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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, D14110, doi:10.1029/2005JD006858, 2006

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A study of the lightning channel corona sheath Grzegorz Maslowski1,2 and Vladimir A. Rakov1 Received 7 November 2005; revised 28 February 2006; accepted 4 April 2006; published 29 July 2006.

[1] Dynamics of lightning channel corona sheath surrounding thin channel core is

examined on the basis of three transmission-line-type models of the return stroke that specify different attenuation of longitudinal current with height. The corona sheath conductivity is estimated using measured radial electric field in the immediate vicinity of the lightning channel and measured channel base current. The corona sheath radius, velocity of corona sheath radial expansion, and corona current are examined using a multiexponential approximation of the lightning channel base current waveform. Additionally, energy dissipated in the corona sheath is estimated and compared to that dissipated in the channel core. Citation: Maslowski, G., and V. A. Rakov (2006), A study of the lightning channel corona sheath, J. Geophys. Res., 111, D14110, doi:10.1029/2005JD006858.

1. Introduction [2] There is presently no consensus on the value of energy associated with the lightning return stroke. Various estimates vary by 1 – 2 orders of magnitude [e.g., Rakov and Uman, 2003]. Further, there are no estimates of energy dissipated in the lightning channel corona sheath. Such estimates require characterization of the dynamics of and currents involved in the corona formation process. Knowledge of lightning energy is needed, for example, in determining the amount of NO produced by lightning and in the testing of proposed thunder generation mechanisms. [3] When leader charge is deposited on a thin lightning channel core, the deposited charge will create a radial electric field that exceeds the breakdown value and pushes the charge away from the core. As a result, the leader channel consists of a thin core surrounded by a radially formed corona sheath. The corona sheath expands outward from the channel core until the radial electric field is less than the breakdown value, assumed to be about 2 MV/m by Baum and Baker [1990] and 1 MV/m by Kodali et al. [2005]. It is generally thought [e.g., Baum and Baker, 1990; Rakov, 1998] that the bulk of the leader charge is stored in the corona sheath whose radius is of the order of meters, while the highly conductive channel core (probably less than 0.5 cm in radius) carries essentially all the longitudinal current. [4] The return stroke current wave traverses the leader channel core and serves to bring it to ground potential. As a result, the leader charge stored in the corona sheath collapses into the channel core and is transferred to ground. The return stroke process in negative lightning can be visualized as a positive current (and charge density) wave that prop1 Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida, USA. 2 Permanently at Department of Electrical and Computer Engineering, Rzeszow University of Technology, Rzeszow, Poland.

Copyright 2006 by the American Geophysical Union. 0148-0227/06/2005JD006858$09.00

agates upward along the leader channel and deposits positive charge in the corona sheath to neutralize the negative charge of the preceding leader. [5] Thottappillil et al. [1997], in their analysis of return stroke models, defined two components of the charge density at a given channel section, one component being associated with the return stroke charge transferred through the channel section and the other with the charge deposited at the channel section. The deposited charge density component is spent to neutralize the leader charge and is the source of radial corona current during the return stroke process. [6] More information on the role of corona envelopes in various lightning processes is given by Heckman and Williams [1989]. A detailed description of the so-called reverse corona associated with the return stroke process is given by Gorin [1985]. [7] In this paper, we consider the return stroke corona current that is implicitly specified by three transmissionline-type models with different longitudinal current attenuation with height. Further, we evaluate the conductivity (using experimental data on the radial electric field and channel current, obtained by Miki et al. [2002], in conjunction with the models) and expansion of the radial corona sheath predicted by these models. Finally, we estimate energy dissipated in the corona sheath during the return stroke process and compare this energy with that dissipated in the channel core.

2. Theory [8] In this section, we derive equations for corona current (per unit length) and corona conductivity using an idealized representation of the lightning channel shown in Figure 1. We assume that longitudinal current flowing in the channel core decreases with increasing height and that this current attenuation is due to the neutralization of the charge deposited by the preceding leader in the radially formed corona sheath surrounding the highly conducting channel

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MASLOWSKI AND RAKOV: LIGHTNING CHANNEL CORONA SHEATH

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Figure 1. Representation of a negative return stroke channel segment composed of a highly conducting channel core (r rcore) surrounded by a corona sheath (rcore < r router). The longitudinal current is attenuated because of radial leakage (conduction) current in the corona sheath.

core. Following Baum and Baker [1990], we assume that the leakage (conduction) current is the dominant current component (much greater than the displacement current component) within the corona sheath, which is equivalent to the assumption that the corona sheath conductivity, scor, is much greater than we0, where w is the angular frequency and e0 = 8.85 1012 F/m is the permittivity of air, for all frequencies of interest. In fact, the condition scor we0 is not satisfied only during the initial rising portion of the return stroke current when the radial electric field changes very rapidly (with typical average values being equal to a fraction of MV/m/ms), in which case the displacement current component is comparable to the conduction current component. However, after a few microseconds, the radial electric field typically flattens (see Figure 2b), and the conduction current becomes dominant, that is, the condition

scor we0 is satisfied. Note that all results presented in this paper are independent of our neglecting the displacement current, except for the conductivity and energy estimates. [9] For a channel segment of length dz0 located at height z0 (above ground surface and below the upward moving return stroke front) that is composed of a thin core of radius rcore and corona sheath of outer radius router (see Figure 1) the charge conservation principle can be expressed as dQin dQout ¼ dQleak

ð1Þ

where dQin is the input positive charge at the bottom of the segment, dQout is the output positive charge at the top of the segment, and dQleak is the leakage positive charge neutralizing the corona sheath charge of opposite (negative)

Figure 2. (a) Return stroke current at the channel base and (b) corresponding horizontal (radial) electric field 0.1 m from the triggered lightning channel core for stroke 1 in flash S0033. In both Figures 2a and 2b, return stroke begins at time equal to zero (t = 0). Adapted from Miki et al. [2002]. 2 of 16


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Table 1. Corona Sheath Conductivity for t > z0/v and Three Different Current Attenuation Factors Return Stroke Model

Current Attenuation Factor 0

Corona Sheath Conductivity

0

MTLL model [Rakov and Dulzon, 1987]

P(z ) = 1 z /H

scor =

MTLP model [Rakov and Dulzon, 1991]

P(z0) = (1 z0/H)2

scor =

MTLE model [Nucci et al., 1988]

P(z0) = exp(z0/l)

scor =

polarity deposited on the channel segment by preceding leader. Expressing the input and output charges in terms of corresponding longitudinal currents flowing in the channel core, one can write equation (1) in the form 0

0

0

0

dQleak ¼ iðz ; t Þdt iðz þ dz ; t þ dz =vÞdt

0

iðz ; tÞ ¼

8 < Pðz0 Þið0; t z0 =vÞ

t z0 =v

:

t < z0 =v:

0

ð3Þ

where i(0, t z0/v) is the channel base current and P(z0) is the height-dependent current attenuation factor. Then the output current can be expressed as dz0 dz0 z0 þ dz0 ¼ Pðz0 þ dz0 Þi 0; t þ i z0 þ dz0 ; t þ v v v z0 0 0 ¼ Pðz þ dz Þi 0; t v

scor ¼

ð5Þ

Dividing both sides by dt, we can obtain an equation for leakage (conduction) current ileak, which flows radially from the lightning channel core dQleak ileak ðz ; t Þ ¼ ¼ Pðz0 Þið0; t z0 =vÞ Pðz0 þ dz0 Þið0; t z0 =vÞ dt ð6Þ

Pðz0 Þið0; t z0 =vÞ Pðz0 þ dz0 Þið0; t z0 =vÞ 2p rEr ðz0 ; r; t Þ dz0

ð9Þ

Assuming, as the first approximation, that P(z + dz) ﬃ P(z) + [dP(z)/dz]dz, we can finally express the corona sheath conductivity as follows: scor ¼

dPðz0 Þ ið0; t z0 =vÞ dz0 2p rEr ðz0 ; r; t Þ

ð10Þ

Note that the dependence of scor on z0, r and t in equations (8), (9), and (10) (also in Table 1) is not explicitly indicated for brevity. If Er is inversely proportional to r, scor is independent of r (within the corona sheath). [11] We now show that the leakage current ileak given by equation (6) corresponds to the return stroke charge deposited on the channel that is introduced by Thottappillil et al. [1997]. Indeed, ileak can be expressed, using equation (6) 0 and approximation P(z0 + dz0) ﬃ P(z0) + dPdzðz0 Þdz0, as ileak ðz0 ; tÞ ¼

ð4Þ

Substituting equations (3) and (4) into equation (2), we get dQleak ¼ ½Pðz0 Þ Pðz0 þ dz0 Þ ið0; t z0 =vÞdt

where Er (r, z0, t) is the radial electric field strength within the corona sheath. Substituting equation (6) into equation (7) and then equation (7) into equation (8), we get

ð2Þ

where v is the return stroke speed, so that the propagation time of current wave from the bottom to the top of the segment is dt = dz0/v. [10] For the generalized transmission-line-type model [Rakov and Dulzon, 1991] longitudinal current distribution along the channel is specified as

ið0;tz0 =vÞ 2p rHEr ðz0 ;r;t Þ ð1z0 =H Þið0;tz0 =vÞ p rHEr ðz0 ;r;t Þ expðz0 =lÞ ið0;tz0 =vÞ 2p rlEr ðz0 ;r;t Þ

dPðz0 Þ ið0; t z0 =vÞdz0 dz0

ð11Þ

(Note that for the original transmission line model [Uman and McLain, 1969], for which P(z0) = 1, ileak = 0, that is, there is no corona sheath, as expected.) Dividing equation (11) by dz0, the length of the channel segment under consideration, we obtain an equation for ileak/dz0, the leakage current per unit channel length at height z0 and time t ileak ðz0 ; t Þ dPðz0 Þ ¼ ið0; t z0 =vÞ 0 dz dz0

0

ð12Þ

Taking the time integral from 1 to t > z0/v on both sides of equation (12), we can write

Owing to the axial symmetry of the considered channel segment, the leakage current density jleak can be expressed as

rleak ðz0 ; t Þ ¼

dPðz0 Þ dz0

Zt

ið0; t z0 =vÞdt

ð13Þ

z0 =v

jleak ðz0 ; r; t Þ ¼

ileak ðz0 ; t Þ 2p rdz0

ð7Þ

where 2prdz0 is the lateral surface of the cylinder of length dz0 and radius rcore < r router. According to the point form of Ohm’s law the conductivity in the corona sheath is given by scor

jleak ðr; z0 ; tÞ ¼ Er ðr; z0 ; t Þ

where rleak is the charge density deposited on the channel by the return stroke. Equation (13) is identical to the second term, which represents the deposited charge density component, of charge density equation (20) derived by Thottappillil et al. [1997] for transmission-line-type models.

3. Conductivity of the Corona Sheath ð8Þ

[12] We now apply equation (10) to specific return stroke models. We consider three transmission-line-type models

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[Rakov and Uman, 1998] that are characterized by different current attenuation factors P(z0). Note that P(z0) is determined by the explicitly or implicitly assumed leader charge distribution along the channel [Rakov and Dulzon, 1991]. Uniform charge density distribution corresponds to linear current attenuation with height, P(z0) = 1 z0/H, which was used in the MTLL model by Rakov and Dulzon [1987]. Further, charge density linearly decaying with height corresponds to parabolic current attenuation with height, P(z0) = (1 z0/H)2. This latter attenuation factor has been considered, among other factors, by Rakov and Dulzon [1991] and is consistent with theoretical predictions for a vertical conductor (representing the leader channel) in an external electric field [e.g., Cooray et al., 2004]. Here we will refer to such a transmission-line-type model as the MTLP model (Modified Transmission Line model with Parabolic current decay with height). Finally, exponential charge density distribution corresponds to exponential current attenuation with height, P(z0) = exp(z0/l), as assumed by Nucci et al. [1988] in their MTLE model. All three models predict remote electric fields that are generally in fairly good agreement with observations [e.g., Rakov and Dulzon, 1991, Figure 4]. Table 1 summarizes formulas for the corona sheath conductivity for the three models described above, MTLL, MTLP, and MTLE. [13] Equations for scor in Table 1 enable one to determine variation of corona sheath conductivity with time at a given radial distance r and at a given height z0 for the three transmission-line-type models when the radial electric field, Er, at those r and z0 is known. The only available measurements of Er in the immediate vicinity of channel core are due to Miki et al. [2002], who measured, using Pockels sensors, both the vertical and the horizontal electric fields 0.1 m to 1.6 m from the triggered-lightning channel attachment point at Camp Blanding, Florida. Both vertical and horizontal electric field waveforms appear as negative pulses with the leading edge of the pulse being due to the leader and trailing edge due to the return stroke. Horizontal (predominantly radial) electric field pulse peaks were obtained for 8 strokes and ranged from 495 kV/m to 1.2 MV/m with the median value being 821 kV/m. The event with the largest value of the horizontal (radial) electric field, 1.2 MV/m, is presented in Figure 2. [14] Using measured values of the radial electric field at r = 0.1 m (Figure 2b), measured values of current (Figure 2a), and typical values of parameters of the MTLL, MTLP, and MTLE models, one can calculate the corona sheath conductivity as a function of time at z0 = 0 m and r = 0.1 m from equations found in Table 1. Results for typical values of model parameters (H = 7500 m, l = 2000 m, v = 130 m/ms) are shown in Figure 3. [15] Note that we consider here only the return stroke current and radial electric field associated with this current. This means that the waveform in Figure 2b is adjusted so that Er = 0 at t = 0 and increases with time. Positive values of Er correspond to the positive (outward) direction of the electric field vector during the return stroke process. The total electric field can be obtained using the principle of superposition, as the algebraic sum of this positive field and the negative electric field generated by the charge of preceding leader. Conductivity was computed starting from time t = 1 ms, although the radial electric field values exceed

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Figure 3. Corona sheath conductivity at z0 = 0 m and r = 0.1 m as a function of time for three return stroke models, based on measured current and radial electric field shown in Figures 2a and 2b, respectively. 1 MV/m, the often assumed value of breakdown electric field, only after t = 4 ms. We roughly estimate from Figure 3 that the values of corona sheath conductivity are of the order of 106 – 105 S/m. For comparison, the electrical conductivity of the virgin air at sea level is about 1014 S/m, and the average conductivity of the Earth is about 103 S/m. Further, the conductivity of the lightning channel core is estimated to be of the order of 104 S/m (comparable to that of carbon) [e.g., Rakov, 1998]. [16] It follows from Figure 3 (see also Figure 2b and discussion in section 2) that the assumption scor we0 is satisfied for frequencies lower than 100 kHz, that is, a few microseconds after the beginning of the return stroke.

4. Expansion of the Corona Sheath 4.1. Corona Sheath Radius [17] The lightning corona sheath expands outward from the channel core during the return stroke process, and it is of interest to estimate the radial extent and rate of this expansion. Consider a closed cylindrical surface (Gaussian cylinder) S that is coaxial with and surrounding a segment of channel core whose length is dz0. According to Gauss’ law, I

e0

E dS ¼ Q

ð14Þ

S

where E is the electric field on closed surface S, and Q is the total charge inside this surface. Surface S consists of the lateral surface of the cylinder, which we choose to be such that it coincides with the outer boundary of the corona sheath, and the bottom and top faces of the cylinder. Thus equation (14) can be expanded as þ

þ 2prouter e0 Erþ dz0

2pe0

Zrouter

Ez ðz0 ; r; tÞrdr þ 2pe0

0

þ Zrouter

Ez ðz0 þ dz0 ; r; t Þrdr ¼ Q

ð15Þ

0

+ where router is the outer radius of the corona sheath containing positive charge deposited by the radial conduction current flowing during the return stroke stage, E+r is the constant radial

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Table 2. Parameters of Multiexponential Function Given by Equation (21) n An, kA an, s1

1

2

3

4

5

6.75 0.3 104

6.75 0.28 106

13.5 6.4 106

42.17 9.1 106

42.17 7.1 106

electric field on the lateral surface, while Ez(z0, r, t) and Ez (z0 + dz0, r, t) are the vertical electric fields at the bottom and at the top faces of the Gaussian cylinder, respectively. In order to + , the radial electric field E+r must be chosen. We estimate router will assume that the corona sheath extends outward from the channel core until the field becomes less than some positive breakdown electric field. Note that the radial electric field cannot be established instantaneously, but we will use a constant value of breakdown field for simplicity. As noted in section 1, Baum and Baker [1990] assumed that this breakdown electric field value was equal to 2 MV/m, and Kodali et al. [2005] adopted a value of 1 MV/m. In this paper, we assume that the positive breakdown electric field is equal to E+r = 1.0 MV/m. [18] We show in Appendix A that the second and third terms on the left-hand side of equation (15) are negligible compared to the first term, so that equation (15) can be rewritten as þ 2prouter e0 Erþ dz0

¼Q

ð16Þ

The first term of equation (18) is the charge transferred through the channel segment, and the second term represents the deposited charge that is spent to neutralize the leader charge deposited in the corona sheath of this segment. [19] Assuming a uniform radial distribution of the negative leader charge, we show in Appendix B that Q, which is a portion of the total negative charge stored in the corona + , of positive sheath located within the radial extent, router + charge Q , can be expressed as Q ¼

ð17Þ

The latter charge is predicted by transmission-line-type models and described by Thottappillil et al. [1997] as Qþ ¼ r2tran dz0 þ rleak dz0 0

0

ið0; t z =vÞ dPðz Þ 6 ¼ 4Pðz0 Þ v dz0

Zt z0 =v

3 7 ið0; t z0 =vÞdt5dz0 ð18Þ

2 þ 2pe0 router Er dz0 rleader ðz0 Þ

ð19Þ

where E r is the negative breakdown electric field, which is assumed to be greater (in absolute value) than E+r and equal to 1.5 MV/m, and rleader(z0) is the negative charge density per unit channel length prior to the return stroke + is necessarily smaller than router , the stage. Note that router radial extent of the negative leader corona sheath (as illustrated in Figure 8). [20] Substituting equations (19) and (18) into equation (17) and then equation (17) into equation (16), we find, after dividing both sides of the resultant equation by dz0, Zt ið0; t z0 =vÞ dPðz0 Þ ið0; t z0 =vÞdt v dz0 z0 =v þ 2 2pe0 router Er þ rleader ðz0 Þ ð20Þ

þ 2prouter e0 Erþ ¼ Pðz0 Þ

where the total charge Q enclosed by S consists of the negative charge Q deposited by the preceding leader and the positive charge Q+ associated with the return stroke, that is, Q ¼ Q þ Qþ

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+ The positive corona sheath radius router is a solution of quadratic equation (20). [21] Estimation of the corona sheath radius involves integration of the channel base current over time. One analytical current waveshape, which is in good agreement with experimental data, was proposed by Nucci et al. [1993]. This approximation involves Heidler’s functions that are not convenient for analytical integration. Therefore, in the following, a different approximation, which is based

Figure 4. Typical subsequent return stroke current at the channel base. (a) Waveform adopted by Nucci et al. [1993] (solid line), peak current of 12 kA, maximum current derivative of 40 kA/ms, and its approximation by equation (21) (dashed line) on a 100 ms timescale. (b) Same as Figure 4a, but shown on a 1 ms timescale. 5 of 16


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Figure 5. Corona sheath radius r+outer versus time and height for three return stroke models, MTLL, MTLP, and MTLE (H = 7500 m, l = 2000 m, v = 130 m/ms). on the multiexponential formula proposed by Bajorek et al. [2004], will be adopted: iðt Þ ¼

5 X n¼1

An expðan t Þ

ð21Þ

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with the parameters specified in Table 2. Figure 4 demonstrates that the channel base current waveform adopted by Nucci et al. [1993] and that described by equation (21) are very similar. The positive charge transferred by both currents up to the time 100 ms is practically the same (0.61 C for Nucci et al.’s [1993] approximation and 0.60 C for the multiexponential approximation, respectively). Small differences occur at late times, and hence the overall charge transferred by the current adopted by Nucci et al. [1993] is equal to 1.7 C while the overall charge transferred by the current described by equation (21) is slightly larger and equal to 2.3 C. [22] Evolutions of the corona sheath in space and time at different heights for the three return stroke models, MTLL, MTLP, and MTLE, are shown in Figure 5. It is worth noting that the variation of corona radius in space and time in our simulations is the same as that of the total charge density illustrated by Thottappillil et al. [1997, Figures 3a and 3b]. Note also that in this analysis we ignored any reflections of the longitudinal current wave from the upper end of the lightning channel. Such reflections are expected only for the MTLE model (as discussed by Rakov and Dulzon [1991]), since for the MTLL and MTLP models the longitudinal current is equal to zero at z0 = H. [23] Figure 6 shows the corona sheath radius as a function of time at different heights for the same three return stroke models. As seen in Figure 6, the corona sheath expands approximately up to t = 1 ms, when the longitudinal current described by equation (21) approaches zero and can be neglected, and remains approximately constant after that time. Note that this constant corona radius value varies with height for the MTLE and MTLP models, but is independent on height for the MTLL model (see Figure 7). It follows + , of fully developed from Figure 7b that the radius, router corona sheath near ground is 2.5 m, 5.0 m, and 9.5 m for the MTLL, MTLP, and MTLE models, respectively. In fact, the dependence of corona sheath radius on height for late times (in this case after 1 ms) is the same as that of the leader charge density just prior to the return stroke, that is, no dependence (uniform charge density distribution) for MTLL, linear decay with height for MTLP, and exponential decay for MTLE. Crawford et al. [2001] inferred, from multiple-station electric field measurements, a more or less uniform distribution of charge along the bottom kilometer or so of the dart leader channel. Cooray et al. [2004] showed theoretically that charge on the bottom half of a vertical conductor in an external electric field decreases linearly with height except for the small region in the immediate vicinity of its bottom end. [24] In the following, we illustrate the structure of corona sheath both before and after (t > 1 ms) the return stroke stage (see Figure 8). As noted earlier, the radial extent of + , is smaller than positive return stroke corona sheath, router . The negative that of negative leader corona sheath, router + is overcompensated by the leader charge within router + is positive return stroke charge (the net charge within router + positive), and the negative leader charge between router and remains uncompensated. The net charge within router in router Figure 8b is equal to zero, as expected for the transmissionline-type models. + being less than router is the fact [25] The reason for router that the negative leader charge near the channel core

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Figure 6. Comparisons of the corona sheath radius r+outer versus time at three different heights, (a and b) 10 m, (c and d) 2 km, and (e and f) 5 km, for three return stroke models, MTLL, MTLP, and MTLE (H = 7500 m, l = 2000 m, v = 130 m/ms), shown on two different timescales, 100 ms (Figures 6a, 6c, and 6e) and 4 ms (Figures 6b, 6d, and 6f). partially compensates the outward moving positive return stroke charge and thereby reduces the radial electric field at its outer boundary to a lower than breakdown value before . the positive charge can reach router [26] We will examine in Appendix C the sensitivity of the + to the variation in assumed value corona sheath radius router + of Er , keeping the negative breakdown electric field associated with the leader stage constant and equal to E r = 1.5 MV/m (absolute value). 4.2. Corona Sheath Expansion Velocity [27] Taking time derivative of the corona sheath radius, + , (as a function of time) obtained from equation (20), router one can find the velocity of radial corona sheath expansion. This velocity is shown as a function of time at different heights in Figure 9 for the three transmission-line-type models. [28] It is worth noting that the corona sheath expansion velocity is significantly less than the lightning return stroke

velocity, that is, the velocity of the longitudinal current wave that traverses the highly conducting channel core. Further, three stages, initial, intermediate, and final, of the corona sheath expansion are identifiable in Figure 9. The initial stage, seen in Figures 9a and 9b (on 5 and 2 ms timescales, respectively), is associated with the return stroke current front, when the transferred charge dominates. This stage resembles a shock wave which expands up to approximately 1 m from the channel core with a velocity that is much grater than the corona sheath expansion velocity during the final stage, seen in Figures 9e and 9f (on a 1000-ms timescale). After the initial stage, the longitudinal current decreases, but the deposited charge associated with the corona current is still very small. Therefore, during the intermediate stage, seen in Figures 9c and 9d (on a tens of microseconds timescale), which lasts only for a few microseconds, the velocity is negative because of a small decrease of corona sheath radius. During the final stage, the depos-

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ited charge becomes dominant, and the corona sheath radius increases again. [29] The velocity profiles within the first microsecond at z0 = 10 m (see Figure 9a) are very similar for all three models but differ at z0 = 2 km (see Figure 9b). At z0 = 10 m, the velocity peak is about 4.0 106 m/s for the MTLL model, 4.4 106 m/s for the MTLP model, and 4.6 106 m/s for the MTLE model, while at z0 = 2 km it is 3.1 106 m/s for the MTLL model, 2.5 106 m/s for the MTLP model, and 1.7 106 m/s for the MTLE model. The secondary velocity peak that occurs at later times (see Figures 9e and 9f) is of the order of 104 m/s for all three models. For comparison, the velocity of radial corona streamers from conductors subjected to negative high voltage in the laboratory is about 105 m/s [Cabrera and Cooray, 1992]. Also, Heckman and Williams [1989] give a characteristic value of radial velocity for positive corona streamers of about 105 m/s.

5. Corona Current [30] Corona current (conduction component only) per unit length at height z0 at time t for transmission-line-type models is given by equation (12). Table 3 summarizes equations for the corona current per unit length, i0cor = i0leak/dz0, for three return stroke models, MTLL, MTLP, and MTLE, derived from equation (12). [31] The cumulative corona current, which flows radially from the channel core between the upward moving return stroke front at z0 = vt (or arbitrary height z0 < vt) and ground surface (z0 = 0) can be found by integrating the right-hand side of equation (12) over z0,

icor ¼

+ router

Figure 7. Corona sheath radius versus height at (a)25 ms and (b)1000 ms for three return stroke models, MTLL, MTLP, and MTLE (H = 7500 m, l = 2000 m, v = 130 m/ms).

Zz0

dPðxÞ ið0; t x=vÞdx dx

ð22Þ

0

Figure 10 (left) shows corona current per unit length versus time at different heights for the MTLL, MTLP,

Figure 8. Illustration of the structure of the corona sheath (a) just prior to the return stroke stage and is the total negative leader charge (absolute (b) in the later part of the return stroke stage (t > 1 ms). Qtotal + , the maximum radial extent of value), Qr+outer is the negative leader charge (absolute value) within router + + is the total positive return stroke charge (Qtotal = Qtotal ), the positive return stroke corona sheath, Qtotal and router is the maximum radial extent of the negative leader corona sheath. 8 of 16


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Figure 9. Variation of the corona sheath expansion velocity on three different timescales, (a, c, e) 10 m and (b, d, f) 2 km above ground for three return stroke models, MTLL, MTLP, and MTLE (H = 7500 m, l = 2000 m, v = 130 m/ms). Downward and upward directed arrows indicate positions of positive and negative velocity peaks (some of which are clipped or/and unresolved), respectively. Note two positive peaks (initial and much smaller secondary) in Figures 9c – 9f.

and MTLE return stroke models (H = 7500 m, l = 2000 m, v = 130 m/ms), computed using equations given in Table 3. Figure 10 (right) shows corresponding cumulative corona current profiles along the channel at different instants of time, computed using equation (22). Note that the height on the horizontal axes of the right panel

represents the upper integration limit, so that the cumulative corona current monotonically increases from the ground surface (z0 = 0) to the height (z0 = vt) of the return stroke front. As seen in Figure 10, the cumulative corona current at times of the order of tens of microseconds is of the order of kiloamperes, consistent with the corona

Table 3. Corona Current Per Unit Length for Three Transmission-Line-Type Models of the Return Stroke Derived From Equation (12) Return Stroke Model

Current Attenuation Factor P(z ) = 1 z /H

i0cor =

MTLP model [Rakov and Dulzon, 1991]

P(z0) = (1 z0/H)2

i0cor =

0

0

Corona Current per Unit Length

MTLL model [Rakov and Dulzon, 1987] MTLE model [Nucci et al., 1988]

0

0

P(z ) = exp(z /l)

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i0cor =

ið0;tz0 =vÞ H 2ð1z0 =H Þ i(0, t z0/v) H expðz0 =lÞ i(0, t z0/v) l

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Figure 10. (left) Corona current per unit length versus time at different heights for the (top) MTLL, (middle) MTLP, and (bottom) MTLE models (H = 7500 m, l = 2000 m, v = 130 m/ms) computed using equations given in Table 3. (right) Corresponding cumulative corona current that flows between ground surface and height z0 at different instants of time, computed using equation (22). current estimates given by Heckman and Williams [1989]. Recall that the longitudinal current peak at z0 = 0 is 12 kA (see Figure 4).

inside the corona sheath is neglected, which is justified + , except for very early times). The since rcore router resultant power per unit length is given by 2 þ 2 Plosses ¼ scor Erþ p router dz0

6. Energy Dissipated in the Corona Sheath [32] We can roughly estimate the energy dissipated during the return stroke stage in the corona sheath assuming reasonable constant values of conductivity scor and radial electric field E+r within the sheath. Consider a channel segment of length dz0. Power that is spent to heat the air inside the corona sheath of this segment can be expressed as 2 2 þ 2 0 Plosses ¼ scor Erþ dV scor Erþ p router dz

ð24Þ

Taking the time integral on both sides of equation (24), we obtain the energy per unit length that is spent for heating air within the corona sheath up to time t:

W¼

Zt

2 þ 2 scor Erþ p router dt

ð25Þ

0

ð23Þ

where the corona sheath volume dV is assumed to be + )2 dz0 (the presence of the thin core roughly equal to p(router

+ The outer corona sheath radius router depends on time and height above ground, and this dependence is different for the three return stroke models considered here. In our

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Figure 11. Corona energy per unit length versus time 10 m, 1 km, and 2 km above ground for three return stroke models, (a and b) MTLL, (c and d) MTLP, and (e and f) MTLE (H = 7500 m, l = 2000 m, v = 130 m/ms). (left) Energy variation on a 100-ms timescale, and (right) energy variation on a 1-ms timescale.

energy calculations we assumed that the corona sheath conductivity is 106 S/m (see section 3) and the radial electric field within the sheath is 106 V/m. (According to Gorin [1985], the average electric field in a positive corona sheath should be about 0.5– 1.0 MV/m.) The resultant energy per unit length versus time at different heights is shown in Figure 11 and versus height at t = 100 ms and t = 1 ms in Figure 12, for the three return stroke models. One can see from Figures 11 and 12 that this energy near the ground level is of the order of 102 – 103 J/m at t = 100 ms (3.9 102, 1.0 103, and 2.7 103 J/m for the MTLL, MTLP, and MTLE models, respectively) and of the order of 104 – 105 J/m at t = 1 ms (1.5 104, 5.6 104, and 2.0 105 J/m for the MTLL, MTLP, and MTLE models, respectively). It

is clear from Figure 12 that the MTLL model predicts a more uniform distribution of energy along the channel than the MTLP or MTLE model. In the simplified approach to computing energy adopted in this paper, the energy per unit length is proportional to scor and to (E+r )2, and thus it can be easily scaled to different assumed values of these two parameters. For the assumed values scor = 106 S/m and E+r = 106 V/m, the corona current density is of the order of 1 A/m2, and the corresponding power density is of the order of 106 W/m3. For comparison, the longitudinal current density and power density in the channel core are of the order of 107 A/m2 and 1010 W/m3, respectively (assuming that rcore = 0.01 m, conductivity of the channel core sz = 104 S/m, and longitudinal current i = 3 kA).

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Figure 12. Distribution of corona energy per unit length along the lightning channel at (left) t = 100 ms and (right) t = 1 ms for (a and b) MTLL, (c and d) MTLP, and (e and f) MTLE return stroke models (H = 7500 m, l = 2000 m, v = 130 m/ms). The total energy dissipated in the corona sheath over the entire channel of height H = 7.5 km is indicated for each plot. [33] We can also calculate the total energy that is dissipated in the corona sheath over the entire channel height by integrating equation (25) over z0 from 0 to H,

Wtotal ¼

ZH Z t 0

2 þ 2 scor Erþ p router dtdz0

ð26Þ

0

As expected, the total energy Wtotal is similar for all three return stroke models, between 106 and 107 J at t = 100 ms and between 108 and 109 J at t = 1 ms. For comparison, the total energy dissipated by a cloud-to-ground flash (typically composed of three to five strokes; all lightning processes included) is 109 to 1010 J [e.g., Rakov and Uman, 2003]. Using Wtotal and H = 7.5 km, we find that the average energy per unit length, which is dissipated in the corona sheath up to t = 1 ms, is 1.3 104 J/m for

the MTLL model, 1.7 104 J/m for the MTLP model and 2.7 104 J/m for the MTLE model, respectively. The corresponding values for t = 100 ms are 1.6 102, 2.1 102, and 3.2 102 J/m. [34] We can compare the above average values of corona energy per unit length with the energy per unit length dissipated in the channel core. The latter energy was computed, typically up to some tens of microseconds, using gas dynamic models and was referred to as the return stroke input energy. Results of such computations are summarized in Table 12.1 of Rakov and Uman [2003]. According to gas dynamic models of Hill [1971, 1977], Plooster [1971], Paxton et al. [1986, 1990], and Dubovoy et al. [1991a, 1991b, 1995], the energy per unit length dissipated in the channel core is of the order of 103 J/m. On the basis of electrostatic consideration, Borovsky [1998] predicted that this energy should be between 2 102 and 1 104 J/m, and

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Krider et al. [1968] calculated the value of 2.3 105 J/m for a single-stroke flash, using field and laboratory experimental data. From electrostatic consideration, Uman [1987] obtained the possible range of values between 105 and 106 J/m. Note that the energy estimates obtained by Krider et al. [1968] and Uman [1987] probably include the energy dissipated during both the leader and return stroke processes and hence are not directly comparable to other estimates based on the models that describe only the return stroke process. It appears that the energy dissipated inside the corona sheath (see Figures 11 and 12) can be comparable with that dissipated in the channel core. On the other hand, the corona energy is deposited in a much larger volume than the channel core energy, with the energy rate of change (power) per unit volume being 4 orders of magnitude lower in the corona sheath than in the core. This probably explains the fact that the lightning channel corona sheath produces insufficient luminosity to be detectable with optical techniques.

7. Summary [35] Return stroke models are traditionally formulated in terms of the longitudinal current in the channel core. However, these models also contain information about the radial corona current. We found the following properties of the radial corona sheath, based on three transmission-linetype models that specify linear (MTLL), parabolic (MTLP), and exponential (MTLE) current decays with height. (1) Electrical conductivity in the return stroke corona sheath is of the order of 106 – 105 S/m, which is comparable to the conductivity of very poorly conducting soil. (2) The return stroke corona sheath expands, at a velocity that is between 104 m/s and 105 m/s (except for the first microsecond when this velocity briefly exceeds 106 m/s), to a radial distance of several meters (close to 10 m near ground for the MTLE model) in a millisecond or so. (3) Energy dissipated in the return stroke corona sheath can be comparable to that dissipated in the channel core, although the rate of energy deposition (power) per unit volume in the corona sheath is 4 orders of magnitude lower than in the core. Knowledge of lightning energy is needed, for example, in determining the amount of NO produced by lightning and in the testing of proposed thunder generation mechanisms.

Appendix A: Vertical Electric Field Flux Versus Radial Electric Field Flux [36] In Appendix A, we will show that the second and third terms (the sum of which represents the net vertical electric field flux) on the left-hand side of equation (15) are negligible compared to the first term (radial electric field flux). Using the Taylor’s expansion and simplifying, we can rewrite equation (15) as þ

þ 2prouter e0 Erþ dz0

þ 2pe0 dz

0

Zrouter

@Ez ðz0 ; r; t Þ rdr ¼ Q @z0

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the spatial derivative @Ez/@z0 within the channel segment, we can find an overestimated vertical electric field flux þ

Zrouter 0 þ 2 @Ez ðz0 ; r; tÞ 0 @Ez ðz ; r; t Þ 2pe0 dz rdr ¼ p r e dz 0 outer @z0 @z0 max max 0

0

ðA2Þ

The overestimated vertical electric field flux given by equation (A2) is much smaller than the radial electric field flux (the first term on the left-hand side of equation (A1)) when @Ez 2Erþ @z0 rþ outer max

ðA3Þ

In the following, we will evaluate the maximum value of the spatial derivative @Ez/@z0 and show that its absolute value is + . The maximum typically much smaller than 2E+r /router derivative of the vertical electric field is within the channel core. According to the point form of Ohm’s law, Ez ¼

jz sz

ðA4Þ

where sz is conductivity of the lightning channel core and jz is the longitudinal current density. The latter for transmission-line-type models can be expressed as jz ¼

Pðz0 Þið0; t z0 =vÞ 2 prcore

ðA5Þ

where rcore is the core radius. Substituting equation (A5) in equation (A4) and taking the spatial derivative with respect to z0 on both sides, we find @Ez Pðz0 Þ @ið0; t z0 =vÞ dPðz0 Þ ið0; t z0 =vÞ ¼ 2 ðA6Þ þ 0 2 s @z prcore sz v @t dz0 prcore z

The absolute value of this spatial derivative is maximal at early times, when z0 = 0, i = imax 105 A, and @i/@t = (@i/ @t)max 1011 A/s [e.g., Rakov and Uman, 2003]. Further, taking sz = 104 S/m, v = 108 m/s, rcore = 0.01 m (all typical values) we calculate j@Ez/@z0jmax = 323 V/m2 for the MTLL model, 327 V/m2 for the MTLP model, and 334 V/m2 for the MTLE model. Since E+r is of the order of 106 V/m and the + is of the order of 10 m, j@Ez/@z0jmax maximum value of router + + 2Er /router (300 V/m2 2 105 V/m2). Since the maximum vertical electric field flux (represented by j@Ez/ @z0jmax) is 2 to 3 orders of magnitude or so smaller than the + ) we can radial electric field flux (represented by 2E+r /router neglect the second term on the left-hand side of equation (A1), which corresponds to the algebraic sum of the second and third terms on the left-hand side of equation (15).

ðA1Þ

0

where the first term on the left-hand side represents the radial electric field flux and the second term represents the vertical electric field flux. Taking the maximum value of

Appendix B: Radial Distribution of the Negative Leader Charge Along the Channel Just Prior to the Return Stroke [37] One can calculate the outer radius router of the negative leader corona sheath just prior to the return stroke

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Figure C1. Corona sheath radius at heights of (left) 10 m and (right) 2 km versus assumed positive breakdown electric field Er+ for the (a and b) MTLL, (c and d) MTLP, and (e and f) MTLE return stroke models (H = 7500 m, l = 2000 m, v = 130 m/ms, Er = 1.5 MV/m). stage using Gauss’ law. For a channel segment dz0, using a and length dz0, we can Gaussian cylinder of radius router write 2prouter e0 Er dz0 ¼ rleader ðz0 Þdz0

ðB1Þ

where E r is the negative breakdown electric field, and rleader(z0) is the line charge density (charge per unit length) on the leader channel. Note that in writing equation (B1) we assumed that the radial electric field flux through the overall lateral surface of the Gaussian cylinder is much larger than the vertical electric field flux through its bottom (z0 = 0 m) and top (z0 = H) faces. From equation (B1), router ¼

rleader ðz0 Þ 2pe0 Er

Thus the negative charge of fully developed leader is . distributed up to a radial distance of router [38] Assuming uniform radial distribution of volume charge density, rVleader, of negative charge within the leader corona sheath of channel segment dz0, we can find the ratio, + to the k, of the negative charge distributed within router overall negative charge stored on channel segment dz0 (within router ), k¼

þ 2 0 þ 2 rVleader p router dz router 2 ¼ 2 rVleader p router dz0 router

ðB3Þ

+ Then, the negative charge distributed within router is equal to

ðB2Þ 14 of 16

Q ¼ krleader dz0

ðB4Þ


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Finally, substituting equation (B2) into equation (B3) and then equation (B3) into equation (B4), we find that Q ¼

2 þ 2pe0 router Er dz0 rleader ðz0 Þ

ðB5Þ

which is reproduced as equation (19) in section 4.1. Note that in deriving equation (B3) we neglected the presence of the thin core inside the corona sheath, assuming that all leader charge is stored in the sheath. [39] The leader line charge density can be obtained for the three return stroke models using MTLL model 1 þ Q H total

ðB6aÞ

2ð1 z0 =H Þ þ Qtotal H

ðB6bÞ

rleader ðz0 Þ ¼

MTLP model rleader ðz0 Þ ¼

MTLE model 0

rleader ðz0 Þ ¼

ez =l þ Qtotal l

ðB6cÞ

where Q+total is the total positive charge that is transferred by the longitudinal current given by equation (21) and that can be calculated as Qþ total

¼

Z1

ið0; t Þdt ¼ 2:3 C

ðB7Þ

0

Note that leader charge density equations (B6a) and (B6b), for the MTLL and MTLP models, respectively, give the total negative charge stored within the corona sheath prior to the return stroke, which is exactly the same as the total charge transferred by the longitudinal current during the return stroke, while, for the MTLE model, the leader charge density given by equation (B6c) is a little bit smaller (0.976 Q+total = 2.2 C for H = 7500 m). Thus for the MTLE model the return stroke current at the channel top is not equal to zero, and we can expects some reflections of the upward propagating current wave from the channel top. Such reflections were neglected in our analysis presented here.

Appendix C: Sensitivity of the Corona Sheath Radius to the Assumed Breakdown Electric Field [40] Figure C1 shows curves illustrating the dependence of corona sheath radius on the assumed positive breakdown electric field E+r . The curves are calculated at heights z = 10 m and z = 2 km at different times. The value of E+r varies from 0.5 to 1.5 MV/m. It follows from Figure C1 that the corona sheath radius computed for different values of E+r varies within a factor of 2 or so, which should be regarded

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as a relatively small variation, given other uncertainties involved in this study. [41] Acknowledgment. This research was supported in part by the State Committee for Scientific Research in Poland and by NSF grant ATM0346164.

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Rakov, V. A., and A. A. Dulzon (1987), Calculated electromagnetic fields of lightning return stroke (in Russian), Tekh. Elektrodinam., no. 1, 87 – 89. Rakov, V. A., and A. A. Dulzon (1991), A modified transmission line model for lightning return stroke field calculation, paper presented at 9th International Zurich Symposium on Electromagnetic Compatibility, ETH Zentrum-IKT, Zurich, Switzerland, March. Rakov, V. A., and M. A. Uman (1998), Review and evaluation of lightning return stroke models including some aspects of their application, IEEE Trans. Electromagn. Compat., 40, 403 – 426. Rakov, V. A., and M. A. Uman (2003), Lightning: Physics and Effects, Cambridge Univ. Press, New York. Thottappillil, R., V. A. Rakov, and M. A. Uman (1997), Distribution of charge along the lightning channel: Relation to remote electric and magnetic fields and to return-stroke models, J. Geophys. Res., 102, 6987 – 7006.

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Uman, M. A. (1987), The Lightning Discharge, 377 pp., Elsevier, New York. Uman, M. A., and D. K. McLain (1969), Magnetic field of the lightning return stroke, J. Geophys. Res., 74, 6899 – 6910.

G. Maslowski, Department of Electrical and Computer Engineering, Rzeszow University of Technology, ul. W. Pola 2, 35-959 Rzeszow, Poland. ([email protected]) V. A. Rakov, Department of Electrical and Computer Engineering, University of Florida, 553 Engineering Building #33, P. O. Box 116130, Gainesville, FL 32611, USA. ([email protected])

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