The change in electric potential due to lightning
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 5, p. 887-901

The change in the electric potential due to lightning is evaluated. The potential along the lightning channel is a constant which is the projection of the pre-flash potential along a piecewise harmonic eigenfunction which is constant along the lightning channel. The change in the potential outside the lightning channel is a harmonic function whose boundary conditions are expressed in terms of the pre-flash potential and the post-flash potential along the lightning channel. The expression for the lightning induced electric potential change is derived both for the continuous equations, and for a spatially discretized formulation of the continuous equations. The results for the continuous equations are based on the properties of the eigenvalues and eigenfunctions of the following generalized eigenproblem: Find $u\in {H}_{0}^{1}\left(\Omega \right)$, $u\ne 0$, and $\lambda \in ℝ$ such that ${〈\nabla u,\nabla v〉}_{ℒ}=\lambda {〈\nabla u,\nabla v〉}_{\Omega }$ for all $v\in {H}_{0}^{1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^{n}$ is a bounded domain (a box containing the thunderstorm), $ℒ$ is a subdomain (the lightning channel), and ${〈·,·〉}_{\Omega }$ is the inner product ${〈\nabla u,\nabla v〉}_{\Omega }={\int }_{\Omega }\nabla u·\nabla v\phantom{\rule{0.277778em}{0ex}}\mathrm{d}x.$

DOI : https://doi.org/10.1051/m2an:2008026
Classification:  35J25,  35Q60,  35A20,  35P10
Keywords: lightning, electric potential, Ampere's law, Maxwell's equations, laplacian, generalized eigenproblem, double layer potential, complete eigenbasis
@article{M2AN_2008__42_5_887_0,
author = {Hager, William W. and Aslan, Beyza Caliskan},
title = {The change in electric potential due to lightning},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {42},
number = {5},
year = {2008},
pages = {887-901},
doi = {10.1051/m2an:2008026},
zbl = {1152.35027},
mrnumber = {2454626},
language = {en},
url = {http://www.numdam.org/item/M2AN_2008__42_5_887_0}
}

Hager, William W.; Aslan, Beyza Caliskan. The change in electric potential due to lightning. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 5, pp. 887-901. doi : 10.1051/m2an:2008026. http://www.numdam.org/item/M2AN_2008__42_5_887_0/

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