Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 290

r@ r V C 2V in Eq. ( 7.38 ) depends on the distribution of the mass velocity, which
becomes zero ahead of the acoustic wave front. So, the region of integration is
restricted, in fact, by the radii within interval R 0 <r<R 0 C C l t. Moreover,
far away from the seismic source, when C l t R 0 , the mass velocity V reaches a
peak value within a short distance/interval near the wave front. The characteristic
longitudinal scale/wavelength of this interval is supposed to be much smaller than
the distance r and the diffusion radius r d . In fact, this means the validity of the short
wavelength approximation in evaluating the integral in Eq. ( 7.38 ). It follows from
Appendix G that the approximate expressions for the GMP are given by (Surkov
2000a , b )

@ r G 1
r

;

u 0 SB 0 cos
4r

ıB r D (7.43)

@ r G 1

r @ r G 1 ;

u 0 SB 0 sin
8r

ıB D (7.44)

@ r 1

r @ r G 1

u 0 SB 0 m sin
8

E 0 D

(7.45)

where

erfc r C 2r l
r d

C exp r l r
m

erfc r 2r l
r d

G 1 .r;t/ D exp r l C r
m

2erfc r
r d

C 2 1 exp r l r
m

.r l r/:

(7.46)

is the area of source surface, r d D 2. m t/ 1=2 is
the radius of the diffusion front, r l D C l t is the radius of the seismic wave front, and
m D m =C l is the length of the magnetic forerunner, .x/ is the step-function, i.e.,
.x/ D 1 if x 0, otherwise .x/ D 0, and erfc .x/ D 1 erf .x/ where erf .x/
denotes the error function, i.e.,

Here u 0 is the static displacement, S

Z

x

exp y 2 dy:

2
1=2

erf .x/ D (7.47)

0

As is seen from Eqs. ( 7.43 )-( 7.45 ), all the components of electromagnetic
perturbations are proportional to the value of u 0 S, that equals the increment of the
source volume. The last value is in turn proportional to the seismic energy emitted
by the source in the form of acoustic waves. So, the magnitude of electromagnetic
perturbations caused by the spherically symmetric longitudinal wave is directly
proportional to the seismic energy. In what follows we will show that this conclusion
is valid for the sources with other, i.e., nonspherical, shapes.