Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 298

so-called seismic moment tensor, M , which is considered in Appendix H. In the
case of tension cracks only three diagonal components of the seismic moment tensor
are nonzero (Aki and Richards 2002 ). The components of medium displacement
around the crack are given by Eq. ( 10.59 ). For convenience, we also introduce the
spherical coordinate system r;;', where the polar angle and azimuthal angle '
are measured from axes z 0 and x 0 , respectively. Taking time-derivative of Eq. ( 10.59 )
and rearranging this equations, we obtain the radial and transverse components of
mass velocity:

r ǚ g 1 .r;t/ C 2 w 2 g 2 .r;t/ cos 2 ;

A w

V r D

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

V D (7.70)

Here we made use of the following abbreviations:
g 1 D R u l z 1 2 w 2 C P u l z 1 4 w 2 C l

r C P u t z 2C t

C t
C l ; r w ; w D

g 2 D R u l z C P u l z 4C l

r P u t z 3C t r w 3 ;

g D R u t z C P u t z

r P u l z

2 w 3 C l
r

3C t

;

u z t ; u t z D u z t ;A D

u l z D

r
C l

r
C t

S
4C t ;

(7.71)

where S is the crack area and the dots above the symbols denote the time-
derivatives. These equations are valid in wave and intermediate acoustic zones. In
order to take into account the attenuation of acoustic waves we multiply the velocity
components by the acoustic damping factor T a .r;R/ D exp . r=L.R//, which
depends on the distance r and the crack radius R. The characteristic length of the
acoustic waves attenuation, L, is given by Eq. ( 10.16 ).
Substituting Eqs. ( 7.70 ) and ( 7.71 ) into Eq. ( 7.55 ), we obtain the effective
magnetic moment M of the induction currents generated in the conducting medium.
Performing integration over the region with radius r l D C l t and ignoring the near-
field contribution to the integral yields

w g 1 C

Z

r l

1 C sin 2 0

M D B 0 A 4
3

2 w 3 g 2
5

0

5 2 cos 2 0 sin 2 0 T a r 2 dr:

g

(7.72)