Geoscience Reference
In-Depth Information
5.8 Uncoupled Alfven and FMS-Modes
Alfven Modes
At
k
y
= 0 from (5.62b) for the Alfven waves we obtain
ξ
yωn
=
L
−
1
ωn
f
yωn
.
Applying to it the inverse Laplace transform, we have
+
∞
+
iσ
0
1
2
π
iω ξ
yn
0
(
x
)
−
ξ
yn
1
(
x
)
ξ
yn
(
x, t
)=
exp (
−
iωt
)d
ω.
(5.67)
k
n
(
ω
)
c
2
A
(
x
)
ω
2
−
−∞
+
iσ
0
Here the integral external currents are omitted and only initial plasma pertur-
bations are left. For
t>
0 the integral is along the path
Γ
(
+
iσ
0
)
(5.67). If we close the integration path by semicircle
Γ
in the low half-plane
and consider the integral over the contour
Γ
+
Γ
, then it can be solved using
the residuum calculus. Tend the semicircle radius to
−∞
+
iσ
0
,
∞
∞
, then the contribution
on the semicircle
Γ
vanishes exponentially. The integral is equal to the sum
of residues of the integrand in poles which are determined by zeros of the
denominator of integrand expression (5.67):
ω
2
k
n
(
ω
)
c
2
A
(
x
)=0
.
−
(5.68)
A solution of (5.68) at some
x
has two sets of roots
ω
(
±
)
(
x
)
.
Then (5.67)
n
reduces to
ξ
yn
(
x, t
)=
ξ
yn
(
x, t
)+
ξ
yn
(
x, t
)
,
iω
(
±
)
ξ
yn
1
(
x
)
c
A
(
x
)exp
(
x
)
t
iω
(
±
)
ξ
yn
0
(
x
)
−
−
n
n
ξ
(
±
)
(
x
)
1
yn
(
x, t
)=
−
i
,
(5.69)
ω
=
ω
(
±
)
n
d
k
n
(
ω
)
d
ω
2
ω
(
±
)
−
c
A
(
x
)
n
(
x
)
where
ω
(
±
n
are found in (5.20), (5.25) and
k
n
(
ω
) is obtained from (5.13). For
instance, at
Σ
P
→∞
from (5.15) we find that
ω
=
ω
(
±
)
n
ω
A
ω
2
d
k
n
(
ω
)
d
ω
2
n
π Σ
P
1
−
c
A
(
x
)
(
x
)
≈
1
−
i
.
Then Alfven oscillations in the box may be considered as a superposition
of non-interacting damped oscillations. Each elementary oscillation can be
presented as
ξ
yn
(
x, z, t
)=
Q
n
(
z
)
u
n
(
x, t
)
,
with
u
n
(
x, t
)=
ξ
yn
(
x, t
)+
ξ
yn
(
x, t
)
.
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