Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 120

4.3 Sources of Resonator Excitation

4.3.1 Lightning Discharges Treated as a Stochastic Process

To study the Earth-Ionosphere cavity resonator in a little more detail, it is necessary
at this point to construct a suitably idealized model of the spatial and temporal
source distribution that is a reasonable approximation to the variation of the
global thunderstorm parameters. In the simple model the global thunderstorm
activity is modeled as a point constant-amplitude emitter located in the vicinity
of equator near the nighttime terminator at 17-18 LT (Galejs 1972 ). Following
the sun this emitter runs around the Earth for the day. Bliokh et al. ( 1980 )have
approximated the actual situation of the lightning activity with the configuration,
which includes three or more fixed world thunderstorm centers. In a more complete
theory, however, thunderstorm distribution on the Earth surface in accordance with
climatological data may be included (Ogawa and Murakami 1973 ), that results in
some complications of the theory.
Actually, the lightning discharge in the thunderstorm region should be considered
most likely as random events described by a sequence of randomly occurring pulses
of random amplitudes (Raemer 1961a , b ; Galejs 1965 ; Polk 1969 ; Nickolaenko
1981 ; Surkov et al. 2005 , 2006 ; Surkov and Hayakawa 2010 ). In what follows
we consider the global lightning activity, whose electromagnetic signals are being
recorded by a ground-based station which is far away from thunderstorms. Let N be
the number of thunderstorms that are in progress around the world at the moment.
Typical thunderstorm area is of the order of 10 3 km 2 . This means that a typical
size of thunderstorm area is smaller than the distance from the recording station
so that we can ignore the lightning spatial distribution within the thunderstorm area.
In this notation all the lightning related to the given thunderstorm will have the same
coordinates as those of the given thunderstorm.
We introduce a coordinate system fixed to the center of the Earth in such a way
that the recording station is situated on the polar axis z , as shown in Fig. 4.3 .Letr ,
, and ' be the spherical coordinates of a thunderstorm, where the subscript
stands for the number of thunderstorms, i.e., D 1;2;:::;N. Notice that in fact r
equals to the Earth's radius R e .

Let b r ;t t n be the random magnetic field at the ground-based station
generated by an individual CG lightning discharge occurring at the point r at a
random moment t n , where n D 1;2;::: is a number of the lightning discharge
happened at this point. Here r denotes a set of the thunderstorm's spherical

coordinates r , , and ' . Notice that b r ;t t n vanishes at t<t n .
In standard meteorological practice a local coordinate system fixed at the
recording station has the x axis eastward, the y axis to the north, and z axis vertically
upward. For convenience, this local coordinate system is shown in Fig. 4.3 from the
right of the main image.
In Sects. 4.1 and 4.2 the electromagnetic field of vertical CG lightning has been
expressed through the spherical components, b r , b , and b ' , in the source-local