Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 180

0.065

2

1

0.06

0.055

3

0.05

5

0.045

4

0.04

0.035

6

0.03

0.025

0.02

0 2 4 6 8 10

r , 10 3 km

Fig. 5.18 Spectra of correlation function . xx =2/ 1=2 as a function of distance r at fixed
frequencies. The distance r denotes the radius which confines the area occupied by randomly
distributed thunderstorms that makes a contribution to the spectrum. The lines 1-3 correspond to
the fixed resonant frequencies f D 0:5, 1:0,and1:5 Hz, respectively. In making these plots we

have used the distance r 1 400 km between circumferences shown in Fig. 5.15 . The lines 4-6

correspond to the same frequencies and distance r 1

D

D 550 km. Taken from Surkov et al. ( 2006 )

World thunderstorm center located far from the observation cite. In such a case we
can neglect the flash distribution inside this thunderstorm center. Thus, the spectral
density of the correlation matrix is simplified to

and '' D 2 ǝ M 2 Ǜ LJ LJ g ' .r;!/ LJ LJ rr D 2 ǝ M 2 Ǜ j g r .r;!/ j

2 ;

2

(5.65)

where is the average rate of the lightning discharges in the area occupied by the
World thunderstorm center. Substituting approximating Eqs. ( 5.59 ) and ( 5.61 )for
the functions g r and g ' into Eq. ( 5.65 ) yields

ǝ M 2 Ǜ
2

! 1=2

LJ LJ LJ LJ LJ LJ LJ LJ

0 Ǜ H L
rd

F .!/ 1 .!/
Œ1 C 1 .!/dq 1 .!/

Πrr .!/ 1=2 D ; (5.66)

ǝ M 2 Ǜ
2

! 1=2

'' .!/ 1=2

0 j F .!/ j
rd

D : (5.67)