[1] Recently, wide band measurements of the electric field near a lightning flash have been obtained by a balloon-borne electric field sonde or Esonde. This paper develops new techniques for analyzing lightning-associated charge transport in a thundercloud by using both the Esonde data and simultaneous Lightning Mapping Array (LMA) measurements of VHF pulses emitted during lightning breakdown processes. Innovations in this paper include the following: (1) A filtering procedure is developed to separate the background field associated with instrument rotation and cloud charging processes from the lightning-induced electric field change. Because of the abrupt change in the signal caused by lightning, standard filtering techniques do not apply. A new mathematical procedure is developed to estimate the background electric field that would have existed if the lightning had not occurred. The estimated background field is subtracted from the measured field to obtain the lightning-induced field change. (2) Techniques are developed to estimate the charge transport due to lightning. At any instant of time during a cloud-toground (CG) flash, we estimate the charge transport by a monopole. During an intracloud (IC) flash, we estimate the charge transport by a dipole. Since the location of the monopole and dipole changes with time, they are referred to as a dynamic monopole and a dynamic dipole. The following physical constraints are used to achieve a unique fit: charge conservation during an IC flash, separation (distance between the CG monopole charge center and the ground and separation between IC dipole charge centers both exceed a minimum threshold), location (charge is placed on lightning channel), and likelihood (after a statistical analysis based on instrument uncertainty, highly unlikely charge locations are excluded). To implement the constraint that the charge is located on the lightning channel, we develop a mathematical object called the ''pulse graph.'' Vertices in the graph are pulse locations obtained from the Lightning Mapping Array. Edges in the graph (that is, the pairs of vertices which are connected by line segments) are obtained by joining, in a systematic way, neighboring vertices. One CG and two IC flashes observed on 18 August 2004 near Langmuir Laboratory are analyzed. In the CG flash, initial strokes drained 12 C charge from an altitude of 5 km, while an intermediate stroke discharged 12 C from a higher charge center at 8 km. For the IC flashes, the current flow lagged behind the channel formation by time intervals on the order of 0.1 s, roughly the same time delay observed for lightning optical signals detected by NASA's Lightning Imaging Sensor.

Lightning charge transport is analyzed for a thunderstorm which occurred on 18 August 2004 near Langmuir Laboratory in New Mexico. The analysis employs wide band measurements of the electric field by a balloon‐borne electric field sonde or Esonde, simultaneous Lightning Mapping Array measurements of VHF pulses emitted during lightning breakdown, and Next Generation Weather Radar data. The thunderstorm was composed of two principal updrafts. In the stronger updraft the positively charged particles reached altitudes up to 14 km, and in the weaker updraft the positive particles reached 11 km altitude. The negatively charged particles generated in the updraft appeared to reach altitudes up to 10 km in the strong updraft and 8 km in the weaker updraft. Just outside the updrafts the positive and negative particles drop sharply; thereafter, they drop down at a nearly linear rate, between 1 and 2 km in altitude per 10 km in horizontal distance. Initially, as the updraft developed, most charge was transported by updraft flashes; later, after about 15 to 20 min, extensive flashes were predominant. Most cloud‐to‐ground (CG) flashes transported negative charge from outside the updraft at 6 km altitude down to ground; however, some strokes of a CG reached into a higher negative charge region closer to the updraft. Nearly 6 times as much charge was transported by intracloud (IC) flashes when compared to CG flashes. The ratio of the average charge transport for an IC flash to the average charge transport for a CG flash was 1.6, while the average generator current associated with the combined updrafts was 2.3 amperes for 40 min.

Journal of Mathematical Analysis and Applications 341 (2008) 1028-1041. doi:10.1016/j.jmaa.2007.11.007Received by publisher: 2007-10-05Harvest Date: 2016-01-04 12:20:16DOI: 10.1016/j.jmaa.2007.11.007Page Range: 1028-104

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 0 (Ω), u = 0, and λ ∈ R such that ∇u, ∇v L = λ ∇u, ∇v Ω for all v ∈ H 1 0 (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.Mathematics Subject Classification. 35J25, 35Q60, 35A20, 35P10.

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