The change in electric potential due to lightning

Issue		ESAIM: M2AN Volume 42, Number 5, September-October 2008


Page(s)		887 - 901
DOI		10.1051/m2an:2008026
Published online		04 July 2008

ESAIM: M2AN 42 (2008) 887-901
DOI: 10.1051/m2an:2008026

The change in electric potential due to lightning

William W. Hager¹ and Beyza Caliskan Aslan²

¹ Department of Mathematics, University of Florida, PO Box 118105, 32611-8105 Gainesville, Florida, USA. hager@math.ufl.edu; http://www.math.ufl.edu/~hager
² Department of Mathematics and Statistics, University of North Florida, 32224 Jacksonville, Florida, USA. aslan@unf.edu; http://www.unf.edu/coas/math-stat/ aslan

Received September 28, 2007. Revised March 16, 2008. Published online July 4, 2008.

Abstract
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 $\langle \nabla u,\nabla v\rangle_\Omega =\int_{\Omega} \nabla u\cdot\nabla v \; {{\rm d}x}.$

Mathematics Subject Classification. 35J25, 35Q60, 35A20, 35P10.

Key words: Lightning, electric potential, Ampere's law, Maxwell's equations, Laplacian, generalized eigenproblem, double layer potential, complete eigenbasis.

© EDP Sciences, SMAI 2008

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