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coordinates we have
b
ν
=
b
ν
/h
ν
,
b
ϕ
=
b
ϕ
/h
ϕ
,
b
µ
=
b
µ
/h
µ
,
E
ν
=
E
ν
/h
ν
,
E
ϕ
=
E
ϕ
/h
ϕ
,
E
µ
=
E
µ
/h
µ
.
Then (6.59) can be rewritten
b
ϕ
, E
ν
, b
ν
, b
µ
, E
ϕ
tr
α
0
=
+
α
1
+
···
ν
−
ν
r
+ log
im
h
ν
ν
r
)
β
0
+
β
1
(
ν
.
h
ϕ
(
ν
−
−
ν
r
)+
···
(6.88)
Equations (6.70)-(6.71) for the vector-functions
α
0
and
β
0
in the dipole co-
ordinates become
α
0
(
w
)=
h
−
1
ϕ
e
1
(
w
)
,h
−
ν
e
2
(
w
)
,
0
,
0
,
0
tr
,
β
0
(
w
)=
C
0
e
1
(
w
)
,C
0
e
2
(
w
)
,
e
2
(
w
)
tr
im
h
ν
h
ϕ
e
1
(
w
)
,
0
,
im
h
ϕ
−
.
Substitution of Lame coecients of the dipole coordinates (6.35) gives
w
2
)
3
/
2
e
1
(
w
)
,ν
r
(1 + 3
w
2
)
1
/
2
e
2
(
w
)
,
0
,
0
,
0
tr
ν
r
α
0
(
w
)=
−
,
(6.89)
(1
−
0
,
0
,
(1 + 3
w
2
)
1
/
2
,
0
,ν
r
e
2
(
w
)
tr
im
e
1
(
w
)
β
0
(
w
)=
C
0
α
0
(
w
)+
−
.
(1
−
w
2
)
3
/
2
(6.90)
The boundary problem (6.79)-(6.80) determines
e
1
(
w
), and
e
2
(
w
) accurate
to an arbitrary constant factor. It can be found from the distribution of the
longitudinal magnetic field
b
along a resonance field-line. We obtain from the
conditions of solvability for (6.63) and (6.64):
d
ω
r
d
ν
−
1
w
0
w
0
ν
r
c
2
Ω
2
mc
2
ν
r
w
)
6
−p
e
2
d
w
=
(1 + 3
w
2
)
b
e
2
d
w.
(6.91)
−
−
−
(1
ν
=
ν
r
−w
0
−w
0
Here the small terms relate to the Pedersen and Hall conductivities are ne-
glected. Equation (6.91) defines the transformation of the FMS-mode into
FLR-oscillations. If the longitudinal magnetic field
b
at the resonance field
line is known, then the amplitude of the resonance Alfven oscillations can be
found from (6.91).
The Dipole Dispersion Equation Special Case
For the power distribution of the plasma density defined by (6.76) and dipole
magnetic field, the boundary problem (6.81) can be solved explicitly. In the
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