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Equation (6.57) should be supplemented with the appropriate boundary
conditions. A principal diculty stems from the fact that the Alfven and
FMS-waves are coupled because of the field-lines curvature and plasma inho-
mogeneity. As this takes place, these modes respond differently to the non-
conductive atmosphere and highly conductive Earth. The boundary condition
for the FMS is rather complicated. However, as it will be shown below, the
only condition of the current non-penetration into the atmosphere is sucient
for the determination of the field in the FLR-region.
The non-penetration condition for the electric current
c
4
π
∇×
b
+
iω
j
=
c
E
is written as
j
3
x
3
=
±d/
2
=
j
e
3
|
x
3
=
±d/
2
=0
.
Neglecting the displacement current, we rewrite it as
∂b
2
·
∂x
1
+
imb
1
=0 at
x
3
=
±
d/
2
.
(6.58)
The second boundary condition is non-local but it is not necessary to know
its exact form to get the resonance structure of the field.
Let us describe the solution of (6.57) with the boundary conditions (6.58)
in a rectangle
Π
(see Fig. 6.2(b)), corresponding to the region between two
magnetic shells
L
1
and
L
2
and the ionospheres. It is appropriate to our prob-
lem to seek solutions, by analogy with the straight field problem, in the form
U
(
x
1
,x
3
)=
α
(
x
1
,x
3
)
x
1
+
β
(
x
1
,x
3
) log(
x
1
−
z
0
)
,
(6.59)
−
z
0
where
α
(
x
1
,x
3
)
,
β
(
x
1
,x
3
) are analytical vector-functions of
x
1
:
α
(
x
1
,x
3
)=
∞
α
j
(
x
3
)(
x
1
z
0
)
j
,
−
(6.60)
j
=0
β
(
x
1
,x
3
)=
∞
β
j
(
x
3
)(
x
1
z
0
)
j
,
−
(6.61)
j
=0
and
z
0
=
x
0
+
iδ
is a free parameter.
By substituting (6.59) into (6.57) we obtain the chain recurrent boundary
problems
A
α
0
=0
,
(6.62)
j
(
j
+1)
A
β
j
+1
+
C
k
β
j−k
=0
,
(6.63)
k
=0
j
j
A
α
j
+1
+
C
k
α
j−k
+
A
β
j
=0
,
(6.64)
k
=0
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