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 such that u , v = λ u , v Ω for all v H 0 1 ( Ω ) , where Ω n is a bounded domain (a box containing the thunderstorm), 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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  15. W. Rison, R.J. Thomas, P.R. Krehbiel, T. Hamlin and J. Harlin, A GPS-based three-dimensional lightning mapping system: Initial observations in central New Mexico. Geophys. Res. Lett.26 (1999) 3573–3576.  
  16. G. Strang, Linear Algebra and Its Applications. Thomson, Belmont, CA, 4th edn. (2006).  Zbl0338.15001
  17. R.J. Thomas, P.R. Krehbiel, W. Rison, T. Hamlin, J. Harlin and D. Shown, Observations of VHF source powers radiated by lightning. Geophys. Res. Lett.28 (2001) 143–146.  
  18. R.J. Thomas, P.R. Krehbiel, W. Rison, S.J. Hunyady, W.P. Winn, T. Hamlin and J. Harlin, Accuracy of the lightning mapping array. J. Geophys. Res.109 (2004) D14207.  

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