Geoscience Reference
In-Depth Information
The oscillation caused by an individual field line
Q
n
(
z
) can be studied as an
eigenvalue of the Dungey's problem (5.9)-(5.10).
For a non-dissipative 1D system
u
n
(
x, t
) is found from the equation of the
oscillations of an ideal harmonic oscillator
d
2
d
t
2
u
(
x, t
)+
ω
An
(
x
)
u
(
x, t
)=
j
xω
(
x, t
)
B
0
4
π
c
−
.
where
u
is the initial displacement and d
u/
d
t
is plasma velocity at
t
=0.
Standing Alfven oscillations of a field line with account taken of the losses
at the confined boundaries, are similar to oscillations of a stretched string with
energy absorption at the clamped end points. The analogy between field line
Alfven oscillations and oscillations of a string is not only formal in character,
but it has a clear physical sense. The restoring force in Alfven oscillations
arises due to the tension of magnetic field-lines (see (4.14)). With dissipation
taken into account, the evolution of
n
-th harmonic of the field line oscillation
can be found from the equation of an oscillator with losses
d
2
j
xω
(
x, t
)
B
0
d
t
2
u
(
x, t
)+2
iγ
n
d
4
π
c
d
t
u
(
x, t
)+
ω
An
(
x
)=
−
,
where the resonance frequency
ω
An
and the decrement are determined by
(5.20)-(5.21).
Cavity Modes
In a cold homogeneous plasma, the absolute value of phase velocity of FMS-
waves is independent of the angle between the main magnetic field (see, e.g.,
(4.41b)). It follows that FMS-waves are not guided by the field-lines and can
fill the whole resonance region. Therefore, the normal modes of FMS-wave
resonator are often called cavity modes.
Differentiating (5.59) and substituting (5.60) for displacements
ξ
xωn
,we
obtain the equation for FMS-waves in the box model:
+
ω
2
k
n
(
ω
)
ξ
xωn
=
d
2
ξ
xωn
d
x
2
c
2
A
(
x
)
−
−
f
ξωn
,
(5.70)
ξ
xωn
|
x
=0;
l
x
=0
.
(5.71)
The corresponding uniform boundary problem
+
ω
2
k
n
(
ω
)
ξ
xωn
=0
,
d
2
ξ
xωn
d
x
2
c
2
A
(
x
)
−
(5.72)
ξ
xωn
|
x
=0;
l
x
=0
.
(5.73)
is the generalized Sturm-Liouville problem for determining the resonance fre-
quencies of the cavity modes.
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