The characteristics and simulation of close leader/return stroke field change waveforms

Authors

  • Qilin Zhang,

    1. Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China
    2. College of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing, China
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  • Xiaodong Liu,

    1. Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China
    2. College of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing, China
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  • Jing Yang,

    1. Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China
    2. College of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing, China
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  • Rubin Jiang,

    1. Laboratory for Middle Atmosphere and Global Environment Observation, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
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  • Zhenhui Wang,

    1. Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China
    2. College of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing, China
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  • Jianchun Bian

    1. Laboratory for Middle Atmosphere and Global Environment Observation, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
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Abstract

[1] In order to reproduce the V-shape structure characteristics of the close dart leader/return stroke field change, we employ two existing models, one for the “source charge” leader model and the other for MTLL return stroke model, both based on the assumption of uniform leader charge distribution along the channel and the charges deposited by the dart leader are completely neutralized by the following return stroke process. The simulated results show that the return stroke electric field is inversely related with the return stroke speed at early times (within few tens of microseconds of the beginning of the return stroke), while at later times (after 100 μs or so) the field is dominated by the deposited charge density component, the close electric field is independent of speed and the ratio of the leader field to the corresponding return stroke field tends to be equal to −1. Therefore, at early times (within few tens of microseconds) there is often some uncertainty regarding whether the charges deposited by the dart leader are completely neutralized by the following return stroke process based on the difference between the return stroke and the leader field on the ground. However, although several other return strokes models (TL, MTLE, TCS, BG and DU) exist, the results shown in the paper are strictly valid only for the used MTLL model.

1. Introduction

[2] The electric field changes produced by the close dart leader/return stroke sequences appear as asymmetrical V-shape pulses, the bottom of the V being the transition from leader to return stroke [e.g., Rubenstein et al., 1995; Crawford et al., 2001; Rakov et al., 2005; Qie et al., 2007; Zhang et al., 2009a]. It is generally believed if the charge deposited by the dart leader is completely neutralized by the return stroke process, the magnitude of the return stroke field ΔERS (the trailing edge of the V-shaped pulse) is expected to be equal to the corresponding leader field magnitude ΔEL (the leading edge of the V-shaped pulse) within a few tens of microseconds of the beginning of the return stroke, at distances of the order of a few tens of meters or less [Thottappillil et al., 1997]. Rakov et al. [2005] observed that some return stroke electric field is appreciably smaller than that of the leader at 15 m and 30 m from the lightning channel, this difference between the return stroke field and the dart leader field was referred to as the residual electric field RE = ΔEL − ΔERS.They examined the value of the residual field (RE) at 20 μs after the beginning of the return stroke, found the RE decreases with the increasing of distance, and the ratio of REs at 15 m and 30 m is 3. Further, Rakov et al. [2005] specially stated that “for many strokes analyzed here, the residual field was observed at 15 m but was undetected at 30 m.” They considered that the RE implies that some charges deposited in the lightning channel by the leader but left unneutralized by the return stroke, which is found from modeling to be associated with an equivalent point charge of the order of the hundreds of microcoulombs to a few millicoulombs at a height of 15 m to 30 m deposited by the leader but presumably left unneutralized by the return stroke, and the nature of the residual charge is unknown, it could be associated with small branches formed near the descending leader tip just prior to or during the attachment process (such branches have been observed in long laboratory spark experiment) [Rakov et al., 2005].

[3] In the paper, we analyze in detail the V-shape structure characteristics of the close dart leader/return stroke field change at 30 m and 60 m in Shandong Artificially Triggered Lightning Experiment (SHATLE) in 2005 and 2009, and find the magnitude of the return stroke electric field may be smaller, or equal to, or larger than that of the dart leader on tens of microseconds timescale, and these observed results are consistent with Rakov et al. [2005]. In order to study the RE nature, based on the assumption of uniform leader charge distribution along the channel and the charges deposited by the dart leader are completely neutralized by the following return stroke process, we will employ two existing models, one for the “source charge” leader model and the other for MTLL return stroke model, to simulate the V-shape structure characteristics and study the effect of the return stroke speed on the RE.

2. Data Analysis

[4] Shandong Artificially Triggered Lightning Experiment (SHATLE) has been conducted since the summer of 2005 in Shandong Province(117°48′E, 37°42′N), middle latitude region of eastern China [Qie et al., 2007, 2009; Yang et al., 2008a, 2008b, 2009, 2010; Zhang et al., 2009a, 2009b; Zhao et al., 2009; Jiang et al., 2011]. Figure 1 shows the return stroke current waveform for stroke 4 of flash number 0901 and the corresponding leader/return stroke electric field change at 30 m. The waveform of the electric field change is characterized by an asymmetrical V-shaped structure, the electric field begins with a relatively slow negative field changes consistent with the lowering of negative change of the dart leader, this slow change is followed by a fast positive field change due to the removal of dart leader charge by the return stroke. The bottom of asymmetrical V-shape is corresponding to the end of dart leader process and the start of return stroke, and the bottom of V-shape is assumed to be the moment t = 0.

Figure 1.

(a) The leader/return stroke electric field change waveform at 30 m and (b) the corresponding return stroke current waveform at the channel bottom for stroke 4 of flash number 0901. The start time of the return stroke is assumed to be t = 0.

[5] In measuring fields from microsecond-scale electric field records, there is often some uncertainty regarding the actual leader starting point. We assume that the leader field change starts from the zero field level, which implies that there is no electric field offset at the time of leader beginning. In order to determine the end time of return stroke, the height of the discharge charge source is assumed to be from 3 km to 5 km and the return stroke speed is about 108 m/s, thus the return stroke field is measured from the V-shape bottom to a point on the electric field waveform from 30 μs to 50 μs after the fast positive field change begins. However, note in Figure 1 within a few microseconds after the beginning of the return stroke (before the return stroke reaches the lightning channel top), although the channel-base current waveform in excess of 9 kA continues to flow to ground, the return stroke electric field is approximately maximum and “flattens” at a level that is a bit larger than the prestroke level.

[6] Figure 2 shows other four leader/return stroke electric field change waveforms at 30 m and 60 m, and Figure 3 further gives the statistical results of the ratio of the leader to return stroke field. The magnitude of the return stroke electric field is measured at 30 μs and 50 μs after the return stroke begins, respectively, and the 30 m mean values of the ratio of the leader to return stroke field at 30 μs and 50 μs is −0.89 (ranging from −0.8 to −1.1) and −0.85 (ranging from −0.8 to −1.02), respectively. The 60 m mean values of the ratio at 30 μs and 50 μs is −0.93 (ranging from −0.74 to −1.18) and −0.86 (ranging from −0.66 to −1.0), respectively. In addition, we found that the ratio of the leader to return stroke field at 20 μs (the time in which Rakov et al. [2005] examined the RE) approximately equal to that at 30 μs and 50 μs, because within tens of microseconds after the beginning of the return stroke, the return stroke electric field is approximately maximum and “flattens” at a level. Comparing our single-station field measurements with multiple-station ones of Rakov et al. [2005], in our experiment only one of eight (about 13%) events at 30 m do show RE features, and about 87% show that return stroke field is equal or larger than the leader fields, while in the latter study, 84% of events at 30 m show RE features, and 16% equal or larger than the leader fields (maybe our samples are limited). If we assume that the RE (the leader electric field is larger than that of return stroke) implies some charges deposited in the lightning channel by the leader but left unneutralized by the return stroke, however, how to explain that the leader electric field magnitude is smaller than that of the corresponding return stroke (seen as Figures 2b and 2c)?

Figure 2.

Vertical electric field waveform for (a and b) strokes 1 and 2 of flash number 0901 at 30 m, and for (c and d) stroke 1 of flash number 0501 and stroke 2 of flash number 0504 at 60 m.

Figure 3.

Histograms of the ratio of the leader field to the corresponding return stroke field for (a) 30 m and (b) 60 m from the lightning channel. Solid bars indicate the return stroke field is measured from the V-shape bottom to a point at 30 us after the fast positive field change begins. Open bars indicate the return stroke field is measured from the V-shape bottom to a point at 50 us after the fast positive field change begins.

3. The Formulation of the Dart Leader/Return Stroke Model

[7] Assumed that the charge density distribution of the dart leader is uniform along the lightning channel (the “source charge” leader model [Schonland, 1953; Kasemir, 1960]) and the charges deposited by the dart leader are completely neutralized by the following return stroke process. The return stroke current waveform is assumed to be independent on height (no distortion, only attenuation), and this implies that in return stroke process only charge stored above the given channel section (no more, no less) is to be transferred through this section to ground, for example, total channel charge will flow through the channel base (z = 0), and no charge will flow through the effective channel top (z = H).

[8] Hence, the return stroke current attenuation factor P(z) at height z simply is a ratio of the leader charge distributed along the channel above the channel section at height z and the total leader charge deposited into the channel of effective height H [Rakov and Dulzon, 1991]:

equation image

If the leader charge is assumed to be uniform distribution along the channel (q(z) = const), which is consistent with the observed results of many researches [Rubenstein et al., 1995; Crawford et al., 2001; Kodali et al., 2005], then it follows from equation (1) that

equation image

This is just right the modified transmission line model MTLL with a linear decay of current with height (i (z, t) = P(z)i(0, tz/v)) [Rakov and Dulzon, 1991]. The combination of the MTLL model and the “source charge” leader model implies that the charges deposited by the dart leader are completely neutralized by the following return stroke processes, the inputted parameters in this model are the measured channel-base return stroke current and the return stroke speed. However, the combination of the “source charge” leader model and other return stroke models (for example, BG model [Bruce and Golde, 1941]; TL model [Uman et al., 1973]; TCS model [Heilder, 1985]; MTLE model [Nucci et al., 1988] and DU model [Diendorfer and Uman, 1990]) implies that the leader charges may be left unneutralized or “overneutralized” by the following return stroke processes, because the charge density deposited by other return stroke models are not uniform along the channel (no deposited charge component for TL model).

4. Result and Analysis

4.1. The Simulation of the Channel-Base Current Waveform

[9] Assumed that the lightning channel is composed of the central core having a diameter of not more than a few centimeters and the corona sheath with a diameter of about 2 m, and the discharging process includes two components, a breakdown current from discharging the central core charges, and a corona current from discharging the corona sheath [Cabrera and Cooray, 1992]. Each of the two components is calculated using the analytical expression suggested by Heilder [1985].

equation image

where i0 is the amplitude, u the amplitude correction factor, τ1 the current risetime constant, and τ2 the current decay time constant. Figure 4 shows the comparison of the simulated and measured return stroke current waveforms for stroke 4 of flash number 0901 (seen as Figure 1b), the dashed line is the simulated value, and the solid line is the measured value. iBD is the breakdown current from discharging the leader core, ic is the corona current from discharging the corona sheath, and I is total current I = ic + iBD).The mean fitted discharging parameters of six return stroke current waveforms for breakdown current iBD:i0 = 11.8 kA, u = 0.83, τ1 = 0.5 μs, τ2 = 13 μs; for corona current ic:i0 = 4.5 kA, u = 0.84, τ1 = 10 μs, τ2 = 80 μs.

Figure 4.

Illustration of the components of the measured channel-base current waveforms for stroke 4 of flash number 0901.

4.2. The Effects of the Return Stroke Speed on the Charge Density Distribution

[10] Using the continuity current equation, distribution of the charge density along the return stroke channel in MTLL model can be expressed as [Thottappillil et al., 1997]

equation image

Where, the first term of (4) represents transferred charges, this term diminishes with the diminishing current. The second term, representing deposited charge, approaches a constant value, seen as Figure 5, which implies a uniform distribution of the leader charge previously stored on the channel (assuming that the return stroke neutralizes all leader charges in the channel). With equation (4), Figure 6 further shows the effect of return stroke speed on the charge density distribution versus 100 m (Figure 6a) and 2 km (Figure 6b) height along the channel for MTLL model, and the total charge density includes the deposited charge and transferred charge. Solid line and dotted line correspond to the return stroke speed v = 1.3 × 108 m/s and v = 1.7 × 108 m/s, respectively. We can see that the transferred charge density decreases gradually, and an increase of speed causes a decrease of this charge density component; the deposited charge density increases gradually, and the speed has no effect on this component. Within a few tens of microseconds of the beginning of the return stroke, the total charge density is inversely related with the return stroke speed.

Figure 5.

Distribution of the return stroke charge density (the total charge density) versus height along the channel at different times (10 μs, 20 μs, 40 μs, 100 μs and 200 μs after return stroke begins) for MTLL model. The total length H of the lightning channel is assumed to be 7.5 km, and the speed of the return stroke is 1.3 × 108 m/s.

Figure 6.

Effect of the return stroke speed on the charge density distributions (the total charge density, the transferred charge and deposited charge density components) versus (a) 100 m height and (b) 2 km height along the channel for MTLL model.

4.3. The Effects of the Return Stroke Speed on the Return Stroke Field Component

[11] The ground is considered a perfectly conducting plane, the lightning channel is straight and vertical to the ground, the geometry is shown as Figure 7, and vertical electric field at any point on the ground can be expressed as follows [Thottappillil et al., 1997; Thottappillil and Rakov, 2001].

equation image

Where, tb(z′)is the time at which the return stroke current pulse front is “seen” by an observer on the ground, tb(z′) = (z2 + r2)1/2/c. L′(t) is the length of the channel which an observer “sees” at time t, t = L′(t)/v + (L′(t)2 + r2)1/2/c, v is the return stroke speed, c is velocity of light, and i(z′, t) is the return stroke current distribution along the channel, the other parameters are shown as Figure 7.

Figure 7.

Geometry used in deriving the expressions for the return stroke electric field at any point on the ground.

[12] The first term of (5) is the electrostatic field component, the second term is the induction field component, and the third term is the radiation field component. Based on the return stroke current waveform measured at the bottom of the channel shown as Figure 4 and the MTLL return stroke model, the structure component of the return stroke electric field on the ground at 30 m and 60 m are calculated by equation (5), and the results are shown as Figure 8. Solid line and dotted line show the electric field components corresponding to the return stroke speed v = 1.3 × 108 m/s and v = 1.7 × 108 m/s, respectively. The labels EQ, EI, ER, and ET indicate the electrostatic, induction, radiation component, and total field. It can be seen that EQ component is very closely related with the return stroke speed (ER is very little and nearly identical for two speeds at very close distance). The faster the return stroke speed, the less EQ component. Within distance 60 m the return stroke field is primary electrostatic field directly related with the charge density distribution along the channel, within a few microseconds after the beginning of the return stroke, the increase of return stroke speed causes the decrease of the total charge density (seen as Figure 6), and the decrease of the charge density furthers causes the decrease of return stroke field at very close distance.

Figure 8.

Effect of the return stroke speed on the electric field component. Solid line and dotted line show the electric field components corresponding to the return stroke speed v = 1.3 × 108 m/s and v = 1.7 × 108 m/s, respectively.

4.4. The Simulation of the V-Shape Structure of the Dart Leader/Return Stroke Field Change

[13] Based on the fitted return stroke current waveform (seen as Figure 4), the charge density of the leader model can be calculated with the second term of equation (4). The close whole electric field changes radiated both by the dart leader and the return stroke can be simulated with the “source charge” leader model and MTLL return stroke model. Figure 9 shows the simulation results of the V-shape structure of the dart leader/return stroke field change at 30 m and 60 m, and the height of the charge source is assumed to be H = 7.5 km, the leader speed is v = 106 m/s (the leader speed has no effect on the magnitude of the leader field), and curves 1, 2, and 3 correspond to the return stroke speed v = 1.3 × 108 m/s, 1.5 × 108 m/s and 1.7 × 108 m/s, respectively.

Figure 9.

Simulation of close leader/return stroke electric field change waveforms at (a) 30 m and (b) 60 m from the lightning channel. Curves 1, 2, and 3 correspond to the return stroke speed v = 1.3 × 108 m/s, 1.5 × 108 m/s and 1.7 × 108 m/s, respectively.

[14] Comparing Figure 2 with Figure 9, it can be seen that the simulated V-shape structure waveform is consistent with the observed results. In this dart leader/return stroke model, although we assume the charges in the leader channel are totally neutralized by the following return stroke process, the simulated leader field magnitude may be equal or unequal to that of the return stroke field, and the ratio of the leader field to the return stroke field is closely associated with the return stroke speed within a few tens of microseconds. The faster the return stroke speed, the less the magnitude of the return stroke field. While at later times (after 100 μs or so), the field is dominated by the deposited charge density component, the close field is independent of speed and the ratio of the leader field to the corresponding return stroke field tends to be equal to −1. Therefore, at early times (within few tens of microseconds) there is often some uncertainty regarding whether the charges deposited by the dart leader are completely neutralized by the following return stroke process based on the difference between the return stroke and the leader field on the ground.

5. Conclusion and Discussion

[15] In triggered-lightning, the electric field changes produced by the close dart leader/return stroke sequences appear as asymmetrical V-shape pulses, the bottom of the V being the transition from leader to return stroke. In the paper we give the statistical results of the ratio of the leader to return stroke field at 30 m and 60 m in Shandong Artificially Triggered Lightning Experiment (SHATLE). The magnitude of the return stroke electric field is measured at 30 μs and 50 μs after the return stroke begins, respectively, and the 30 m mean values of the ratio of the leader to return stroke field at 30 μs and 50 μs is −0.89 (ranging from −0.8 to −1.1) and −0.85 (ranging from −0.8 to −1.02), respectively. The 60 m mean values of the ratio at 30 μs and 50 μs is −0.93 (ranging from −0.74 to −1.18) and −0.86 (ranging from −0.66 to −1.0), respectively.

[16] Generally speaking, at very close distances, any return stroke electric field change value that is smaller than the preceding leader field change is indicative of the presence of the leader charge unneutralized by the return stroke in the channel section “visible” from the field measured point. However, if we assume that the RE (the leader electric field is larger than that of return stroke) implies some charges deposited in the lightning channel by the leader but left unneutralized by the return stroke, how to explain that the leader electric field magnitude is smaller than that of the corresponding return stroke? In order to study the RE nature, based on the assumption of uniform leader charge distribution along the channel and the charges deposited by the dart leader are completely neutralized by the following return stroke process, we employ two existing models, one for the “source charge” leader model and the other for MTLL return stroke model, to simulate the V-shape structure characteristics and study the effect of return stroke speed on the RE.

[17] The result shows that the ratio of the leader to return stroke field is closely associated with the return stroke speed at early times (within few tens of microseconds of the beginning of the return stroke), while at later times (after 100 μs or so, near the diminishing time of return stroke current) the field is dominated by the deposited charge density component, the close electric field is independent of speed and the ratio of the leader field to the corresponding return stroke field tends to be equal to −1. Therefore, at early times there is often some uncertainty regarding whether the charges deposited by the dart leader are completely neutralized by the following return stroke process based on the difference between the return stroke field and the leader field on the ground. Also, it is found that the leader/return stroke fields at 30 m and 60 m are predominantly determined by the channel section below the height of about 120 m and 200 m, respectively, which close fields only response to the spatiotemporal distribution change of the charge density within the channel section near ground level.

[18] Although several other return strokes models (for example, TL, MTLE, TCS, BG and DU) exist, we have no choice but to employ the MTLL model, because we assume that the dart leader density is uniform along the channel (referred to as the “source charge” leader model) and that the leader charges are completely neutralized by the following return stroke process. The combination of the “source charge” leader model and other return stroke models implies that the leader charges may be left unneutralized or ‘overneutralized’ by the following return stroke processes, and the charge density deposited by other return stroke models is not uniform along the channel (no deposited charge component for TL model). An additional advantage of the MTLL model is that it is the only one for which the current is zero at the top of the channel. For all other return stroke models, it would be necessary to consider boundary conditions at the top of the channel of limit the field computation time to the channel traversal time.

Acknowledgments

[19] The research was supported by the China National Science Fund (40975002), the China Commonweal Industry Research Project (GYHY200806014), the One Hundred Person Project of the Chinese Academy of Sciences, and the Knowledge Innovation Program of the Chinese Academy of Sciences (IAP09313).

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