Geoscience Reference
In-Depth Information
After a change of variables
kl
0
−
,
i
√
2
ix
−
h
|
k
|
/k
p
=
l
0
the integral reduces to
exp
i
√
2
l
0
i∞
(
x
+
ih
)
2
2
l
0
exp
z
2
dz
+c.c.
−
−
x
+
ih
√
2
l
0
and the magnetic field on the ground may be written as
Yb
0
sin
I
X
b
(
g
)
SA
(
x
)=
b
(
g
x
(
x
)=
−
Re
w
(
p
0
)
,
(8.33)
where
w
(
p
0
)is
p
0
=
x
+
ih
p
0
)erfc(
w
(
p
0
) = exp(
−
−
ip
0
)
,
√
2
l
0
.
(8.34)
The complementary error function
p
0
ip
0
)=1+
2
i
exp(
p
2
)
dp.
√
π
erfc(
−
0
Approximation (8.33) can be used up to a few thousands kilometers from
the beam axes for ground conductivities larger than 10
7
s
−
1
. Figure 8.3 shows
the
b
(
g
x
(
x
)-component for different values of periods and ground conductivities
calculated by approximation (8.12), (8.19), and (8.33). The computations are
carried out for the Alfven velocity
c
A
= 1000 km
/
s
.
Respectively, the wave
Alfven conductivity
Σ
A
is
Σ
A
=
c
2
/
(4
πc
A
)=7
.
16
10
6
km/s. Then for
the normalized dayside Pedersen
Σ
P
and Hall
Σ
H
conductivities, we have
Σ
P
=
Σ
P
/Σ
A
= 10,
Σ
H
=
Σ
H
/Σ
A
= 14, and for the nightside
Σ
P
=0
.
1,
Σ
H
=0
.
14. The corresponding dimensional conductivities are
Σ
P
=7
.
16
×
×
10
7
km/s,
Σ
H
=1
10
6
km/s,
for the day and night ionospheres, respectively. The last parameters are given
in Table 8.2. The dashed line in Fig. 8.3 shows the initial field (8.32). The
thick solid line is the spatial distribution for the high conductive ground is
given by (8.33). The other curves are the result of numerical integration for
the parameters shown in the Table 8.2.
One can see that approximation (8.33) adequately describes the decreasing
of the field for the day conditions and ground conductivities
σ
g
=10
6
10
8
km/s and
Σ
P
=7
.
16
10
5
km/s,
Σ
H
=1
×
×
×
−
10
7
s
−
1
up to 2000 km
.
However, for the Pedersen conductivities typical for
the night ionosphere, the approximation (8.33) is applicable till
≈
500 km for
σ
g
=10
6
s
−
1
and till
1000 km for
σ
g
=10
7
s
−
1
.
≈
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