Key Laboratory of Middle Atmosphere and Global Environment Observation (LAGEO), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
Faculty of Science, Beijing Information Science and Technology University, Beijing, China
Graduate School, Chinese Academy of Sciences, Beijing, China
Corresponding author: C. Wang, Key Laboratory of Middle Atmosphere and Global Environment Observation (LAGEO), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China. (firstname.lastname@example.org)
 Mainly based on the channel base current of the 0803 and 0902 flashes detected in the Shandong Artificial Triggering Lightning Experiment (SHATLE) from 2005 to 2010, the current waveforms of six return strokes and an RM process (a process characterized by both a return stroke and an M-component) containing subsidiary peaks were analyzed. The percentage of strokes with subsidiary peak is approximately 17%. The amplitudes of the stroke primary and subsidiary peaks varied from 8.53 to 16.34 kA and 5.40 to 15.86 kA, respectively. The time intervals between the primary and the secondary peak varied from 6.3 to 12.0 μs. The strokes of the 0803 and 0902 flashes have different current waveform characteristics, indicating that the mechanisms generating the subsidiary peaks in the two lightning may be different. To learn more about the causes of forming the subsidiary peaks based on the assumption that channel branches and every peak corresponds to a physical process, a Heidler function model was used to simulate the current waveforms of four strokes and an RM in flash 0902. The simulated results indicated that the Heidler function component could reflect the physical characteristics of the peaks to some extent. Based on simulated results and comparison of waveforms, four possibilities of forming subsidiary peaks were proposed: channel branching, flashover along a triggering wire from a previous unsuccessful launch, the corona current or reflection of the current. Branching and flashover may be the main causes of generating the subsidiary peaks in 0902 and 0803 flashes, respectively.
 The return stroke is the biggest concern during lightning discharges because it causes the most destruction and electromagnetic disturbance of ground objectives, such as electrical and telecommunication systems. Performing direct measurements of the return stroke current of rocket-triggered lightning and tall structure-induced lightning is the main approach for obtaining and accumulating return stroke current waveforms. The characteristics, parameters and physical processes associated with lightning currents are important in research and applications related to protection against lightning [e.g.,Depasse,1994; Liu et al., 1994; Pinto et al., 2005; Rakov et al., 2005; Zhang et al., 2009; Yang et al., 2009; Jiang et al., 2011]. The normal current waveform of the return stroke is characterized by only one peak [Fisher et al., 1993; Uman et al., 2002; Qie et al., 2007, 2011], though a secondary peak following the primary peak has occasionally been reported by several researchers. To analyze the physical mechanism generating the subsidiary peak, some researchers have studied the characteristics and causes of pronounced subsidiary peaks that are superimposed on the falling portion of the return stroke pulse. Weidman and Krider  summarized the time intervals between the adjacent distinct subsidiary peaks, which varied from 10 to 30 μs. They also studied the luminous characteristics and branching time of the channel and concluded that the subsidiary peak was caused by channel branching. LeVine and Meneghini  and Le Vine and Willett found that the current waveforms of the strokes within a given flash were similar. They hypothesized that the subsidiary peak coming from the current direction changing and the channel geometry would influence the fine structure of the return stroke waveform. More than one peak was observed in the first return stroke current of tall tower-induced lightning, which was primarily due to reflection caused by the tower's main structural discontinuities [Milewski and Hussein, 2008]. However, there have been few reports addressing the subsidiary peak and its origin in triggered lightning. Jerauld et al.  once observed a dip in the first return stroke current waveform (Figure 2) of triggered lightning, which was not observed in the corresponding magnetic field and luminosity waveform. These researchers considered the dip to likely be due to a flashover and the resultant current missing. However, it is not clear whether the above reasons are all of the actual causes of the subsidiary peak or whether other explanations exist.
 Lightning currents are often used as an input function in many types of lightning models because they are an electromagnetic field source [Master et al., 1981]. Accurately assessing lightning effects mainly depends on the appropriate analytical representation of the lightning current waveshape. Bruce and Golde  presented the sum of two exponential functions for this purpose. This expression was shown to be inconvenient because it exhibited a discontinuity in its time derivative at t = 0. There is currently a trend of using the Heidler function [Heidler, 1985] to express the characteristics of a measured current with better accuracy. The important improvement of the Heidler function is that its first time derivation for t = 0 equals to zero and does not exhibit a discontinuity at the beginning. In general, two Heidler functions are sufficient to represent the subsequent stroke currents [Nucci et al., 1993]. However, two Heidler functions cannot reproduce a current waveform with more than one pronounced subsidiary peak. De Conti and Visacro  used the sum of Heidler functions to express the first and subsequent stroke currents characterized by a concave profile at the front of the first return stroke current. An appropriate description of a return stroke current should include not just one peak, but multiple peaks.
 In this report, the current characteristics of the subsidiary peak in triggered lightning are summarized, and possible causes producing current subsidiary peaks are analyzed. To learn more about the causes of the subsidiary peak, return stroke current waveforms with subsidiary peaks were reconstructed by summing Heidler functions.
2. The Experiments
 The data used in this study mainly come from the 0803 and 0902 flashes triggered using classical techniques during the SHATLE [Qie et al., 2009] in 2008 and 2009, respectively. The rocket launching facilities and channel base current sensors used in 2009 are shown in Figure 1. The channel base current was measured using a 0.5 mΩ shunt with a bandwidth of 0–3.2 MHz, as illustrated in the upper-left picture inFigure 1. The current signals were transmitted via an ISOBE5600 fiber-optic system linked to a control room located 70 m away from the launching facilities and recorded with a DL750 digitizing oscilloscope. The sampling interval was 0.1 μs, and the record length was one second. In 2008, the channel base current was measured using a Rogowski coil with bandwidth of 300 Hz–1 MHz. The current signals were recorded with a DL708 digitizing oscilloscope. The sampling interval was 2 μs, and the record length was one second. Additional descriptions of SHATLE can be found in the literature [Yang et al., 2010, Wang et al., 2010, 2012].
3. Return Stroke Current Subsidiary Peak Characteristics
 To learn about the statistical characteristics of the return stroke current subsidiary peak in triggered lightning, all 22 triggered lightning and return stroke current pulses were carefully examined in SHATLE from 2005 to 2010. The conditions regarding the return stroke number with current recording and the current subsidiary peak number are shown in Table 1. Current measurements were obtained for eight of these 22 triggered lightning. For two of these eight lightning, the 0803 and 0902 flashes, subsidiary peaks were detected in the current waveforms of the return strokes. The 22 triggered lightning contain 88 return strokes, and current measurements are available for 36 of these 88 return strokes. Six of these 36 return strokes contain subsidiary peaks. The percentage of return strokes with more than one peak is approximately 17%.
Table 1. Numbers of Return Strokes Having Current Recordings and Current Subsidiary Peaks in SHATLE From 2005 to 2010
Time (Beijing Time)
Flash Serial Number
Flash duration (ms)
Numbers of Return Strokes for Current Record
Numbers of Return Strokes With Subsidiary Peak
 Six and four return strokes occurred in flashes 0803 and 0902, respectively. Figure 2 shows the overall current waveform of flash 0902. The four return strokes are labeled as R1, R2, R3 and R4. One distinct subsidiary peak followed the primary stroke peaks in R1 and R2, while two subsidiary peaks were observed in R3 and R4. The inset in Figure 2shows the current waveform of R2. A current pulse characterized by both a return stroke and M-component were also detected in flash 0902, which was designated an RM-event byQie et al.  and is indicated with RM in Figure 2. The RM pulse will be discussed in detail later. Flash 0803 contains six return strokes, but only the first two return strokes (R1 and R2) exhibit a subsidiary peak. There are a total of six return strokes with more than one peak (4 from flash 0902 and 2 from flash 0803).
 The peak amplitudes of the six return strokes with subsidiary peaks are summarized in Table 2. It was found that the maximum amplitude of the first peak was 16.34 kA, and the minimum was 8.53 kA. The amplitudes of the subsidiary peaks vary from 5.40 to 15.86 kA. The first subsidiary peaks of the six strokes were smaller than their primary peaks, and their second subsidiary peaks were smaller than their first subsidiary peaks. The time intervals between the adjacent peaks of the six return strokes are summarized in Table 3. The intervals between the primary peak and the first subsidiary peak varied from 6.3 to 12.0 μs. However, the intervals between the first and the second subsidiary peaks of R3 and R4 from flash 0902 are 11.0 and 10.9 μs, respectively, which is consistent with the results reported by Weidman and Krider  of 10–30 μs.
Table 2. Statistics for the Peak Amplitudes of the Six Return Strokes With Subsidiary Peaks From Flashes 0803 and 0902
Flash Serial Number
Table 3. Intervals Between the Adjacent Peaks of the Return Stroke Currents
Flash Serial Number
4. Current Subsidiary Peak Analysis and Results
 More than one peak was detected in the first return stroke current of the tall structure-induced lightning mainly due to reflection. However, there are few reports addressing the subsidiary peak and its origin in triggered lightning return stroke currents. We analyze and compare the subsidiary peaks of flashes 0803 and 0902 in the following section to determine possible reasons for the return stroke current subsidiary peak.
4.1. Observational Results of Flash 0902 and Flash 0803
 Flash 0902 is particularly special and has many characteristics that differ from those of normal triggered lightning. Observational results from video images showed that there was faint luminosity in the channel preceding all four return strokes, which implied that a channel current existed in the channel preceding all of the return strokes, which was confirmed by the channel base current observations. One or two peaks followed the primary peak in all of the return stroke pulses, and there was an RM pulse, which is characterized by both a return stroke and M-component.Figure 3 shows the current waveform of the RM pulse. Qie et al. analyzed video, channel base current and E-field observation data, and their results indicated that two branches in the upper channel may result in all of the above phenomena and that the RM pulse was produced by a return stroke and an M-component that occurred almost simultaneously in the two branches. A more detailed description of the RM pulse can be found in their paper [Qie et al., 2011]. Therefore, the subsidiary peaks may also be correlated with the observed channel branching or with intracloud events producing the M-components.
 Flash 0803 also contained return strokes exhibiting subsidiary peaks. The return stroke waveform was compared between flashes 0803 and 0902. Figure 4 shows the channel base current waveforms of six return strokes exhibiting subsidiary peaks from flashes 0902 and 0803 at same time scale. First, all of the return strokes of flash 0902 have more than one peak, while only two return strokes of 0803 exhibit a subsidiary peak. Second, the waveform of flash 0902 is different from that of 0803. The bottom of the primary peak of flash 0803 is sharper than that of 0902.
 With respect to the four return strokes in flash 0902, does their subsidiary peak also come from the M-component, like the RM, or from other processes? First, we believe that every subsidiary peak corresponds to a physical process or a pulse (an M-component or other pulse), while the primary peak corresponds to a normal return stroke pulse. Based on this assumption, we are of the opinion that each return stroke waveform can be reconstructed using a normal return stroke pulse and M-component or other pulse. The Heidler function can easily represent the return stroke pulse. During the process of searching for an appropriate function to represent the pulse of the subsidiary peak (an M-component or other pulse), we found that the Heidler function could also describe it very well. To obtain more information about the subsidiary peak and to seek the possible physical mechanism involved in the return stroke discharge process, we simulated the return stroke waveform by summing Heidler functions, as described as follow.
4.2. Method of Heidler Function Model Representation
 To generate an appropriate analytical representation of a lightning current waveshape with one or multiple peaks and to examine the electromagnetic characteristics of lightning, a method for analytical representation of a Heidler function was proposed, as follows.
 The Heidler function expression is as follows:
where i0 is the current peak; n is a current steepness factor; and η = exp[−(τ1/τ2)(nτ2/τ1)1/n] is a correction factor. (t/τ1)n/[1 + (t/τ1)n] represents the current rise function, and exp[−(t/τ2)] is the current decay function. τ1 and τ2 are the time constants determining the current rise and decay time, respectively. To obtain better simulation results, n was not limited to being an integer and can be less than 1. Based on Shao's observation [Shao et al., 1995], it is possible that the pulses forming the subsidiary peaks of currents occurred at different times. A time delay parameter, td, was introduced into the Heidler function. Presuming that a Heidler function corresponded to a peak or a subsidiary peak of a current, we simulated a return stroke current with m peaks using the sum of m Heidler functions. The simulation expression is as follows:
where m is the number of peaks in the current waveform. A nonlinear least squares fitting method was adopted to simulate the current waveform. The objective function was defined as follows:
where i˜(t) represents the observed current; i¯(X, t) is the current calculated using formula (2); T is the total observed time; and X is a vector representing the coefficients (−i0, η, τ1, τ2, n, td). The equation is solved using the Gauss-Newton iteration method. After several hundred or even more iterations, the solver is stopped at the best simulated result,X, due to satisfaction of the termination tolerance of the function value. To investigate the convergence character of the objective function, we calculated the distribution of the objective function, which showed only one minimum. The interesting single minimum observed in this scenario leads to the conjecture that the result might be close to a global minimum in the least squares sense.
4.3. Simulated Results for Return Strokes in Flash 0902
 The methodology proposed above was applied to calculate the representative current waveshapes for the four return strokes of lightning flash 0902, and m equals two, two, three and three for the four return strokes, respectively. The simulated current parameters for the four return strokes are shown in Table 4. For each current waveform, the Heidler function components were expressed as HF1, HF2 and HF3. The total simulated curve (FC), the Heidler function component curves (HF1, HF2 and HF3) and the measured data (Data) corresponding to each current waveform are all indicated in Figure 5. The data curve is gray, which can be distinguished from the black simulated FC curve. As shown in Figure 5, the simulated curve is in good agreement with the measured data.
Table 4. Simulated Parameters for the Four Return Strokes of Flash 0902
 Each Heidler function component waveform of every return stroke, especially that corresponding to the primary peak, is almost invariable based on the condition that the objective function is in convergence, and the value of the objective function is the same when the solver is stopped. Because (t/τ1)n/[1 + (t/τ1)n] represents the waveform's rising function, the rising time of the Heidler function component is correlated not only with τ1, but also with n. The simulated parameters τ1 and τ2 cannot directly reflect the rising time and falling time of the Heidler function component waveform very well. Therefore, we summarized the Heidler function component parameter directly from the Heidler function component waveform. Table 5shows the peak, 10–90% rising time and half-peak width of all of the Heidler function components, as shown inTable 4 and Figure 5. The 10–90% rising time and half-peak widths of HF2 components are approximately one microsecond and several microseconds, respectively. These parameters are typical for a normal return stroke pulse. The 10–90% rising time and half-peak width of HF1 are approximately several microseconds and several tens of microseconds, respectively. The HF3 of R3 and R4 exhibits a greater 10–90% rising time and half-peak width compared to HF1 and HF2. HF1 and HF3 could be an M-component or other pulse based on their parameters. The above findings indicated that the simulated results are agreement with the assumption that each return stroke waveform can be reconstructed using a normal return stroke pulse and M-component or other pulse because all parameters involved in the calculations are optimized using the least squares method in the program. These results also indicate the rationality of the simulated method, and the results may reflect a physical process related to the subsidiary peak to some extent. Therefore, we discuss the possible causes of subsidiary peak formation in the following section, combining with the above simulated results.
Table 5. Statistical Characteristics of the Heidler Function Components of the Four Return Strokes in Flash 0902
5. Conclusions and Discussions
 Subsidiary peak in return stroke current was found in the triggered lightning in SHATLE from 2005 to 2010. Among the 22 triggered lightning flashes, there are two flashes, 0902 and 0803, contained six return strokes and a stroke-M component (RM) process with subsidiary peaks in their current waveforms. The amplitude of current subsidiary peaks ranged from 5.4 kA to 15.86 kA, and peak interval from 6.3 μs to 12.0 μs.
 The 0902 and 0803 flashes exhibit some different waveform characteristics, possibly indicating that the subsidiary peaks of these two lightning flashes are generated via different mechanisms. To obtain additional information on the lightning subsidiary peaks, based on the assumption that channel branches and a current peak corresponds to a physical process, a method for analytically representing a current curve with one or several subsidiary peaks was applied. Each return stroke with multiple peaks can easily be properly restructured using the sum of several Heidler functions. The results demonstrated that the waveshape and characteristics of the simulated Heidler function components can reveal the physical properties of the lightning process to some extent. According to the simulating results and the waveform characteristics, four kinds of possible causes forming subsidiary peak are suggested, with three for flash 0902 and one for flash 0803.
 The first is that the return stroke subsidiary peak is correlated with the channel branching or with intracloud events that produce the M-components, like the RM pulse in flash 0902 described above. That is to say, each return stroke is produced by a return stroke and an M-component that occur almost simultaneously in the different branches. It is necessary to explain that the RM is a special pulse and cannot be simply classified as a return stroke or M-component. First, the high speed video and current recordings showed no obvious current cutoff for the channel. A current cutoff is a standard for distinguishing the return stroke from the M-component. The current waveform of the RM pulse presented a fairly short shift time from the first peak to the second peak. Second, the rising time of the electric field change for the RM was much longer than that of the dart leader-return stroke sequence and even longer than that of some large M-components. This indicated that the rising edge of the E-field change for the RM was not caused only by the dart leader, as the rising time would not be as long. Finally, the ratio of the magnitude of the E-field pulse to the current pulse for the RM was 10.7, while those of four return strokes and five M-components in this flash were 6–7.3 and 2–3, respectively. This further suggested the possibility of the superposition of the dart leader and the downward M-component incident process for the RM event. Therefore, the RM is different from the M-component or return stroke.
 To compare the RM and return stroke, we summarized the subsidiary peak parameters for the RM pulse and simulated it using the Heidler function representation method described above. The first and second peaks of the RM pulse are 2.67 kA and 3.38 kA, respectively. The RM pulse exhibits a subsidiary peak larger than the primary peak, as shown in Figure 3. The interval between the two peaks is greater than 20 μs. The RM pulse presents one subsidiary peak and is concave at the front of the current curve, which may correspond to a small amplitude pulse, so m was assigned a value of three during the simulation of RM. The simulated parameters for the RM current are shown in Table 6. The Heidler function components (HF1, HF2, HF3), simulated curve (FC) and measured data (Data) are shown in Figure 6. The characteristics of the three Heidler function components are summarized in Table 7. Based on Table 6, the originating times of the three Heidler function components of the RM are almost zero. It is shown in Table 7 and Figure 6 that the HF1 component, with a 10–90% rising time of 3.73 μs, is very similar to a return stroke pulse, while the HF3 component, with a very small peak, is very similar to an M-event.
Table 6. Simulated Parameters for the RM Pulse in Flash 0902
Table 7. Statistical Characteristics of the Heidler Function Component for the RM Pulse
 Based on the above discussion regarding the RM and section 4.3addressing the return stroke, it is inferred that the RM pulse and four return strokes of 0902 are produced by a return stroke and M-component that are quite possibly initiated almost simultaneously from different upper branches. This conclusion supports the reports ofWeidman and Krider  and Le Vine and Willett . The almost simultaneous occurrence of these events leads to the superposition of the leader and the M-component incident wave in the common lower channel. However, the forms of formation of the RM and return stroke are different. For the return stroke, the M-component in one branch is much smaller than the return stroke pulse in the other branch. The M-component and return stroke pulses occurred simultaneously in different branches. In contrast, in the RM-event, the M-component in one branch is almost the same as the return stroke pulse in the other branch, and there is a small pulse just preceding the return stroke pulse of the RM, which leads to a concave region at the front of the RM (seen inFigure 6).
 It is not clear whether the return stroke subsidiary peaks of 0902 come from M-components or other pulse. If they are not M-components, there are other possible ways of producing subsidiary peaks, which are described below.
 The second possible cause of flash 0902 is reflection of the current at the far end of the channel. This phenomenon is similar to the reflection that occurs during intracloud discharge [Hamlin et al., 2007; Nag and Rakov, 2010] when the forward current pulse encounters an impedance discontinuity at the far side of the channel. However, only a few currents with subsidiary peaks are observed, which may be related to the reflection ratio and the signal amplitude decay as a function of length/time [Hamlin et al., 2007]. Flash 0902 occurs at a small thundercloud cell, and the bottom of the cloud is only approximately 800 m above the ground based on radar observations. Therefore, the channel length of flash 0902 is relatively short, which results in the reflected current pulse having sufficient remaining amplitude when it reaches the channel base so that the subsidiary peak can be distinctly shown. Based on the wave superposition analysis, whether a subsidiary peak is observed depends on many factors, such as the amplitude, rising time, steepness factor and time delay of the reflection waveform. The HF3 components of R3 and R4 exhibit obvious time delays of 10.5 and 9.48 μs, respectively, which are much larger than other Heidler function components. The results indicated the possibility that the pulse corresponding to the third peaks of R3 and R4 occur later than the first two. This finding is showed that the third peaks of R3 and R4 were possibly due to reflection of the current at the far end of the channel.
 Third, the current subsidiary peak in flash 0902 is possibly caused by the corona current. Based on the simulated results, each return stroke current contains a Heidler function component with a fast rising time, which corresponds to the first peak, and one with a slow rising time, which corresponds to the first subsidiary peak. In Table 4, it can be observed that the time delays of the HF2 and HF1components in each return stroke are less than 0.4 μs, indicating that the two pulse components occur almost simultaneously. Based on Cabrera and Cooray's  channel structure model, the return stroke current can be divided into the corona current and the breakdown current [Lin et al.,1980, Diendorfer and Uman, 1990]. On the basis of the simulated results, it is found that the HF2 and HF1 components of each return stroke correspond to the breakdown current and the corona current of a normal return stroke, respectively. Therefore, the subsidiary peak of the return stroke may be caused by the corona current, as illustrated in the report of Lin et al. [1980, Figures 9–11].
 Based on our observed results and the above discussion, we were of the opinion that the first possibility was more reasonable than the other two for flash 0902. Whether the subsidiary peak is really related to the above three causes will require additional measurements and analyses.
 As to the cause of subsidiary peak of flash 0803, we notice that the waveforms of flash 0803 were very similar to that in Figure 2 presented by Jerauld et al. . They considered it possible that a flashover resulted in a dip in the normal current waveform, causing some current pass through a triggering wire from a previous unsuccessful launch, instead of the current-measuring resistor. They reconstructed the missing current in waveform by adding a trend line. The dotted lines shown inFigure 4 represent a reconstruction of every current waveform using the method of Jerauld et al. . The reconstruction indicates that the return stroke waveform of 0803 is a combination of the normal return stroke waveform and a missing current pulse. However, each return stroke of 0902 is more similar to a superposition of a normal return stroke waveform and an increasing current pulse. Moreover, there was an unsuccessful launch before flash 0803 occurred, but not before flash 0902.
 Based on the above discussion, the reasons for existence of the subsidiary peak in flash 0803 may be different from those for flash 0902. There is a small possibility that the subsidiary peaks of flash 0902 are produced by a flashover. However, it is possible that the flashover contributes to producing the second peak of flash 0803.
 Thanks all the workmates who took part in and supported the SHATLE from 2005 to 2010. The authors also thank anonymous reviewers for their comments and suggestions which helped us improve the manuscript. The research was supported by National Natural Science Foundation of China (grant 40774083, 40930949), One Hundred Person Project of the Chinese Academy of Sciences, and partially supported by PHR (IHLB) (grant PHR201008435).