The change in electric potential due to lightning

Hager, William W. ; Aslan, Beyza Caliskan

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008), p. 887-901 / Harvested from Numdam

Résumé

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 $u \in H_0^1 (\Omega )$, $u \ne 0$, and $\lambda \in \mathbb {R}$ such that $ \langle \nabla u, \nabla v \rangle _{\mathcal {L}} = \lambda \langle \nabla u, \nabla v \rangle _{\Omega } $ for all $v \in H_0^1 (\Omega )$, where $\Omega \subset \mathbb {R}^n$ is a bounded domain (a box containing the thunderstorm), $\mathcal {L}$ is a subdomain (the lightning channel), and $\langle \cdot , \cdot \rangle _{\Omega }$ is the inner product $ \langle \nabla u,\nabla v\rangle _\Omega =\int _{\Omega } \nabla u\cdot \nabla v \; {{\rm d}x}. $

Détail

Publié le : 2008-01-01

MR 2454626 | Zbl 1152.35027

DOI : https://doi.org/10.1051/m2an:2008026
Classification:  35J25,  35Q60,  35A20,  35P10

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