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whereas (6.50), after the substitution d
η
=
h
2
h
3
h
−
1
dx
3
,
becomes
∂
2
2
E
2
=0
.
∂η
2
+
ω
(6.56)
c
A
The modified Alfven velocities
c
A
('toroidal') and
c
A
('poloidal') in (6.55) and
(6.56)) are
c
A
=
h
1
c
A
=
h
2
h
1
c
A
.
Thus, the propagation of both modes of Alfven waves in a curvilinear magnetic
field can be described by (6.55) and (6.56), similar to the equation in a straight
field, but with
c
A
replaced with the modified Alfven velocities
c
A
and
c
A
.
A simplified derivation of equations for Alfven waves in the present chapter
does not give information about the transversal distribution of the amplitude
of Alfven waves. To obtain them on approximation over the small transversal
scale is required. This will be done in the next section with the aid of a
generalization of the method used in order to study FLR in a straight magnetic
field (see Section 6.2).
h
2
c
A
,
FLR Rigorous Solution
In this section the problem of excitation of forced low frequency oscillations
of the magnetosphere-ionosphere resonator is considered within the axially
symmetric model with the plasma parameters independent on the azimuth.
Let
x
2
=
ϕ
. Then
g
12
=
g
21
=0,
g
12
=
g
21
=0,and
g
23
=
g
32
= 0. We shall
restrict ourselves to the analysis of the oscillations
exp(
imϕ
) with a fixed
azimuthal wavenumber
m
. Then from (6.39) we obtain the 2D equations:
A
∝
∂x
1
+
C
(
x
1
,x
3
)
U
=0
,
∂
(6.57)
where
A
=
A
1
from (6.39),
C
(
x
1
,x
3
) at the fixed
x
1
is a differential operator
with respect to the coordinate
x
3
:
⎛
⎞
∂
∂x
3
ag
22
−
0
0
0
⎝
⎠
∂
∂x
3
aε
11
aε
12
−
0
im
C
(
x
1
,x
3
)=
∂
∂x
3
ag
11
ag
13
0
0
ag
31
ag
33
0
im
0
∂
∂x
3
aε
21
aε
22
0
0
with
a
=
ik
0
√
g
. Equation (6.57) describes monochromatic small oscillations of
an axially symmetrical magnetosphere-ionosphere resonator with an arbitrary
angle between the field-lines and ionosphere.
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