# A 3D Numerical Model to Estimate Lightning Types for PyroCb Thundercloud
## 2 Thundercloud Model Description
In this model, the cloud-base height of tripole structure-based pyroCb thunderclouds is represented by the altitude of its lower charge layers estimated from the ground surface. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Config.**} & \multirow{2}{*}{\begin{tabular}{c} **Charge** \\ **Layer** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Altitude,** \\ \(h_{n}\) **(km)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Lateral** \\ **Range,** \\ \(r_{x}\) **\& \(r_{y}\) **\\ **(km)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Vertical** \\ **Range,** \\ \(r_{z}\) **(km)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Lateral** \\ **Centre,** \\ \(x_{0}\) **\& \(y_{0}\) **\\ **(km)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Charge** \\ **Density,** \\ **(mC/m3)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Charge,** \\ **Q (C)** \\ \end{tabular} } \\ \hline \multirow{4}{*}{1} & SC & 11.75 & 6 & 0.5 & 15 & \(-\)1.1 & \(-\)19 \\  & UP & 9.75 & 6 & 1.5 & 15 & 2.2 & 77 \\  & MN & 6.75 & 5 & 1.5 & 15 & \(-\)2.5 & \(-\)73 \\  & LP & 4.25 & 3 & 1 & 15 & 1.0 & 18 \\ \hline \multirow{4}{*}{2} & SC & 11.75 & 6 & 0.5 & 15 & \(-\)1.1 & \(-\)19 \\  & UP & 9.75 & 6 & 1.5 & 15 & 2.2 & 77 \\  & MN & 6.75 & 5 & 1.5 & 15 & \(-\)2.5 & \(-\)73 \\  & LP & 4.25 & 4 & 1.2 & 15 & 1.5 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters related to the dimensions and electrical properties of tripole structured pyroCb thundercloud. To model the impact of wind shear extension on tripole structure in pyroCb, the positioning and lateral extents of both SC and UP charge layers are consistently adapted while keeping their left boundary fixed. The electrical charge within the anvil section of the pyroCb is assumed to have initially formed within the central region of the thundercloud and subsequently moved towards the forward flank due to lateral air currents originating from the electrified area. Throughout this adjustment, the extension parameter \(d\) has been changed by 2, 4, and 8 km while the charge distribution within the upper thundercloud regions remains unchanged. The simulation assumes the ground surface (positioned at  
Figure 2: PyroCb thunderclouds exhibit the following: (**a**) a tripole structure characterized by a prevailing upper positive (UP) charge layer, a prominent middle negative (MN) and a minor lower positive (LP) charge layers, accompanied by an extra negative screening layer (SC) positioned at the top, and (**b**) the effect of wind shear to create the titled structure. \(x=0\) and \(y=0\)) beneath the pyroCb thundercloud as an unconstrained conducting plane with uniform potential. Electric potential, denoted as \(V(x,y,z)\) produced by the n\({}^{th}\) charge region in the thundercloud represented by their respective charge density \(\rho_{n}(x,y,z)\), can be determined by solving the Poisson equation:  
\[\
abla^{2}V=-\frac{\rho_{n}(x,y,z)}{\epsilon_{0}} \tag{2}\]  
where \(\epsilon_{0}\) represents the free space's permittivity. In computational terms, the total potential for the pyroCb thundercloud can be achieved by combining the influences from all individual nodes \((i,j,k)\) within the computational domain, along with their respective reflections or images given in (3). \[ \begin{split} V_{i,j,k}{}^{(m)}=V_{i,j,k}{}^{(m-1)}+\omega\left[ A(V_{i+1,j,k}{}^{(m-1)}+V_{i-1,j,k}{}^{(m)})+B(V_{i,j+1,k}{}^{(m-1)}+V_{i,j-1,k}{}^{(m )})\right.\\ \left.+C(V_{i,j,k+1}{}^{(m-1)}+V_{i,j,k-1}{}^{(m)})+\frac{D}{ \epsilon_{0}}\sum_{n}\rho_{n}(x,y,z)\right]\end{split}