Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 301

A w 3
r

A
r g .r;t/ cos 2 0 cos ' 0 ;

g 2 .r;t/ sin 2 0 cos ' 0 ;V D

V r D

A
r g .r;t/ cos 0 sin ' 0 ;

V ' D (7.76)

where r, 0 , and ' 0 are spherical coordinates. The angles 0 and ' 0 are measured
from axis z 0 and x 0 , respectively. The functions g 2 , g and constant A are given
by Eq. ( 7.71 ) where Πu z should be replaced by Πu x . The expressions for the
velocity components become more complicated in the general coordinate system
.x;y; z /.The velocity transform from the local coordinate system to the general one
can be written as follows:

V x ;V y ;V z D O F V r ;V 0 ;V ' 0 :

(7.77)

Here the components of the transfer matrix O F are given by

F 11 D F 1 sin 0 cos 0 sin Ǜ sin ;
F 12 D F 2 sin 0 C cos 0 cos Ǜ sin ;
F 13 D cos 0 cos C sin 0 sin sin LJ ' 0 ;
F 21 D F 1 cos 0 C sin 0 sin Ǜ sin ;
F 22 D F 2 cos 0 sin 0 cos Ǜ sin ;
F 23 D cos 0 sin sin LJ ' 0 sin 0 cos ;
F 31 D cos Ǜ sin LJ ' 0 sin Ǜ cos cos LJ ' 0 ;
F 32 D cos Ǜ cos cos LJ ' 0 C sin Ǜ sin LJ ' 0 ;
F 33 D cos cos LJ ' 0 ; (7.78)

where

F 1 D cos Ǜ cos LJ ' 0 C sin Ǜ cos sin LJ ' 0 ;
F 2 D sin Ǜ cos LJ ' 0 cos Ǜ cos sin LJ ' 0 : (7.79)

In order to find the effective magnetic moment of the induction currents
generated by the shear crack one should substitute Eqs. ( 7.76 )-( 7.78 ) for the mass
velocity into Eq. ( 7.55 ). Performing integration with respect to angles 0 and ' 0 and
neglecting the near-field contribution to the integral, we obtain

M x Df sin I 2 cos I .cos Ǜ cos LJ C sin Ǜ sin LJ cos / g ‰;
M y Df sin I .sin Ǜ cos LJ cos cos Ǜ sin LJ cos 2/
C cos I sin .sin 2Ǜ sin LJ cos C cos 2Ǜ cos LJ/ g ‰;