The change in electric potential due to lightning

Hager, William W.; Aslan, Beyza Caliskan

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 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 ${\langle \nabla u,\nabla v\rangle }_{ℒ}=\lambda {\langle \nabla u,\nabla v\rangle }_{\Omega }$ 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 ${\langle ·,·\rangle }_{\Omega }$ is the inner product ${\langle \nabla u,\nabla v\rangle }_{\Omega }={\int }_{\Omega }\nabla u·\nabla v\mathrm{d}x.$

Classification : 35J25, 35Q60, 35A20, 35P10
Keywords: lightning, electric potential, Ampere's law, Maxwell's equations, laplacian, generalized eigenproblem, double layer potential, complete eigenbasis

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