Geoscience Reference
In-Depth Information
Fig. 5.3.
The decrement
γ
1
/ω
A
and the Q-factor
Q
1
=
ω
A
/
2
γ
1
of the fundamental
harmonic depending on the normalized integral conductivity
5.5 FLR-Equations
Let us discuss now the propagation of a hydromagnetic wave excited by ex-
ternal sources. The displacement
ξ
⊥
and longitudinal magnetic field
b
are
completely determined by (5.7a), (5.7b), (5.7c) and boundary conditions (5.3),
(5.6). We assume that field sources in (5.7a)-(5.7c) are near the magnetopause,
while the inner regions of the magnetosphere are free of external currents.
The simplest way to deduce the solution is by expanding it into a series of the
eigenfunctions
Q
n
(
x
). Substituting the series
ξ
y
=
m
x
=
m
a
m
(
x
)
Q
m
(
z
)
,
c
m
(
x
)
Q
m
(
z
)
,
=
m
b
B
0
b
m
(
x
)
Q
m
(
z
)
,
into (5.7a)-(5.7c) and noting that
L
A
Q
m
(
z
)=
k
A
(
x
)
k
2
m
Q
m
(
z
)
,
−
we obtain
k
2
m
a
m
(
x
)
Q
m
(
z
)=
ik
y
m
j
(
d
)
k
A
(
x
)
4
π
c
x
B
0
,
−
b
m
(
x
)
Q
m
(
z
)
−
m
b
m
(
x
)
Q
m
(
z
)=
m
j
(
d
)
k
A
(
x
)
k
2
m
c
m
(
x
)
Q
m
(
z
)
−
4
π
c
y
B
0
,
−
m
m
{
c
m
(
x
)
Q
m
(
z
)=
−
ik
y
a
m
(
x
)+
b
m
(
x
)
}
Q
m
(
z
)
,
m
where the prime denotes differentiation with respect to
x
.
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