I_{0}(\sqrt{k^{2}+m^{2} \pi^{2}/l^{2}}\rho). \tag{27}\]  
Finally, we obtain the expression of \(n(\rho,\,z)\) in a product of \(R(\rho)\) and \(Z_{m}(z)\) in Equation (28). \[ \begin{array}{l}n(\rho,\,z)=R(\rho)Z_{m}(z)=\sum\limits_{m=0}^{\infty}c_{m} \sin(m\pi z/l)\cdot d_{m}I_{0}(\sqrt{k^{2}+{\
u_{m}}^{2}}\rho)\\ =\sum\limits_{m=0}^{\infty}a_{m}\sin(m\pi z/l)\cdot I_{0}(\sqrt{k^{2}+m^{2}\pi ^{2}/l^{2}}\rho).\end{array}  \tag{28}\]  
Next, some special mathematical skills, i.e., the limit idea, are used in Equation (29) and a delta distribution that is independent on the spatial coordinates is obtained, which interprets the self-coagulation idea. \[ \begin{array}{l}n(\rho,\,z)=R(\rho)Z_{m}(z)=\sum\limits_{m=0}^{\infty}a_{m} \sin(m\pi z/l)\cdot I_{0}(\sqrt{k^{2}+m^{2}\pi^{2}/l^{2}}\rho)\\ =\lim\limits_{m\rightarrow\infty}[a_{m}\sin(m\pi z/l)\cdot\infty]=\lim\limits_ {m\rightarrow\infty}\left[a_{m}\sin(m\pi z/l)\cdot\lim\limits_{z\to 0} \frac{1}{z}\right]\\ =\lim\limits_{z\to 0}\left[\lim\limits_{m\rightarrow\infty}a_{m} \sin(m\pi z/l)\cdot\frac{1}{z}\right]=\lim\limits_{z\to 0}\left[\lim \limits_{m\rightarrow\infty}a_{m}\cdot\frac{\sin(m\pi z/l)}{z\pi/l}\cdot \frac{\pi}{l}\right]\\ =\lim\limits_{\zeta\to 0}\left[\lim\limits_{m\rightarrow\infty}a_{m} ^{\prime}\cdot\frac{1}{\pi}\frac{\sin(m\zeta)}{\zeta}\right]=\lim\limits_{ \zeta\to 0}\left[a_{\infty}^{\prime}\lim\limits_{m\rightarrow\infty}\frac{1}{ \pi}\frac{\sin(m\zeta)}{\zeta}\right]\\ =a_{\infty}^{\prime}\lim\limits_{\zeta\to 0}\delta(\zeta).\end{array}