Geoscience Reference
In-Depth Information
Substituting the solution (7.76), we obtain
0) =
D
(0)
i
+
D
(0
r
R
b
(
i
)
,
b
τ
(
−
(7.81)
where
=
b
(
α
)
A
GE
(
α
S
,
GE
(
α
)
A
,
b
(
α
)
S
D
(0)
α
−
−
α
=
i
or
r.
The covariant components
b
1
,b
2
and
E
1
,E
2
coincide with the horizontal
magnetic
b
x
,b
y
and electric
E
x
,E
y
, that is
b
x
=
b
1
,
y
=
b
2
,
x
=
E
1
,
E
y
=
E
2
,
while for
b
z
we get
b
1
cot
I
+
b
3
.
The formulae obtained relate the total MHD-wave field above and under the
thin ionosphere with an Alfven or FMS-wave incident upon it from a homo-
geneous half-space filled with cold plasma.
Components of
E
(
z
)and
b
(
z
) in the atmosphere can be easily calculated
from (7.81). Explicit expressions will be given here permitting
E
(
z
)and
b
(
z
)
to be found in the most important case when displacement currents and con-
ductivity in the atmosphere can be neglected.
It is convenient to write the field in the atmosphere and on the ground
surface in the Cartesian-altitude system
b
z
=
−
{
x, y, z
}
. For horizontal wavenumbers
1
/
2
from (7.41), (7.42), and (7.45), it follows that the atmospheric
fields are given by
|
k
|
k
0
|
ε
a
|
b
τ
(
z
)=
1+
R
g
e
−
2
k
(
z
+
h
)
1+
R
g
e
−
2
kh
e
kz
b
τ
(0)
,
(7.82)
i
k
0
k
1
−
R
g
e
−
2
k
(
z
+
h
)
1+
R
g
e
−
2
kh
e
kz
z
E
τ
(
z
)=
−
×
b
τ
(0)
,
b
z
(
z
)=
k
x
E
y
(
z
)
−
k
y
E
x
(
z
)
,
(7.83)
k
0
where
ikZ
(
m
)
R
g
=
1
−
/k
0
g
.
1+
ikZ
(
m
)
/k
0
g
The expressions obtained for fields are rather cumbersome and their appli-
cation requires numerical calculations. Nevertheless, the utilization of these
general equations permits several useful results to be obtained. In particu-
lar, no special diculties are involved in carrying out the inverse numerical
Fourier transformation, which allows field distribution to be obtained both
above the ionosphere and on the ground from MHD-wave beam.
The approximation used in this section is applicable only to waves with
small horizontal wavenumbers
k
1
and
k
2
. The condition
k
1
l
I
1and
k
2
l
I
1
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