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potential ‰.k;
z
;!/. It is interesting to note also that these equations completely
coincide with Eqs. (
5.112
) and (
5.113
) for the atmosphere and ground derived for
the “plane symmetry” problem. The only difference is that the “wave number” k
?
in Eqs. (
5.112
) and (
5.113
) should be replaced by the factor k. This means that
the solution of the axially symmetrical problem for ‰.k;
z
;!/ is the same as that
given by Eqs. (
5.29
)-(
5.31
). This analogy can be extended to the ionosphere and
magnetosphere equations.
Before discussing the wave equation for the plasma, it is useful to give some
concerns about the axially symmetrical problem. In our model the Maxwell
equations (
4.2
) and (
5.2
) for the magnetosphere are split into two independent set of
equations in analogy to the plane symmetry problem. Depending on the components
E
r
, E
z
and ıB
'
, the first set of equations describes the shear Alfvén wave and thus is
related to the TM mode in the atmosphere. The second equation set, which contains
the components ıB
r
, ıB
z
and E
'
, corresponds to the FMS wave and is related to the
TE mode in the atmosphere.
As before we assume that the parallel plasma conductivity is infinite so that
the electromagnetic field can be presented via two scalar potential, ‰ and dž.
In greater details this problem is examined in Appendix E where the Bessel
transform (
5.134
) is applied to the Maxwell equations to yield the equations for
scalar potentials. These equations can be reduced to the form that completely
coincide with Eqs. (
5.12
) and (
5.13
) for the shear Alfvén and FMS waves in the
plane problem. In other words, after applying the Fourier and Bessel transforms to
the Maxwell equations, both the problems, plane and axially symmetrical, lead to
the same wave equations for the potentials dž and ‰, and thus both the problems can
be studied jointly. This conclusion holds true for all the problems we are going to
study in this section.
Some complication appears in the E layer studies due to the importance of mode
coupling to the shear Alfvén and FMS modes. For simplicity, the neutral wind
velocity is assumed to be independent of azimuthal angle ' in the ionosphere, so
that we have an axially symmetrical problem. In the thin E layer approximation,
which is valid as l
l
s
, where l is the thickness of the E layer and l
s
is the skin-
depth, we obtain the boundary conditions (
5.167
) and (
5.168
)at
z
D
0 for the jump
of the potentials and their derivatives across E layer.
The solution of the problem for the neutral atmosphere and the ground is treated
in any detail in Appendix E. The potential functions in the ionosphere and the
atmosphere is matched by virtue boundary conditions (
5.167
)-(
5.168
) to yield the
general solution of the problem. To an accuracy of the factor .V
AI
=c/
2
10
6
1,
the electromagnetic field at the ground is given by Eqs. (
5.181
)-(
5.182
). Taking the
inverse Bessel transform of the functions b
r
and b
'
, these equations are reduced to
the following:
iLB
0
V
AI
w
r
.!;r/;
ıB
r
.!;r;
d/
D
Mg
r
.!;r/
C
(5.54)
ıB
'
.!;r;
d/
D
Mg
'
.!;r/:
(5.55)
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