Secondary Fast Breakdown in Narrow Bipolar Events.

The physical mechanism of Narrow bipolar events (NBEs) has been studied for decades but it still holds many mysteries. Recent observations indicate that the fast breakdown discharges that produce NBEs sometimes contain a secondary fast breakdown that propagates back in the opposite direction but this has not been fully addressed so far in electromagnetic models. In this study, we investigate fast breakdown using different approaches that employ a Modified Transmission Line with Exponential decay (MTLE) model and propose a new model, named "rebounding MTLE model," which reproduces the secondary fast breakdown current in NBEs. The model provides new insights into the physics of the fast breakdown mechanism.

Introduction

Narrow bipolar events (NBEs), sometimes also known as narrow bipolar pulses (NBPs), are impulsive and powerful radio emissions from intracloud discharges characterized by intense very high frequency (VHF) radiation, fast propagation speed and short‐duration bipolar sferic waveforms in the Low Frequency band (Smith et al., 1999, 2002, 2004). NBEs have received great attention since first discovered in the 1980s (Le Vine, 1980; Willett et al., 1989) but their physical mechanism, possibly related to how lightning is initiated inside thunderstorms, remains poorly understood.

Over the past decades, several physical mechanisms have been put forward to describe NBEs. One proposal argued that NBEs are caused by energetic particles from cosmic ray air showers triggering relativistic runaway electron avalanches (RREA), a model known as the relativistic runaway electron avalanches‐extensive air showers (RREA‐EAS) model or as the Runaway Breakdown (RB)‐EAS model (e.g., Gurevich et al., 1999; Gurevich & Zybin, 2001). Other explanations require or suggest the presence of leaders before the NBE source current (Da Silva & Pasko, 2015; Karunarathne et al., 2016).

These models are challenged by the latest observations enabled by broadband digital interferometry of lightning (Stock et al., 2014). By combining an interferometer (INTF) with a fast antenna (FA) and a lightning mapping array (LMA), Rison et al. (2016) found that positive‐polarity NBEs are associated with a new type of discharge which they named fast positive breakdown (FPB), with features incompatible either with the RREA‐EAS model or the existence of highly conducting channels. Tilles et al. (2019) extended these observations to negative‐polarity NBE, which they found to be similarly caused by Fast Negative Breakdown (FNB). Lyu et al. (2019) found NBEs to initiate some but far from the majority of lightning discharges, the remaining fraction being initiated by an unknown process emitting weak, extremely short VHF pulses, named initial events (IEs) or the electromagnetic activities before the initial breakdown pulses (IBPs) by some authors (e.g., Karunarathne et al., 2014; Kostinskiy et al., 2020; Marshall et al., 2019, 2014). It is interesting to note that both the IEs and the fast breakdowns of the initiation‐type NBEs (INBEs; Wu et al., 2014) or weak/less‐strong NBEs (Rison et al., 2016) can potentially produce an ionized path to further initiate a stepped leader (Karunarathne et al., 2014; Stolzenburg et al., 2013).

As a consequence of these observations, it is generally accepted that Fast Breakdown (FB, encompassing both FPB and FNB) is the source of all NBEs and that it involves the propagation of a system of streamers (Attanasio et al., 2019; Cooray et al., 2020; Griffiths & Phelps, 1976; Luque & Ebert, 2014; Phelps, 1974). These streamers, possibly initiated by ice hydrometeors (Petersen et al., 2006, 2015), propagate hundreds of meters at a speed of a few times 107 m/s (although recently [Liu et al., 2021] pointed out that the observations are also compatible with a series of streamer bursts at fixed locations) and intensify the electric field in the starting region. Besides the observations listed above, additional studies also support this conclusion, including the analysis of radio spectra (Liu et al., 2019) and space‐based optical observations (Li et al., 2021; Soler et al., 2020).

Some observations show that NBE‐producing fast breakdowns sometimes contain a secondary fast breakdown that propagates in the opposite direction along the previous path (Attanasio et al., 2021; Rison et al., 2016; Tilles et al., 2019). Hamlin et al. (2007) discussed secondary pulses in the NBE signatures and interpreted them as a signature of reflection off the far end of the channel. Leal et al. (2019) examined more than 1000 NBE waveforms and found a majority of them exhibiting secondary peaks and oscillations, suggesting that these structures are a fundamental property of NBE waveforms.

Recently, Tilles et al. (2020) found alternating‐polarity streamer fronts associated to energetic in‐cloud pulses (EIPs). Huang et al. (2021) also found fast breakdown events consisting of simultaneous upward and downward streamer fronts, with the trajectory of the later streamer development pointing back to the initial source location. Most recently, Attanasio et al. (2021) discussed the physical mechanism of this secondary fast breakdown of NBEs based on an improved version of the Griffths and Phelps model (Griffiths & Phelps, 1976).

Simplifying the NBE source as an infinitesimally short dipole (Nag & Rakov, 2010a; Smith et al., 1999, 2004) or as a more complex Transmission Line (Smith et al., 1999, 2004; Watson & Marshall, 2007; Nag & Rakov, 2010a,b), allows inferring properties of the source current from ground‐based electromagnetic measurements. Using a TL representation of NBEs, Nag and Rakov (2010a) proposed a bouncing‐wave model, where a current pulse travels consecutively and repeatedly upward and downward. As discussed later, a TL representation of the current pulse can, with appropriate modifications, explain the existing electromagnetic observations. Often TL models suppose the existence of a conducting channel but, given the evidence provided by Rison et al. (2016), where no activity is observed before the initiation of the main current pulse, it seems unlikely that a conducting channel is established before the NBE current source. In addition, the recent space‐based optical observations by Soler et al. (2020); Li et al. (2021) indicate that hot channels are absent during isolated NBEs (Kostinskiy et al., 2020; Rison et al., 2016) emissions. However, the physical mechanism of fast breakdowns of the initial type NBEs (Rison et al., 2016; Wu et al., 2014) remains poorly understood. Stolzenburg et al. (2021) analyzed five IC flashes and in four of them reported faint light emissions (⩽5% above background cumulative intensity) at the time of the IEs. Recently López et al. (2022) investigated the initiation of four lightning flashes detected from LMA and simultaneous optical satellite observations and found that, in three of the flashes, the detection of blue emissions without any red luminosity supports the pure streamer nature of fast breakdown processes at the flash initiation. However, the onset of the fourth flash was associated with both blue and red radiation and accompanied by an NBE with weak VHF power.

In their work, Rison et al. (2016) also applied a TL model to explain their FA observations and inferred a current profile that matched the observations without reflected current pulses. However, as we discuss below, this agreement resulted from the use of Shao's equation (Shao et al., 2004, 2005) for the electromagnetic radiation field. The validity of Shao's equation rests on the assumption that the current pulse has decayed completely as it reaches the endpoint of the discharge. The equation works well for a return stroke since its channel is typically long enough to justify this assumption (Shao et al., 2012), but it remains unclear whether this condition can be safely assumed in the current profile inferred by Rison et al. (2016).

In this letter, we re‐analyze the data presented by Rison et al. (2016). We compute, without Shao's approximation, the electromagnetic emissions from the current profile proposed by Rison et al. (2016) and show that they exhibit a radiation peak that is missing in the observed sferic. This raises the question of why Shao's expression works better than the full Maxwell equations in fitting the data. We show below that this has a physical explanation and unveils a secondary, counter‐propagating current pulse, which is consistent with the recently reported secondary fast breakdown of NBEs. We propose a new model called rebounding Modified Transmission Line with Exponential decay (MTLE) model, to explain the secondary fast breakdown current in NBEs which is likely driven by counter‐propagating streamers triggered by the primary fast breakdown pulse. The existence of this secondary streamer wave, suggesting that the fast breakdown does not completely dissipate its driving electric field, is a new key to our understanding of this phenomenon.

Sferic Waveform Analysis

In their study, Rison et al. (2016) recorded sferics from NBEs using an FA and evaluated the sferic waveforms using a MTLE with height model, which was previously employed for return strokes by Nucci and Rachidi (1989) and Rachidi and Nucci (1990). Adopting the sign convention where a negative current moves positive charge downward (i.e., electrons flow upward), this transmission line scheme is sketched in Figure 1. A positive NBE current front propagates downward from an altitude H 2 to H 1, with a channel length of L = H 2 − H 1. The downward current I (red curve in the figure) decreases exponentially along its propagation channel with an attenuation rate λ : where ν is the downward propagation velocity, related to the downward propagation time t by v  = L/t . The injected current I has a double‐exponential waveform where α = 1/τ 1, β = 1/τ 2 are the rise and fall time constants. The amplitude I 0 can be normalized to the peak current I by setting Let us focus on the events NBE1 and NBE3 analyzed by Rison et al. (2016) and initially assume that the current dies out as it reaches the end of the channel at H 1. In the following, we adopt the same geometry as Rison et al. (2016), derived from their observations. This includes the observation distance ρ, maximum channel altitude H 2, channel length L and the velocity v listed in Table 1 for the case NBE1 and NBE3, respectively.

Figure 1

The rebounding MTLE model of streamer‐based NBEs, (I)–(III) are different growth stages of the streamer corona system of NBEs. We model the NBE discharge channel as a system of positive streamer coronas that propagate downward from an altitude H 2 to H 1 with a channel length L, followed by upward negative streamer corona discharges that propagate back along the same path. Here, I is the downward current and I is the rebounding‐wave current.

Table 1

The Parameters of the NBE‐Producing Current Used in the Simulation for Cases NBE1 and NBE3 Reported by Rison et al. (2016)

ID	Method	Parameters adopted in Rison et al. (2016)
		I peak (kA)	τ 1 (μs)	τ 2 (μs)	λ (m)		ρ (km)	H 2 (m)	L (m)	v (m/s)
NBE1	Uman's eq/Shao's eq with MTLE model	−55.2	0.8	6.0	900		5.5	6,000a	455	3.5 × 107
NBE3	Uman's eq/Shao's eq with MTLE model	−63.4	0.3	2.3	900		3.3	6,600	560	3.5 × 107
ID	Method	Simulation‐determined parameters b					Interferometer‐determined parameters c
		I peak (kA)	τ 1 (μs)	τ 2 (μs)	λ d (m)	λ u (m)	ρ (km)	H 2 (m)	L (m)	t d (μs)	t u (μs)
NBE1	Uman's eq with rebounding MTLE model	−30.5	0.8	7.0	374.9	857.6	5.5	6,700	720	12	13
NBE3	Uman's eq with rebounding MTLE model	−61.7	0.3	3.4	378.7	113.7	3.3	6,600	412	11	6

Note. Three different models were used in simulation: Uman's equation (Uman et al., 1975) with the MTLE model, Shao's equation (Shao et al., 2004, 2005) with the MTLE model, and Uman's equation with the rebounding MTLE model.

The altitude H 2 is derived from the LMA data, see Rison et al. (2016) for details.

The current amplitude I , rise and fall time constants (τ 1, τ 2), as well as the downward and upward exponential attenuation rates (λ , λ ) are best‐fit parameters defining I (z, t) to the sferic waveforms of NBE1 and NBE3.

The observation distance ρ, altitude H 2, length L, the downward and upward propagation times (t , t ) are determined by the interferometer data in Rison et al. (2016) (see Figure 2).

Figure 2

The current distribution of rebounding MTLE model for NBE1 and NBE3 along with the interferometer data observed by Rison et al. (2016). NBE1 and NBE3: (a), (d) downward current I (positive streamer propagates downward), (b), (e) upward rebounding current I (negative streamer propagates upward) and (c), (f) total current I . Note that the time of interferometer data here has been corrected to the source time. The detailed interferometer data can be found in Rison et al. (2016).

We compute the electric field at ground level at a location that follows the source‐observer geometry of the events (see Table 1) by employing two approaches: Uman's equation (Uman et al., 1975) and a solution of the complete Maxwell's equations using the Finite‐Difference Time‐Domain (FDTD) method (Li et al., 2020). We use Uman's equation with the same current parameters used by Rison et al. (2016). The adopted parameters are listed in table 1, and the current pulse as a function of height resulting from these parameters is shown in Figure S1 in Supporting Information S1, panels (a) and (b), whereas Figure 3 contains the computational results (blue line and blue triangles), as well as the measured sferic observations (black line).

Figure 3

Comparison between simulation and measurement corresponding to the case NBE1 (a, c, e) and NBE3 (b, d, f) in Rison et al. (2016). (a), (b) Different approaches: Uman's equation (Uman et al., 1975) and full‐wave FDTD method with the MTLE model, Uman's equation and full‐wave FDTD method with the rebounding MTLE model, and Shao's equation with the MTLE model. The electrostatic, induction and radiation components of the total electric field calculated by using Uman's equation with the MTLE model (c), (d) and the rebounding MTLE model (e), (f).

The two computational approaches agree with each other to very good accuracy but they differ from the measured waveforms. At around 17 µs for NBE1 and, more markedly, at around 21 µs for NBE3, the model predicts negative deviations that are conspicuously absent in the observations. These deviations result from a peak in the radiation component, proportional to the time‐derivative of the current, emitted as the current terminates abruptly at the end of the channel. This peak has previously been referred to as the “mirror image” effect (Shoory et al., 2009; Uman et al., 1975).

In their work, Rison et al. (2016) used a different approach to compute the radiation field. Instead of Uman's equation, which is a particularization of the retarded time, integral formulation of Maxwell's equations (called Jefimenko's or Schott's equations [Zangwill, 2013, p. 726]; a more general version of Uman's equation valid for arbitrary time‐dependent current density can be found in Shao (2016)), they applied Shao's expression (Shao et al., 2005, equation 11), which disregards the current discontinuity at the end of the TL and applies only to cases where the current is sufficiently attenuated before it reaches that point. This approach produces the green curves in panels (a) and (b) of Figure 3. The late radiative peak is absent and the calculations agree reasonably well with the observations.

This raises the question of why reproducing the observations requires suppressing the radiative peak. We considered the possibility that the current does not disappear abruptly at the end of the TL but instead vanishes gradually, damping the radiation peak below the instrument's sensitivity. To investigate this, we extended the MTLE model with an extra region where the current decays smoothly but we found that an unrealistically long extension with d ≥ 5 km is required to sufficiently attenuate the radiative peaks (see Figure S3 and accompanying text in Supporting Information S1 for further details).

On the basis of the idea of the bouncing‐wave proposed by Nag and Rakov (2010a) and the rebounding fast breakdown waves discussed by Attanasio et al. (2021), we assume that the current, instead of vanishing, reverses direction and heads upward when it reaches the end of the TL channel. This eliminates the current discontinuity at the end of the channel. Besides, the existence of an upward‐directed pulse is supported by the interferometer traces observed by Rison et al. (2016) (see Figures 2a and 2b in Rison et al. (2016) for details). The uncertainties of the elevation and azimuth source locations measured by the high‐speed broadband VHF interferometer are presented in supplementary figures 15–17 in Rison et al. (2016).

One difference of the proposed model with the one proposed by Nag and Rakov (2010a) is that, to ensure the current continuity, we prescribe that the current is attenuated as it propagates, instead of having a discrete attenuation factor at the ends of the TL. Because of this difference and because we assume only one direction reversal, we refer to our model as “rebounding wave” although it could also arguably be called “modified bouncing wave.”

In our model, we include this upward rebounding current I (marked as a blue curve in Figure 1) as a pulse with a velocity v along the previous path and also following the MTLE model but with a different attenuation rate λ : where t is the upward propagation time related to the upward velocity by v  = L/t and the factor ensures the continuity between the downward and the upward‐propagating pulses.

The total current I is the sum of the downward current I and the upward rebounding current I :

As shown in Figure 2, the downward and upward propagation time (t and t ) are obtained by fitting the interferometer traces for both NBE1 and NBE3 with the best‐fit lines shown in panels (a,b) for NBE1 and panels (d,e) for NBE3. We fit the parameters defining the current I (z, t) to the sferic waveforms of NBE1 and NBE3, with the best‐fit results listed in Table 1. The downward, upward and total current as a function of height for the rebounding MTLE model are represented in Figure 2 along with the interferometer data, panels (a–c) for NBE1 and (d–f) for NBE3. In all cases discussed here, the upward current pulse is attenuated to a negligible value before it reaches the upper top boundary. The resulting waveforms derived from Uman's equation with the rebounding MTLE model and the FDTD model are plotted with a red line and red triangles in panels (a) and (b) of Figure 3. Panels (c) and (d) of that figure show the three components of the electric field resulting from Uman's equation using both the MTLE model and the rebounding MTLE model. The late radiation peak is almost completely suppressed and the model predictions match the observations with the same accuracy as Shao's equation. It is shown that the rebounding currents propagate upward following the previous path, which agrees with the interferometer traces for both NBE1 and NBE3. Note that for NBE1 the absence of the radiation peak can also be explained by a strong attenuation of the downward current; with the parameters of Table 1 the sferic is weakly sensitive to the parameters of the upward current. This is not the case for NBE3, which cannot be reasonably explained without an upward current and therefore provides the strongest evidence for our conclusion.

Both Shao's expression and the rebounding MTLE model underestimate the electric field in the tail of the sferic waveform. As discussed by Rison et al. (2016), NBE3 is more complex than NBE1 and contains features such as a small tilt and a substantial azimuthal spread (see figure 9 in the supplementary material of Rison et al. (2016)). This deserves further analysis but falls out of the scope of the present work.

Discussion and Conclusions

Our results show that the NBE sferics published by Rison et al. (2016) are better explained if one assumes that once the primary current pulse has propagated about 400–700 m, it triggers a counter‐propagating current pulse. The data are best fit when downward and upward currents rebound continuously; otherwise, the predicted waveform partly recovers the radiation peak that is absent in the observations. This secondary pulse is also observed in the interferometer data reported by Rison et al. (2016); Tilles et al. (2019).

The results also indicate that the measured NBE sferics are consistent with a current distribution with a significant spatial extent where FPB and FNB overlap within a significative volume. Although one cannot exclude the possibility that other current distributions may also reproduce the sferics, we believe that the existence of a secondary fast breakdown current is a plausible explanation for the absence of the radiation peak to better agree with the observations.

In their original bouncing‐wave model, Nag and Rakov (2010a) explained the secondary pulse as the reflection of a wave as it reaches the channel ends. The simulated results by using the bouncing‐wave model based on “Uman's equation with the MTLE model” along with the parameters shown in Table 1 as well as adopting different sets of values for the current reflection coefficients are discussed in the Supplemental Materials which do not match well with the measurements (see Figures S4 and S5, and the text there in Supporting Information S1). In addition, we further compared the results of the rebounding MTLE model with the well‐known two‐station measurements reported by Eack (2004). The simulations agree well with the measurements for both near and far stations (see Figure S6 and Table S1 in Supporting Information S1). Note that, in the rebounding MTLE model, the downward and upward propagation times (t and t ) are determined by the modeling since there is no available interferometer observation. Assuming a downward attenuation rate λ  = 1 × 105 m and an upward attenuation rate λ  = 100 m, the results corresponding to the rebounding MTLE model are found to be similar to those calculated by the bouncing‐wave model in Nag and Rakov (2010a).

On the other hand, the interferometer observations suggest that both the downward and upward pulses share the same nature, most likely being systems of streamers as discussed by Attanasio et al. (2021). The triggering of streamer channels by streamers of opposite polarity is not uncommon. Kochkin et al. (2012) captured images where a positive streamer corona triggers negative streamers as it approaches an electrode. The same group found positive streamers emerging from the channels of preceding negative streamers (Kochkin et al., 2014) and a similar process explains the shape of carrot sprites in the upper atmosphere (Malagón‐Romero et al., 2020). In the numerical simulations of Malagón‐Romero et al. (2020), similar currents flow through the channels of the downward and upward streamers because they are connected through electrically conducting regions.

We also note that collisions between streamers of opposite polarities have been proposed as a source of X‐rays and as precursors of Terrestrial Gamma‐ray Flashes (TGF; Babich & Bochkov, 2017; Cooray et al., 2009; Ihaddadene & Celestin, 2015; Köhn et al., 2017; Luque, 2017). Tilles et al. (2020) presented a case where alternating fast positive and negative breakdown precedes a large current pulse (EIP) which is likely associated with a TGF (Pu et al., 2019). Note that the rebounding‐wave model proposed here is different from the physical picture described by Huang et al. (2021): in the cases that they analyzed, negative and positive fast breakdowns appeared to be simultaneously launched from the same region.

A final conclusion concerns the persistence of electric fields after the fast breakdown discharges. That a secondary discharge is allowed to propagate suggests an incomplete screening of the electric field. Furthermore, the alternating fast positive and negative breakdown reported by Tilles et al. (2020) hints at the possibility of very weak screening of the electric fields, a conclusion that is relevant to estimates of the driving field as analyzed by Cummer (2020) and has key implications for physical models of fast breakdown.

Supporting Information S1

Click here for additional data file.