Atmospheric Electricity - Ultra and Extremely Low Frequency Electromagnetic Fields - page 83

In the vicinity of the axis of symmetry .r z / the relationship between r and z can
be simplified in such a way that we come to the explicit dependence r . z / which is
valid as z z :
r D 4 z " 0 E 0
15d

z 3

exp z 3 exp

1=2

z H a z

H a

: (3.16)

The wave front given by Eq. ( 3.16 ) is schematically shown in Fig. 3.21 with
red line. It follows from Eq. ( 3.15 ) that an increase in dipole moment d results
in a decrease of z . This means that the enhancement of thundercloud field is
accompanied by downward propagation of the point z and the wave front given
by Eq. ( 3.16 ). Although this qualitative analysis is consistent with the results of
numerical simulations reported by Luque and Ebert ( 2009 ), the above approach
cannot predict the sharp prominence arising in the center of the wave surface
because we have ignored the electric field of charges accumulated at the wave front.
This point requires the precise analytical analysis because this effect can be due to
plasma or other kind of instabilities (e.g., Derks et al. 2008 ).

3.2.4 Runaway Electron Breakdown

As has already been discussed, the conventional streamer-leader mechanism for air
breakdown can explain, in principle, the basic properties of the TLEs (e.g., see
Rioussetetal. 2010a , b ; Raizer et al. 2010 ). An alternative approach assumes the
relativistic runaway electron avalanches as the proper candidate for producing the air
breakdown at stratospheric and mesospheric altitudes (e.g., see Gurevich et al. 1992 ,
1994 ; Roussel-Dupré and Gurevich 1996 ; Lehtinen et al. 1997 , 1999 ; Babich et al.
1998 , 2008 ; Gurevich and Zybin 2001 ; Lehtinen 2000 ; Füllekrug et al. 2010 , 2011 ).
One of the merits of this mechanism is that the electric field threshold required for air
breakdown may be one order of magnitude lower than that due to the conventional
breakdown.
In the course of this text, the runaway breakdown is only treated in a sketchy
fashion. First of all we note that if the electron energy greater than 50 eV then there
prevails the electron forward scattering at small angles. In this notation we consider
the simple one-dimensional model in which all the high-energy electrons can move
only along z axis parallel to the constant electric field E . The electron collisions are
taken into account by means of the so-called dynamical friction force F fr which is
pointed oppositely to the vector of electron momentum p . In such a case the equation
of electron motion is reduced to the following (Gurevich et al. 1992 , 1994 )

dp
dt D eE F fr :

(3.17)