Geoscience Reference
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Substituting Eqs. ( 10.21 ) and ( 10.23 )for b and g into Eq. ( 10.25 ) and performing
integration over l gives the sought value of the mean magnetic perturbations. The
result of integration can be written in the form of a quasi-steady field of magnetic
dipole (Eq. ( 7.55 )) whose effective magnetic moment, M , is given by
1
B 0 :
LJkbǛ 2 l 52b
max
6.5 2b/
4 w 2
3
M D
(10.26)
The vector M is directed oppositely to the unperturbed geomagnetic field B 0 .Itis
not surprising since, as we have noted above, the conductor motion in external
magnetic fields must result in the usual diamagnetic effect, which makes for
formation of the effective magnetic moment with negative sign.
As is seen from Eq. ( 7.5 ) which describes the solution of problem, the average
level of the electromagnetic noise at far distances decreases as r 3 . This case that
is r L, corresponds to strong attenuation of acoustic emissions followed by
strong attenuation of GMPs. It follows from the numerical estimation (Surkov and
Hayakawa 2006 ) that at such far distances the crack-generated electromagnetic
noise is practically undetectable and thus this case-study is of little importance.
At short distances of interest here, that is, as r Ǜl max 50l max ,the
contribution of the small and large cracks to the integral ( 10.25 ) should be estimated
separately. As it follows from Eq. ( 10.15 ) and Eq. ( 10.23 ), which is valid for the
small cracks with length l r=Ǜ, the function b . r ;l/ in Eq. ( 10.21 ) is proportional
to l 5 and hence the integrand in Eq. ( 10.25 ) is proportional to l 42b
l 1:8 , where b
is the fractal dimension in Eq. ( 10.24 ). In the case of the large cracks when x 1,
Eq. ( 10.22 )forg is transformed to
u 0 B 0
2 m
g
:
(10.27)
In this case g is not a function of distance. This implies that the attenuation of the
acoustic waves radiated by large cracks is nearly unimportant.
It follows from Eqs. ( 10.21 ) and ( 10.27 ) that as long as l r=Ǜ the function
b . r ;l/ / l 3 and therefore the integrand in Eq. ( 10.25 ) is proportional to l 22b
l 0:2 . The rough estimate of the contributions to the integral ( 10.25 ) due to the small
and large cracks gives the ratio Œr=.Ǜl max / 32b . This means that at the distance r
Ǜl max the small cracks make a little contribution to the integral sum in Eq. ( 10.25 )
compared to that due to the large cracks. Taking the notice of this fact, substituting
Eqs. ( 10.21 ) and ( 10.27 )for b and g into Eq. ( 10.25 ), and performing integration,
we obtain
1
2 O r cos O
sin :
0 B 0 LJkbl 32b
4 w 2
3
max
4r .3 2b/
h ı B t . r / iD
(10.28)
It should be emphasized that Eq. ( 10.28 ) determines, as a matter of fact,
only a statistical average, which indicates the mean level of the crack-induced
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