Time-domain solution to Maxwell's equations for a lightning dart leader and subsequent return stroke leader and subsequent return stroke

Abstract

1.IntroductionIn this study we present an analytic solution to Maxwell's Equations for the lightning leader followed by a novel return stroke model. We model the leader as a downward propagating boxcar function of uniform charge density and constant velocity, and the subsequent return stroke is modeled as an upward propagating boxcar with a time dependent velocity. Charge conservation is applied to ensure self-consistency of the driving current and charge sources, and physical observations are used to support model development. The resulting transient electric and magnetic fields are presented at various distances and delay times and compared with measured waveforms and previously published models. 2.The Lightning Dart LeaderEquations (1) through (4) below are the exact time-domain solutions to Maxwell's Equations for the scalar and vector potentials and the electric and magnetic fields of a charge and current distribution commonly derived in elementary electromagnetics texts [1]. Here, φ is the scalar potential, A is the vector potential, tr is the retarded time, E is the electric field, B is the magnetic field and ρ and J are the charge and current densities respectively.The downward propagating lightning dart leader is defined in Figure 1 as a propagating uniform current pulse originating from a negative point charge; the point charge, Q − C , represents the lower charge distribution in a typical thundercloud and Q − C represents the positive charge. The leader propagates downward at a constant velocity, vL, which is consistent with observation for a dart leader. This model considers the more common downward negative lightning flash, therefore the charge distribution in the leader will have a net negative charge. Figure 1: Leader source current and charge distribution.The leader current as a function of height and time can be written in cylindrical coordinates as:We can find the charge distribution directly from the continuity equation for charge conservation which has the added benefit of ensuring our charge and current distributions are self-consistent. The Continuity Equation:− ∂ρ ∂t = ∇ · JBy integrating (6) with respect to space and time while using (5) for the current density, the charge density can be shown to be equal to (7).