Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 118

where ! p is the plasma freque ncy given by Eq. ( 2.20 ). Here we have neglected the
small terms of the order of m e =m i .
In the lower ionosphere and D layer the total electron collision frequency, e ,is
mainly determined by the electron-neutral collisions. Note that e and ! p are much
larger than ! for all wave frequencies of interest here, so that we obtain the pure
imaginary value " p i! p =.! e /. At the height about 60 km the typical daytime
parameters are as follows (Nickolaenko and Hayakawa 2002 ) ! p 5 10 5 s 1 and
e 5 10 7 s 1 , while ! 50 s 1 . Substituting these values into Eq. ( 4.34 ) yields
" p 10 2 i.
Now we chose the simplest way to estimate the effect of wave energy absorption,
aiming at physical intuition rather than detailed analysis. In order to take into
account the damping factor of the electromagnetic waves, one should formally
remove the poles ! D ! n from real axes in the complex plane !. From physical
viewpoint, the real part of the poles ! n defines eigenfrequencies of the resonator
while the imaginary part of ! n defines the damping factors of the eigenmodes. In a
more accurate model of dissipative resonator the eigenfrequencies can be found
from the following equation (e.g., see Nickolaenko and Hayakawa ( 2002 ), and
references therein)

c 2 n.n C 1/
R e

ic!Z.!/
d

! 2

C D 0;

(4.35)

instead of Eq. ( 4.25 ). Here Z.!/ stands for the surface impedance of the

ionosphere. In the simple model of the isotropic ionosphere and perfectly

conducting Earth this impedance is given by Z D " 1= p .
When comparing the rigorous solution of the problem (e.g., see monographs by
Wait ( 1972 ), Galejs ( 1972 )) with the approximate solution given by Eq. ( 4.32 ), the
difference in the denominators of these solutions comes into particular prominence.
To make Eq. ( 4.32 ) identical to the rigorous solution, one should replace the term
! 2 in the denominator of the sum in Eq. ( 4.32 ) by the imaginary expression ! 2

C

ic!Z.!/=d. As a result we obtain that

B ' D Mg ' .;!/: (4.36)

Here we made use of the following abbreviation:

X

F .!/
4" 0 R e d

.2n C 1/@ P n .cos /
! 2

C ic!Z.!/=d ! n

g ' .;!/ D : (4.37)

n D 1

where ! n is given by Eq. ( 4.26 ). As before, the spectrum of magnetic variations
produced by the lightning discharge is proportional to the spectrum of the lightning
current moment m.!/ D MF .!/.