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equality from the other, we have
d
2
V
nl
d
x
2
d
x
+
l
x
l
x
V
nl
d
2
ϕ
(1)
ωn
d
x
2
ϕ
(1)
ωn
K
(
x, ω
))
V
nl
ϕ
(1)
−
(
K
(
x, ω
nl
)
−
ωn
d
x
=0
.
0
0
Integration by parts with boundary conditions (5.74), (5.75) yields
l
x
ϕ
(1)
K
(
x, ω
))
V
nl
ϕ
(1)
ωn
(
l
x
)
−
V
nl
(
l
x
)=
−
(
K
(
x, ω
nl
)
−
ωn
d
x.
0
Dividing both parts of the relation over
ω
−
ω
nl
and tending
ω
→
ω
nl
,we
have, for the derivative of the Wronskian,
ω
nl
=2
ω
nl
N
(1)
ω
nl
ω
nl
ϕ
(1)
d
W
ωn
d
ω
ωn
(
l
x
)
∂ω
d
k
n
(
ω
)
d
ω
N
(2)
nl
=
nl
−
k
n
(
ω
nl
)
and for displacements
∞
i
2
V
nl
(
x
)
V
nl
(
x
)exp(
−
iω
nl
t
)
ω
nl
ξ
xn
(
x, t
)=
−
.
d
k
n
(
ω
)
d
ω
ω
nl
N
(1)
N
(2)
nl
−
k
n
(
ω
nl
)
l
=1
nl
The relationship of the longitudinal magnetic field and plasma displacement
is given by
b
ω
B
0
d
d
x
ξ
xω
.
For the Alfven FLR-periods
T
A
(
x
) equal to those shown in Figure 5.9,
the cavity resonance periods of the first two harmonics are
T
1
=64
.
38 s,
T
2
=31
.
16 s. The amplitude distribution of these cavity resonance oscillations
is shown in Fig. 5.6. Their decrements
γ
1
≈
−
=
0
.
41
×
10
−
3
s
−
1
and
γ
2
≈
0
.
18
×
10
−
3
s
−
1
and the Q-factor are 120 and 575, correspondingly.
Such Q-factors are at least 1.5-2 orders higher than the observed
Q
-factors
of the resonance magnetospheric cavity modes. Within the Box model the
dissipation is mostly caused by the Joule dissipations at the ends of field-lines
(this corresponds to the Joule dissipation in the ionospheres due to Pedersen
conductivity). This model is adequate to estimate the Q-factor of FLR, with
the disturbances localized in the vicinity of resonance field-lines.
As for the cavity modes, these oscillations cover the whole magnetosphere
or a significant part. Azimuthal propagation leads to the dissipation in the
magnetotail. The energy also lost itself at the magnetopause, as it only partly
reflects the waves and they transmit through the magnetopause into the mag-
netosheath and then into the interplanetary space.
Thus, FMS-oscillations non-interacting with Alfven waves are global and
fill the whole magnetosphere. In a non-dissipative medium they appear as a
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