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Hence, as it follows from (6.26), when
ω
s
(
x
0
)
>
0
,
the parameter
δ
s
>
0
.
Then the integration path goes below a resonance point and we have
ln(
x
−
x
0
)
|
x
=
x
0
+0
=ln(
x
−
x
0
)
|
x
=
x
0
−
0
+
iπ
for weak losses. In the opposite case, when
ω
s
(
x
0
)
<
0
,
and
δ
s
<
0
,
the integration path goes above the resonance point
x
0
,and
ln(
x
−
x
0
)
|
x
=
x
0
+0
=ln(
x
−
x
0
)
|
x
=
x
0
−
0
−
iπ.
Above, we assumed that for each
z
, there exists a complex function
c
A
(
w
=
x
+
iv, z
) regular in O
µ
such that
c
A
(
w
c
A
(
x, z
). The developed
theory is, therefore, not valid when function
c
A
(
x, z
) contains a discontinuity
or a steep slope near
x
0
. One of example of such problems is a problem of
surface waves on discontinuities of Alfven velocity which requires a particular
consideration (see, e.g., [61]).
→
x, z
)
→
6.3 MHD-Waves in a Curvilinear Magnetic Field
In the previous section, Alfven wave resonators have been described using
models with straight magnetic field-lines. In the present section, we will show
that in the curved magnetic field, not only the basic features of FLR are
conserved but also some new features associated with the field curvature occur.
First, we shall write the ideal MHD-equations in the curvilinear coordi-
nates connected with the magnetic field and obtain the FLR-equations from
simple qualitative considerations. Then the scheme of calculations described in
detail in Section 6.2, will be generalized to the plasma with Hall conductivity
in a curve magnetic field.
Coordinate System Potentials
General Case
In the cold magnetized plasma, a convenient description of MHD-oscillations
can be obtained in a curvilinear system of coordinates
x
1
,x
2
,x
3
satisfying two
conditions:
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