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• Recent progress in studying the breakthrough phase of the attachment process is reviewed. • During the BTP, hot channels of both the DL and UCL extend toward each other inside the CSZ. • During the BTP, the current rises from the UCL level to tens of kA (for negative first strokes). • Initial conductivity of common streamer zone is expected to be of the order of 10−5 S/m. Recent progress in studying the breakthrough phase (BTP) of the attachment process in lightning and long sparks is reviewed. The main focus is on the new insights gained from recent observations for 3 types of electric discharges: natural lightning (Nag et al., 2012; Tran and Rakov, 2017a,c), rocket-and-wire triggered lightning (Howard et al., 2010; Hill et al., 2016), and long sparks (Kostinskiy et al., 2016). The BTP (also known as the final jump) starts when the poorly-conducting streamer zones, developing ahead of the hot channels of negative downward leader (DL) and positive upward connecting leader (UCL), come in contact and a common streamer zone (CSZ) is formed. The beginning of BTP (establishment of CSZ) is usually marked by an abrupt current rise, a burst of dE/dt pulses (referred to as leader burst or LB), and hard X-ray emission. During the BTP, hot channels of both DL and UCL extend toward each other inside the CSZ, resulting in its shrinking, until the high-impedance CSZ is eliminated and low-impedance connection of DL to the grounded object is established. The process of bridging of CSZ by hot leader channels is accompanied by the formation of slow front (SF) in the channel current and in electric and magnetic field waveforms at both close and far distances from the channel. Attempted or relatively weak hot-channel connections producing current surges and associated field pulses superimposed on the SF (SF pulses) can occur. The SF lasts some microseconds and ends at the onset of the submicrosecond-scale fast transition (FT), which signifies the end of BTP. During the BTP, the current rises from the UCL level of the order of tens to hundreds of amperes to about 50% of the overall (SF + FT) current peak, which is of the order of tens of kiloamperes (for negative first strokes). This two orders of magnitude current rise during the BTP occurs before the collision of hot leader channels inside the CSZ; that is, before the onset of return stroke proper. [ABSTRACT FROM AUTHOR]

• Expressions to estimate the current peak from the far-field peak considering channel tilt are derived. • Expressions to reconstruct the entire current waveform from the field waveform are derived. • The expressions are derived for three most widely used engineering return-stroke models. • This is the first attempt to solve the inverse problem (infer sources from fields) for inclined channels. The inclination of a lightning return-stroke channel significantly affects the waveform of the associated electromagnetic field. In this paper, we have derived mathematical expressions to estimate the peak and waveshape of lightning return-stroke current at the channel base from the waveform of electromagnetic field observed more than some tens of kilometers away from an inclined lightning channel. Three models of the lightning return stroke are considered, which are the transmission-line (TL) model, the modified TL model with linear current decay with height (MTLL), the modified TL model with exponential current decay with height (MTLE). The peak of the lightning return-stroke current at the channel base is estimated from the peak of observed lightning electric or magnetic field using an analytical relation between the peak of channel-base current and that of the radiation field component for an inclined channel. The waveform of the channel-base current is reconstructed from the electric or magnetic field waveform using a relation between the channel-base current and the sum of induction and radiation field components for an inclined channel. It has been shown that the peak and waveshape of current at the base of an inclined channel are sufficiently accurately estimated with the proposed expressions. This work is the first attempt to solve the inverse problem (infer source parameters from electromagnetic fields) for the case of inclined lightning channel. Solving this problem is important for developing methodology of remote measurements of lightning currents. [ABSTRACT FROM AUTHOR]

Although, due to the high variability of key parameters such as the return-stroke speed, it is impossible to determine the lightning current accurately from the remotely measured electric or magnetic field for a given event, we show in this paper that, for an assumed return-stroke model, the statistical estimation (e.g. in terms of mean values and standard deviations) is possible. We show additionally that for the transmission line (TL) model, the equation permitting to infer the mean value of the return-stroke current from the mean value of electric or magnetic field and the mean value of speed has the same functional form as the well-known TL current—far field relationship. This result gives to some extent a theoretical justification to the use of lightning location systems to infer parameters of lightning current statistical distributions from measured fields alone. [Copyright &y& Elsevier]