The change in electric potential due to lightning

William W. Hager; Beyza Caliskan Aslan

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 5, page 887-901
  • ISSN: 0764-583X

Abstract

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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 ∈ H 0 1 ( Ω ) , u ≠ 0 , and λ ∈ ℝ such that 〈 ∇ u , ∇ v 〉 ℒ = λ 〈 ∇ u , ∇ v 〉 Ω for all v ∈ H 0 1 ( Ω ) , where Ω ⊂ ℝ n is a bounded domain (a box containing the thunderstorm), ℒ is a subdomain (the lightning channel), and 〈 · , · 〉 Ω is the inner product 〈 ∇ u , ∇ v 〉 Ω = ∫ Ω ∇ u · ∇ v d x .

How to cite

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Hager, William W., and Aslan, Beyza Caliskan. "The change in electric potential due to lightning." ESAIM: Mathematical Modelling and Numerical Analysis 42.5 (2008): 887-901. <http://eudml.org/doc/250344>.

@article{Hager2008,
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\}. $},
author = {Hager, William W., Aslan, Beyza Caliskan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Lightning; electric potential; Ampere's law; Maxwell's equations; Laplacian; generalized eigenproblem; double layer potential; complete eigenbasis.; lightning; Laplacian; complete eigenbasis},
language = {eng},
month = {7},
number = {5},
pages = {887-901},
publisher = {EDP Sciences},
title = {The change in electric potential due to lightning},
url = {http://eudml.org/doc/250344},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Hager, William W.
AU - Aslan, Beyza Caliskan
TI - The change in electric potential due to lightning
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/7//
PB - EDP Sciences
VL - 42
IS - 5
SP - 887
EP - 901
AB - 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}. $
LA - eng
KW - Lightning; electric potential; Ampere's law; Maxwell's equations; Laplacian; generalized eigenproblem; double layer potential; complete eigenbasis.; lightning; Laplacian; complete eigenbasis
UR - http://eudml.org/doc/250344
ER -

References

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  16. G. Strang, Linear Algebra and Its Applications. Thomson, Belmont, CA, 4th edn. (2006).  
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