Geoscience Reference
In-Depth Information
Consider the disturbances of
E
and
b
exp[
iΦ
(
x
1
,x
2
)], where
Φ
(
x
1
,x
2
)is
the wave phase varying quickly in the transverse direction. Let also the cross-
components
k
1
and
k
2
of the wave vector
k
:
∝
∂Φ
∂x
1
,
∂Φ
∂x
2
,
k
1
=
2
=
such that
∂/∂x
1
ik
2
. Assume, as is typical in the magne-
tosphere, that the Alfven velocity varies more rapidly along, for example,
the coordinate
x
1
≈
ik
1
,∂/∂x
2
≈
as compared to
x
2
. In this case, one can extract two wave
classes.
The first class corresponds to the case in which the wave field varies along
x
1
much more rapidly than along
x
2
. Since the element lengths in the direction
x
1
,x
2
are
dx
1
/h
1
,
dx
2
/h
2
, respectively, then the wavelength along
x
1
is shorter
than along
x
2
that results in
.
In the opposite case, in which the wavelength is shorter along
x
2
,wehave
|
h
1
k
1
||
h
2
k
2
|
|
.
In these important limiting cases, a set of equations describing Alfven
modes with
b
3
→
h
2
k
2
||
h
1
k
1
|
0 can be separated out from the complete set (6.40)-(6.44)
and from compressional FMS oscillations, in which the field-aligned magnetic
component
b
3
is finite. Assuming that
k
1
→∞
, we find from (6.42) and (6.44)
that the components
E
2
→
0,
b
3
→
0 (while
k
1
E
2
,
k
1
b
3
should not necessarily
tend to zero).
Thus, by virtue of (6.43), we also have
b
1
→
0. As a result, (6.40) and
(6.41) yield a closed set of equations for the components
E
1
and
b
2
:
1
h
3
∂E
1
∂x
3
−
ik
0
h
1
h
2
b
2
=0
,
(6.45)
1
h
3
∂b
2
∂x
3
−
h
2
h
1
E
1
=0
,
ik
0
ε
⊥
(6.46)
which, after eliminating
b
2
, reduces to one second-order equation
h
1
h
2
h
3
∂
∂x
3
h
2
h
1
h
3
∂E
1
∂x
3
+
ω
2
c
2
ε
⊥
E
1
=0
.
(6.47)
Alfven oscillations described by (6.45), (6.46) are polarized so that the
E
-field
is directed along the coordinate
x
1
, while the
b
-field perturbations and plasma
displacements are along the coordinate
x
2
. These oscillations are primarily
excited by large-scale sources (with small
k
2
). This mode corresponds to the
toroidal mode, according to geophysical terminology.
For the small-scale perturbations along the coordinate
x
2
(
k
2
→∞
), com-
ponents
E
1
and
b
3
vanish in the system (6.40)-(6.44) while
k
2
E
1
,
k
2
b
3
should
not necessarily tend to zero. In that case, by virtue of the second equation of
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