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z
is the unit vector of the
z
-axis,
E
τ
,
b
τ
are the horizontal components of the
E
and
b
vectors. For the potential
Ψ
, from (7.138) we have
d
2
Ψ
dz
2
+
k
0
ε
⊥
−
k
2
Ψ
=
4
πσ
H
ik
1
c
−
E
A
,
(7.143)
where
ϕ
=
E
A
sin
I/ik
1
,
ε
⊥
=
i
4
πσ
P
/ω
in the ionosphere. Equation (7.143)
is supplemented by the boundary conditions of the potential
Ψ
at
z
−
→∞
and
the impedance condition at the ground surface
dΨ
dz
Ψ
z
=
−h
ik
0
Z
(
m
)
+
=0
,
(7.144)
g
where
Z
(
m
g
is the surface impedance (7.27).
We will examine disturbances with a horizontal scale less than the wave-
lengths in the magnetosphere (
m
), ionosphere (
i
) and the atmosphere (
a
),
that is, with the wavenumbers
k
satisfying the conditions
1
/
2
.
k
k
0
|
ε
m,i,a
|
(7.145)
For the io
nosphe
re, it can be rewritten in terms of the Pedersen skin depth
d
P
=
c/
√
2
πωσ
P
as
kd
P
1
.
10
−
1
s
−
1
and
σ
P
=10
7
s
−
1
the skin depth
d
P
For frequency
ω<
3
×
100 km
10
−
2
km
−
1
. Similar reasonings for
the atmosphere lead us to the conclusion that on numerous occasions the
magnetostatic approximation is adequate for the analysis of propagation of
FMS-waves in the atmosphere-ionosphere-magnetosphere system.
Thus, computations of the field with an explicit indication of the ionospheric
source can be carried out in the following way:
and the last inequality is reduced to
k
d
2
Ψ
dz
2
−
4
πσ
H
ik
1
c
k
2
Ψ
=
−
E
A
,
(7.146)
with the boundary conditions (7.144) on the ground surface. The Green's
function
G
(
z, z
1
) of (7.146) is
G
(
z, z
1
)=
1
R
g
exp [
−
k
(
z
+
z
1
+2
h
)]
2
k
2
k
exp (
−
k
|
z
−
z
1
|
)
−
,
(7.147)
where
ikZ
(
m
)
R
g
=
1
−
/k
0
g
.
1+
ikZ
(
m
)
/k
0
g
Then the solution of (7.146) is given by
σ
H
(
z
1
)
e
−k|z−z
1
|
−
R
g
e
−k
(
z
+2
h
+
z
1
)
dz
1
,
Ψ
(
z
)=
i
2
π
c
E
A
k
1
k
(7.148)
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