Geoscience Reference
In-Depth Information
As before all the perturbed quantities are considered to vary as exp .
i!t/,so
the Maxwell equations (
4.1
), (
4.2
) for the TM mode in the neutral atmosphere
.
d<
z
<0/ can be written as
i!
c
2
E
r
;
@
z
ıB
'
D
(5.48)
r
@
r
rıB
'
D
1
i!
c
2
E
z
C
0
m.!/
2r
ı.
z
C
d
h/ı.r/;
(5.49)
@
z
E
r
@
r
E
z
D
i!ıB
'
:
(5.50)
where m.!/ is the Fourier transform of the lightning current moment and ı.x/
denotes Dirac's delta-function. The second term on the right-hand side of Eq. (
5.49
)
describes the current density due to the lightning discharge.
As before the ground is assumed to be a uniform conductor with constant
conductivity
g
. In the case of the axially symmetrical problem Eq. (
5.28
)forthe
TM mode in the ground can be written as
@
z
ıB
'
D
0
g
E
r
;
(5.51)
r
@
r
rıB
'
D
0
g
E
z
:
1
(5.52)
We recall that the TM mode components can be expressed through the scalar
potentials dž and A via Eqs. (
5.96
), (
5.98
), and (
5.100
). It is customary to seek for
the solution of the axially symmetrical problem in the form of Bessel transform.
Substituting the potentials dž and A into the Maxwell's equations for the neutral
atmosphere and for the ground and applying the Bessel transform to those equations,
we obtain a set of equations for the functions dž.k;
z
;!/ and A.k;
z
;!/, where
k is the parameter of the Bessel transform given by Eq. (
5.134
). These equations
should be supplemented by the proper boundary conditions at
z
D
0. A detailed
solution of this problem is found in Appendix E. In the atmosphere, altitude range
0>
z
>h
d, the solution (
5.145
) is reduced to the following:
0
m.!/
2k
A
D
A.
d/ cosh
f
k.d
C
z
/
g
sinh
f
k.
z
C
d
h/
g
;
(5.53)
where A.
d/ is the value of the potential A at the ground surface while the potential
dž is derivable from A through Eq. (
5.132
).
As we have noted above, the TE mode can also arise in the atmosphere due
to the mode coupling via the Hall conductivity in the ionosphere. A detailed
analysis of Maxwell's equation for the TE mode in the neutral atmosphere and
in the ground is given in Appendix E. The TE mode components, ıB
r
, ıB
z
, and
E
'
, can be expressed through the potential function ‰ via Eq. (
5.131
). Applying a
Bessel transform to Maxwell's equation for the atmosphere and conducting ground
with the proper boundary condition at
z
D
d gives a set of equations for the
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