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The second one is for the three components of the FMS wave, i.e., E
'
, ıB
r
, and ıB
z
i!
V
A
@
r
ıB
z
@
z
ıB
r
D
E
'
;
(5.158)
@
z
E
'
D
i!ıB
r
;
(5.159)
@
r
E
'
D
i!ıB
z
:
(5.160)
Since "
k
has been assumed to be infinite, the parallel electric field in the
magnetosphere becomes infinitesimal, i.e., E
z
!
0. As before, the electromagnetic
field is derivable by the scalar potentials A, dž, and ‰ through Eqs. (
5.130
), (
5.131
),
and (
5.80
). Substituting these equations into the set of Eqs. (
5.155
)-(
5.160
), and
rearranging under the requirement that i!A
D
@
z
dž, yields
!
2
V
A
@
z
dž
C
dž
D
0;
(5.161)
!
2
V
A
1
r
@
r
.r@
r
‰/
D
@
zz
‰
C
‰:
(5.162)
Applying Bessel transforms to Eqs. (
5.161
) and (
5.162
), we come to the equations
that are completely similar to Eqs. (
5.12
) and (
5.13
) for the shear Alfvén and
compressional waves in the plane problem.
E
Layer of the Ionosphere
In order to obtain the boundary conditions at the bottom of the ionosphere we
now consider the conductive E layer of the ionosphere. In the framework of the
axially symmetrical problem the neutral wind velocity is assumed to be independent
of azimuthal angle ' in the ionosphere. Substituting Eq. (
5.24
) for the current
density into Ampere's equation (
1.5
), taking the notice of
B
D
ı
B
C
B
0
, and using
cylindrical coordinates, we obtain
0
@
z
ıB
'
D
H
E
'
V
r
B
0
P
E
r
C
V
'
B
0
;
(5.163)
0
.@
z
ıB
r
@
r
ıB
z
/
D
P
E
'
V
r
B
0
C
H
E
r
C
V
'
B
0
;
(5.164)
where
P
and
H
are the Pedersen and Hall conductivities, and V
r
and V
'
are the
components of the neutral flow velocity. In what follows we use a thin E layer
approximation, which is valid if a typical thickness of the E layer, l, is much smaller
than the skin-depth in the ionosphere l
s
.
0
P
!/
1=2
. Integrating Eqs. (
5.163
)
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