Estimation of channel characteristics of narrow bipolar events based on the transmission line model

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009jd012021, 2010 Estimation of channel characteristics of narrow bipolar events based on the transmission line model Baoyou Zhu, 1 Helin Zhou, 1,2 Ming Ma, 1 Fanchao Lv, 1 and Shanchang Tao 1 Received 6 March 2009; revised 29 March 2010; accepted 17 May 2010; published 1 October [1] Narrow bipolar event (NBE) is a distinct class of intracloud lightning discharge, which is associated with the strongest radio frequency emissions and produces typical narrow bipolar radiation field waveforms. On the basis of the transmission line model, we introduce a direct technique to measure the time taken by the current front to propagate along the channel from distant radiation field pulses; the channel length of the NBE can then be estimated by multiplying this time by an assumed propagation speed. Our method involves integrating over the initial half cycle of narrow bipolar waveform of the NBE. The ratio of the integral result to the initial peak amplitude makes a good approximation to the time taken by the current front to travel along the channel, even though the current amplitude suffers heavy attenuation along the propagating channel. This method can be applied to all NBEs which produce narrow bipolar radiation field waveforms. Besides, if both the far radiation field and the near electrostatic field measurements were available, one could combine the method here and that of Eack (2004) to obtain the channel length of the NBE. Citation: Zhu, B., H. Zhou, M. Ma, F. Lv, and S. Tao (2010), Estimation of channel characteristics of narrow bipolar events based on the transmission line model, J. Geophys. Res., 115,, doi: /2009jd Introduction [2] The narrow bipolar event (NBE) refers to a distinct class of intracloud discharge characterized by the most powerful radio frequency (RF) emissions, a narrow bipolar electric field in the very low frequency (VLF)/low frequency (LF) bands, and temporal isolation from common lightning discharges [Smith, 1998]. NBEs were first reported by Le Vine [1980] to be the sources of strongest RF radiation from lightning and exhibited narrow bipolar radiation field waveforms of ms duration. The peak powers of NBE associated very high frequency (VHF) emission (66 60 MHz) can be greater than 300 kw [Thomas et al., 2001]. Recently, NBEs have received a great deal of attention, as they are considered as the most promising candidate of spacebased lightning detection [Suszcynsky et al., 2001; Jacobson and Light, 2003; Suszcynsky and Heavner, 2003]. [3] To date, the physical mechanism of NBEs remains unsolved to the lightning community. Some authors attempted to explain the NBE discharge using the runaway breakdown mechanism [e.g., Tierney et al., 2005; Gurevich et al., 2006; Watson and Marshall, 2007]. Gurevich et al. [2006] proposed that the NBE phenomena could be fairly explained by the runaway breakdown extensive atmospheric shower discharge stimulated by high energy cosmic ray particles, and their theoretical work was in agreement with the observations 1 School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui, China. 2 Now at School of Electrical Engineering, KTH Royal Institute of Technology, Stockholm, Sweden. Copyright 2010 by the American Geophysical Union /10/2009JD of NBEs. However, the knowledge on electrical/channel characteristics of NBEs is still limited. Several other authors tried to gain some insights into electrical/channel characteristics of NBEs from electric field measurements, as described below. [4] Smith [1998] made a comprehensive study on the NBE discharge. He multiplied the duration of RF emissions by a presumed streamer velocity to estimate the channel length of the NBE discharge. The relative short channel length of m was possibly the reason why Smith [1998] termed this distinct type of discharge as compact intracloud discharges (CIDs). Eack [2004] introduced a method to obtain the current propagation speed of NBEs by using close range observations in conjunction with distant radiation field measurements. The derived propagation speed was multiplied by the event duration to estimate the average length of the discharge channel of 3.2 km. On the basis of a vertical oriented dipole assumption of the NBE discharge, the dipole moment change of the NBE can also be estimated from single far field/near field measurements [Smith et al., 1999; Eack, 2004]. [5] Recently, Hamlin et al. [2007] suggested a technique to directly measure the time it takes the current front to propagate along the channel for those positive NBEs whose electric field waveforms show a secondary positive peak following the primary positive peak. They presumed the secondary peak to be the signature of the forward traveling inbound current pulse s reflection off the far end of the channel; therefore, the time difference between the two peaks provides a direct technique to estimate the length of the discharge channel if the propagation speed was assumed. Although Hamlin et al. [2007] had no knowledge of the current propagation speed in their study, they stated that the upper limit on the channel 1of6

2 Note that equation (2) expects a simple linear relationship between the electric field and the current for the earlier time of the discharge, and it provides a simple and convenient way to determine the peak current I peak of the return stroke from the remote peak electric field E peak [Willet et al., 1989, 1998; Krider et al., 1996]: E peak ¼ I peakv 2" 0 c 2 r : ð3þ [9] We then rewrite equation (1) into the frequency domain by the Fourier transformation E ð! Þe j!r=c v ¼ 2" 0 c 2 r I ð! Þ 1 e j!l=v ; ð4þ Figure 1. Dependence of F on frequency for different current propagation times (1 4 ms). length of the NBE would be obtained if the speed of light was assumed. It was still a puzzle why the reflection signatures were only seen in a small fraction (12%) of positive NBEs and, certainly, this method cannot be applied to all NBEs. [6] In this paper, we introduce a different method to directly measure the time taken by the current front to propagate along the channel from the single station radiation field measurement of the NBE. Our method involves integrating the narrow bipolar field waveform over the initial half cycle, and it can be applied to all NBEs. 2. Methodology [7] The transmission line model was an engineering model which was proposed by Uman and McClain [1969] to relate the longitudinal channel current to the base current of lightning return strokes through a simple relationship. In the transmission line model, a current pulse is assumed to propagate along the conducting channel at a speed comparable to the speed of light. For a vertical lightning channel above the perfectly conducting ground, a current pulse (I(t)) propagating without distortion up the channel at a constant speed will yield the far electric field in the following way [Uman et al.,1975]: Etþ ð r=cþ ¼ v 2" 0 c 2 ½ r ðþ It ð l=v Þ Š; ð1þ where l is the channel length, v is the current propagation speed, r is the distance between the channel and the observer, c is the speed of light, and " 0 is the air permittivity. The term l/v denotes the time it takes the current front to propagate along the channel, hereafter referred to as the current propagation time (CPT). [8] In equation (1), the term I(t l/v) is equal to zero for t < l/v, and it can be written in an alternative form [Uman et al.,1975]: v Eðt þ r=cþ ¼ 2" 0 c 2 r It ðþ; t < l=v: ð2þ where w is the angular frequency, E(w) and I(w) are the electric field and current in the frequency domain, and j is defined as the square root of 1. The bracketed term on the right hand side of equation (4) can be extended as follows: 1 e j!l=v ¼ 1 cos ð!l=vþþj sin ð!l=vþ ¼ 2 sin 2 ð!l=2vþþ2j sin ð!l=2vþcos ð!l=2vþ ¼ 2j sin ð!l=2vþe j!l=2v : ð5þ Using this equality, we rewrite equation (4) to get E ð! Þe j!r=c ¼ j!e j!l=2v I ð! Þl 2" 0 c 2 r F ð6þ where the factor F is given by F ¼ sin! l! l : ð7þ 2v 2v Note that, if we let F = 1 and eliminate the term F, equation (6) will reduce to the radiation field pattern of the current dipole in the frequency domain. [10] Figure 1 illustrates the relationship between the factor F and the frequencies w/2p. For any given value of CPT (l/v), the factor F will approach unity as the frequency extends toward the low frequency. It is better to take the F expression as the frequency response of a low pass filter, whose cutoff frequency covers the VLF/LF bandwidths if the value of the CPT is on the order of several microseconds. Figure 1 also suggests that the transmission line model output will result in the attenuation of higher frequency components. [11] In the case of a NBE discharge, the VLF/LF narrow bipolar electric field is supposed to produce by the charge flow along the conducting channel (see further discussion in section 5). This process can be similar to the return stroke process in a cloud to ground lightning flash. [12] As reported by Orville and Idone [1982], the breakdown velocity of virgin air in the stepped leaders was on the order of m/s, while the speed of the current flow along the already conducting channel was as much as m/s [Eack, 2004]. Given the channel length, the time for the current front to propagate along the channel (CPT) will be less than the RF emitting time. The RF emission of the NBE was observed to last 3 5 ms[smith et al., 1999; Holden et al., 1995], so the CPT for the NBE discharge will be a few microseconds or less. Therefore, a NBE discharge can be 2of6

3 Refer to equation (3) to see a simple relationship between the peak current I peak and the peak electric field E peak. One can combine equations (3) and (11) to eliminate the term I peak and reach a direct method to measure the CPT of the NBE discharge from the radiation field waveform: CPT ¼ l v S peak E peak : ð13þ Figure 2. (a) The electric field waveform produced by the transmission line (TL) model at 100 km with an arbitrary current pulse, as illustrated in (b), and (b) the TL model injected current (solid) waveform and the retrieved current (dashed) waveform (both are normalized to unity amplitude). approximated as a current dipole with respect to the VLF/LF radiation. By letting F = 1 in equation (6), one can obtain the approximated current dipole radiation form for the NBE discharge: E ð! Þe j!r=c j!e j!l=2v I ð! Þl 2" 0 c 2 r : ð8þ Dividing each side of equation (8) by jw and retransferring it into the time domain, we obtain a linear relationship between the current and the time integral of the electric field (S(t)) as Sðt þ r=cþ SðÞ¼ t It ð l=2v Þl 2" 0 c 2 ; and ð9þ r Z t 0 EðÞd: ð10þ Equation (9) offers a promising way to retrieve the current profile of the NBE directly from the radiation field measurement. But this method will result in the attenuation of higherfrequency components of the current due to the low pass nature of F (see Figure 1). In fact, if the CPT was known, one can refer to equations (6) and (7) to compensate for the loss of higher frequency components. Meanwhile, the integral of the current moment over the entire duration makes the charge moment DQl R I (t)ldt; thus, similar to Smith et al. [1999], one can integrate each side of equation (9) over the entire event duration to estimate the charge moment of the NBE from the distant radiation field: DQl 2p" 0 c 2 r RR E(t)dtdt. [13] In view of the bipolar nature of the NBE electric field waveform, it is inferred from equation (9) that the electric field crosses zero when the current waveform reaches its maximum. Thus, one can estimate the peak current moment (PCM = I peak l) of the NBE discharge by integrating over the initial half cycle of the narrow bipolar waveform S peak ¼ I peak l 2" 0 c 2 rs peak ; and Z zero-crossing 0 ð11þ Eð þ r=cþd: ð12þ As a matter of fact, S peak represents the area covered by the initial half cycle of the electric field waveform. It follows from equation (13) that the CPT of the NBE is only determined by the initial half cycle, other than the overall waveform of the NBE. Thus, equation (13) can be applied to all NBEs. The advantage of our method is that the accurate calibration of the radiation field measurement is not very essential. [14] The channel length of the NBE can be obtained by multiplying the CPT by an assumed propagation speed. After the channel length is estimated, the peak current amplitude can also be estimated via equation (11). 3. Model Test [15] Unfortunately, no direct measurements of the channel length, the current waveform, and the current propagation speed were available to test the algorithm described above. In this section, we perform such a validity test with model simulation by applying the algorithm to the electric field calculated under the transmission line (TL) model with a given channel length, a current waveform, and a current propagation speed. [16] Figure 2a shows the TL model electric field produced by an arbitrary current waveform at 100 km away, with the CPT = 2.84 ms and the PCM = 20 ka km. The calculated electric field waveform was a typical narrow bipolar pulse of the NBE. Equation (13) was applied to the calculated electric field to obtain a CPT of 2.71 ms, which was in quite good agreement with the original value of 2.84 ms. Figure 2b represents the TL model injected current waveform (solid line) and the retrieved current waveform (dashed line) from the calculated electric field following equation (9). Both original and retrieved current waveforms are normalized to unity amplitude, and they agree quite well with each other, except the retrieved current waveform broadens in the initial rise in response to the low pass nature of F (see Figure 1). We believe equation (9) provides a promising way to deduce the intracloud current waveform from the radiation field measurement. [17] For the current waveform, as illustrated in Figure 2b, electric fields were calculated under the TL model for several CPTs, and our algorithms were applied to each of the model electric fields to derive the corresponding CPT. We do this in two situations: the original TL model, namely, the current travels along the channel with no decay of amplitude; and the modified TL model, namely, the current travels along the channel with an amplitude decay rate constant of 5.0 ms, close to 4.0 ms estimated by Hamlin et al. [2007]. Our results are listed in Table 1. It shows that when the original CPT is chosen in the range of ms, the estimate error is a fraction of a microsecond. Specifically, even when the current travels along the channel with decaying amplitude, 3of6

4 Table 1. Comparison of original current propagation times (CPTs) and retrieved CPTs from model electric fields Original CPTs (ms) Retrieved CPTs (ms) a Retrieved CPTs (ms) b a A current pulse propagates along the channel without decay of amplitude. b A current pulse propagates along the channel with an amplitude decay rate constant of 5.0 ms. equation (13) still reliably estimates the time it takes the current front to propagate along the channel of the NBE. However, Table 1 also indicates that when the original CPT goes toward large values, retrieved CPT values will be less than the original values, and the estimate error will increase correspondingly. 4. Application to an Observed NBE [18] Eack [2004] presented both close and far electric field data for a positive NBE. In his study, Eack equated the electrostatic and radiation dipole moment changes from the near field and far field observations to obtain a current propagation speed of m/s for that NBE. In this paper, the near and far electric field waveforms (dashed lines) from Figure 1 of Eack [2004] are reproduced in Figure 3. We use the TL model to give a fit to both far and near electric field data, especially the small hump at the beginning of the electric field, and the best fit fields (solid lines) are also illustrated in Figure 3 for comparison. Note that a nearer horizontal distance of 1.8 km, other than the 2.8 km claimed by Eack [2004], is assumed for the better fit of the small hump at the beginning of the near electric field. [19] A summary of results is given in Table 2. The CPT for this NBE is calculated to be 4.9 ms with the far radiation field, using equation (13). A channel length of 824 m is obtained by multiplying the CPT by the current propagation speed of Eack Figure 3. Comparison between simulated (solid) under the transmission line model and measured (dashed) (left) far and (right) near electric field waveforms. Table 2. Comparison of results derived from three methods for a positive narrow bipolar event (NBE) a Channel Characteristics Parameters Current propagation speed b Channel length b Positive NBE event duration Current propagation speed used in the transmission line (TL) model perfect fit Channel length used in the TL model perfect fit Current propagation time (CPT) obtained by the TL model perfect fit CPT derived from radiation field in this study Channel length obtained by multiplying the above CPT derived from the radiation field in this study by Eack s propagation speed a Of Eack [2004]. b Estimated by Eack [2004]. Values m/s 4km 24 ms m/s 1000 m 5.6 ms 4.9 ms 824 m [2004]. As shown in Table 2, although the values derived from the radiation field in this study are smaller than those obtained by fitting both far fields and near fields of the NBE under the TL model, two results are consistent with each other. Furthermore, our CPT is only one fifth of the event duration (24 ms) of this NBE. Thus, when estimating the discharge size (channel length) of this NBE, our method gives a channel length of about one fifth of the result given by Eack [2004], who estimated the channel length via multiplying the propagation speed by the event duration. 5. Discussion [20] The channel length of NBEs can be estimated by multiplying the current propagation speed by the time taken by the current front to travel through the channel (CPT). The narrow bipolar pulse duration was arbitrarily chosen as the value of the CPT by Eack [2004] to estimate the channel length of the NBE. Coupled with the current propagation speed derived from a combination of the far fields and near fields, Eack gave an average NBE channel length of 3.2 km, which was a factor of three larger than the 1 km upper limit on the NBE channel length, as estimated by Smith [1998]. Recently, Hamlin et al. [2007] suggested a direct technique to measure the CPT, referring to the intrachannel current pulse reflection signature on the electric field waveform of the NBE. For 133 events, they found an average peak toreflected peak time of 6.7 ms, which was about one half of the typical event duration [Smith, 1998]. It seems that the narrow bipolar pulse event duration may be an overestimation of the CPT. However, it still remains a puzzle why only a small fraction (12%) of positive NBEs show such signatures; hence, this method cannot work on the overwhelming majority of NBEs without any reflection signature on the electric field waveform. [21] Similar to Hamlin et al. [2007], we developed a method to directly measure the CPT from the narrow bipolar electric field waveform of the NBE. The narrow bipolar electric field waveform of the NBE was supposed to generate by the current flow along the already conducting channel. We obtain the CPT of the NBE by integrating the narrow bipolar electric field waveform over the initial half cycle rather than referring to the intrachannel current pulse reflection signature 4of6

5 proposed by Hamlin et al. [2007]. Our method depends only on the initial half cycle of the narrow bipolar electric field waveform, and it can be applied to all NBEs. In section 4, we applied our method to an event studied by Eack [2004], and the current propagation time result (4.9 ms) was only one fifth of the event duration (24 ms). For comparison, we hope that, in the future, our method can be applied to the same data set as studied by Hamlin et al. [2007]. Note that, even when the current amplitude suffers a heavy distortion along the propagating channel, our method can still make a good estimation of the current propagation time of the NBE. [22] For the most realistic estimate, however, one can take the speed of light as an upper limit on the current propagation speed; therefore, the method of this study can be used to obtain the upper limit on the channel length of the NBE. Interestingly, Eack [2004] introduced a method to estimate the current propagation speed of the NBE from a combination of the near electrostatic field and the far radiation field measurements. If one combines the method here to obtain the current propagation time and that of Eack [2004] to derive the current propagation speed, then the discharge size (channel length) of the NBE can be determined. After the channel length had been determined, one can also refer to equation (11) to further estimate the peak current amplitude of the NBE. [23] The method developed here was based on the lightning transmission line model; namely, the narrow bipolar electric field waveform of the NBE was supposed to produce by the current flow along the existing conducting channel. As we know, the intracloud flashes are different from cloud toground flashes, since (1) there is no conducting ground (or other low impedance source) for the leader to make attachment to; and (2) the vertical extent is maximum at the time of peak current, which occurs in cloud to ground flashes, since the leader is essentially at the level of the ground. [24] On one hand, the charge within the cloud can be diffuse and cannot move easily. On the other hand, the charge with an opposite polarity can be induced and distributed along the preceding initial breakdown ionized channel (i.e., a leader channel) in one charge region. These breakdown processes can happen simultaneously in the charge regions of opposite polarity at different altitudes inside the cloud, so the corresponding induced charge around the channel will increase accordingly as the leader extends. Here, the induced charge in the leader can move easily along the pre existing conducting channel. If the potential between two leader channels of opposite polarity is large enough, a vertical breakdown will occur, with two independent leaders coming in contact; either of the charges distributed in a leader channel will be a source or sink of charge transferred to another leader channel in a cloud with opposite polarity. Then, the channel extent is maximum, and a large current pulse will propagate along the conducting channel at a speed on the order of 10 8 m/s. This charge transfer process in the cloud may be similar to the return stroke process in a cloud to ground lightning flash. Although the chances of two leaders coming in contact seem small, perhaps they would offer an explanation of the lower occurrence rate of NBEs compared to other lightning activities. [25] We speculate this process is possible in a NBE discharge process, since the observed current of an intracloud streamer or leader process is involved in the order of 10 to 100 A [Proctor, 1997], which cannot account for the narrow bipolar electric field normalized to 100 km. A large current pulse (the primary peak of the NBE) process with amplitudes on the order of 10 ka, comparable to the return stroke process, is probably responsible for the large electric field occurrence. [26] The reflections may also occur when the large current pulse reaches to the far end of the channel (secondary peaks have been used by Hamlin et al. [2007] and Nag and Rakov [2009] to infer the electrical characteristics of NBEs); this will likely result in coronalike electrical breakdown at the channel extremity [Nag and Rakov, 2009]; furthermore, the streamer/leader extends away from the channel end. This process will produce the VHF emissions and the secondary peaks in the NBE waveform as expected. [27] Tierney et al. [2005] explained the NBE with the runaway breakdown mechanism, and their simulation results indicated that the HF/VHF radiation produced by the breakdown process were consistent with the observations; however, the corresponding electric field was lower than the observations, indicating that the narrow bipolar electric field waveforms of the NBEs may not be produced by the breakdown process. So, we suggest that the VLF/LF narrow bipolar pulse may be produced by the current flow along the conducting channel. [28] There are additional arguments to support the hypothesis mentioned above. The current propagation speed of the NBE estimated by Eack [2004] was quite in agreement with the return stroke speeds [Wang et al., 1999], indicating the current channel was in a good conducting state. On the basis of the transmission line model, Le Vine and Willett [1992] developed a formula to calculate the radiation from lightning. Utilizing this formula, Hager and Wang[1995] developed a method to derive the location, current, and velocity of the in cloud discharge from multiple station measurements. Their algorithms were applied to an observed in cloud pulse to obtain the current propagation speed of 0.61 c, indicative of a conducting channel. We are unaware whether the in cloud pulse studied by Hager and Wang [1995] was produced by a NBE discharge. However, the high propagation speed of that in cloud pulse suggests that the field pulse of the in cloud discharge was produced by the current flow along the conducting channel. Besides, under the assumption that the charge flows along the conducting channel, the narrow bipolar pulse zero crossing is indicative of the time of the peak current [see equation (9)]. The supported evidence can be found from Figure 1 of Eack [2004], where the zero crossing in the radiation field corresponds to a peak in the induction component dominated near field, while the induction component of the electric field resembles the current in waveform. [29] As discussed above, the TL model was an approximation to the actual physical process of NBEs; we hope our TL model based method can help gain some insights into electrical/channel characteristics of NBEs. However, the physical mechanism of NBEs remains unresolved; for example, what is the source of charge for the current flow? It is suggested that VHF mapping combined with VLF/LF electric field observations is needed in a future study for better understanding of the physical process, giving rise to 5of6

6 the powerful RF emission and narrow bipolar electric field waveform during a NBE discharge. 6. Conclusions [30] Compared to other lightning discharge types, the narrow bipolar event is a relative newcomer, and the knowledge on electrical/channel characteristics of this distinct type of lightning discharge is limited. In this paper, we introduced a direct technique to measure the time taken by the current to propagate along the channel from single station radiation field measurements of NBEs based on the transmission line model. The narrow bipolar field of the NBE was supposed to be produced by the current flow along the conducting channel, and the channel length of the NBE can be estimated by multiplying the current propagation time by an assumed propagation speed. If the speed of light is chosen as the upper bound on the propagation speed, then the upper limit on the channel length can be obtained. Our method only involves integrating over the initial half cycle of the narrow bipolar pulses, and it can be applied to all NBEs. [31] If both the far radiation field and the near electrostatic field measurements were available, one could estimate the channel length of the NBE base on the current propagation speed determined with the method of Eack [2004] and the current propagation time calculated with the method here. Then, the NBE charge transfer can also be estimated by dividing the dipole moment change by the channel length, while the dipole moment change of the NBE can be derived from either the near electrostatic field or the far radiation field [Eack, 2004]. One can even use equation (11) to estimate the peak current amplitude of NBEs. [32] Acknowledgments. This study was supported in part by the Knowledge Innovation Program of the Chinese Academy of Sciences under grant KZCX2 YW 206, the National Natural Science Foundation of China under grant , and the R and D Special Fund for Public Welfare Industry under grant GYHY (QX) The authors would like to thank two anonymous reviewers for their valuable comments. References Eack, K. B. (2004), Electrical characteristics of narrow bipolar events, Geophys. Res. Lett., 31, L20102, doi: /2004gl Gurevich, A. V., K. P. Zybin, and Y. V. Medvedev (2006), Amplification and nonlinear modification of runaway breakdown, Phys. Lett. A, 349, Hager, W. W., and D. Wang (1995), An analysis of errors in the location, current, and velocity of lightning, J. Geophys. Res., 100(D12), 25,721 25,729, doi: /95jd Hamlin, T., T. E. Light, X. M. Shao, K. B. Eack, and J. D. Harlin (2007), Estimating lightning channel characteristics of positive narrow bipolar events using intrachannel current reflection signatures, J. 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Zhou, Division of Electromagnetic Engineering, School of Electrical Engineering, KTH Royal Institute of Technology, SE Stockholm, Sweden. (zhuby@ustc.edu.cn) 6of6