Geoscience Reference
In-Depth Information
must be satisfied, where
l
I
is the thickness of the conductive region of the
ionosphere. Since
l
I
≈
30 km, it means that rather strict conditions are ob-
tained on the horizontal scale
L
= min
k
−
1
,k
−
2
30 km.
This condition may not hold, for instance, for the fields near the FLR.
In that case, as is known from Chapter 6, small-scale approximation is
valid. Since uncoupled equations for Alfven waves do not include horizon-
tal wavenumbers
k
1
,
k
2
, the thin ionosphere approximation is valid for them
even at
k
1
l
I
1and
k
2
l
I
1.
It it worth noting that the large
k
approximation and the thin ionosphere
approximation taken together cover almost all the virtually important cases.
7.6 Propagation Along a Meridian
R and T Matrices
Consider MHD-waves propagating in a meridional plane,
k
2
= 0. In that
case, the reflection and transmission matrices have rather simple analytical
expressions. In this section, expressions will be given for matrix
R
connecting
horizontal magnetic components in reflected
b
(
r
τ
and incident waves
b
(
i
τ
above
the ionosphere as well as for matrix
T
connecting the ground
b
(
g
)
and
b
(
i
τ
.
Equations (7.41) and (7.42) are simplified for
k
2
= 0 and become
Y
(
m
)
Y
(
e
)
g
ζ
4
g
ζ
1
b
(
g
)
1
(
g
)
2
=
−
b
1
(
−
0)
,
=
−
b
2
(
−
0)
.
(7.84)
From (7.36)-(7.38) with (7.47), it follows that
ζ
4
ζ
3
E
2
(0)
,
0) =
ζ
1
b
1
(
−
0) =
−
2
(
−
ζ
2
E
1
(0)
.
(7.85)
Combining (7.84) and (7.85), we obtain
=
Y
(
m
)
Y
(
e
)
g
ζ
3
g
ζ
2
b
(
g
)
1
(
g
)
2
E
2
(0)
,
=
−
E
1
(0)
.
(7.86)
The magnetic field
b
(
i
)
in incident and
b
(
r
)
in the reflected waves and the
horizontal components
b
(
g
)
on the ground can be written as
=
T
b
(
i
)
.
S
b
(
i
A
b
(
r
)
τ
=
R
b
(
i
τ
,
b
(
g
)
=
T
b
(
i
)
τ
and
(7.87)
Substituting (7.76) into (7.87), at
x
3
= 0 we find
R
=
b
(
r
)
S
,
b
(
r
A
R
b
(
i
)
,
b
(
i
A
−
1
,
S
T
=
b
(
i
)
S
,
b
(
i
A
T
.
(7.88)
Search WWH ::
Custom Search