## 2 Leader Model Description
(2007) [14] have shown that the maximum line charge density \(\rho_{L0}(z)\) for \(z_{tip}\geq 10\) m can be expressed based on the following empirical formula:  
\[\rho_{L0}(z)=k\cdot\left(1-\frac{z}{H-z_{tip}}\right)\cdot G\left(z_{tip} \right)\cdot I_{P}+\frac{I_{P}\cdot\left(a+b\cdot z\right)}{1+c\cdot z+d\cdot z ^{2}}\cdot L\left(z_{tip}\right) \tag{1}\]  
where \(H\) is the cloud charge source height in [m], \(z\) and \(z_{tip}\) are in [m], and \(I_{P}\) is the return-stroke peak current in [A]. The other symbols represent empirical constants or functions determined in [PERSON] et al. (2007) [14] and given below:  
\[ \begin{array}{c}k=5.09\times 10^{-6}\ \frac{s}{m},\ a=1.325\times 10^{-8}\ \frac{s}{m},\ b=7.06\times 10^{-9}\ \frac{s}{m^{2}},\\ c=2.089\ \frac{1}{m},\ \text{and}\ d=1.492\times 10^{-2}\ \frac{1}{m^{2}} \end{array}  \tag{2}\]  
\[G\left(z_{tip}\right)=1-\frac{z_{tip}}{H},\ \ \ L\left(z_{tip}\right)=0.3\cdot \alpha\left(z_{tip}\right)+0.7\cdot\beta\left(z_{tip}\right) \tag{3}\]  
\[\alpha\left(z_{tip}\right)=e^{-\frac{z_{tip}-10\ m}{\gamma\otimes\alpha}},\ \ \ \beta\left(z_{tip}\right)=1-\frac{z_{tip}}{H} \tag{4}\]  
Leader channel consists of a narrow hot core of radius \(R_{c}\) and a radial corona sheath of thickness \(R_{cor}\) (see inset in Figure 2) with \(R_{cor}\) increasing with time. Based on [PERSON]'s  
Figure 2: The downward-extending leader geometry used in the modeling. The leader moves along a previously created, but decayed channel, not shown in this Figure. Adapted from [PERSON] et al. (2010) [9]. Law, the maximum thickness of corona sheath at height \(z\), \(R_{cor}(z)\), can be found using the line charge density, \(\rho_{L0}(z)\), the breakdown electric field, (assumed to be \(E_{c}=3\times 10^{6}\) V/m), and the electric permittivity of air, \(\varepsilon_{0}=8.854\times 10^{-12}\) F/m, as follows:  
\[R_{cor}(z)=\frac{\rho_{L0}(z)}{2\cdot\pi\cdot\varepsilon_{0}\cdot E_{c}} \tag{5}\]  
Assuming that the charge at \(z\) exponentially increases from 0 to \(\rho_{L0}(z)\) with time constant \(\tau=0.5\) us (See [PERSON] et al. (2010) [9]), we express the line charge density at height \(z\) seen at the observation point P (see Figure 2) as:  
\[\rho_{L}(z,\ t^{\prime})=\rho_{L0}(z)\cdot\left(1-e^{-t^{\prime}/\tau}\ \right) \tag{6}\]  
where \(t^{\prime}\) is the retarded time required for the electric field to propagate from the source element \(dz\) to the observation point P when the downward-moving leader tip has arrived at height \(z_{tip}\). The retarded time, \(t^{\prime}\), can be computed as:  
\[t^{\prime}=\frac{z}{v}-\frac{z+z_{0}}{c} \tag{7}\]  
where \(v\) is the downward leader speed and \(c\) is the speed of light (\(c=3\times 10^{8}\) m/s). The total radius of the leader channel, \(R_{S}\) (see inset in Figure 2), at height \(z\) increases from \(R_{c}\) (assumed to be equal to 1 mm according to [PERSON] (1996) [15]) to \(R_{cor}(z)\), computed using Equation (5). Following [PERSON] et al. (2010) [9], we assumed that the expansion of corona sheath is exponential and the total leader channel radius is given by:  
\[R_{S}(z,\ t^{\prime})=R_{c}+R_{cor}(z)\cdot\left(1-e^{-\frac{2\cdot 2t^{\prime} }{R_{S}(z)}}\ \right) \tag{8}\]  
where \(t^{\prime}\) is the retarded time computed using Equation (7), 2.3 is an empirical constant determined in [PERSON] et al. (2010) [9], and \(T_{S}(z)\) is time during which the corona sheath attains its maximum extent at height \(z\), estimated as:  
\[T_{s}(z)=\frac{R_{cor}(z)}{v_{s}} \tag{9}\]  
with \(v_{s}\) being the speed of the streamers forming the corona sheath, which was set to \(2\times 10^{5}\) m/s, according to [PERSON] (2014) [16].