Atmospheric Electricity - Ultra and Extremely Low Frequency Electromagnetic Fields - page 86

Runaway electron propagation through the air is accompanied by the generation
of large amount of secondary low-energy electrons due to the neutral molecule
ionization by runaway electron impact. Although a majority of secondary electrons
have a small energy, a portion of such electrons may gain energy " which is greater
than the threshold value; that is, ">" r . The ambient electric field will accelerate
these energetic electrons, so that they may also become runaway electrons, which
in turn results in additional ionization of air and the generation of a new portion
of secondary and runaway electrons. The exponentially increasing avalanche of
runaway electrons is a crucial factor in the development of air breakdown since
a great deal of the secondary slow electrons is produced along with the runaway
electrons (e.g., see Colman et al. 2010 ).
It follows from Eq. ( 3.21 ) that the runaway breakdown field E r is proportional to
the neutral number density. Taking the notice of Eq. ( 2.3 )forn m gives the following
approximation (Gurevich and Zybin 2001 ):

E r D 2:16exp . z =H a /; kV/cm. (3.23)

It should be emphasized that the value of the runaway breakdown threshold
E r is one order of magnitude smaller than the conventional breakdown E c .This
important fact follows from a comparison of Eqs. ( 2.3 ) and ( 3.23 ). The runaway
breakdown field E r is shown in Figs. 3.4 , 3.15 and 3.17 with line 3. On the other
hand, the thermal runaway breakdown threshold E th 260 kV/cm is approximately
100 times greater than the runaway threshold E r . Under such a strong electric field
all the thermal electrons become runaway ones since the electric force acting on
electrons becomes greater than the maximal dynamical friction force shown in
Fig. 3.22 . A more sophisticated treatment has shown that as the electric field E
is close to the breakdown threshold then the characteristic length l r of runaway
electron avalanches is inversely proportional to n m (Gurevich and Zybin 2001 ).
Since the neutral number density decreases with altitude, the value of l r ,onthe
contrary, increases from several tens meters at the ground surface level to a few
km at the mesospheric altitudes. It is obvious from Fig. 3.20 that the thundercloud
electric field arising after a CG discharge (lines 1-3) can exceed the runaway
breakdown threshold (line 7) in the tens km altitude range which is greater than
the length l r of exponential growth of runaway electron avalanche.
Radiations of relativistic electrons give rise to the generation of Roentgen and
gamma quanta which in turn are able both to ionize the molecules and to generate
electron-positron pairs when interacting with nuclei of molecules. In the course of
this text, we cannot come close to exploring these topics owing to the complexity of
this problem. In a more accurate theory, the Boltzmann transport equation is used to
describe the runaway electron distribution function f . r ; p ;t/ in phase space (e.g.,
see recent reviews by Roussel-Dupré et al. ( 2008 ) and by Milikh and Roussel-Dupré
( 2010 ))

@ t f C V r f C e E r p f D S .f;f n /; (3.24)