Geoscience Reference
In-Depth Information
In the same approximation the equation for the atmosphere .
l
0
<
z
<0/ reads:
d
2
b
z
=d
z
2
D
0. If the ground is considered as a perfect conductor, the solution of
this equation is given by
b
z
a
D
C
0
.
z
C
l
0
/:
(11.51)
The arbitrary constants C
0
, C
1
, and C
2
can be found from Eqs. (
11.48
)-(
11.51
) and
the boundary conditions given by Eq. (
11.39
).
Now consider the low frequency perturbations when the corresponding skin-
depth in the ground is greater than the depth l
1-2 km of upper layer of
sedimentary rocks which possess a high conductivity. The lower boundary of the
sedimentary rock layer is shown in Fig.
11.15
with wavy line. In this extreme case
we can consider this layer as if it were transparent for the GMP. To simplify the
problem, we assume that formally l
0
goes to infinity. Then the inverse Laplace
transformation of the solution can be reduced to the simple quadratures. For
example, in the ionosphere .
z
0/ the result can be written as follows (Surkov
1996
)
B
z
i
.
z
;r;t/
D
b
0
2
H
1=2
Z
t
G
z
z
;t
0
J
H
r;t
t
0
dt
0
; (11.52)
r
1
@
r
r
0
Z
t
0
H
2
1=2
G
'
z
;t
0
J
H
r;t
t
0
dt
0
;
0
z
b
2
B
'i
.
z
;r;t/
D
(11.53)
0
where the functions G
z
and G
'
are given by
t
1=2
C
cos
Ǜ
z
2
exp
;
t
C
sin
Ǜ
z
2
LJ
z
2
t
1
G
z
D
(11.54)
t
C
sin
Ǜ
z
2
exp
:
t
cos
Ǜ
z
2
LJ
z
2
t
1
t
3=2
G
'
D
(11.55)
t
Here the following abbreviations are introduced
n
1
C
m
2
1=2
˙
m
o
1=2
˙
D
m
˙
;
˙
D
;
0
H
4
0
P
4
P
H
:
Ǜ
D
; LJ
D
; m
D
(11.56)
In order to analyze the features of this solution, we choose the pulsed source as a
simple model of extrinsic current, that is J
H
.r;t/
D
H
B
0
V
r.a
r/Tı.t/=a,
where ı.t/ denotes ı-function, .a
r/ is the step-function, V
is the amplitude
of mass velocity at the lower boundary of the ionosphere, and T is the typical time
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