Compact intracloud lightning discharges: 2. Estimation of

Compact intracloud lightning discharges:2. Estimation of electrical parameters

Amitabh Nag1 and Vladimir A. Rakov1

Received 19 March 2010; revised 14 June 2010; accepted 19 July 2010; published 20 October 2010.

[1] We estimated electrical parameters of 48 located compact intracloud discharges(CIDs) using their measured electric fields and vertical Hertzian dipole approximation.This approximation is consistent with the more general bouncing‐wave CID model for areasonably large subset of allowed combinations of propagation speed and channel length.For nine events, we estimated channel lengths from observed reflection signatures inmeasured dE/dt waveforms and assumed propagation speeds between 2 × 108 m/s and 3 ×108 m/s, which limit the range of allowed values. For a speed of 2.5 × 108 m/s (averagevalue), the channel lengths for these nine events ranged from 108 to 142 m. Thecorresponding geometric mean values (GM) of peak current, zero‐to‐peak current risetime,and charge transfer in the first 5 ms are 143 kA, 5.4 ms, and 303 mC, respectively. The GMpeak radiated power and energy radiated in the first 5 ms are 29 GW and 24 kJ,respectively. For the remaining 39 events, there were no reflection signatures observed,and the channel length was assumed to be 350 m, for which the Hertzian dipoleapproximation is valid for speeds in the range of 2 to 3 × 108 m/s. In this case, GM valuesof peak current, zero‐to‐peak current risetime, and charge transfer in the first 5 ms are 64 kA,4.9 ms, and 142 mC, respectively. The GM peak radiated power and energy radiated in thefirst 5 ms are 28 GW and 32 kJ, respectively.

Citation: Nag, A., and V. A. Rakov (2010), Compact intracloud lightning discharges: 2. Estimation of electrical parameters,J. Geophys. Res., 115, D20103, doi:10.1029/2010JD014237.

1. Introduction

[2] Nag and Rakov [2010] (part 1) showed that the currentdistribution along the CID channel is often not much dif-ferent from uniform, because of relatively short channellength, Dh, relatively long current waveform, and relativelyhigh propagation speed, v. This observation suggests that atleast for some “allowed” combinations of v and Dh, we canreasonably approximate the CID channel by a Hertziandipole. This approximation will enable us to simplify thefield equations and use measured CID fields to infer variousparameters of CIDs. Within the Hertzian dipole approxi-mation, the propagation speed is not an input parameter andthe current waveshape is the same as that of the time integralof CID radiation field signature.[3] In this paper, we determine the limits of validity of the

Hertzian dipole approximation as applied to CIDs, and usethis approximation to infer the peak current, current rise-time, charge transfer, radiated power, and radiated energyfor the 48 located CIDs previously studied by Nag et al.[2010]. Additionally, we estimate the upper bound on

cloud electric field prior to CID and total energy dissipatedby this type of lightning discharge.

2. Hertzian Dipole Approximation

[4] The general time domain equation for computing thevertical electric field dEz due to a vertical differential currentelement idz (vertical dipole of length dz carrying a uniformcurrent i(t)) at a height z above a perfectly conductingground plane for the case of an observation point P being onthe plane at a horizontal distance r from the dipole is givenby [e.g., Uman, 1987; Thottappillil et al., 1997]

dEz r; tð Þ ¼ 1

2�"0

2z2 � r2ð ÞR5 zð Þ dz

Z t

tb zð Þ

i z; � � R zð Þc

� �d� þ 2z2 � r2ð Þ

cR4 zð Þ

264

� i z; t � R zð Þc

� �dz� r2

c2R3 zð Þdi z; t � R zð Þ

c

� �dt

dz

35; ð1Þ

where "0 is the electric permittivity of free space, tb(z) is thetime at which the current is “seen” by an observer to beginat height z, c is the free‐space speed of light, and R is theinclined distance from the dipole to the observation point,which is given by R(z) =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ r2

p. From equation (1) for the

geometry shown in Figure 1, the total electric field at theobservation point for a finite‐length channel whose lower

1Department of Electrical and Computer Engineering, University ofFlorida, Gainesville, Florida, USA.

Copyright 2010 by the American Geophysical Union.0148‐0227/10/2010JD014237

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, D20103, doi:10.1029/2010JD014237, 2010

D20103 1 of 15

http://dx.doi.org/10.1029/2010JD014237

and upper ends are at altitudes of h1 and h2, respectively, isgiven by

Ez r; tð Þ ¼ 1

2�"0

Zh2h1

2z2 � r2ð ÞR5 zð Þ

Z t

tb zð Þ

i z; � � R zð Þc

� �d�dz

264

þ 2z2 � r2ð ÞcR4 zð Þ i z; t � R zð Þ

c

� �dz

� r2

c2R3 zð Þdi z; t � R zð Þ

c

� �dt

dz

35; ð2Þ

where h2 is a function of time during the first traversal of thechannel and constant afterward.[5] Equation (1) also applies to a vertical dipole of finite

length Dh at height h, provided that Dh is very shortcompared to the shortest significant wavelength l (Hertzianor electrically short dipole approximation). For example, adipole of length, Dh = 500 m can be considered Hertzian ifl � 500 m. This means that the above approximation isvalid for frequencies f � 600 kHz. For a vertical Hertziandipole we can write

Ez r; tð Þ ¼ 1

2�"0

2h2 � r2ð ÞDh

R5

Z t

tb

i � � R=cð Þd�24

þ 2h2 � r2ð ÞDh

cR4i t � R=cð Þ � r2Dh

c2R3

di t � R=cð Þdt

�; ð3Þ

where R =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2 þ r2

p. Note that current i in equation (3)

varies only as a function of time, with all the geometricalparameters being fixed.[6] In section 3, we will show that the vertical Hertzian

dipole approximation is consistent with the more generalCID bouncing‐wave model for a reasonably large subset of

“allowed” combinations (established in part 1 [Nag andRakov, 2010]) of v and Dh.[7] Equation (3) can be rewritten as a second‐order dif-

ferential equation,

dEz

dt¼ Dh

2�"0

2h2 � r2ð ÞR5

iþ 2h2 � r2ð ÞcR4

di

dt� r2

c2R3

d2i

dt2

� �; ð4Þ

where arguments of Ez and i have been dropped to simplifynotation. For known Ez and the geometrical parameters (Dh,h, and r) this equation can be numerically solved for i.[8] We employed the Runge‐Kutta method of order three

(with four stages and an embedded second‐order method,also known as the Bogacki–Shampine method [Bogacki andShampine, 1989]) to solve equation (4) for i using measuredelectric fields Ez (which had a better signal‐to‐noise ratiothan the measured electric field derivative waveforms) of 48located CIDs that occurred at horizontal distances r rangingfrom 12 to 89 km and heights h ranging from 8.8 to 29 km[Nag et al., 2010]. The initial and final values of currentwere required to be zero, and the error tolerance of thenumerical solution was set to 10−6. Channel lengths Dh for9 CIDs were estimated from reflection signatures in electricfield derivative (dE/dt) waveforms and assumed propagationspeeds representing the entire range of their allowed values[see Nag and Rakov, 2010] (part 1). For the remaining 39CIDs there were no reflection signatures observed, and areasonable value of Dh = 350 m was assumed. This value isconsistent with the Hertzian dipole approximations forspeeds in the range of 2 to 3 × 108 m/s. We also consideredother values of Dh, which are consistent with the Hertziandipole approximation. Note that for Ez measured at fardistances, the peak current can also be estimated using theradiation field approximation, given by

dEz

dt¼ Dh

2�"0� r2

c2R3

d2i

dt2

� �; ð5Þ

Figure 1. Geometrical parameters needed for calculating the vertical electric field at observation point Pon perfectly conducting ground at horizontal distance r from the vertical CID channel.

NAG AND RAKOV: COMPACT INTRACLOUD DISCHARGE ELECTRICAL PARAMETERS D20103D20103

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from which it follows that Ez is proportional todi

dt.

Equations (4) and (5) represent two ways to solve the“inverse source problem,” with equation (5) assuming thatcontributions from the electrostatic and induction fieldcomponents (the first and second terms in equation (4)) arenegligible. In order to compare equations (4) and (5), we(step a) assumed a CID current waveform, (step b) computedelectric field at a horizontal distance of 200 km, where itis essentially radiation, using general equation (3), (step c)solved the inverse source problem using the computed elec-tric field and equations (4) and (5), and (step d) comparedthe results with the assumed current. The computed electricfield waveform at 200 km is shown in Figure 2a and thethree current waveforms, the assumed one and those inferredfrom equations (4) and (5), are shown in Figure 2b. Thebottom of the CID channel was assumed to be at 15 km andits length was set to 100 m. It is clear from Figure 2b thatthe three current waveforms are very similar to each other.For the 48CIDs examined in this paper, occurring at horizontaldistances ranging from 12 to 89 km, current peaks estimatedusing the radiation field approximation (equation (5)) were,on average, 15% greater than those obtained by solving exact

equation (4). The differences in peak ranged from about 1%to 31% for 47 CIDs and for one CID the difference was47%. We attributed the discrepancy to contributions fromthe electrostatic and induction field components at closerdistances and to better constrained (regularized) solutionsyielded by the Runge‐Kutta procedure that required zerovalues for initial and final currents. We employed equation(4) for estimating CID currents presented in this paper.[9] In Appendix A, we estimated errors in current inferred

from equation (5) due to uncertainties in CID heights andtheir horizontal distances from the field‐measuring station[see Nag et al., 2010]. We assume that errors in current (dueto uncertainties in the same two parameters) inferred fromequation (4) are similar.[10] The charge transferred up to time t can be obtained by

integrating the current with respect to time,

Q ¼Z t

0i �ð Þd�: ð6Þ

Further, the source current can be used to find the radiationcomponents of E� and H� and hence the Poynting vector,total radiated power, and energy. The magnitude of thePoynting vector, which has the meaning of the radiatedpower density, can be obtained as

S ¼ E� � H�

; ð7Þ

where E� =Dh

4�"0sin �

1

c2R

di

dtand H� =

Dh

4�sin �

1

cR

di

dt[e.g.,

Uman, 1987]. After substituting the latter two field expres-sions in (7) we get

S ¼ 1

"0c3Dh sin �

4�R

di

dt

� �2

: ð8Þ

[11] The total radiated power, obtained by integrating (8)over a spherical surface of radius R, whose center is at theposition of the dipole, is (presence of ground is not takeninto account)

Prad ¼Z 2�

�¼0

Z �

�¼0S R2 sin � d� d� ¼ Dh2

6�"0c3di

dt

� �2

; ð9Þ

where � and � are the polar and azimuthal angles of thespherical coordinate system. Note that Prad is proportional todi

dt

� �2

. The total energy dissipated up to time t can be ob-

tained by integrating the radiated power with respect to time,

W ¼Z t

0Prad �ð Þd�: ð10Þ

Sincedi

dtis inversely proportional to Dh (see equation (5)),

|S|, Prad, and W are each independent of Dh.

3. Limits of Validity of the Hertzian DipoleApproximation

[12] In this section, we compare electric fields produced at200 km by a CID at a height of 15 km having a zero‐to‐peakcurrent risetime RT of 6 ms computed using the verticalHertzian dipole approximation with their counterparts pre-

Figure 2. Comparison of equations (4) and (5). (a) Bouncing‐wave‐model‐predicted electric field at a horizontal distanceof 200 km for a CID with its channel bottom at a height of15 km,Dh = 100 m, v = 2 × 108 m/s, r = 0, and the assumedinput current shown in Figure 2b (black line). (b) Assumedcurrent waveform (black line), used as an input in comput-ing the electric field shown in Figure 2a, and those obtainedby solving equations (4) (blue line) and (5) (red line). Seetext for details.


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dicted by the bouncing‐waveCIDmodel [seeNag and Rakov,2010] (part 1). Calculations were performed for differentcombinations of effective current reflection coefficients atchannel ends, channel lengths, and propagation speeds, eachwithin the bounds (“allowed” ranges) established in part 1[Nag and Rakov, 2010]. The Hertzian dipole was excited bythe current found, using the bouncing wave model, for themiddle of the channel (z = h1 + Dh/2) [see Nag and Rakov,2010, Figure 4a] (part 1). The bouncing‐wave model pre-dicted waveforms were used as the ground truth, and electricfield waveforms based on the Hertzian dipole approximationwith initial peaks within about 15% of ground truth peakswere considered as confirming the validity of the approxi-mation. The influence of RT was also considered.[13] Figure 3 shows the electric fields for the Hertzian

dipole approximation (dashed line) and bouncing‐wavemodel (solid line) for r = 0, v = 2 × 108 m/s, RT = 6 ms, andchannel lengths of 100, 350, and 700 m. While the Hertziandipole approximation is acceptable for the lengths in therange of 100–350 m, it is not for 700 m. Figure 4 shows theelectric fields for the Hertzian dipole approximation (dashedline) and bouncing‐wave model (solid line) for r = −0.5, v =2 × 108 m/s, RT = 6 ms, and channel lengths of 100 and350 m. In both cases the Hertzian dipole approximation isacceptable. The results are summarized and compared withthe “allowed” ranges of variation of v and Dh in Figure 5,from which one can see that the Hertzian dipole approxi-mation is consistent with the bouncing‐wave model for areasonably large subset of “allowed” combinations ofpropagation speed and channel length. Specifically, it can be

seen from Figure 5 that for r = 0 the Hertzian dipoleapproximation is valid for Dh ranging from about 100 to550 m and for speeds ranging from about 0.7 × 108 m/s to 3 ×108 m/s, while the “allowed” ranges are from about 100 to1000 m for Dh and from about 0.3 × 108 m/s to 3 × 108 m/sfor v. For r = −0.5, the Hertzian dipole approximation isvalid for the entire “allowed” domain (from about 100 to500 m for Dh and from about 0.7 × 108 m/s to 3 × 108 m/sfor v [see Nag and Rakov, 2010, Figure A4b] (part 1). Notethat for RT = 6 ms the maximum value of Dh/v (channeltraversal time), for which the Hertzian dipole approximationis acceptable, is almost constant and equal to about 1.9 msand 1.7 ms for r = 0 and r = −0.5, respectively.[14] Channel length values for which the Hertzian dipole

approximation is valid (initial field peaks are within about15% of bouncing‐wave model predicted peaks) for differentvalues of propagation speed and for RT = 6 ms are given inTable 1. Note that the errors for the opposite polarityovershoot were larger, so the Hertzian dipole approximationmay be invalid after the initial half cycle, whose duration inour data set varies from 2.8 to 13 ms with a GM of 5.6 ms[Nag et al., 2010]. Since the errors at later times (during theopposite polarity overshoot of the electric field waveform) arelarger, the waveforms for current, charge, radiated power, andenergy (see, for example, Figure 6) may be invalid at thosetimes. Additionally, for some of the electric field waveformsafter the initial half cycle and particularly toward the tail of theopposite polarity overshoot the signal‐to‐noise ratio wasrather poor. This is why we limited our estimates of chargetransfer and energy to the initial 5 ms of the process. Note

Figure 3. Vertical electric fields at 200 km for the bouncing‐wave model (solid line) for r = 0, v = 2 ×108 m/s, RT = 6 ms, and channel lengths of (a) 100, (b) 350, and (c) 700 m versus those for the Hertziandipole approximation (dashed line). The Hertzian dipole approximation is acceptable for channel lengthsin the range of 100–350 m, but not for 700 m.


4 of 15

that the peak current, current risetimes, and peak power aregenerally unaffected by the uncertainties encountered at latertimes.[15] We now consider the influence of zero‐to‐peak rise-

time, RT. Nag and Rakov [2010] (part 1) determined that theCID zero‐to‐peak current risetime is likely to be in the rangefrom about 2 to 8.5 ms. For RT = 2 ms, the Hertzian dipoleapproximation is not valid for any allowed combination ofparameters. For RT = 3 ms, the approximation is validwhen v = 2 to 3 × 108 m/s and Dh = 100 to 200 m (channeltraversal time,Dh/v = 0.3 to 1 ms) for both r = 0 and r = −0.5.For RT = 8.5 ms, the approximation is valid for Dh = 100 mand v = 1 to 3 × 108 m/s (channel traversal time,Dh/v = 0.3 to1 ms), which cover the entire range of “allowed” values forboth r = 0 and r = −0.5. In summary, the shorter the currentrisetime, the smaller the Hertzian dipole domain relative tothe allowed one, as expected.

4. Electrical Parameters of CIDs

[16] Of the 48 located CIDs studied by Nag et al. [2010],for nine events, we were able to estimate channel lengthsusing reflection signatures in their measured dE/dt wave-forms and assumed speeds within their limiting values of 2 ×108 and 3 × 108 m/s. Speeds lower than 2 × 108 m/s are notconsidered, since, within our model, for measured traversaltimes they would result in unreasonably small radiatorlengths. For v = 2.5 × 108 m/s (average speed value),channel lengths for the nine events range from 108 to 142 m.These events occurred at horizontal distances of 19 to 89 kmand estimated heights of 8.8 to 19 km, and had electric fieldpeaks normalized to 100 km ranging from 14 to 35 V/m.Figure 6 shows the measured electric field and electric field

derivative along with the inferred current, charge trans-ferred, and radiated power and energy, each as a function oftime, for one such event. Vertical arrows indicate reflectionsignatures in the dE/dt waveform. The estimated channellength (for v = 2.5 × 108 m/s) was 138 m. Table 2 sum-marizes the estimated parameters for all nine events. Thegeometric mean (GM) peak current is 143 kA with a rangeof 87 to 259 kA. The zero‐to‐peak current risetimes rangefrom 3.0 to 9.5 ms with a GM of 5.4 ms. The charge transferfor the first 5 ms ranges from 79 to 496 mC with the GM

Figure 5. Comparison of the Hertzian dipole validitydomain (combinations of propagation speed and channellength) with the “allowed” one for RT = 6 ms and r = 0.See text for details.

Figure 4. Same as Figure 3 but for r = −0.5 and channel lengths of (a) 100 and (b) 350 m, for both ofwhich the Hertzian dipole approximation is acceptable.


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being 303 mC. The GM peak radiated power and energyradiated for the first 5 ms are 29 GW (ranging from 12 to70 GW) and 24 kJ (ranging from 7.5 to 52 kJ), respectively.[17] We now discuss uncertainties in the estimated elec-

trical parameters for the nine events. For the lower bound onspeed (2 × 108 m/s), all radiator lengths (86 to 114 m) arenear the assumed lower bound, 100 m [Nag and Rakov,2010, Appendix B] (part 1), which is expected becausemultiple reflection signatures are likely to be pronouncedonly for the shortest channels. For the upper bound on speed(3 × 108 m/s), the channel lengths would increase by a factorof 1.5 to range from 129 to 171 m. Since the inferred peakcurrent and charge transfer are inversely proportional tochannel length within the Hertzian dipole approximation (seesection 2), these two parameters would decrease by a factorof 1.5. Thus, the uncertainty in our current and chargetransfer estimates for the 9 events is ≤ 25% for the assumedv = 2.5 × 108 m/s. The current risetimes, peak power, andenergy values, on the other hand, are independent of channellength (and hence of assumed speed) and will remain thesame as those for 2.5 × 108 m/s. Table 3 summarizes thepeak current and charge transfer at 5 ms each scaled to dif-ferent channel lengths (inferred using measured channeltraversal times and different assumed propagation speeds)for the nine CIDs.[18] Note that since all Dh values are less than 200 m

(even for v = 3 × 108 m/s), all nine events can be reasonablyapproximated by Hertzian dipoles for almost the entiretypical range of current zero‐to‐peak risetimes (about 2 to8.5 ms). As a result, the inferred current risetimes (rangingfrom 3 to 9.5 ms with the GM being 5.4 ms) should be closeto their true values, confirming our assumed typical zero‐to‐peak risetime of 6 ms.

Table 1. Comparison of Electric Fields Based on the HertzianDipole Approximation and Bouncing‐Wave Model for a Zero‐to‐Peak Current Risetime of 6 ms by Different Combinations ofCurrents Reflection Coefficients, Speeds, and Channel Lengthsa

Speed(m/s)

ChannelLength(m)

Difference inInitial Peak

(%)

Difference inOpposite PolarityOvershoot (%)

Reflection Coefficient r = −0.50.7 × 108 100 8.7 58

125 15 371 × 108 100 3.2 48

170 13 392 × 108 100 −0.76 −1.9

350 15 393 × 108 100 −1.3 −1.9

500 14 45

Reflection Coefficient r = 00.3 × 108 100 51 1.10.7 × 108 100 7.4 −1.4

135 15 −0.77250 61 1.6

1 × 108 100 2.7 −1.7170 11 −0.7190 15 −0.3350 60 1.6

2 × 108 100 −0.62 −1.9350 13 −1.0375 15 −0.9700 63 1.8

3 × 108 100 −1.3 −1.9500 12 −1.1550 15 −0.891000 60 1.6

aCombinations of speed and channel length in boldface are considered tobe consistent with the Hertzian dipole approximation.

Figure 6. (a) Measured electric field, (b) measured electric field derivative, (c) inferred current,(d) inferred charge transfer, (e) inferred radiated power, and (f) inferred radiated energy, each as a functionof time, for a CID that occurred at a horizontal distance of 19 km, at an inferred height of 15 km, and had aninferred channel length of 138 m (v = 2.5 × 108 m/s). The current and power values given in the parenthesesare the peaks, and the charge and energy values are estimated at 5 ms.


6 of 15

[19] For the remaining 39 events, which did not exhibitreflection signatures, the electrical parameters were esti-mated for three assumed values of channel length, 170, 350,and 500 m, for which the Hertzian dipole approximation isvalid (for RT = 6 ms) if the implied propagation speeds are≥108, ≥2 × 108, and 3 × 108 m/s, respectively. CID electricalparameters for the three values ofDh are compared in Table 4.The geometric mean values of peak current for the assumedchannel length values ofDh = 170 m, 350 m, and 500 mwere132, 64, and 45 kA, respectively. The corresponding geo-metric mean charge transfers at 5 ms were 293, 142, and100 mC. Note that for the 39 events the Hertzian dipolevalidity domain is smaller than the “allowed” domain. As aresult, we cannot assign any specific uncertainty to the esti-mated parameters in this case.[20] Figures 7, 8, 9, 10, 11, and 12 show histograms for

the peak current, zero‐to‐peak current risetime, 10−90%current risetime, charge transfer at 5 ms, peak radiatedpower, and energy at 5 ms, respectively, for all 48 events,including 9 events with channel lengths estimated fromreflection signatures and 39 events with an assumed channellength of 350 m. The minimum, maximum, arithmetic, andgeometric mean values for each parameter are given for the9 and 39 events individually and for all 48 events combined.As noted earlier, current risetimes, peak radiated power, andradiated energy are independent of channel length, whilepeak current and charge transfer can be scaled to otherchannel lengths, provided that they are consistent with theHertzian dipole approximation.[21] For the 39 CIDs, the geometric mean values of zero‐

to‐peak current risetime and 10−90% current risetime are4.9 and 2.5 ms, respectively. The geometric mean peakradiated power and energy radiated for the first 5 ms are

28 GW and 32 kJ, respectively. For all 48 events, GM va-lues of peak current, zero‐to‐peak current risetime, 10−90%current risetime, and charge transfer for the first 5 ms are74 kA, 5.0 ms, 2.5 ms, and 164 mC, respectively. Thecorresponding geometric mean peak radiated power andenergy radiated for the first 5 ms are 29 GW and 31 kJ. Notethat in the distributions of peak current and charge transfer(Figures 7 and 10, respectively) the nine CIDs with reflectionsignatures constitute the tail of the histogram, which is relatedto their shorter inferred channel lengths. Overall, the CIDcurrent parameters are comparable to their counterparts forfirst return strokes in cloud‐to‐ground lightning, while theirpeak radiated electromagnetic power appears to be consid-erably higher, as discussed in section 7.

5. Upper Bound on Electric Field Prior to CID

[22] In this section, we will consider the CID as a dis-charge between two spherical charge regions. Such simpli-fied configuration is similar to that employed by Cooray[1997] for studying regular cloud discharges. CIDs wereobserved to occur both inside clouds and above cloud tops[e.g., Smith et al., 2004; Nag et al., 2010]. The chargeconfiguration considered in this section, as well as in section6, is meant to apply to the in‐cloud variety.[23] For a CID current pulse with a peak current of 50 kA,

total duration of 30 ms, and zero‐to‐peak risetime of 6 ms[see Nag and Rakov, 2010] (part 1) the total CID chargetransfer (which is the time integral of current), Q, is 0.44 C.Let us consider two spherical volumes, one containing +0.44 Cof charge and the other −0.44 C of charge, each with a uniformvolume charge density, rv, of 20 nC/m3 (absolute value)[e.g., Rakov and Uman, 2003, section 3.2.5]. Then, from

Table 2. Parameters of Nine Located CIDs With Channel Lengths Estimated Using Reflections in dE/dt Waveforms and Assumed Prop-agation Speed of 2.5 × 108 m/sa

EventIdentification

Distance(km)

RadiatorHeight(km)

Electric Fieldat 100 km(V/m)

RadiatorLength(m)

PeakCurrent(kA)

Zero‐to‐PeakCurrent Risetime

(ms)

10–90% CurrentRisetime(ms)

Charge Transferat 5 ms(mC)

PeakPower(GW)

Energyat

5 ms (kJ)

082408_427 19 15 14 138 140 7.9 3.9 243 18 17083008_009 89 19 35 108 197 3.8 1.5 496 70 52083008_031 52 16 28 138 259 5.9 3.4 441 28 39083008_045 50 16 30 128 142 3.4 1.5 412 50 40083008_052 50 17 30 139 148 3.0 1.4 493 59 50091008_138 33 16 14 134 118 9.5 4.9 79 12 9.1091008_140 36 13 22 142 104 4.7 1.6 272 28 23091108_161 30 10 21 123 157 6.9 3.6 412 36 25091108_176 43 8.8 16 113 87 7.0 3.9 230 12 7.5AM 45 14 24 129 150 5.8 2.9 342 35 29GM 41 14 22 129 143 5.4 2.6 303 29 24Min 19 8.8 14 108 87 3.0 1.4 79 12 7.5Max 89 19 35 142 259 9.5 4.9 496 70 52

aAM, arithmetic mean; GM, geometric mean; Min, minimum value; and Max, maximum value.

Table 3. Peak Current and Charge Transfer at 5 ms Scaled to Different Channel Lengths That Were Inferred for Nine CIDs UsingReflection Signatures in Measured dE/dt Waveforms and Assumed Propagation Speeds

v (m/s)

Dh (m) Peak Current (kA) Charge Transfer at 5 ms (mC)

AM GM Min Max AM GM Min Max AM GM Min Max

2 × 108 (lower bound) 103 103 86 114 188 179 108 324 427 379 98 6213 × 108 (upper bound) 155 154 129 171 125 119 72 216 285 253 65 4142.5 × 108 (average) 129 129 108 142 150 143 87 259 342 303 79 496


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Q = 43 pb

3 rv, the radius, b, of each sphere is 174 m. Let usassume that the vertical distance between the surfaces ofthese spherical regions is 100 m (assumed to be approxi-mately equal to the assumed shortest CID channel length), asshown in Figure 13a, and that themedium outside the chargedspheres contains zero net charge. Presence of ground andother charges in the cloud is neglected. The primary questionto be answered here is: Can the positive and negative chargesestimated for CIDs be accumulated 100 m apart withoutexceeding the conventional breakdown electric field? The100 m assumption allows us to obtain the upper bound onelectric field, since, all other conditions being the same, largerseparations between the spheres will result in lower electricfields.[24] For each sphere, the electric field intensity at any

point at a distance r from the center of the sphere can beobtained using Gauss’s law as

E ¼

r�v3"0

r � b

b3�v3"0r2

r � b:

8>><>>:

ð11Þ

For the positively charged sphere rv is positive and theelectric field is directed radially outward, while for thenegatively charged one it is negative and directed radiallyinward. For the configuration shown in Figure 13a, the totalelectric field at any point along the line joining the centers ofthe two spheres (x = 0) is given by superposition of the fieldcontributions (both directed upward) from the two spheres.The total electric field intensity profile (solid line) andcontributions from the individual spheres (dashed line) areshown in Figure 13b. Inside each of the charged spheres, thetotal electric field intensity increases almost linearly withdistance from 2 × 104 V/m at the center to a maximum of1.8 × 105 V/m at the surface. Outside the charged spheres,the electric field intensity is minimum (1.6 × 105 V/m) at apoint equidistant from the two spheres. A CID channel islikely to be formed primarily between the two sphericalvolumes of charge, where the net electric charge is assumedto be zero and the electric field intensity is near its highestvalues. The charges are apparently collected in the sourceregion and delivered to the vertical channel end via a heavilybranched streamer network pervading this region. Thewideband radiation field signature of such a “volume radi-ator” is presently unknown. We assume here that theobserved wideband vertical electric field signatures of CIDsare essentially due to the current in the vertical channel. Thisassumption would be justified if the feeding streamers in the“source region” were predominantly horizontal (leavingaside the spherical geometry), because such channels would

contribute little to nothing to CID vertical electric fieldwaveforms.[25] We will now examine dependence of electric field

profile on charge transfer and volume charge density byusing the peak currents for the nine events summarized inTable 2, but keeping the same current waveshape (zero‐to‐peak risetime of 6 ms and total duration of 30 ms). In section 4,we estimated the channel lengths for the nine located CIDsusing channel traversal times measured in dE/dt waveformsand assumed propagation speed of 2.5 × 108 m/s to rangefrom 108 to 142 m. The inferred peak currents ranged from87 kA to 259 kA, with the geometric mean being 143 kA. Thetotal charge transferred (over 30 ms current duration) for peakcurrents of 87, 143, and 259 kA are 0.77 C, 1.3 C, and 2.3 C,respectively. For a total charge transfer of 1.3 C andassumed volume charge densities of 2 nC/m3, 20 nC/m3,and 200 nC/m3, the radii of the spherical charge regions are533m, 247m, and 115m, respectively; the maximum electricfield intensities (occurring between the spheres, on theirsurfaces) are 6.9 × 104, 2.8 × 105, and 1.1 × 106 V/m,respectively. The maximum electric field values for differentcombinations of charge transfer and assumed charge densityare summarized in Table 5. Additionally given are minimumand average electric fields between the spherical charge re-

Table 4. Peak Current and Charge Transfer at 5 ms Scaled to Dif-ferent Channel Lengths for 39 CIDs

Dh, m v (Allowed) m/s

Peak Current (kA)Charge Transfer at 5

ms (mC)

AM GM Min Max AM GM Min Max

170 ≥108 142 132 68 328 327 293 45 706350 ≥2 × 108 69 64 33 160 159 142 22 343500 3 × 108 48 45 23 112 111 100 15 240

Figure 7. Histogram of peak currents for 48 CIDs. Fornine events (shaded histogram columns) with reflectionsignatures, channel lengths were inferred using channel tra-versal times measured in dE/dt waveforms and assumedpropagation speed of 2.5 × 108 m/s. For the other 39 events(white histogram columns) an assumed channel length of350 m (implied v ≥ 2 × 108 m/s) was used. Statistics givenare the arithmetic mean (AM), geometric mean (GM), mini-mum value (Min), and maximum value (Max) for the 9 and39 events individually and for all data combined.


8 of 15

gions and estimated electric potential difference between them.Except for the probably unrealistically high rv = 200 nC/m3

cases (for which maximum fields range from 7.3 × 105 V/mto 1.4 × 106 V/m), the maximum electric field for all com-binations of Q and rv considered does not exceed 3.5 ×105 V/m, which is less than the conventional breakdownelectric field in the cloud (on the order 106 V/m) and isgenerally on the order of 104 to 105 V/m. These maximumelectric field estimates are for Dh = 100 m. All other con-ditions being the same, the maximum electric field willdecrease with increasing Dh. The estimated electric fieldsappear to be too low for conventional breakdown, but maybe sufficient for runaway electron breakdown involving anenergetic cosmic ray particle.[26] If the two spherical charge regions were in direct

contact (Dh = 0) the total electric field profile would havemaximum at the point of contact that is given by

Emax ¼ 2b�v3"0

¼ 2Q

4�"0b2¼ Dp

4�"0b3; ð12Þ

where Dp = 2Qb is the dipole moment change for chargetransfer Q over a distance of 2b. Smith et al. [1999] usedsuch a configuration to impose a lower bound on CIDchannel length, which they assumed to extend between thecenters of the charged spheres. In doing so, they computedEmax for their average measured dipole moment change,Dp =0.38 C km, and different b and compared those field valueswith electric fields measured in thunderclouds. For b = 50 m(corresponds to their channel length of 100 m), they found

Emax = 2.7 × 107 V/m, which is about an order of magnitudegreater than the conventional breakdown electric field in thecloud, and concluded that such a short channel was physi-cally unrealistic. Our estimated CID channel lengths on theorder of 100 m, which are inferred from measured dE/dtwaveforms, suggest that Smith et al.’s configuration, inwhich two oppositely charged regions are in direct contact,is inconsistent with at least some of our observations. Inorder to explain such short channel lengths, the chargedregions should be separated by a region of essentially zeronet charge (which can be created due to mixing) or by aregion of very low charge density compared to that in thecharged regions. The question of how exactly the chargesare fed into the vertical CID channel remains open.

6. Total Energy Dissipated by CIDs

[27] In this section, we estimate the total energy dissipatedby a CID with reference to the charge configuration shownin Figure 13a. If we assume that a CID neutralizes all thecharge in each of the two spherical regions, the total energydissipated will be equal to the total electrostatic energystored in this charge configuration. One can find the totalelectrostatic energy as the sum of the electrostatic energiesrequired to individually assemble the two uniformly chargedspheres (self electrostatic energy [Cooray, 1997]) and theelectrostatic energy due to the two spherical charge regionsbeing placed in relatively close proximity to each other(mutual electrostatic energy).[28] The self electrostatic energy, W1, of a uniformly

charged sphere of radius b containing charge Q can be

Figure 8. Histogram of zero‐to‐peak current risetimes for48 CIDs. See also caption of Figure 7.

Figure 9. Histogram of 10−90% current risetimes for 48CIDs. See also caption of Figure 7.


9 of 15

readily found as [Cheng, 1993]

W1 ¼ 3Q2

20�"0b: ð13Þ

This equation can be obtained by integrating the electro-

static energy density, We ¼ "0E2

2, where E is given by

equation (11) for r ≤ b, over a spherical volume. Estimationof the mutual electrostatic energy, W2, is more complicatedand requires additional simplifying assumptions. We willreplace the two spherical charge regions with their equiva-lent point charges of magnitude +Q and −Q located atcorresponding sphere centers, so that they are separated bydistance (2b + Dh). This approximation is similar to thatemployed by Cooray [1997] for studying the energy dissi-pated by regular cloud discharges. The mutual electrostaticenergy W2 of these two point charges is given by [Cheng,1993]

W2 ¼ Q2

4�"0 2bþDhð Þ : ð14Þ

Thus, the total electrostatic energy W of the overall chargeconfiguration, and hence the energy dissipated by a resultantCID can be found as

W ¼ 2W1 þW2: ð15Þ

[29] Estimates of energy dissipated by a CID having a100 m long channel for different values of total chargetransfer Q (computed for a fixed current waveshape anddifferent peak currents) and volume charge density rv are

given in Table 5. The assumed volume charge density rv isused to compute b in equation (13) for a givenQ. For a chargetransfer of 0.44 C (peak current of 50 kA), the total dissipatedenergy values are 7.7 × 106, 1.6 × 107, and 3.3 × 107 J forcharge densities of 2, 20, and 200 nC/m3, respectively. Fora charge density of 20 nC/m3 and charge transfers rangingfrom 0.44 C to 2.3 C the energy values range from 1.6 × 107 to2.6 × 108 J. The corresponding average energy per unitlength ranges from 1.6 × 105 to 2.6 × 106 J/m.[30] It follows from equations (13) to (15) that, all other

conditions being the same, the mutual electrostatic energyW2 due to the two equivalent point charges and hence thetotal electrostatic energy of the charge configuration shownin Figure 13a decreases with increasing the CID channellength. Thus, the values of total CID energy given above forthe lower bound on channel length of 100 m should beviewed as the upper bound. For a CID current pulse with apeak of 50 kA, assumed rv = 20 nC/m3, and channel lengthsof 350, 500, and 1000 m, the total electrostatic energy va-lues will be 1.47 × 107, 1.42 × 107, and 1.35 × 107 J,respectively, versus 1.6 × 107 J for Dh = 100 m.[31] In our calculations of electrostatic energy, we have

neglected the effect of ground. This simplifying assumptionshould not introduce a significant error because (1) CIDsoccur at relatively large altitudes, typically greater than10 km [Smith et al., 2004; Nag et al., 2010] and (2) W2 isinversely proportional to the distance between equivalentpoint charges. Indeed, the mutual electrostatic energybetween a charge and its image separated by more than 20 kmis at least an order of magnitude smaller than that betweenthe two actual charges separated by less than 1.4 km (the

Figure 10. Histogram of charge transferred in 5 ms for 48CIDs. See also caption of Figure 7.

Figure 11. Histogram of peak radiated power for 48 CIDs.See also caption of Figure 7.


10 of 15

limiting case in Table 5; Dh = 100 m). This is true even forCID channel lengths of 1000 m in which case the twoequivalent point charges representing the actual charges areseparated by less than 2.3 km.

[32] The total electrostatic energy of the charge configu-ration shown in Figure 13a can be alternatively estimated asthe sum of energies stored inside the two spheres and energystored between them. The internal energy can be computed

as the volume integral of energy density,"0E2

2, which is

different from the self energy W1 in that E within eachsphere should additionally include the contribution from theother sphere. As a zero approximation, we neglect this dif-ference and assume that the internal energy is not muchdifferent from self energy W1 (the larger the separationbetween the spheres the smaller the difference). Further, wewill roughly estimate the external energy as the product ofcharge transfer Q and potential difference DV between thesurfaces of the two spheres, with DV being found as theaverage electric field intensity Eav between the spheres timesthe distance Dh between their surfaces. Values of Eav, DV,and the total energy, W = 2W1 + QDV, are given in Table 5.[33] The total electrostatic energy values estimated using

the two different approaches are fairly similar, particularlyfor more realistic lower (2 and 20 nC/m3) values of rv. Asexpected, the GM electromagnetic energy of 31 kJ radiatedduring the first 5 ms (see Figure 12) is much lower than thetotal CID energy estimates based on electrostatic con-siderations (see Table 5).

7. Discussion

[34] Smith et al. [1999], using distant (essentially radia-tion) electric field signatures of 15 CIDs from thunderstormsin New Mexico and west Texas, estimated the mean CIDdipole moment change to be 0.38 C km. The minimum andmaximum values were 0.26 and 0.80 C km. The meandipole moment change duration (10–90%) for the 15 CIDswas 13.7 ms. They also inferred that CID channel lengths

Figure 12. Histogram of radiated energy in 5 ms for 48CIDs. See also caption of Figure 7.

Figure 13. (a) Simplified charge configuration giving rise to a CID. The CID channel length,Dh = 100 m,is assumed to be approximately equal to the distance between the two spherical charged regions. The electricfield vector points upward and is assumed to be positive. (b) The electric field intensity as a function of z forx = 0 that is produced by each of the two charged regions (dashed line) shown in Figure 13a and the totalelectric field profile (solid line). Electric field, potential difference, and electrostatic energy values for thisand other configurations are given in Table 5.


11 of 15

should be in the range of 300 to 1000 m. Using the meandipole moment change (0.38 C km) and the limiting channellengths we estimate the range of charge transfers for theirCIDs to be 0.38 C to 1.27 C. Our estimated charge transfersfor the first 5 ms are on the order of tens to hundreds ofmillicoulombs (see Tables 2, 3 and 4 and Figure 10) and areapparently consistent with Smith et al.’s values which cor-respond to the mean charge transfer time of 13.7 ms.[35] Eack [2004], using simultaneous measurements of

electric fields at near and far distances for seven CIDs,estimated that, on average, a CID transferred 0.3 C over adistance of 3.2 km. These results are based on a value ofspeed derived from the misinterpreted dipole approximationequation [see Nag and Rakov, 2010, section 8] (part 1) and,therefore, are invalid. For one event produced by a NewMexico thunderstorm, Eack also estimated the peak current of29 kA from the measured predominantly induction electricfield peak (assuming a uniform current over the entire channellength). For the same event, Watson and Marshall [2007],using the transmission line model, inferred a considerablylarger peak current of 74 kA, which is about the same as ourestimates based on both the bouncing‐wave model andHertzian dipole approximation [see Nag and Rakov, 2010](part 1).[36] For downward negative lightning, Berger et al.

[1975] reported a median total charge transfers of 5.2 Cand 1.4 C for first and subsequent strokes, respectively.Schoene et al. [2009] reported the geometric mean chargetransfer within 1 ms after the beginning of the return strokefor 151 negative rocket‐triggered lightning strokes (whichare similar to subsequent strokes in natural lightning) to be1.0 C. Using in situ measured electric field profiles, Maggioet al. [2009] estimated the average (probably arithmeticmean) charge transferred by 29 IC flashes to be 17.6 C.Thus, charges transferred by individual CG strokes and byregular ICs are considerably larger than those transferredwithin the first 5 ms by CIDs.[37] Rakov et al. [1998], from two‐station measurements

of electric and magnetic fields of a dart‐stepped leader intriggered lightning, estimated that the formation of eachleader step is associated with a charge of a few milli-coulombs and a current of a few kiloamperes. From mea-surements of electric field pulses radiated by leader steps offirst strokes in natural lightning, Krider et al. [1977] inferredthat the minimum charge involved in the formation of a stepis 1–4 mC and the peak step current is at least 2–8 kA. Thecharge transfers for cloud‐to‐ground lightning leader stepsare 1−2 orders of magnitude smaller than those transferredduring the first 5 ms by CIDs. Our estimated CID peakcurrents of tens to hundreds of kiloamperes are at least anorder of magnitude greater than those expected for leadersteps. Note that the step‐formation process is thought tooccur on a time scale on the order of 1 ms, and typical steplengths are 10 and 50m for dart‐stepped and stepped negativeleaders, respectively [Rakov and Uman, 2003].[38] Krider et al. [1968] estimated the peak input power

per unit channel length for a natural lightning first stroke tobe 7.8 × 108 W/m. Jayakumar et al. [2006] found the meanvalue of peak input power per unit length for triggeredlightning strokes to be 9.6 × 108 W/m. Krider and Guo[1983] and Krider [1992] estimated the wideband radio‐frequency electromagnetic power radiated by a subsequentT

able

5.Maxim

umElectricField

attheSurface

ofEachof

theTwoOpp

ositely

Charged

Sph

eres

Separated

byaZeroNet

ChargeRegion(See

Figure13

a)forDifferent

Com

binatio

nsof

ChargeTransferandAssum

edVolum

eChargeDensity

a

Peak

Current

(kA)

Total

Charge

Transfer

(Q)(C)

Volum

eCharge

Density

(rv)(nC/m

3)

Radiusof

Sph

erical

Charge

Regions

(b)(m

)

Maxim

umElectric

Field

Between

Sph

eres

(Emax)(V

/m)

Minim

umElectric

Field

Between

Sph

eres

(Emin)(V

/m)

Average

ElectricField,

Eav

=(E

max

+Emin)/

2(V

/m)

Potential

Difference

Between

Sph

eres,

DV

≈EavDh(V

)

External

Electrostatic

Energy,

(QDV)(J)

Self(≈Internal)

Electrostatic

Energy

(W1)(J)

Mutual

Electrostatic

Energy

(W2)

(J)

Total

Energy

(W=2W

1+W

2)

(J)

Total

Energy

(W=2W

1+

QDV)

(J)

500.44

237

54.6×10

44.4×10

44.5×10

44.5×10

62.0×10

62.8×10

62.1×10

67.7×10

67.6×10

6

2017

41.8×10

51.6×10

51.7×10

51.7×10

70.7×10

76.1×10

63.9×10

61.6×10

71.9×10

7

200

817.3×10

54.7×10

56.0×10

56.0×10

72.6×10

71.3×10

76.7×10

63.3×10

75.2×10

7

87(M

in)

0.77

245

15.7×10

45.5×10

45.6×10

45.6×10

64.3×10

67.1×10

65.3×10

62.0×10

71.9×10

7

2021

02.3×10

52.1×10

52.2×10

52.2×10

71.7×10

71.5×10

71.0×10

74.1×10

74.7×10

7

200

979.1×10

56.4×10

57.8×10

57.8×10

76.0×10

73.3×10

71.8×10

78.4×10

71.3×10

8

143(G

M)

1.3

253

36.9×10

46.7×10

46.8×10

46.8×10

68.8×10

61.6×10

71.2×10

74.5×10

74.1×10

7

2024

72.8×10

52.6×10

52.7×10

52.7×10

73.5×10

73.5×10

72.4×10

79.4×10

71.1×10

8

200

115

1.1×10

68.4×10

51.0×10

61.0×10

81.3×10

87.5×10

74.4×10

71.9×10

82.8×10

8

259(M

ax)

2.3

264

98.6×10

48.4×10

48.5×10

48.5×10

62.0×10

74.4×10

73.4×10

71.2×10

81.1×10

8

2030

13.5×10

53.3×10

53.4×10

53.4×10

77.8×10

79.4×10

76.7×10

72.6×10

82.7×10

8

200

140

1.4×10

61.1×10

61.3×10

61.3×10

83.0×10

82.0×10

81.2×10

85.3×10

87.0×10

8

a The

distance

Dhbetweenthesurfaces

ofchargedspheresisassumed

tobe

100m.A

lsogivenaretheminim

umandaverageelectricfields,potentiald

ifferencebetweenthespheres,thetotalelectrostaticenergy

and

itscompo

nents.See

text

fordetails.


12 of 15

return stroke at the time of the electric field peak to be 3 to5 GW. The average zero‐to‐peak risetime of the subsequentstroke field waveforms was 2.8 ms, so that the radiatingchannel length at the time of field peak was probably somehundreds of meters, which is comparable to the channellength for CIDs. Our arithmetic mean peak radiated power(wideband) of 37 GW found for 48 CIDs is about an orderof magnitude higher than the above estimates for subsequentreturn strokes. For first strokes, Krider and Guo [1983]estimated an arithmetic mean peak electromagnetic powerof 20 GW, which is still lower than our estimate for CIDs.[39] Rison et al. [1999] and Thomas et al. [2001] reported

the peak radiated power in the narrowband VHF (60–66 MHz) frequency range of the Lightning Mapping Array(LMA) for one New Mexico CID to be greater than 100 kWand greater than 300 kW, respectively. Source peak powersradiated at 60–66 MHz by “normal” lightning processesranged from 1 W (minimum locatable value) up to 10–30 kW [Thomas et al., 2001]. Rison et al. [1999] stated thatthe peak VHF radiation from CIDs was typically 30 dB(a factor of 1000) stronger than that from other lightningprocesses.[40] Our estimates of CID energy per unit channel length

(based on values in the next to last column of Table 5 andDh = 100 m) range from 7.7 × 104 to 5.3 × 106 J/m.Interestingly, the input energy per unit length for a naturallightning first stroke of 2.3 × 105 J/m reported by Krider etal. [1968] is within this range. For triggered lightningstrokes, Jayakumar et al. [2006] estimated the mean valueof input energy per unit length for 36 triggered lightningstrokes to be 3.6 × 103 J/m, which is more than an order ofmagnitude smaller than the lower bound estimated here forCIDs.[41] Cooray [1997] estimated the energy dissipated by a

typical first leader/return stroke sequence to be 5.5 × 108 Jfor a 5 km long channel, which corresponds to 1.1 × 105 J/m.Assuming that the first stroke channel length ranges from 5 to8 km [e.g., Rakov and Uman, 2003] and Cooray’s value ofenergy per unit length, we estimated the total energy rangeto be from 5.5 to 8.8 × 108 J. Cooray [1997] estimated the

dissipated energy for regular cloud discharges to be about18 × 108 J for a charge transfer of 8 C. The correspondingchannel length was 2.5 km. Taking the charge transfer of8 C as typical for cloud discharges we estimate the energyper unit channel length to be 7.2 × 105 J/m. Using thislatter value and the range of IC channel lengths of 2 to 5 km[e.g., Shao and Krehbiel, 1996], we estimate the IC energyrange of 1.4 to 3.6 × 109 J. Maggio et al. [2009], usingmedian charge transfers and average in‐cloud potentials,estimated the energy dissipated by 16 IC flashes to be 1.5 ×109 J per flash, which is near the lower bound of our rangefor regular ICs. Channel lengths for CIDs are within therange of 100m to 1000 m [seeNag and Rakov, 2010] (part 1).As seen in Table 5, the energies dissipated by a CID withDh = 100 m and rv = 20 C/m, transferring 1.3 C of charge, isabout 108 J, or 106 J/m. Using this energy per unit lengthand the range of channel lengths, we estimate the range ofvalues of total energy dissipated by CIDs to be 108 to 109 J.Note that the upper end of this range is similar to the energydissipated by a typical multiple‐stroke ground flash [Rakovand Uman, 2003].[42] Figure 14 summarizes the ranges of total energy

dissipated by first strokes in cloud‐to‐ground discharges,regular intracloud discharges, and CIDs along withcorresponding ranges for channel length. It appears thatCIDs typically do not surpass either cloud‐to‐ground firststrokes or regular cloud discharges in terms of energy.Hence, the labels like “energetic intracloud discharges” [e.g.,Eack, 2004; Smith et al., 2004] and “high‐energy discharges”[Watson andMarshall, 2007] that are sometimes used to referto CIDs are probably not justified. As seen in Figure 14, themost distinctive feature of CIDs (besides their very intenseHF‐VHF radiation) is their small spatial extent.

8. Summary

[43] Compact Intracloud Discharges (CIDs) are either in‐or above‐cloud lightning discharges that produce singlebipolar electric field pulses having typical full widths of 10 to30 ms and intense HF‐VHF radiation bursts (much moreintense than those from any other cloud‐to‐ground or“normal” cloud discharge process). We estimated electricalparameters of 48 located CIDs using their measured electricfields and the vertical Hertzian dipole approximation. Thisapproximation is consistent with the more general bouncing‐wave model for a reasonably large subset of allowed com-binations of propagation speed and channel length. Forexample, for zero‐to‐peak current risetime RT = 6 ms andpropagation speed v = 2 × 108 m/s the allowed range ofchannel length, Dh, is from 100 to 700 m, with the Hertziandipole approximation being valid for Dh up to 375 m.[44] For nine events, we estimated CID channel lengths

from channel traversal times measured in dE/dt waveformsand assumed propagation speeds of 2 × 108 m/s to 3 ×108 m/s, which limit the range of allowed speed values. Forv = 2.5 × 108 m/s (average value), the channel lengths forthese nine events ranged from 108 to 142 m. Thecorresponding geometric mean values of peak current, zero‐to‐peak current risetime, 10−90% current risetime, andcharge transfer for the first 5 ms are 143 kA, 5.4 ms, 2.6 ms,and 303 mC, respectively. The uncertainty in our current and

Figure 14. Ranges of dissipated energy and channel lengthfor typical first strokes in cloud‐to‐ground flashes (CGs),regular intracloud flashes (ICs), and CIDs; charge transferswere assumed to be 5 C, 8 C, and 1.3 C, respectively.


13 of 15

charge transfer values for the nine events was estimated tobe ≤ 25%. The geometric mean peak radiated power andenergy radiated for the first 5 ms (both wideband) are 29 GWand 24 kJ, respectively.[45] For the remaining 39 events, there were no reflection

signatures observed, and Dh was assumed to be 350 m, forwhich the Hertzian dipole approximation is valid for speedsin the range of 2 to 3 × 108 m/s. In this case, geometricmean values of peak current, zero‐to‐peak current risetime,10−90% current risetime, and charge transfer for the first 5 msare 64 kA, 2.5 ms, and 142 mC, respectively. The geometricmean peak radiated power and energy radiated for the first5 ms are 28 GW and 32 kJ, respectively.[46] The radiated power and energy are independent of

Dh, while peak current and charge transfer can be scaled toother channel lengths, provided that they are consistent withthe Hertzian dipole approximation. For the 39 events, weadditionally considered Dh = 170 m, which is consistentwith v ≥ 108 m/s, and Dh = 500 m, which is consistent withv = 3 × 108 m/s.[47] For all 48 events, GM values of peak current, zero‐

to‐peak current risetime, 10−90% current risetime, andcharge transfer for the first 5 ms are 74 kA, 5 ms, 2.5 ms, and164 mC, respectively. The geometric mean peak radiatedpower, and energy radiated for the first 5 ms are 29 GW and31 kJ, respectively. Overall, the estimated CID currentwaveform parameters are comparable to their counterpartsfor first strokes in cloud‐to‐ground lightning, while theirpeak radiated electromagnetic power appears to be consid-erably higher.[48] The upper bound on electric field prior to CIDs is

generally on the order of 104 to 105 V/m. The total energydissipated by a CID typically ranges from 108 to 109 J,which is comparable to the energy dissipated by a typical

multiple‐stroke ground flash and lower than that dissipatedby a regular cloud discharge.

Appendix A: Errors in CID Electrical ParametersDue to Uncertainties in CID Locations

[49] Uncertainties in CID locations include errors inheights, h, and horizontal distances, r, from the field mea-suring station. Nag et al. [2010] estimated errors in heightsof 48 CIDs considered here to range from 4.7 to 95% with amean of 17%. Most of the events (42 of 48) had heighterrors less than 30% and 39 had errors less than 25%. Theyfound that the error in source height is a sensitive function

ofEz

cB�¼ cos� ¼ r

Rwhen this ratio exceeds 0.95 (a < 18°)

or so [see Nag et al., 2010, Figure A1]. Horizontal dis-tances to the 48 CIDs from the field‐measuring stationwere estimated using NLDN data. Errors in r ranged from0.63 to 21%. In this Appendix, we examine the effect onthe inferred CID parameters of errors in h and r. We willshow that, with one exception, errors in CID currents dueto errors in h and r do not exceed 15%, even for eventswhose height errors are in the 30 to 95% range.[50] One can see from equation (5) that Ez is proportional

todi

dt, the proportionality constant being � Dh

2�"0c2r2

R3, where

R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2 þ r2

p. Any error in estimation of i from Ez due to

uncertainties in CID location is associated with the ratior2

R3,

which is a function of two variables, h and r. The error inr2

R3

was estimated from errors in r and h, using the derivativemethod [Taylor, 1997], which was also employed by Nag etal. [2010] to estimate errors in CID heights. The error in

y ¼ r2

R3was defined as

Dy

y= |

@y

@h|Dh

h+ |

@y

@r|Dr

r, and values of

Figure A2. Same as Figure A1 but versus errors in thehorizontal distances to CIDs from the field‐measuringstation.

Figure A1. Errors in ratior2

R3which is needed in solving

the inverse source problem (finding i from measured Ez)versus errors in CID heights.


14 of 15

Dh

hand Dr

r for the 48 CIDs were taken from Nag et al.

[2010].[51] The computed errors in

r2

R3versus errors in h and r are

shown in Figures A1 and A2, respectively. One can see inFigure A1 that even though errors in height for six events

are in the 30 to 95% range, errors inr2

R3are less than 15%

with one exception for which the error is 23%. These resultscan be explained as follows. Events with larger (up to 95%)height errors correspond to larger horizontal distances r, sothat R only weakly depends on h. Further, errors in r arerelatively small, mostly less than 5% (see Figure A2). As aresult, relatively large errors in h do not lead to large errors

in the ratior2

R3which is needed to find i from Ez. The outlier

in each of Figures A1 and A2, which is characterized by the

largest error inr2

R3of 23%, is associated with an unusually

large error in r.[52] The error analysis above applies to equation (5), but

we assume that errors in currents inferred from equation (4),due to uncertainties in CID locations, are similar. We con-clude that the errors in h and r are not expected to have asignificant effect on the inferred parameters of the 48 CIDsconsidered in this paper.

[53] Acknowledgments. The authors would like to thank D. Tsalikisfor his help in developing instrumentation and performing measurementsand J. Cramer of Vaisala for providing NLDN data. This research was sup-ported in part by NSF grants ATM‐0346164 and ATM‐0852869 and byDARPA.

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A. Nag and V. A. Rakov, Department of Electrical and ComputerEngineering, University of Florida, 553 Engineering Bldg. 33, PO Box116130, Gainesville, FL 32611‐6130, USA. ([email protected]; [email protected])


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