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1.2
200
1
(a)
(b)
100
3
2
0.8
1
4
0
4
0.4
2
-100
3
0.0
-200
1E-6
1E-5
1E-4
1E-3
1E-2
1E-1
1E-6
1E-5
1E-4
1E-3
1E-2
1E-1
k
x
1/km
k
x
1/km
2.0
1.2
1.6
(c)
(d)
1
2
0.8
4
1.2
2
0.8
1
0.4
3
0.4
0.0
0.0
1E-4
1E-3
1E-2
1E-1
1E-6
1E-5
1E-4
1E-3
1E-2
1E-1
k
x
1/km
k
x
1/km
Fig. 7.5.
Longitudinal magnetic field above the ionosphere (1
,
2) and the vertical
on the ground (3
,
4) as a function of the horizontal wavenumber. The amplitude
(a) and phase (b) for
T
= 30 s. Curves 1, 3 are numerical results, 2, 4 is by the
small-scale approximation formula. (c) is the same as (a) but for
T
= 300 s. (d) is
the relative error
solution. An error of the small-scale approximation is shown in Fig. 7.5d as
well. The other parameters are indicated in Table 7.3. It can be seen that at
k
x
k
0
X
the approximation equation yields good results. For instance, at
k
x
>
10
−
2
km
−
1
,
k
0
X
=3
10
−
2
km
−
1
for
T
= 30 s, the error less than a few
×
percents.
The calculations conducted for different models showed that the small-
scale approximation can be used in calculating electric and magnetic fields in
the ULF for wavenumbers
k
x
10
−
2
km
−
1
.
Table 7.3.
Magnetospheric wavelength
l
m
=(
k
0
√
ε
m
)
−
1
km, ground skin depth
d
g
and wavenumber
k
∗
for periods
T
=30
,
120 and 600 s
0
.
116
×
10
9
σ
g
s
−
1
1
×
10
6
Σ
P
km/s
0
.
136
×
10
9
1
×
10
3
Σ
H
km/s
c
A
km/s
h
km
I
π/
2
100
T,
s
30
120
600
1
.
4
×
10
6
5
.
7
×
10
6
,
km
2
.
9
×
10
7
l
m
,
km
d
g
,
km
260
520
1170
k
∗
,
km
−
1
1
.
2
×
10
−
6
6
×
10
−
7
2
.
6
×
10
−
7
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