Geoscience Reference
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The coecient of transformation of an FMS-wave into a reflected Alfven
wave (
R
A
S
) i
s small in the entire range of wavenumbers and is of the order of
k
A
=
k
0
√
ε
m
h
at small
k
and
k
A
/k
at large
k
.
Further,
R
SS
=1and
R
SA
=0at
k
=
k
A
. At larger wavenumbers
R
SS
approaches zero as 1
/k
. The coecient
R
SA
∼
k
A
at small
k
(
k
k
A
). At
large
k
2
Y
∆
(0)
A
R
SA
≈−
sign
I.
(7.130)
The transition from small values
R
SA
to (7.130) is determined by the
concrete model of the ground. For instance, in the case when the skin depth
d
g
is larger than the atmosphere thickness
h
,
R
SA
begins to reach (7.130) at
|
kd
g
|≈
1.
For transmitted Alfven waves (electric modes)
|
T
AA
|
1and
|
T
AS
|
1
.
For instance,
sin
2
I
Xk
sinh
k
2
k
sinh
k
.
2
k
0
h
|
ε
a
|
=2
|
k
∗
h
|
|
T
AA
|≈
(7.131)
For estimates of
see (7.121).
Let the Earth's conductivity be high, so that the skin depth
d
g
is small
in comparison with the atmosphere's height
h
. Consider the wavenumbers
meet
i
ng the condition
h
−
1
|
k
∗
|
k
A
. Then, tanh
k/k
d
−
1
g
k
and
k
≈
1
,
(
k
A
−
k
2
)=
ik, Z
(
m
)
≈
0. In this approximation (7.117) gives
ζ
4
/ζ
3
≈
g
−
ik/k
0
. Substituting last equations into (7.125) we get
2
ke
−k
T
(0)
SS
=
(7.132)
k
−
iτ
K
/
2
T
SA
(
k
) differs from
T
SS
(
k
) only by the factor sign
IY/∆
(0)
. Hence,
A
ke
−k
2
Y
sin
I
X
+
√
ε
m
|
T
SA
≈−
iτ
K
/
2
.
(7.133)
k
sin
I
|
−
goes down exponentially at large
k
.
It follows from (7.131) and (7.133) that an incident Alfven wave is trans-
formed to a magnetic mode. This means that the initial magnetic field vec-
tor turns by
π/
2 counterclockwise (as seen along
B
0
). Recall that this con-
clusion at
k
y
= 0 is independent on the inclination angle of
B
0
. One can
argue that in the limit
ω
It can be seen that
|
T
SA
|
0, the system of 2D-inclined field-aligned cur-
rents does not contribute to the magnetic field under the ionosphere at
Σ
H
=0.
The obtained results are important in interpreting observations of the
ULF electromagnetic fields conducted by low-orbit satellites: a longitudinal
component appears in the reflected FMS-waves whose amplitude can become
commensurate with the incident magnetic field of an Alfven wave (see also
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