Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 117

C n ! n n C

P n .cos /;

X

0 m.!/
2i!r

1
2

E r D (4.30)

n D 0

where the coefficients C n are given by Eq. ( 4.20 ). Here we have taken into account
that the summand with number n D 0 in Eq. ( 4.29 ) is equal to zero since
@ P 0 .cos / D 0.
The implication here is that the electromagnetic field inside the Earth-Ionosphere
cavity is represented as a linear superposition of normal modes excited by the ver-
tical lightning discharge with the current moment m.!/. Each normal mode has
a specific angular structure, which is determined by the Legendre polynomial
associated with the mode number. In practice, the angular dependence determines
the decrease of the normal mode amplitude with distance between the source and
observation point.
The coefficient C n in Eqs. ( 4.29 ) and ( 4.30 ) can be expanded in a power series of
a small parameter kd. Using the first order approximation similar to that given by
Eq. ( 4.22 ), one can reduce Eq. ( 4.20 ) to the following:

c 2
R e d ! 2

R e
d k 2 R e n.n C 1/ D

C n

! n

: (4.31)

On the ground surface r D R e we thus get

X

1

m.!/
4" 0 R e d

.2n C 1/
! 2

! n

B ' D @ P n .cos /; (4.32)

n D 1

X

1

! n .2n C 1/

im.!/
4" 0 !R e d

! n ! 2

E r D P n .cos /: (4.33)

n D 0

We recall that this form of solution is based on the idealized model of the Earth-
Ionosphere resonance cavity which includes the perfect conducting walls.
The finite conductivity of the ionospheric plasma is believed to be the major
cause for wave energy losses. At the altitudes of the gyrotropic E layer the
ionospheric plasma is described by the conductivity tensor of Eq. ( 2.5 )orby
the plasma dielectric permittivity tensor of Eq. ( 2.16 ). In the D layer, the altitude
range 50-70 km, the total electron and ion collision frequencies, i.e., e D ei C en
and i D in C ie , are much greater than the corresponding gyrofrequencies, ! H
and H , of the electrons and ions. This implies that all diagonal components of
the conductivity/dielectric permittivity tensor [see Eqs. ( 2.7 )-( 2.9 )] become equal
to each other in such a way that this tensor can be replaced by the scalar value

! p
!.! C i e / ;

" p D 1 (4.34)