Distribution of charge along the lightning channel: Relation to remote electric and magnetic fields and to return-stroke models

We derive exact expressions for remote electric and magnetic fields as a function of the time- and height-varying charge density on the lightning channel for both leader and return-stroke processes. Further, we determine the charge density distributions for six return-stroke models. The charge density during the return-stroke process is expressed as the sum of two components, one component being associated with the return-stroke charge transferred through a given channel section and the other component with the charge deposited by the return stroke on this channel section. After the return-stroke process has been completed, the total charge density on the channel is equal to the deposited charge density component. The charge density distribution along the channel corresponding to the original transmission line (TL) model has only a transferred charge density component so that the charge density is everywhere zero after the wave has traversed the channel. For the Bruce-Golde (BG) model there is no transferred, only a deposited, charge density component. The total charge density distribution for the version of the modified transmission line model that is characterized by an exponential current decay with height (MTLE) is unrealistically skewed toward the bottom of the channel, as evidenced by field calculations using this distribution that yield (1) a large electric field ramp at ranges of the order of some tens of meters not observed in the measured electric fields from triggered-lightning return strokes and (2) a ratio of leader-to-return-stroke electric field at far distances that is about 3 times larger than typically observed. The BG model, the traveling current source (TCS) model, the version of the modified transmission line model that is characterized by a linear current decay with height (MTLL), and the Diendorfer-Uman (DU) model appear to be consistent with the available experimental data on very close electric fields from triggered-lightning return strokes and predict a distant leader-to-return-stroke electric field ratio not far from unity, in keeping with the observations. In the TCS and DU models the distribution of total charge density along the channel during the return-stroke process is influenced by the inherent assumption that the current reflection coefficient at ground is equal to zero, the latter condition being invalid for the case of a lightning strike to a well-grounded object where an appreciable reflection is expected from ground.