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where
c
.!/ stands for the correlation radius. The spectral density of this random
process is given by
lp
k
x
;!;k
0
x
;!
0
D
ı
!
!
0
ı
k
x
k
0
x
G
lp
.k
x
;!/;
.
w
/
(6.94)
where ı denotes Dirac's function and
exp
:
k
x
c
.!/
4
F
lp
.!/
c
.!/
2
1=2
G
lp
.k
x
;!/
D
(6.95)
Combining these equations, applying an inverse Fourier transform, and performing
the integration over k
0
x
and k
x
, leads to the spatial distribution of the spectral density
of the horizontal magnetic field variations. Setting
D
x
x
0
D
0 yields
exp
4d
2
c
.!/
1
erf
2d
c
.!/
;
0
‚
1
.!/
4
‰
.B/
xx
.!/
D
(6.96)
where erf .x/ denotes the error function. Here the function ‚
1
.!/ is given by
X
3
X
3
1
m
.
w
/
xl
m
.
w
/
‚
1
.!/
D
xp
F
lp
.!/;
(6.97)
p
D
l
D
1
where m
.
w
/
lp
stands for the matrix elements appearing in Eq. (
6.78
). When consider-
ing the extreme case 2d
c
,Eq.(
6.96
) is simplified to
0
‚
1
.!/
c
.!/
8
1=2
d
‰
.B/
xx
.!/
D
:
(6.98)
Substituting Eq. (
6.91
)for
c
.!/ into Eq. (
6.98
) gives the estimate of the power
spectrum on the ground surface for the case of 1D distributions of the ionospheric
current fluctuations
0
1=2
‚
1
.!/V
a
.!/
4!d
‰
.B/
xx
.!/
:
(6.99)
When this result is compared with Eq. (
6.92
), it is apparent that the 2D-case
correlation function falls off more rapidly with frequency than does the 1D-case
correlation function. In the analysis that follows, we show that Eq. (
6.92
)isbetter
consistent in magnitude with the observations than does Eq. (
6.99
).
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