Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 108

z

EMW

m( t )

B j

E r

d

R e

E ʸ

ʸ r

R i

Fig. 4.1 A simple model of the Earth-Ionosphere resonance cavity. Here R e and R i are the radii
of the Earth and the ionosphere, d is the height of the non-conductive atmosphere, EMW denotes
electromagnetic wave radiated by the vertical current moment m .t/,andB , E r ,andE are the
components of the TM mode

and

r E D i! B : (4.2)

The current density j s due to the vertical lightning discharge can serve as a source
function for the electromagnetic waves inside the resonance cavity.
The conduction current in the atmosphere is much smaller than the displacement
one under the requirement of a E " 0 !E. In what follows we show that the
Schumann resonances lie in the frequency range f>7:5Hz, that is ! D 2f >
47 Hz. Whence it follows that the above requirement reduces to the following a <
" 0 ! D 4 10 10 S/m. Taking into account that the atmospheric conductivity a
increases exponentially with altitude, one can find that this requirement is valid in
the lower atmosphere up to the altitude 40-50 km at the daytime and up to 60-70 km
at night. These altitudes may serve as an estimate of the upper boundary of the
“spherical capacitor.” In order to make our consideration as transparent as possible,
we assume a constant distance between the resonator walls thereby disregarding
the difference between daytime and nighttime conditions.
We shall use spherical variables r, , and ' and a coordinate system in which
the vertical current of the return stroke is in the direction of the polar axis z .By
symmetry of the problem the vertical current produces the so-called transverse
magnetic mode (TM mode), which contains only three components B ' , E r , and E ,
as shown in Fig. 4.1 . The transverse electric mode (TE mode), E ' , B r , and B , can
be excited by virtue of the mode coupling at the boundary between the atmosphere