Geoscience Reference
In-Depth Information
integration of Eqs. (
11.36
) and (
11.37
) over
z
from zero to " and then assuming
"
!
0 we come to the following boundary conditions at
z
D
0
B
'a
B
'i
D
0
bJ
H
; B
ra
B
ri
D
0
bJ
P
; B
z
a
D
B
z
i
;
(11.39)
where the subscripts a and i are related to the atmosphere and ionosphere,
respectively. The extrinsic current densities, J
H
D
H
V
r
B
0
and J
P
D
P
V
r
B
0
,
are assumed to be given functions.
Equations (
11.36
) and (
11.37
) can be solved for E
r
and E
'
. Substituting these
components of electric field into Maxwell equation
r
E
D
@
t
B
and rearranging
yields
r
@
r
r@
z
B
'
;
D
H
2
B
z
@
t
B
z
D
D
P
r
(11.40)
@
t
B
r
D
D
P
B
r
r
2
2
B
r
C
D
H
@
zz
B
'
;
r
(11.41)
2
B
'
D
H
;
B
r
r
2
2
B
r
@
t
B
'
D
D
P
r
r
(11.42)
where
P
0
P
C
H
H
0
P
C
H
D
P
D
; D
H
D
;
(11.43)
2
is
are the coefficients of diffusion in a gyrotropic medium and the operator
r
given by
2
D
r
1
@
r
C
@
rr
C
@
zz
:
r
(11.44)
The set of Eqs. (
11.40
)-(
11.42
) can be solved with respect to each component of the
magnetic perturbation. For example, excluding B
'
from Eq. (
11.40
) gives (Surkov
1996
)
@
t
@
t
B
z
D
P
r
2
B
z
D
@
zz
ǚ
D
p
@
t
B
z
D
P
C
D
H
r
2
B
z
B
z
=r
2
: (11.45)
Applying Eq. (
11.36
) to the atmosphere and taking into account that the compo-
nents of the conductivity tensor are equal to zero, we obtain that B
'
D
0 everywhere
in the atmosphere. Other components of the GMP can be determined from the
Laplace equation:
2
B
r
D
B
r
=r
2
;
r
2
B
z
D
0:
r
(11.46)
Suppose that the acoustic wave reaches the lower boundary of the ionosphere at
the moment t
D
0. The region of interaction between the wave and ionosphere
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