Atmospheric Electricity - Ultra and Extremely Low Frequency Electromagnetic Fields - page 68

number n by the fixed value l,thatis,asl n D l 1 C .n 1/l. The increase in
the lightning channel length results in gradual enhancement of the electric current
moment.
Actually the current profiles of the individual strokes can be different whereas
in this model all the current impulses are assumed to have the same shape. The
inverse time parameter, n , of the return stroke is related to the channel length, l n ,
as follows

l n l

V 0 .n 1/;

n D

V 0 D 1 C

(3.9)

where 1 D V 0 =l 1 . The net magnetic moment of the flash which contains the
multiple discharge can be described by an equation analogous to Eq. ( 3.7 ), that
is m.t/ D MF .t/, where M D l 1 j I 1 j is the “magnitude” of the current
moment while the dimensionless function F .t/ determines the shape of the multiple
discharge

n 0

X

l 1 1 exp n t n

l n

F .t/ D

nD1

X

4

t n

exp ! m t n ;

I m
j I 1 j

(3.10)

m D 1

where t n D t .n 1/t 0 and .x/ denotes the step-function, i.e., D 1 if x 0
and D 0 if x<0.
Taking the numerical values l 1 D 4 km, l D 1 km and above-mentioned
parameters of the return stroke, one can estimate the typical magnitude of the
magnetic moment of the multiple return stroke as M 10-10 2 kA km. Model
calculation of the flash current moment for n 0 D 3 is shown in Fig. 3.9 .
The spectrum of the current moment MF .t/ is given by

X

n 0

X

4

I m exp Œi!t 0 .n 1/
.! m i!/. n C ! m i!/ :

MF . ! / D V 0

(3.11)

n D 1 m D 1

Figure 3.10 shows model calculation of the absolute value of the spectrum MF .!/
originated from n 0 D 3 return strokes, which follow one by one with equal interval
t 0 D 40 ms. In such a case the spectrum has approximately a quasi-oscillatory
profile with maximum repetition period about 25 Hz. This peculiarity of the
spectrum is due to the presence of imaginary exponents in Eq. ( 3.11 ) and thus it
follows from the interference between the fields of individual return strokes (e.g.,
see Surkov et al. 2010 ). Actually the number of strokes and intervals between
their occurrence are rather random values that may result in randomizing of these
oscillations.