Geoscience Reference
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dž is described by Poisson equation. Considering the thundercloud as a point current
source located on
z
-axis at the altitude h, we come to the following equation:
a
r
@
r
.r@
r
dž/
C
@
z
.
a
@
z
dž/
D
Iı.r
/
4r
2
;
(3.12)
where I denotes the total current flowing from the source, ı stands for Dirac delta-
function, and r
D
n
r
2
C
.
z
h/
2
o
1=2
. This equation should be supplemented by
the proper boundary conditions for the conducting ground and at the infinity, that
is, dž
D
0 at
z
D
0 and dž
!
0 when
z
!1
. Substituting the relationship
a
D
0
exp .Ǜ
z
/ into Eq. (
3.12
) and solving the problem gives (e.g., see Soloviev
and Surkov
2000
)
4
0
exp
h
2
.
z
C
h/
in
r
1
exp
Ǜr
2
exp
Ǜr
2
o
: (3.13)
I
Ǜ
r
1
C
dž
D
The total charge q of the source/thundercloud can be related to the source
current I through the Gauss theorem and Ohm law whence it follows that I
D
.q
0
="
0
/ exp .Ǜh/. When Ǜ
D
0,Eq.(
3.13
) describes a potential of point charge
q and its mirror image in the perfectly conducting ground. In general case the
exponential factors in Eq. (
3.13
) lead to the strong field attenuation with altitude
due to air conductivity.
This model with the parameter Ǜ
D
0:15 km
1
and q
D
150 C was used for
numerical calculation of the vertical component of the thundercloud electric field
E
z
D
@
z
dž along
z
-axis in the presence of atmospheric conductivity as shown in
Fig.
3.20
with dash-and-dot line 8. The role of the atmospheric conductivity comes
into particular prominence when comparing this graph with that calculated at the
same thundercloud charge and zero atmospheric conductivity (line 3). It is obvious
from Fig.
3.20
that the conduction current due to the atmospheric conductivity may
decrease the thunderstorm field to such an extent that it makes impossible the air
breakdown in the mesosphere. In a more accurate model which takes into account
both the time-dependent CC and the air conductivity, the altitude profile of the
electric field be situated between lines 3 and 8 (Mareev and Trakhtengerts
2007
). In
this notation, the sprites must build up very quickly just after the causative lightning
discharge for the short period limited by the relaxation time that varies within
1-100 ms in the altitude range 60-80 km.
The sprite initiation, visible evolution, streamer structure, and their relationship
with IC process are so complex that any quantitative theory of the sprites has not
been established yet except a number of numerical simulations (e.g., Pasko et al.
2000
,
2001
; van der Velde et al.
2006
,
2007
; Asano et al.
2009a
,
b
;Ebertetal.
2010
).
Luque and Ebert (
2009
,
2010
,
2012
) have recently developed a numerical model
of sprite initiation that takes into account the photoionization effect and altitude-
dependent transport and ionization parameters of electrons and neutrals. This model
does not require any kind of seed electrons since the primary streamer is assumed to
be due to drift of the background electrons subjected to thundercloud electric field.
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