Geoscience Reference
In-Depth Information
As before we may apply a Fourier transform to Eqs. (
5.27
) and (
5.28
). Taking
into account Eqs. (
5.7
) and (
5.8
) we arrive at equations for the potential function
‰. These equations should be supplemented by the proper boundary conditions at
z
D
d that follow from the Ampére's law. On account of the continuity of the
magnetic and horizontal electric fields at the boundary between the atmosphere
and the ground we obtain that the scalar potential ‰ and its derivative @
z
‰ to
be continuous at this boundary. In greater detail the solution of this problem
is examined in Appendix D. Combining Eqs. (
5.115
) and (
5.116
) we obtain the
solution of problem in the atmosphere .
d<
z
<0/
1
C
exp Œk
?
.
z
C
d/
‰.0/
2LJ
3
k
?
‰
D
1
exp Œ
k
?
.
z
C
d/
;
k
?
C
(5.29)
where the parameter
D
k
2
?
i
0
g
!
1=2
(5.30)
plays a role of the propagation factor/“wave number” in the ground and
k
?
LJ
3
D
cosh.k
?
d/
C
sinh.k
?
d/:
(5.31)
Taking the electromagnetic field representation (
5.7
)-(
5.8
) through the scalar
potentials .A;dž;‰/, and using the solutions (
5.18
), (
5.19
), and (
5.29
)forthe
potentials, we can find the fields
b
and
e
both in the ionosphere and atmosphere.
Substituting these fields into the boundary condition (
5.26
) gives the set of
Eqs. (
5.128
) and (
5.129
) for the undetermined constants ‰.0/ and dž.0/.The
potential ‰ on the ground is derivable from ‰.0/ through Eq. (
5.29
) as follows:
‰.
d/
D
‰.0/=LJ
3:
: The solution of these equations is found in Appendix D.
Finally, we obtain
B
0
F
0
k
2
?
‰.
d/
D
;
(5.32)
qLJ
3:
where the following abbreviations are introduced
AI
.
k
?
v
/
Ǜ
2
H
C
Ǜ
P
C
LJ
1
Ǜ
P
.
k
?
v
/
z
LJ
1
Ǜ
H
;
F
0
D
LV
1
(5.33)
q
D
.is
C
x
0
Ǜ
P
/.LJ
1
C
Ǜ
P
/
C
x
0
Ǜ
2
H
;
(5.34)
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