Electromagnetic Effects Resulted from Explosions - Ultra and Extremely Low Frequency Electromagnetic Fields - page 455

increases in time, but we shall assume the simultaneous arrival of waves at
the ionospheric boundary. Owing to the wave refraction in the atmosphere, the
ionospheric region interacting with the wave is limited by the radius a l 0
100 km. As alluded to earlier in Chap. 7 , the GMP will propagate into the ionosphere
in accordance with the diffusion law. For the time t the diffusion front climbs to the

altitude z d 2 ǚ t= 0 p 1=2 and thus the front will reach the upper boundary of
the ionospheric E-layer at the moment t d 0 p l 2 =4. In what follows we restrict
our study to the short interval 0<t<t d which corresponds to the initial stage of
signal. During this interval the solution of Eqs. ( 11.40 )-( 11.42 ) weakly depends on
the boundary conditions of the problem at z D l. Hence, we replace this condition
by the requirement that the solution must be finite when z !1 .
Considering the GMP diffusion in a horizontal direction, we note that for the
time t<t d the diffusion front propagates at the distance much smaller than a. So,
we will neglect the lateral expansion of the diffusion region in the ionosphere and
focus on the vertical propagation of the GMP. This implies that in the region r<a
the terms @ rr B z , r 1 @ r B z , and B z =r 2 in Eq. ( 11.45 ) are much smaller than @ zz B z .
As a first approximation, we assume that B z is not a function of r. Notice that the
components B r and B ' are equal to zero at r D 0 and their dependence on r should
already be taken into account in the first approximation.
Laplace transformation with respect to time can be applied to all the equations
with boundary conditions. Let b z , b r , b ' , and j be Laplace transforms of the
magnetic perturbations and extrinsic current density, respectively. Taking the notice
of the above approximations, one can reduce Eq. ( 11.45 ) to the following:

d z 2 C p 2 0 P C H b z D 0;

d 4 b z

d z 4 2p 0 P d 2 b z

(11.47)

where p denotes the parameter of Laplace transformation. When the finiteness of b z
at z !1 is taken into account, the solution of ( 11.47 ) is given by

1=2 ; (11.48)

b z i D C 1 exp . C z / C C 2 exp . z / I ˙ Df 0 p. P ˙ i H / g

where C 1 and C 2 are the arbitrary constants, and Re ˙ >0. Other components of
the magnetic perturbations in the ionosphere can be expressed through b z i via

Z

r

1
r

db z i

r 0

d z dr 0 ;

b ri D (11.49)

0

Z

r 0

dr 0 ;

r

d 3 b z i
d z 3

1
r

1
p 0 H

P
H

db z i
d z

b 'i D

(11.50)

0