Geoscience Reference
In-Depth Information
It is appropriate to our problem to use the periodicity condition on coordinate
y
corresponding to geomagnetic longitude
ξ
⊥
(
x, y, z
)
|
y
=0
=
ξ
⊥
(
x, y, z
)
|
y
=
l
.
(5.6)
We shall discuss the applicability of the boundary conditions (5.3), (5.4), and
(5.5) in Chapter 7.
Box Equations
As usual, using the methods of Fourier analysis, we can decompose the signal
into its sinusoidal components in time and space and study each component
separately. Let us consider perturbations
∝
exp(
ik
y
y
) with
k
y
=2
πm/l
y
,
m
=0
,
2
....
Equation (4.42a), with the field sources and displacements
(
ξ
x
,ξ
y
) reduces to
±
1
,
±
j
(
d
)
b
B
0
−
4
π
c
x
B
0
,
L
A
ξ
y
=
ik
y
(5.7a)
j
(
d
)
b
B
0
∂
∂x
4
π
c
y
B
0
,
=
L
A
ξ
x
−
(5.7b)
b
B
0
−
∂
∂x
ξ
x
=
−
ik
y
ξ
y
,
(5.7c)
where
j
(
d
)
are driving currents. Recall that the operator
L
A
is
∂
2
∂z
2
−
∂
2
∂t
2
1
c
2
A
(
x
)
L
A
=
(5.8)
and for monochromatic oscillations exp(
−
iωt
)
∂
2
∂z
2
+
k
A
(
x
)
,
k
A
(
x
)=
ω
2
c
2
A
(
x
)
.
A perturbation propagating from a harmonic source in the hydromagnetic
box can be presented as a superposition of normal oscillations. Consider first
of all the problem of Alfven oscillations of a field-lines with the periodical
conditions (5.6).
L
A
=
5.2 Dungey's Problem
In order to study the standing free oscillations, we assume that exciting sources
are absent, that is
j
(
d
)
= 0 in (5.7a) and
k
y
=0
.
Denote by
Q
(
x
) the amplitude
distribution of displacement in the standing Alfven oscillations. At a fixed
frequency
ω
, this function is determined from the boundary problem on the
eigenvalues of the equation
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