Geoscience Reference
In-Depth Information
1+
k
y
x
1
)]
.
b
B
0
≈
x
1
)
2
2
b
0
B
0
(
x
−
ln [
k
y
(
x
−
(4.61)
Then plasma displacements are given by
k
y
k
A
(
x
1
)
b
0
B
0
b
0
B
0
ik
y
k
A
(
x
1
)
1
ξ
x
≈
ln [
k
y
(
x
−
x
1
)]
,
y
≈
x
1
.
(4.62)
x
−
Contrary to the longitudinal magnetic component which is finite when
approaching the resonance field-line, transversal displacements, electric and
magnetic components tend to infinity.
Let us include a small dissipation in order to determine the rules for by-
passing the singular point and for estimating energy losses at it. Consider, for
instance, plasma with small but finite transverse conductivity
σ
⊥
. The Alfven
wavenumber in a lossless plasma is
k
A
(
x
)=
ω
2
c
2
A
(
x
)
.
Take into account transverse conductivity replacing
k
A
(
x
)by
k
A
+
iκ
2
,
with
κ
2
=
ω
4
πσ
⊥
c
2
.
FLR-condition (4.56) can be rewritten
k
A
(
x
)+
iκ
2
=
k
2
.
(4.63)
Condition (4.63) cannot be satisfied with real
ω
,
k
and
x
.If
ω
and
k
are
real variables, then (4.63) has no solution under real
x
and it is necessary
to extend the function
k
A
(
x
) from the real axis
x
onto the complex plane
w
=
x
+
iv.
Thus, there should exist a complex function
k
A
(
w
) such that
k
A
(
w
k
A
(
x
)
.
Substitute
w
1
=
x
1
+
iδ
into (4.63), assuming
κ
2
and
δ
small. Expanding
k
A
(
x
1
+
iδ
) in a power series of
δ
and leaving only terms of the first order, we
obtain
→
x
)
→
x
=
x
1
dk
A
(
x
)
dx
k
A
(
x
1
+
iδ
)
k
A
(
x
1
)+
i
≈
δ,
(4.64)
and from (4.63) find
κ
2
dk
A
(
x
)
dx
κ
2
c
4
A
(
x
1
)
ω
2
dc
2
A
(
x
)
dx
δ
≈−
x
=
x
1
=
x
=
x
1
.
(4.65)
Replacing
x
1
by
x
1
+
iδ
in (4.60a) - (4.60c) and (4.61) - (4.62), we obtain
expressions for plasma displacements and magnetic field near the resonance
Search WWH ::
Custom Search