Geoscience Reference
In-Depth Information
Magnetic Field Perturbations in the Atmosphere
and in the Solid Earth
Since there are no sources in the neutral atmosphere .
d<
z
<0/,theULF
electromagnetic perturbations excited by the ionospheric current in the atmosphere
are described by Laplace equation (
5.27
). A spatial Fourier transform of this
equation is given by
@
zz
b
k
2
b
D
0:
(5.109)
?
A vertical electric current j
z
flowing from the conducting ionosphere must be
zero at the boundary
z
D
0 and everywhere in the layer
d<
z
<0because the
atmosphere is an insulator. Taking Ampere's law, applying a Fourier transform to
this equation, using Eq. (
5.87
) and the representation (
5.7
) of the magnetic field via
the potentials, we obtain
0
j
z
D
.
r
b
/
z
D
.
k
?
b
?
/
z
D
k
2
A
D
0;
(5.110)
?
whence it follows that A
D
0 in the atmosphere including the upper boundary
z
D
0.
Thus the magnetic field in the atmosphere is derivable from only the potential ‰
b
D
k
2
?
O
z
C
i
k
?
@
z
‰:
(5.111)
Substituting Eq. (
5.111
)for
b
into Eq. (
5.109
) yields
@
zz
‰
k
2
‰
D
0:
(5.112)
?
The solid Earth .
z
<
d/ is supposed to be a uniform conductor with a
constant conductivity
g
. The low frequency electromagnetic field in the solid
Earth is described by the quasisteady Maxwell equation (
5.28
). Applying a Fourier
transform to this equation, using Eq. (
5.7
), and rearranging, we obtain
@
zz
‰
2
‰
D
0;
(5.113)
where
2
D
k
2
?
i
0
g
! is the squared “wave” number/propagation factor in the
ground.
Now we need to solve Eq. (
5.112
) and (
5.113
) for the atmosphere and for the
solid Earth, respectively, and then match the solutions at the boundary
z
D
d.The
solution of Eq. (
5.113
) decays at infinity .
z
!1
/ and is
‰
D
‰.
d/ exp Π.
z
C
d/; .Re>0/:
(5.114)
The solution of Eq. (
5.112
) can be written as
‰
D
C
C
exp .
k
?
z
/
C
C
exp .k
?
z
/;
(5.115)
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