Electromagnetic Effects Resulted from Explosions - Ultra and Extremely Low Frequency Electromagnetic Fields - page 463

Appendix I: Magnetic Perturbations Caused
by Underground Detonation

High-Heated Plasma Ball Expanding in Ambient
Magnetic Field

Let us consider an expanding homogeneous plasma ball immersed into the uniform
magnetic field with induction B 0 (Ablyazov et al. 1988 ). At the time t D 0 the ball
radius begins to increase in accordance with the dependence R.t/ D R 0 LJ.t/ where
R 0 is the initial ball radius and LJ.t/ is a given function, which is equal to unity
at t D 0. The plasma conductivity obeys the known law p D p .t/ as well. We
assume that the plasma ball is surrounded by the non-magnetic rock . D 1/ whose
conductivity is everywhere negligible compared with the plasma one. As the ball
is situated at the depth which is much greater than the ball radius, one can neglect
the influence of the atmosphere in calculating the field in the vicinity of the ball.
In this approach the magnetic induction B in the plasma ball is described by the
quasi-stationary Maxwell equations .0<r<R/

1
0 p r

2 B ;

@ t B Dr . V B / C r B D 0; (11.65)

where V is the plasma velocity.
Since the rock conductivity is ignored, Maxwell equations outside the ball are
given by r B D 0 and r B D 0. In this region we seek for the solution of these
equations as a sum of the uniform field, B 0 , and of the field of effective magnetic
dipole whose moment is proportional to B 0 . If the origin of spherical coordinate
system is placed in the ball center, the solution of the problem can be represented as
follows .r>R/
B D B 0 cos 2R 0

r 3 C 1

O r C sin R 0

r 3 1

;

O

(11.66)

O

where the angle is measured from the direction of B 0 and O r and denote the

unit vectors of spherical coordinate system. Here the dimensionless function .t/ is
related to the magnetic moment of the plasma ball through the following relationship
M .t/ D 4.t/R 0 B 0 = 0 .
The plasma ball is assumed to expand uniformly so that the plasma moves
in radial directions. Consequently, the radius-vector of an elementary plasma
volume can be written as r D r 0 LJ.t/, where r 0 denotes the initial coordinate
of the elementary volume. Whence it follows that the plasma velocity is given
by: V D r 0 dLJ=dt D r .dLJ=dt/=LJ. Substituting this expression into Eq. ( 11.65 )
and then transforming Euler's variables r;t to Lagrange's ones; that is to r 0 ;t,we
come to