Geoscience Reference
In-Depth Information
The wave can travel also within the magnetosphere along the upper
ionospheric boundary and be re-emitted to the ground observer from above
far from the beam axis. And at last, the wave passing through the ionosphere
and finite conductive atmosphere results in the production of the atmospheric
TEM-mode.
All these questions will be treated in the present chapter. We will develop
a 2D theory of the horizontal propagation of the MHD-wave beams. Much
attention is given to the Alfven beams. Examples of spatial distributions of
various field components are given with particular applications to synchronous
bound Alfven wave beams in which different points of the beam in the plane
transversal to the beam axis oscillate in-phase and non-synchronous beams
like a resonance Alfven shell.
8.2 Coordinate Dependencies
Let a beam of the Alfven waves generated in the magnetospheric equatorial
region propagate along the field-lines. The transversal dispersion of the beam
is a comparatively slow process and only for small spatial scales does the
transversal dispersion and spreading of a beam become significant (decrease
of the spatial scale is due to the phase mixing, see Chapter 5).
The spatial distribution of an electromagnetic field can be found by the
inverse Fourier transform of the electric and magnetic fields in the magne-
tosphere (7.74) and on the ground (7.83) over wavenumbers
k
.
In the mag-
netosphere at
k
y
=0
,
only
E
x
and
b
y
components remain in an Alfven wave
(see (7.95), (7.96)) and
b
x
,b
z
,
and
E
y
components remain in an FMS-wave
(see (7.97), (7.98)). The beam is given by the distribution of incident mag-
netic wave components
b
(
i
x
(
x,
+0) and
b
(
i
y
(
x,
+0) above the ionosphere. The
Fourier transform of the above ionosphere incident field is given by
+
∞
1
√
2
π
b
(
i
τ
(
k
)=
b
(
i
τ
(
x,
+0) exp (
−
ikx
)
dx,
(8.1)
−∞
where
b
(
i
τ
(
x,
+0) is the horizontal vector of the magnetic field. In this chapter,
for notational simplicity, we shall put
k
x
=
k.
In Chapter 7, coecients of
reflection and transmission have been given at
k
0. Coecients
R
(
k
)and
T
(
k
) at arbitrary
k
can be obtained from
R
(
k
)and
T
(
k
) of Chapter 7 by
replacing
k
with
≥
.
The fields reflected from the ground-ionosphere system
b
(
r
)
(
k
) and trans-
mitted to the ground surface
b
(
g
)
(
k
) are determined by
|
k
|
b
(
r
τ
(
k
)=
R
(
k
)
b
(
i
τ
(
k
)
a d
b
(
g
)
(
k
)=
R
(
k
)
b
(
i
τ
(
k
)
,
(8.2)
where matrices
R
and
T
are given by (7.108) and (7.105). For components
of the electric field and current, we have
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