Geoscience Reference
In-Depth Information
where
Z
h
=
Z
(
m
)
(
k
0) is the ground spectral impedance for the vertically
incident plane wave. Relations (8.20)-(8.23) were obtained for the dayside
ionosphere, i.e. it was assumed that
4
πΣ
P
c
≡
→
g
X>>
c
c
A
.
Low Conductive Ground
The scale of FLR on the ground surface is about 100
200 km. There are
regions with the effective skin depth
d
g
in the
Pc
3,4 range comparable to the
horizontal scale
L
of the field
L
−
|
−
1
. In this case, the simple impedance
conditions cease to be true. However, a simple explicit expression can be
obtained in the contrary case of the low-conductive ground when
∼|
k
|
kd
g
|
1
and
κ
g
=(
k
0
ε
g
−
k
2
)
1
/
2
. At so small horizontal scales, the ground
conductivity scarcely affects the field distribution. The surface admittance
Z
(
m
)
≈
i
|
k
|
/k
0
and we have (see (7.128))
∆
(0)
SK
(
k
)
≈
i
|
k
|
/k
0
and
ζ
4
/ζ
3
≈−
i
|
k
|
≈
g
−
2
i
(
|
k
|−
iτ
K
/
2) . Substitution of these expressions into the integrals
Φ
3
A
and
Φ
4
A
gives
∞
k
exp[
k
(
ix
1
2
−
1)]
b
(
i
A
dk,
Φ
3
A
=
−
k
−
iτ
K
/
2
0
∞
exp[
k
(
ix
1
2
ik
0
h
−
1)]
b
(
i
A
dk.
Φ
4
A
=
k
−
iτ
K
/
2
0
1and
δ
i
=
δ
i
/h
Let us estimate
Φ
3
A
for
τ
K
1
.
In this case, one can
neglect the small term
iτ
K
/
2 in the denominator of the integrand. Integrating,
we obtain
i
A
0
2
h
1
x
+
i
(1 +
δ
i
)
.
Φ
3
A
≈−
(8.24)
In estimating
Φ
4
A
,
it is impossible to ignore the small corrected term in
the denominator of the sub-integral expression. We must take into account
the pole at
k
=
iτ
K
/
2 . Then
Φ
4
A
can be written as
i
A
0
k
0
2
Φ
4
A
≈−
exp (
−
Z
0
)Ei (
Z
0
)
,
where
2
x
+
i
(1 +
δ
i
)
.
Ei (
Z
0
) is the exponential integral which can be presented as the series
τ
K
Z
0
=
Z
0
)+
∞
Z
0
Ei (
Z
0
)=
C
+ln(
−
n
!
n
,
n
=1
C
1 we restrict our consideration to
the first two terms in the expansion of Ei (
Z
0
), and exp(
≈
0
.
577 is the Euler constant. At
|
Z
0
|
−
τ
K
x/
2)
≈
1 . Write
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