Geoscience Reference
In-Depth Information
Substituting Eqs. (
5.126
) and (
5.127
) for the jump of functions A and @
z
‰ into
boundary conditions (
5.120
) and (
5.121
), we are thus left with the set
ix
0
Ǜ
H
L
.LJ
1
C
Ǜ
P
/
V
AI
‰.0/
dž.0/
D
f
1
;
(5.128)
Ǜ
H
V
AI
dž.0/
C
.ix
0
Ǜ
P
s/
L
‰.0/
D
f
2
;
(5.129)
where the dimensionless frequency x
0
is again defined in Eq. (
5.17
), and the
functions f
1
and f
2
are given by Eqs. (
5.122
) and (
5.123
). Equations (
5.128
)
and (
5.129
) can be solved for ‰.0/.
Appendix E: Solutions of the Axially Symmetrical Problem
TM Mode in the Neutral Atmosphere and in the Ground
In Sect.
5.3
we study the electromagnetic field excited by the vertical CG lightning
discharge which is located on the vertical
z
axis in the neutral atmosphere. The
problem is axially symmetrical since the geomagnetic field
B
0
is assumed to be
directed vertically upward. The components of the electromagnetic perturbations
can be expressed through potential functions A, dž and ‰ in cylindrical coordinates
z
;r;'via Eqs. (
5.95
)-(
5.100
). For the axially symmetrical problem these equations
are simplified to
ıB
'
D
@
r
A; E
r
D
@
r
dž; E
z
D
@
z
dž
C
i!A;
(5.130)
and
1
r
@
r
.r@
r
‰/:
E
'
D
i!@
r
‰; ıB
r
D
@
r
z
‰; ıB
z
D
(5.131)
To treat the TM mode generated by the vertical CG discharge in the atmosphere,
Maxwell's equations are required, which are given by the set of Eqs. (
5.48
)-(
5.50
).
As is seen from Eq. (
5.130
) the TM mode components ıB
'
, E
r
, and E
z
are
represented by the potentials dž and A and do not depend on ‰. Substituting these
components into Eqs. (
5.48
)-(
5.50
) and rearranging, we obtain that Eq. (
5.50
)is
reduced to identity, while Eqs. (
5.48
) and (
5.49
) take the forms
i!
c
2
dž;
@
z
A
D
(5.132)
!
2
c
2
A
D
1
r
@
r
.r@
r
A/
C
0
m.!/
2r
@
zz
A
C
ı.
z
C
d
h/ı.r/;
(5.133)
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