The change in electric potential due to lightning

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 ∈ H 1 (Ω), u � , andλ ∈ R such that �∇ u, ∇vL = λ�∇ u, ∇v� Ω for all v ∈ H 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 dx.