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where
∆
(0)
SK
−
1
b
(
i
)
(
A,S
)
k
exp
ikx
Φ
3(
A,S
)
Φ
4(
A,S
)
=
1
Z
(
m
)
dk.
+
∞
Z
(
m
)
cosh
(8.14)
k
k
k
/
(
k
0
h
)
−
i
tanh
|
|
/
|
|
|
|
g
g
−∞
For a definition of the impedance
Z
(
m
g
,
see (7.27) and (7.28).
Spatial dependencies of the total fields above the ionosphere and on the
ground surface can be obtained from (8.7)-(8.10) and (8.11)-(8.13) employing
the fact that
(
x
)=
b
(
i,r,g
)
SS
(
x
)+
b
(
i,r,g
)
SA
(
x
)=
b
(
i,r,g
)
AS
(
x
)+
b
(
i,r,g
)
AA
b
(
i,r,g
)
x
b
(
i,r,g
)
y
(
x
)
,
(
x
)
,
(
x
)=
b
(
i,r,g
)
zA
(
x
)+
b
(
i,r,g
)
zS
b
(
i,r,g
)
z
(
x
)
.
Equations (8.7)-(8.10) and (8.11)-(8.13) can be conveniently used in nu-
merical calculations at small distances less than 1000
2000 km. At large dis-
tances, the values of the integrals are determined by the behavior of integrand
expressions in the vicinity of the singular points. Therefore, direct use of the
numerical Fourier transformation can lead to wrong results.
−
8.3 Small Distances
Resonance Magnetic Shell
Let us suppose that an Alfven wave beam excited by the FLR is incident on the
ionosphere. Resonance disturbances above the ionosphere for fixed azimuthal
harmonics
∝
exp(
ik
y
y
) are given by (6.88) which can be rewritten
δ
i
.
b
(
i
y
(
x
)=
b
(
i
)
x
+
iδ
i
+
Ck
y
ln
k
y
(
x
+
iδ
i
)+
···
(8.15)
0
Here
x
is the horizontal coordinate directed southward from the base of the
resonance shell,
b
(
i
)
y
is the azimuthal component above the ionosphere,
b
(
i
)
0
is the intensity of the magnetic component in the maximum of the incident
wave, and
C
is a constant determined by the field line geometry and field
line distribution of the cold plasma.
δ
i
is the half-width of the FLR above the
ionosphere. The azimuthal wavenumber
=
k
, and, thus, approxi-
mate estimates for the reflected and transmitted magnetic and electric fields
are found from the equations (8.7)-(8.14) for the meridional propagation.
|
k
y
||
k
x
|
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