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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, D16102, doi:10.1029/2007JD008559, 2007
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Expressions for far electric fields produced at an arbitrary
altitude by lightning return strokes
Rajeev Thottappillil,1 Vladimir A. Rakov,2 and Nelson Theethayi1
Received 19 February 2007; revised 7 May 2007; accepted 29 May 2007; published 17 August 2007.
[1] Electromagnetic fields produced at high altitudes by return strokes in cloud-to-ground
lightning are needed in studies of transient luminous events in the mesosphere. Such
calculations require the use of a lightning return stroke model. Two of the widely used
return stroke models are (1) the modified transmission line model with exponential
decay (MTLE) of current with height and (2) the modified transmission line model with
linear decay (MTLL) of current with height. In this paper, simplified expressions based on
the MTLE and MTLL models are derived for calculating far (radiation) electric fields
produced at an arbitrary elevation angle by lightning return strokes. It is shown that
different (for example, containing either spatial or time integral), but equivalent equations
can be derived for each of the models. Predictions of simplified expressions are
compared with electric fields computed using exact expressions, including all the field
components, and the validity of simplified expressions for distances that are much greater
than the radiating channel length is confirmed.
Citation: Thottappillil, R., V. A. Rakov, and N. Theethayi (2007), Expressions for far electric fields produced at an arbitrary altitude
by lightning return strokes, J. Geophys. Res., 112, D16102, doi:10.1029/2007JD008559.
1. Introduction
[2] There are some applications where knowledge of
lightning return stroke fields above ground is needed. For
example, modeling of ‘‘elves’’, one type of mesospheric
transient luminous events, requires computation of transient
return stroke fields (and Poynting vectors) at high altitudes
[e.g., Rowland et al., 1995]. Various return stroke models
are used for the calculation of remote fields. A description
of commonly used return stroke models can be found in the
work of Rakov and Uman [1998]. Some of the models have
been validated previously [e.g., Nucci et al., 1990;
Thottappillil and Uman, 1993; Thottappillil et al., 1997]
using measured remote fields, and, when available, optically
measured return stroke speeds and measured currents.
Perhaps the most widely used are the transmission line
(TL), modified transmission line model with exponential
decay (MTLE), and modified transmission line model with
linear decay (MTLL) models. Analytical expressions for far
fields at an arbitrary elevation angle based on the TL model
are well known [e.g., LeVine and Willett, 1992; Krider,
1992; Thottappillil et al., 1998; Rakov and Tuni, 2003].
Deriving similar expressions for more realistic MTLL and
MTLE models is of interest. Nucci et al. [1990] has
presented an analytical expression for far radiation field,
but only at ground, for the MTLE model. Recently, Shao et
1
Division for Electricity and Lightning Research, Uppsala University,
Uppsala, Sweden.
2
Department of Electrical and Computer Engineering, University of
Florida, Gainesville, Florida, USA.
al. [2005] have derived, for the MTLE model, an expression
for radiation field at an arbitrary elevation angle without
taking into account the presence of ground. Shao and
Heavner [2006] used their expression in modeling of
bipolar electric field pulses associated with the lightning
preliminary breakdown process. Electric field expression for
the MTLE model (including the effect of perfectly conducting ground) for the special case of step function current
wave was derived by Wait [1998] and employed by Rakov
and Tuni [2003].
[3] In this paper, we derive expressions for far (radiation)
fields at an arbitrary elevation angle for both the MTLE and
MTLL models and test their validity against exact general
expressions that include all the terms (static, induction, and
radiation). Field expressions are derived for free space; that
is, neglecting interaction of electromagnetic waves with
conducting atmosphere, which should be important at ionospheric altitudes. Additionally, we demonstrate in this paper
that different, containing either spatial or time integral, but
equivalent equations can be derived for each of the models.
We also show that the expression for the MTLE model
derived by Shao et al. [2005] is equivalent to the
corresponding expressions derived here. Note that the fields
above ground presented by Thottappillil and Rakov [2006]
using Shao et al.’s equation are incorrect, because of an
error in the computer code.
2. Far Electric Field Equations for Transmission
Line Type Models
[4] The relationship between the channel-base current
and current at any height along the channel in the trans-
Copyright 2007 by the American Geophysical Union.
0148-0227/07/2007JD008559$09.00
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THOTTAPPILLIL ET AL.: LIGHTNING FAR ELECTRIC FIELD EXPRESSIONS
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between the vertical channel and the inclined distance R(z0)
(see Figure 1), and c is the speed of light. The effect of
ground plane is not included. The exact expression for L0(t)
L0 ðt Þ RðL0 Þ
can be obtained as the solution of t ¼
þ
, where
c
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v
0
R(L ) =
L0 2 ðt Þ þ r2 2L0 ðt Þ r cos q, and is given by
Thottappillil et al. [1998, equation (25)]. Angle q is the
angle between the vertical channel and the line connecting
the channel base and point P (see Figure 1). An approximate
equation for L0(t) that is valid at far distances; that is, when
L0(t) r, is given by [e.g., Thottapillil et al., 1998]
Figure 1. Geometry of the problem defining the symbols
used in field equations of this paper. Note that elevation
angle (not labeled) is defined here as 90° q.
mission line type (TL, MTLE, and MTLL) models is
defined as [e.g., Rakov and Uman, 1998]
iðz0 ; t Þ ¼ ið0; t z0 =vÞ Pðz0 Þ
ð1Þ
0
where P(z ) is the current attenuation factor, and v is the
upward return stroke speed. In the original TL model, the
current is assumed to travel without any attenuation [Uman
and McLain, 1969]. Nucci et al. [1988] proposed an
exponential function for current decay and Rakov and
Dulzon [1987, 1991] proposed a linear function for current
decay with height. Relationships between the channel-base
current and current at any height z0 for the three return
stroke models introduced above are given by
TL
iðz0 ; t Þ ¼ ið0; t z0 =vÞ
MTLE
iðz0 ; t Þ ¼ ið0; t z0 =vÞ ez =l
ð3Þ
MTLL
0
iðz0 ; t Þ ¼ ið0; t z0 =vÞ 1 Hz
ð4Þ
ð2Þ
0
where z0 is the height above ground, t is the time, v is the
return stroke speed, l is the current decay height constant,
and H is the total length of the return stroke channel.
Although the MTLE and MTLL models are the focus of this
paper, we also include equations for the TL model, in order
to facilitate direct comparison.
[5] The exact expression for the electric field at any point
in space has been presented in a generalized form by
Thottappillil et al. [1998]. Far from the channel the field
is dominated by the radiation field component, traditionally
identified as that containing c2 R1 and is given by [e.g.,
Thottappillil et al., 1998]
far
Etotal
L0 ðt Þ v ðt r=cÞ
1 vc cos q
The geometrical parameters used in equations (5a) and (5b)
are illustrated in Figure 1. The second term in (5a) is
nonzero only if there is a current discontinuity at the return
stroke front. The total electric field is given by (B1) in
Appendix B, from which equation (5a) can be obtained
retaining only radiation field terms (fifth and sixth terms of
(B1)), which dominate at far distances and at early times. It
is worth noting that for small values of q and at intermediate
distance the induction field component gives the primary
contribution to the total field. Lu [2006], who computed
electric fields due to return strokes at a height of 90 km,
found that the induction field component dominates within a
radius of about 11 km (depending on the return stroke
speed) of the vertical lightning channel. Also, relative
magnitude of the electrostatic component increases at later
times. Note also that the traditional division of the total field
into static, induction (or intermediate), and radiation field
components, on the basis of distance dependence, is not
unique, especially at close and intermediate distances, as
demonstrated by Thottappillil and Rakov [2001a, 2001b].
[6] Equations (2), (3), and (4) can be substituted in (5a)
and the latter can be simplified to get approximate expressions for far radiation field corresponding to the TL, MTLE,
and MTLL model, respectively. Details of the steps leading
to the simplified expressions are given in Appendix A. The
final approximate expressions, accounting for current discontinuity at the front (if any), are,
EfarTL ¼ ^q
EfarMTLE ¼ ^q
1
v sin q r
i 0; t v
2
4pe0 c r 1 cos q
c
c
vðtr=cÞ
1ðv=cÞ cos q
Z
z0 Rðz0 Þ
z0
0
l
e dz
i 0; t v
c
0
0
ZL ðtÞ
sin aðz0 Þ @iðz0 ; t Rðz0 Þ=cÞ 0
^ ðz Þ 2 0
a
dz
c Rðz Þ
@t
0
1
sin aðL0 Þ
RðL0 Þ dL0 ðt Þ
0
^ ðL0 Þ 2
þ
ð
t
Þ;
t
a
i
L
4pe0
c RðL0 Þ
c
dt
ð7aÞ
0
ð5aÞ
EfarMTLL ¼ ^q
where L0(t) is the length of channel contributing to the field
at point P at time t, R(z0) is the inclined distance between the
channel segment at height z0 and point P, a(z0) is the angle
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ð6aÞ
1
v sin q
4pe0 c2 r 1 v cos q
c
r 1
i 0; t c
l
Erad ðr; q; tÞ
1
¼
4pe0
ð5bÞ
1
v sin q
2
4pe0 c r 1 v cos q
c
r 1
i 0; t c
H
v:ðtr=cÞ
1ðv=cÞ cos q
Z
0
z0 Rðz0 Þ
dz0 ð8aÞ
i 0; t c
v
THOTTAPPILLIL ET AL.: LIGHTNING FAR ELECTRIC FIELD EXPRESSIONS
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Figure 2. Comparison of electric fields (effect of ground
included) at a distance of 100 km for the modified
transmission line model with exponential decay (MTLE)
model, using exact expression (B1) plus corresponding
image expression and approximate far field expression, (7a)
plus (7b), for different polar angles q = 10°, 30°, 60°, and
90°and return stroke speed 1.5 108 m/s. Solid line gives
exact expression and dotted line gives approximate one.
Equations (6a), (7a), and (8a) are derived without including
the effect of the ground plane, which can be easily added by
considering similar expressions for image channel. The
latter are obtained by replacing polar angle q by p q
everywhere in (6a), (7a), and (8a), including R(z0) =
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z0 2 þ r2 2z0 r cos q, and are given below.
i
E farTL ¼ ^q
i
E farMTLE
¼ ^q
1
v sin q r
i
0;
t
v
4pe0 c2 r 1 þ cos q
c
c
1
v sin q
r
i
0;
t
v
4pe0 c2 r 1 þ cos q
c
c
vðtr=cÞ
1þðv=cÞ cos q
1
l
Z
i
the corresponding image expression (not given in this
paper). The channel-base current shown in Nucci et al.
[1990, Figure 4], which is representative of subsequent
return strokes was used in the calculations. Results are
presented for the return stroke speed 1.5 108 m/s in
Figure 2 for the MTLE model and in Figure 3 for the MTLL
model. In the calculations, the current decay height constant
l in the MTLE was assumed to be 2000 m, and the constant
H in MTLE model was 7500 m. One can see that predictions of approximate expressions (7a) and (7b) and (8a) and
(8b) for far electric fields (dotted line) are in very good
agreement with those of the exact expression (solid line). In
fact, the curves based on approximate expressions are
almost indistinguishable from exact ones, which indicate
that (1) equations (7a) and (7b) and (8a) and (8b) are
correctly derived from (B1) (and its image counterpart)
and (2) fields at 100 km and q ranging from 10° to 90°
are indeed dominated by the radiation field component. In
this paper whenever the effect of ground is considered, it is
assumed to be perfectly conducting. Real ground has finite
conductivity, which gives rise to field propagation effects
[e.g., Cooray, 2003; Shoory et al., 2005]. These effects may
be significant at ground surface (q = 90°), but are expected
to be small at high altitudes considered here.
[ 8 ] Equations (7a) and (7b) and (8a) and (8b),
corresponding to the MTLE and MTLL models, respectively, are new. When l = 1 in (7a), it reduces to (6a)
corresponding to the TL model. Similarly, when H = 1 in
(8a), it reduces to (6a). Rachidi and Nucci [1990] had
derived an expression relating the channel-base current
and the far (radiation) fields for the MTLE model (including
the effect of perfectly conducting ground) for the special
case of field point at ground (q = 90°) and assuming no
current discontinuity at time t = 0. Their expression is
equivalent to the sum of (7a) and (7b), if q = 90°.
the special case of a step function
[9] We now consider
0
current wave, I0ez /l u(t z0/v), that travels upward at
z0 R0 ðz0 Þ
z0
i 0; t e l dz0
v
c
0
E farMTLL ¼ ^q
ð6bÞ
D16102
ð7bÞ
1
v sin q
r
i 0; t v
2
4pe0 c r 1 þ cos q
c
c
vðtr=cÞ
1þðv=cÞ cos q
1
H
Z
z0 R0 ðz0 Þ
dz0
i 0; t v
c
ð8bÞ
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 0
In (7b) and (8b), R (z ) = z0 2 þ r2 þ 2z0 r cos q. Equation
(6b) for the TL model was previously given by Krider
[1992].
[7] Calculations based on equations (7a) and (7b) and
(8a) and (8b) were performed for different polar angles q =
10°, 30°, 60°, and 90° and compared with the fields
computed using exact expression (B1) in Appendix B and
Figure 3. Comparison of electric fields (effect of ground
included) at a distance of 100 km for the modified
transmission line model with linear decay (MTLL) model
calculated using exact expression (B1) plus corresponding
image expression and approximate far field expression, (8a)
plus (8b), for different polar angles q = 10°, 30°, 60°, and
90° and return stroke speed 1.5 108 m/s. Solid line gives
exact expression and dotted line gives approximate one.
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THOTTAPPILLIL ET AL.: LIGHTNING FAR ELECTRIC FIELD EXPRESSIONS
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speed v and whose magnitude decreases with height as
ez’/l (the MTLE model). The radiation field (q component)
equation for this special case (including the effect of
perfectly conducting ground) was derived by Wait [1998]
and employed by Rakov and Tuni [2003] and Thottappillil
and Rakov [2005]. That expression can be readily obtained
from (7a) and (7b). A version of that equation, excluding the
effect of ground, is given below.
E
step
vðtr=cÞ
1
v sin q
v
ðt Þ ¼ þ^q
I0 eð1c cos qÞl
v
2
4pe0 c r 1 cos q
c
D16102
Let us define
z
z0 v
1 cos q v
c
v
ð12Þ
z
1 ðv=cÞ cos q
ð13Þ
From which
z0 ¼
ð9Þ
In section 3, we will show how application of the
Duhamel’s integral [Greenwood, 1991] to (9) can be used
to obtain equation (7a), representing the return stroke far
field on the basis of the MTLE model for an arbitrary
current i(0, t r/c) at the channel base.
Equation (13) is similar in form to (5b). Let us now express
z0 and dz0 in (7a) in terms of z and dz. Then for the limits of
the integral in (7a), when z0 = 0, z = 0 and when z0 =
v ðt r=cÞ
, z = v (t r/c). Thus (7a) can be written as
1 ðv=cÞ cos q
1
v sin q
4pe0 c2 r 1 v cos q
c
"
vðZ
tr=cÞ
r
1
z r
i 0; t i 0; t c
l
v c
0
#
z
dz
elð1ðv=cÞ cos q
1 ðv=cÞ cos q
EfarMTLE ¼ ^q
3. Comparison of Far Electric Field Equation
Derived for the MTLE Model by Shao et al. [2005]
With Equation (7a)
[10] Shao et al. [2005], henceforth denoted as SFJ, have
presented an expression for far field (not accounting for
ground effect) for the MTLE model that is reproduced below,
after adapting it to the notation (see Figure 1) of this paper.
1
v sin q
4pe0 c2 r 1 v cos q
c
"
r
1
i 0; t v
c
l 1 cos q
c
#
vðZ
tr=cÞ z0
z0 r
0
lð1vc cos qÞ
e
i 0; t dz
v c
E farSFJ ¼ ^q
ð10Þ
0
Equation (10) is to be compared with equation (7a). The
first term in both equations represents the radiation field
predicted by the TL model (see equation (6a)), and is the
same. However, three differences between the second terms
of (10) and (7a) can be observed. These are (1) SFJ have
l0 = l (1 v cos q/c) in two places, in the multiplying factor
before the integral and in the argument of the exponential
function inside the integral, whereas (7a) has only l in both
these places. (2) In (10) the upper limit of integration L0 =
v (t r/c) is the height of the return stroke channel at retarded
time t r/c, whereas in (7a) it is given by (5b), the height of
the return stroke channel contributing to the field at time t,
(3) In (7a)
an approximation for L0(t) valid at far distances.
0
0
the returnstrokecurrent atheightz0 isgiven byi 0,t Rðcz Þ zv ,
0
whereas in (10) it is given by i(0, t cr zv ). For a field
0
point above ground, r > R(z ), and hence (t r/c z0/v) < (t R(z0)/c z0/v). In spite of the apparent
differences between (7a) and (10), they are equivalent, as
will be shown both analytically and numerically. Let us
first derive (10) starting from (7a).
[11] Far away from the channel, i.e., when z0 r, R(z0) r z0 cos q, and therefore
t
r
z0 Rðz0 Þ
z0 v
1 cos q t
v
v
c
c
c
ð14Þ
Rearranging (14), we get the same expression as (10),
derived by SFJ. Note that z in (14) and z0 in (10) are
variables of integration that vary between the same limits of
integration and hence are interchangeable.
[12] Thus we have shown that SFJ’s expression for far
fields, equation (10), is equivalent to our expression (7a),
even though two different methods of derivation are adopted in SFJ and in this paper. It should be pointed out that the
upper limit of the integral in (10), v (t r/c), gives the
length of the return stroke channel at retarded time, not
the length of the channel actually contributing to the field at
a given time t. On the other hand, the upper limit of the
integral in (7a) gives the length of the channel actually
contributing to the field at a given time t. As an example,
the length of the return stroke channel at a retarded time of
one microsecond for a return stroke speed of 2.7 108 m/s
is 270 m, but the length of the channel contributing to the
field at a far distance at angle q = 10° will be 2375 m.
Therefore the appropriate far field condition is L0(t) r,
where L0(t) is given by (5b), and not v (t r/c) r, as
given by Shao et al. [2004].
[13] Using a procedure similar to that employed in
deriving (10) from (7a), one can get an expression that is
different from but equivalent to (8a) for the MTLL model as
1
v sin q
4pe0 c2 r 1 v cos q
c
"
r
1
i 0; t v
c
H 1 cos q
c
#
vðZ
tr=cÞ z0 r
0
i 0; t dz
v c
EfarMTLL ¼ ^q
ð11Þ
0
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ð15Þ
THOTTAPPILLIL ET AL.: LIGHTNING FAR ELECTRIC FIELD EXPRESSIONS
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time between r/c and t. The unit step response E ustep is given
by equation (9) divided by I0. From (16), we get,
Fv sin q
E ðr; t Þ ¼ ^q
2
4pe
2 0c r
3
Zt
vF
ð
tr=c
Þ
r
vF
6
7
4i 0; t ið0; t t Þ e l dt5 ð17Þ
c
l
r=c
1
Figure 4. Comparison of electric fields (effects of ground
not included) at a distance of 100 km for the MTLE model,
calculated using far field expressions (7a), (10), and (17) for
different polar angles q and return stroke speed 1.5 108 m/s. The three curves for each angle are indistinguishable because of excellent match.
where F = [1 (v/c)cosq] . It is shown in Appendix C that
(17) is analytically equivalent to (7a). It should be also
equivalent to (10), since the equivalence between (7a) and
(10) has been shown in section 3 above.
[15] Equivalence between (17), (7a), and (10) were also
verified numerically, as illustrated in Figure 4. Note that
(7a) and (10) contain spatial integrals (with different
upper limits), while (17) contains a time integral. Nevertheless, fields predicted by these three equations are
indistinguishable.
[16] An equation similar to (17) can be readily derived
(substituting (C1) – (C6) of Appendix C in (8a)) for the
MTLL model, the result being given by
2
Field expressions for image channel corresponding to (10)
and (15) for the MTLE and MTLL models, can be derived
from (7b) and (8b), respectively. In fact, one only needs to
replace 1 (v/c)cosq in (10) and (15), wherever it appears,
with 1 + (v/c)cosq. It is interesting to note that the upper
limit of integration of the second terms in (10) and (15)
remains unchanged in the expressions for image channel,
whereas upper limits of integration in expressions (7b) and
(8b) differ from those in (7a) and (8a).
E farMTLL
3
Zt
Fv sin q 6
r
vF
7
¼ ^q
ið0; t t Þdt5
4i 0; t 4pe0 c2 r
c
H
r=c
ð18Þ
Figure 5 shows, in a format similar to that of Figure 4, a
comparison of three equivalent equations for the MTLL
model. Again, as expected, an excellent agreement is seen.
5. Conclusions
4. Far Field Expressions Containing Time
Integrals for the MTLE and MTLL Models
[14] In the following, we will derive expressions for the
MTLE model containing the time integral and show that it
is equivalent to expressions (7a) and (10) containing the
spatial integral. In doing so, we will use the fact that both
equations (7a) and (10) reduce to equation (9) for the
special case of step function current at the channel base
and Duhamel’s integral. If equation (9) is the electric field
response of a step current input at the bottom of the
channel, then one can get the field response for any other
specified current at the channel base by applying the
Duhamel’s integral, because we are dealing with a linear
system. One form of Duhamel’s integral is given by
[Greenwood, 1991]
E ðr; t Þ ¼ E
ustep
Zt
ðr=cÞ ið0; t r=cÞ þ
dE
ustep
dt
ðt Þ
[17] Far field approximate expressions at any elevation
angle for the MTLE and MTLL return stroke models are
derived and their validity is tested against exact solution. It
is shown that different (for example, containing a spatial or
iðt t Þdt
0
ð16Þ
where E(r, t) is the electric field at distance r (response) due to
a specified channel-base current i(0, t r/c) (input), E ustep is
the field response to a unit step current at the channel base, r/c
is the initial time, t is the present time, and t is an arbitrary
Figure 5. Comparison of electric fields (effects of ground
not included) at a distance of 100 km for the MTLL model,
calculated using far field expressions (8a), (15), and (18) for
different polar angles q and return stroke speed 1.5 108 m/s. The three curves for each angle are indistinguishable because of excellent match.
5 of 8
THOTTAPPILLIL ET AL.: LIGHTNING FAR ELECTRIC FIELD EXPRESSIONS
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time integral), but equivalent approximate expressions, can
be derived for each of the considered models.
Appendix A: Far Electric Field Expressions
Based on the TL, MTLE, and MTLL Models
(Effects of Ground Not Included)
^ ^q. Applying these
L0(t), then R r, a q, a
approximations in the factors that influence the amplitude
and direction of the field in equation (A5), we get
EMTLE ðr; q; t Þ A1. TL Model
[18] The approximation for far field at an arbitrary
elevation angle is found, for example, in the works of
LeVine and Willett [1992], Krider [1992], and Thottappillil
et al. [1998], and the resultant expression is equation (6a).
This equation is valid whether there is current discontinuity
at the upward moving front or not.
A2. MTLE Model
[19] Applying (3) to retarded current, the current derivative at height z0 is given by
0
@iðz0 ; t Rðz0 Þ=cÞ @ ið0; t z0 =v Rðz0 Þ=cÞ ez =l
¼
@t
@t
@ið0; t z0 =v Rðz0 Þ=cÞ z0 =l
¼
e
@t
ðA1Þ
The relationship between the time derivative of retarded
current in (A1) and its spatial derivative is given by [e.g.,
Thottappillil et al., 1998]
@ið0; t z0 =v Rðz0 Þ=cÞ
@ið0; t z0 =v Rðz0 Þ=cÞ
v FTL ðz0 Þ
¼
@t
@z0
ðA2Þ
where
1
FTL ðz0 Þ ¼ 1 vc cos aðz0 Þ
ðA3Þ
is the so called F factor discussed extensively by several
authors [e.g., Rubinstein and Uman, 1990; LeVine and
Willett, 1992; Krider, 1992; Thottappillil et al., 1998; Shao0
et al., 2004; Thottappillil and Rakov, 2005]. The speed, dL
dt ,
of the wave front as ‘‘seen’’ by the observer at P (Figure 1)
is given by Thottappillil et al. [1998, equation [31]] as
dL0
1
¼ v FTL ðL0 Þ
¼v
dt
1 ðv=cÞ cos qðL0 Þ
ðA4Þ
sin q
v
^q
4pe0 c2 r 1 ðv=cÞ cos q
0
ZL ðtÞ
@ið0; t z0 =v Rðz0 Þ=cÞ z0 =l 0
dz
e
@t
0
þ ^q
sin q
v
0
ið0; 0ÞeL ðtÞ=l
4pe0 c2 r 1 ðv=cÞ cos q
A3. MTLL Model
[20] Using a procedure similar to that adopted above for
the MTLE model, one can readily derive equation (8a), the
far electric field equation based on the MTLL model, with
or without discontinuity at the upward moving front.
Appendix B: Exact Electric Field Equation
(Effects of Ground Not Included)
[21] The exact expression for remote electric field for an
arbitrary current distribution along a vertical channel is
presented by Thottappillil et al. [1998], which is reproduced
as equation (B1) below. In the first and third terms of (B1), v
appearing in the lower limits of time integrals is the return
stroke speed. Other symbols used in (B1) are defined in
Figure 1. The effect of ground plane is not included. The
sum of the fifth and sixth terms in (B1) is the radiation field,
which is given by equation (5a) in section 2.
ZL ðtÞ "
0
1
Eðr; t Þ ¼ þ
4pe0
Substituting (A2) in (A1) and the resulting equation and
(A4) in (5a), we get
1
EMTLE ðr; q; t Þ ¼ 4pe0
Rðz0 Þ
z0
0 @i 0; t v
c
sin aðz Þ
^ ðz0 Þ 2 0
a
@z0
c Rðz Þ
0
zl
0
^ ðz0 Þ 2 cos aðz Þ
R
R3 ðz0 Þ
1
þ
4pe0
ZL ðtÞ 0
0
^ ðz0 Þ 2 cos aðz Þ i z0 ; t Rðz Þ dz0
R
cR2 ðz0 Þ
c
0
1
þ
4pe0
0
ZL ðtÞ "
sin aðz0 Þ
^ ðz Þ 3 0 a
R ðz Þ
0
0
0
v FTL ðz Þ e dz
1 v:FTL ðL0 Þ sin aðL0 Þ
0
^ ðL0 Þ
þ
ið0; 0ÞeL ðtÞ=l a
4pe0
c2 RðL0 Þ
ðA5Þ
If the distance to the field point P is very large compared to
the radiating length of the channel at time t, that is if r 6 of 8
#
Zt Rðz0 Þ
0
i z ;t dt dz0
c
0
z0 Rðz Þ
vþ c
0
0
0
ðA6Þ
Expanding the product under the integral in (A6) by partial
integration and simplifying we get equation (7a), the far
field expression based on the MTLE model. Similar to
equation (6a), equation (7a) is valid whether there is current
discontinuity at the upward moving front or not.
0
0
ZL ðtÞ
D16102
z0 Rðz
vþ c
0
1
þ
4pe0
ZL ðtÞ 0
^ ðz0 Þ
a
#
Rðz0 Þ
dt dz0
i z ;t c
Zt
0
0Þ
sin aðz0 Þ
Rðz0 Þ
0
;
t
i
z
dz0
cR2 ðz0 Þ
c
THOTTAPPILLIL ET AL.: LIGHTNING FAR ELECTRIC FIELD EXPRESSIONS
D16102
[23] Acknowledgments. This work was supported by a grant from
Swedish Research Council (VR grant 621-2005-5939), B. John F. and Svea
Andersson donation fund, and by NSF grant ATM-0346164.
0
sin aðz0 Þ @iðz0 ; t Rðz0 Þ=cÞ 0
a
^ ðz0 Þ 2 0 dz
c Rðz Þ
@t
0
1
sin aðL0 Þ
RðL0 Þ dL0
0
^ ðL0 Þ 2
a
þ
;
t
i
L
dt
4pe0
c RðL0 Þ
c
1
þ
4pe0
ZL ðtÞ References
ðB1Þ
In equation (B1), dL0/dt is the speed of the current wavefront
as ‘‘seen’’ by the observer at P, which is different from the real
speed v. The lower limit of the time integral in the first and
third terms of (B1) is the time at which the return stroke
wavefront has reached the height z0 for the first time, as
‘‘seen’’ from the observation point. The time t and the length
of the channel0 contributing
to the field at P are related by the
0
equation t = Lv + RðcL Þ. The sixth term of (B1) containing dL0/dt
will have a nonzero value only if there is a current
discontinuity (nonzero current) at the wavefront. Equation
(B1) can be resolved into components explicitly in the
directions of the spherical coordinate unit vectors ^q and ^r, as
done by Thottappillil and Rakov [2001a, 2001b, 2007]. At a
distance of 100 km, the ^r component is negligible compared
to the ^q component [e.g., Thottappillil and Rakov, 2007].
Appendix C: Equivalence Between Equations
(7a) and (17)
[22] Symbol t in (17) is the variable of integration, and
symbol z0 is the variable of integration in (7a). Therefore we
have certain freedom in defining them, as long as they are
consistent with each other. Let us define t in a physically
meaningful way as the time it takes for a return stroke
wavefront traveling at speed v to reach height z0 plus the
time it takes for a signal traveling from that height at the
speed of light c to reach the field point at distance R(z0).
t¼
z0 Rðz0 Þ
þ
v
c
ðC1Þ
From (C1) we can get [e.g., Thottappillil et al., 1998]
dz0
v
¼
¼ F ðz0 Þ v
dt 1 ðv=cÞ cos aðz0 Þ
ðC2Þ
dz0 ¼ F ðz0 Þ v dt
ðC3Þ
Therefore
Far from the channel,
F ðz0 Þ F ¼
1
;
1 ðv=cÞ cos q
and aðz0 Þ q
ðC4Þ
Also, when
z0 ¼ 0;
t ¼ r=c
z0 ¼ L0 ðtÞ; t ¼ t
ðC5Þ
Therefore from (C4) and (C5),
z0 ¼ F v ðt r=cÞ
Applying (C1) –(C6) to (7a), we get (17).
D16102
ðC6Þ
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V. A. Rakov, Department of Electrical and Computer Engineering,
University of Florida, 553 Engineering Building 33, Gainesville, FL 326116130, USA.
N. Theethayi and R. Thottappillil, Division for Electricity and Lightning
Research, Uppsala University, Box 534, 75121 Uppsala, Sweden.
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