Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 176

functions is defined by the functions g r and g ' given by Eqs. ( 5.56 ) and ( 5.57 ). In the
reference frame fixed to the ground-recording station the Cartesian components of

the field b x ;b y ;b z are related to the cylindrical components b r ;b ' ;b z by virtue
Eq. ( 5.62 ). The net magnetic field is a sum of random fields, b n , originated from
the lightning totality.
Consider now a number of thunderstorm centers distributed around the ground-
recording station. Since the lightning activity is treated as a random process, the
net magnetic field variation, B .t/, is a random quantity as well. The necessary
summation of the random magnetic fields over all the lightning discharges is found
in Sect. 4.3 and the result of this summation is given by Eq. ( 4.43 ). As has already
been stated, the mean value of the net magnetic field is close to zero and thus
is not interesting for practise. More usually we measure the power spectrum or
spectral density of correlation matrix of the random process. A general definition
of this matrix for a steady stochastic process is given by Eq. ( 4.45 ) while the basic
properties of the matrix are examined in Appendix B in greater detail. In what
follows we consider only the correlation between the horizontal component of
magnetic field.
As before, the lightning appearance is assumed to be a Poisson random process.
In such a case the spectral density of the correlation matrix is described by equations
analogous to Eqs. ( 4.47 ) and ( 4.48 ). Only difference is that the functions g and g '
in these equations should be replaced by the function g r and g ' , given by Eqs. ( 5.56 )
and ( 5.57 ).
It follows from these equations that both g r and g ' are proportional to the
function F .!/ while the spectral density of correlation matrix is proportional to
j F .!/ j

2 . In the low-frequency limit the spectral density m.!/ D MF .!/ of
the current moment generated by CG return stroke is given by Eq. ( 4.50 ). The
IAR eigenfrequencies lie in the ULF region where ! ! 4 so that Eq. ( 4.50 )is
simplified to

MF .!/ l I 3

:

I 4
! 4 ! 3 C

(5.63)

The empirical parameters I 3 , I 4 , ! 3 and ! 4 determine, in fact, the magnitude and
duration of CC which follows the return stroke. As is seen from Eq. ( 5.63 ), the
spectrum of single lightning discharge is practically constant in the ULF region.
To study the random magnetic variations in a little more detail, we approximate
the actual thunderstorm distribution around the recording station with the idealized
configuration that is displayed in Fig. 5.15 (Surkov et al. 2006 ). The thunderstorm
sites shown in Fig. 5.15 with little circles are randomly distributed along the
circumferences in such a way that the mean number density of the thunderstorms
or thunderstorm number per unit square, N=S, is approximately constant. In
the model the circumference radii, r n , and the thunderstorm numbers, N n ,onthe
circumference are chosen in the following way r n D r 1 n and N n D 4n, where
n D 1;2;3;::: If n !1 , the summation of the thunderstorm numbers over the