Lightning Return Stroke Model Research Articles

Model of the First Lightning Return Stroke Using Bidirectional Leader Concept

AbstractWe have developed a new lightning return stroke model which takes into account a realistic charge distribution to compute the ambient electric potential. It contains a nonlinear resistance model for the development of the finite conductivity of the lightning channel. It also describes a redistribution of charges after the bottom end of the channel is attached to the potential of the ground. The model is based on solving Maxwell's equations linked to Poisson's equation, and coupled with Ohm's law. The waveshape of the current at the channel base is obtained from the model. The simulated current decreases and its rise time increases with height, which is in accordance with the luminosity observations. The electric and magnetic field waveforms modeled for different distances from the lightning flash show that most of the waveform features typically observed at these distances are well reproduced. We also successfully compare modeled magnetic field waveforms with measurements at the distances larger than 30 km.

Stability and accuracy of the finite difference time domain (FDTD) method to determine transmission line traveling wave voltages and currents: The lightning pulse

The numerical solution of the transmission line wave equation is examined using the transmission line model of a lightning return stroke (LRS) taken as a case study. The LRS is represented by a transverse electromagnetic wave travelling at a velocity close to the velocity of light along a highly ionized lightning leader channel. The leader channel is modeled as a lossy transmission line. Attempts to model the lightning return stroke by an electric circuit in order to determine the currents and radiated electromagnetic fields is categorized as the distributed transmission line model (DTLM). The DTLM method in turn may be subdivided into two categories: Distributed inductance (L), capacitance (C) and resistance (R) model (DLCRM) and the lumped LCR model (LCRM). In this brief communication, we indicate some pitfalls to watch out for in solving for the transmission line wave equationusing numerical methods, and how these may be avoided. Furthermore, we propose tests that may be applied, including the analytical solution of diffusion waves, to ensure the stability and accuracy of the numerical solution DLCRM simulations. Key words: Finite difference time domain method, numerical solution of transmission line wave equation, lightning return stroke model, lightning radiated electromagnetic fields.

A Macroscopic Model for First Return Stroke of Lightning

A lightning return stroke model for a downward flash is proposed. The model includes underlying physical phenomena governing return stroke evolution, namely, electric field due to charge distributed along the leader and cloud, transient enhancement of series channel conductance at the bridging regime, and the nonlinear variation of channel conductance, which supports the return stroke current evolution. Thermal effects of free burning arc at the stroke wave front and its impact on channel conductance are studied. A first-order arc model for determining the dynamic channel conductance along with a field-dependent conductivity for corona sheath is used in the model. The model predicts consistent current propagation along the channel with regard to current amplitude and return stroke velocity. The model is also capable of predicting the remote electromagnetic fields that are consistent with the experimental observations.

Influence of a Nonlinear Channel Resistance on Lightning-Induced Voltages on Overhead Lines

In this paper, a lightning return-stroke model proposed by the authors is used for investigating the effect of a nonlinear (current-dependent) channel resistance on lightning-induced voltages on an overhead wire. Lightning-induced voltages are shown to behave at early times as if they were calculated by assuming a lossy channel and at later times as if they were calculated by assuming a lossless channel, the transition between early and latter stages being smoothed by the continuous time decay of the channel resistance. Despite such a transitional behavior, it is argued that similar induced-voltage waveforms can be obtained if a channel resistance with constant value between the upper and lower limits associated to the leader and return-stroke channels is suitably selected. It is also shown that in the cases evaluated in this paper simple engineering return-stroke models are able to lead to lightning-induced voltages in relatively good agreement with those predicted by the considered nonlinear return-stroke model. Finally, lightning-induced voltages calculated using the nonlinear return-stroke model are shown to be in good agreement with experimental data.

On the use of lumped sources in a nonlinear lightning return stroke model and extension for evaluating strikes to tall objects

The use of lumped current and voltage sources is investigated in a nonlinear return stroke model that includes the dependence of the channel per‐unit‐length resistance on the lightning current and the variation of the channel impedance with height. First, considering the case of lightning strikes to flat ground, it is shown that equivalent channel currents can be obtained regardless of the lumped source type (current or voltage) connected at the channel base. This equivalency is obtained provided the effect of current reflections coming from the upper part of the channel is taken into account in the derivation of the voltage source waveform, which is made according to a procedure proposed by the authors. The nonlinear return stroke model is then extended for evaluating lightning strikes to tall grounded objects. Two different possibilities are considered for selecting the channel impedances in this case. In one approach, the impedance of each channel segment is calculated by considering its total height to ground whereas in the other approach the distance from each channel segment to the excitation point at the tower top is considered. It is shown that currents calculated along a lossy channel are nearly independent of the considered approach, whereas differences of about 10% are expected in the peak values of currents calculated along a 102‐m‐high object. Also, regardless of the selected approach, relatively small errors are observed in the evaluation of the total charge transferred to ground, which is convenient for allowing a consistent comparison between lightning strikes to flat ground and lightning strikes to grounded objects of different heights. Finally, comparisons with results obtained with the transmission line return stroke model show that the nonlinear interaction between the channel resistance and the lightning current has a moderate effect on the predicted tower currents, as opposed to the channel currents, which are greatly affected by such interaction.

A comparison of different approaches to simulate a nonlinear channel resistance in lightning return stroke models

Different physical models that describe the time variation of the channel resistance are investigated in a lightning return stroke model. Such models consider one of the three following hypotheses: (1) the channel resistance decays exponentially with time; (2) the channel resistance decays with the radial expansion of the channel core, which is assumed to be described by the strong‐shock approximation, or (3) the channel resistance varies with time according to three different arc resistance models (defined by Toepler, Barannik and Kushner et al.). Analyses illustrate the effect of a time‐varying channel resistance on channel currents and corresponding electromagnetic fields. It is shown that the strong‐shock approximation is able to predict typical features of experimentally observed lightning electromagnetic fields and return stroke speed profiles. It is also shown that results predicted by the strong‐shock approximation can be qualitatively reproduced by either using simplified arc resistance equations (such as Toepler's and Barannik's ones) or considering an exponential decay of the channel resistance with attenuation constants linearly increasing with height.

Expressions for far electric fields produced at an arbitrary altitude by lightning return strokes

Electromagnetic fields produced at high altitudes by return strokes in cloud‐to‐ground lightning are needed in studies of transient luminous events in the mesosphere. Such calculations require the use of a lightning return stroke model. Two of the widely used return stroke models are (1) the modified transmission line model with exponential decay (MTLE) of current with height and (2) the modified transmission line model with linear decay (MTLL) of current with height. In this paper, simplified expressions based on the MTLE and MTLL models are derived for calculating far (radiation) electric fields produced at an arbitrary elevation angle by lightning return strokes. It is shown that different (for example, containing either spatial or time integral), but equivalent equations can be derived for each of the models. Predictions of simplified expressions are compared with electric fields computed using exact expressions, including all the field components, and the validity of simplified expressions for distances that are much greater than the radiating channel length is confirmed.

Reproduction of Features of Electromagnetic Field Waveforms Associated with Lightning Return Stroke

A lightning return stroke model is indispensable in calculating electromagnetic field associated with a return stroke. A lot of return stroke models have been proposed, however, no model has ever succeeded in reproducing all the features of typical electromagnetic field waveforms at various distances. In this paper, two return stroke models are proposed. One is a Transmission-Line type model, and the other is modification of Diendorfer-Uman model. These models successfully reproduce all the features of typical electromagnetic field observed on ground.

Lightning return-stroke model incorporating current distortion

A lightning return-stroke model of transmission-line (TL) type, which incorporates both the attenuation and distortion of the waveform of the propagating current, is proposed. With the proposed model, the influence of the distortion of the propagating current on associated electromagnetic field waveforms is investigated and found to be small.

A new lightning return stroke model based on antenna theory

A new approach based on antenna theory is presented to describe the lightning returnstroke process. The lightning channel is approximated by a straight and vertical monopole antenna with distributed resistance (a so‐called lossy antenna) above a perfectly conducting ground. The antenna is fed at its lower end by a voltage source such that the antenna input current, which represents the lightning return‐stroke current at the lightning channel base, can be specified. An electric field integral equation (EFIE) in the time domain is employed to describe the electromagnetic behavior of this lossy monopole antenna. The numerical solution of EFIE by the method of moments (MOM) provides the time‐space distribution of the current and line charge density along the antenna. This new antenna‐theory (or electromagnetic) model with specified current at the channel base requires only two adjustable parameters: the return‐stroke propagation speed for a nonresistive channel and the channel resistance per unit length, each assumed to be constant (independent of time and height). The new model is compared to four of the most commonly used “engineering” return‐stroke models in terms of the temporal‐spatial distribution of channel current, the line charge density distribution, and the predicted electromagnetic fields at different distances. A reasonably good agreement is found with the modified transmission line model with linear current decay with height (MTLL) and with the Diendorfer‐Uman (DU) model.

Reconstruction of lightning currents and return stroke model parameters using remote electromagnetic fields

Estimating parameters of a lightning return stroke model and the current at the base of the lightning channel from measurements of remote electromagnetic fields is an important practical problem. In the present paper we apply a genetic algorithm (a global optimization technique) to the above problem. Firstly, we use the genetic algorithm to reconstruct the return stroke model parameters using the channel‐base current and the remote electromagnetic field measured simultaneously. The channel‐base current is used to calculate the electric field at the measurement point. The difference between the calculated and the measured fields is minimized using the genetic algorithm, which finds the optimal solution for the return stroke parameters. The transmission line and Diendorfer‐Uman return stroke models are used in the illustrations. Secondly, we develop a method to reconstruct the return stroke model parameters and channel‐base current from electric or magnetic fields measured at two different distances, one of which is a far‐field and the other is a near field. The Diendorfer‐Uman model is used in the reconstruction. Thirdly, we develop a method to reconstruct the return stroke model parameters and channel‐base current from two magnetic fields at two different distances, one of which is an intermediate field and the other is a near field. For doing the above with the Diendorfer‐Uman model we derive an explicit inversion formula using both the induction and the radiation terms of the magnetic field.

Lightning return stroke model with height‐variable discharge time constant

A new lightning return stroke model is proposed in which the lightning channel, previously charged by the leader, is exponentially discharged with the discharge time constant being a general function of height. The new model is both more general and more physically reasonable than the Diendorfer‐Uman model (Diendorfer and Uman, 1990) and its modification by Thottappillil et al. (1991). In the new model a single height‐variable discharge time constant is used instead of the two height‐invariant discharge time constants used for natural lightning in the Diendorfer‐Uman model. As an illustration of the use of the new model, the measured electric fields for three triggered lightning return strokes about 5‐km distant are compared with the fields calculated from the model, using simultaneously measured channel base current and return stroke speed as inputs to the model.

Extension of the Diendorfer‐Uman lightning return stroke model to the case of a variable upward return stroke speed and a variable downward discharge current speed

A new lightning return stroke model has recently been proposed by Diendorfer and Uman (1990). In this model, if one specifies a current at the channel base (ground), a constant return stroke speed, and a channel discharge time constant, one can derive analytically the current and charge as a function of time and height associated with the channel above ground. Here we present a more general and more straightforward derivation of the Diendorfer‐Uman model. In the new formulation we allow a variable return stroke speed that can be any arbitrary function of height. The influence of a decrease in speed with height, as occurs in nature, on the channel current and charge distributions and on the radiated electric and magnetic fields is determined and compared with the constant speed case. For a given channel base current, a decreasing speed with height does not change the initial peak electric and magnetic fields appreciably from the values found for a constant speed having the same value as the variable speed at the channel base, but a decreasing speed can cause considerable changes in field waveshapes for both the distant radiation fields and the near electrostatic fields. In addition to allowing an arbitrary return stroke speed, the new formulation of the Diendorfer‐Uman model allows the downward propagating current waves released by the return stroke front also to have an arbitrary variable speed, whereas the original model assumed that speed to be constant at the speed of light.

On the Master, Uman, Lin, Standler and the Modified Transmission Line Lightning return stroke current models

We show that the lightning return stroke model by Master, Uman, Lin, and Standler (MULS), a modification of the Lin, Uman, Standler (LUS) model, can be viewed mathematically as having a specified channel‐base current, although in the original formulation it was the channel current sources that were specified. The specification of the MULS channel‐base current is independent of the time dependence of the corona current sources which characterize the MULS model. Further, we show that, when starting from the same channel‐base current, the MULS return stroke model, except for the uniform current component, is mathematically equivalent to the Modified Transmission Line (MTL) model. For both the MULS and the MTL models we present a single analytical expression relating the far fields and the channel‐base current. This formulation provides for a simpler procedure to infer MULS channel‐base currents from remote electric and magnetic field measurements than was previously available and hence to test experimentally the MULS model.

The Limiting Accuracy of Long Wavelength Lightning Flash Location

Abstract Many recently developed systems for lightning flash location use frequencies of hundreds of kHz or higher to achieve enhanced precision, although this reduces potential range. However, for coherent systems operating at high signal-to-noise ratios VLF observing frequencies can reduce the target to a “point image” with typical horizontal dimensions of 10 m or less, potentially reducing relative location errors. Absolute location errors are larger, but may be below 1 km for a well calibrated system. Thus there is no fundamental conflict between potential aims of precision in flash location and long range. A typographical error in Lin et al.'s description of their lightning return stroke model is reported.