Electromagnetic Effects Resulted from Explosions - Ultra and Extremely Low Frequency Electromagnetic Fields - page 432

where .x/ denotes the -function of Riemann. Taking into account that .2/ D
2 =6 we obtain that Eq. ( 11.14 )for M coincides with Eq. ( 11.9 ) which was derived
in the same extreme case. In the opposite case p ! 0 we have a so apparent result
M D 0.
The low-frequency conductivity of the heated plasma is defined by Eqs. ( 11.3 )
and ( 11.4 ) because the electron-ion collisions prevail over other ones at high
temperature. Suppose that the plasma is the perfect gas that expands according to the
adiabatic law ( 11.12 ). Then the plasma temperature varies as T D T 0 =LJ 3.1/ where
T 0 is the initial plasma temperature. Now we first examine the exponent function
under the integral sign in Eq. ( 11.13 ). The expression standing in the index of the
exponent function can be written in the form (Gaussian system of units)

LJ
LJ m

c 2 n 2
4 p R 0 LJ 2 D

n 2
d

; (11.15)

where

16 p 2
5=2

R 0 T 3=2 9 13

2

0
Ze 2 Lc 2 m 1= e LJ m

d D ; D : (11.16)

Here c is the light speed in the free space and LJ m denotes maximum of the function
LJ.t/; that is LJ m D R m =R 0 , where R m is the final radius of the plasma. The
parameter d determines the back diffusion time of the perturbed magnetic field
into the plasma ball. This parameter has the same sense as the relaxation time given
by Eq. ( 11.5 ). It can be shown that Eq. ( 11.16 ) coincides with Eq. ( 11.5 ) within a
constant factor. Substituting the numerical parameters R 0 D 1 m, T 0 D 1 keV,
Z D 2, L D 4, D 5=3 and LJ m D 30 into Eq. ( 11.16 ) we obtain d D 0:55 s. This
value is compatible with the relaxation time of EMP observed during underground
explosions. However the dependence d / Y 2=3 which follows from Eq. ( 11.16 )
contradicts with the empirical dependence d / Y 1=3 displayed in Fig. 11.4 .
There are a lot of factors which may affect the electromagnetic signals under the
explosions and thus may concern this discrepancy. For example, the melting and
evaporation of the surface of an underground chamber subjected to the radiation of
nuclear explosion results in changing the plasma constituents due to the injection of
evaporated particles. A fall in plasma temperature brings the decrease in the plasma
ionization degree due to recombination process. These effects lead to the changes in
the plasma conductivity, adiabatic exponent, and other plasma parameters that leave
out of account in the above models.
The changes in the underground chamber size can be approximated by a smooth
function, for example,

LJ.t/ D 1 C .LJ m 1/Œ1 exp . t= b /; (11.17)