Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 276

and time t. This indicates that variations of acoustic pressure are much greater than
those of magnetic pressure due to perturbations of the external magnetic field B 0
so that the electromagnetic perturbations do not influence acoustic ones. In practice
this is a good approximation, which can be applied to real seismic process (Kaliski
1960 ; Kaliski and Rogula 1960 ; Viktorov 1975 , 1981 ).
There is an exotic case which deserves mention, that is, as the magnitude of
the magnetic perturbation amounts to so large a value that the magnetic pressure
become comparable with the acoustic one. In this extreme case an MHD wave,
which depends on acoustic and electromagnetic parameters, can propagates as it
follows from hydrodynamical and Maxwell equations. A number of researchers
have assumed that such MHD modes can appear in the vicinity of the Earth kernel
boundary (Knopoff 1955 ; Keilis-Borok and Monin 1959 ), where the elastic and
magnetic pressures may be much stronger than that at the ground surface.
To study the generation and propagation of electromagnetic field in the Earth's
crust due to motion of the conductive layer of the ground, we need Maxwell's
electrodynamics equations, which are given by Eqs. ( 1.1 )-( 1.4 ). First of all let
us compare the displacement current with the conduction one in the conducting
ground. The displacement current can be estimated as j @ t D j "" 0 !E, where "
is the dielectric permittivity of the ground. This current is small compared to the
conduction one if E "" 0 !E. The last inequality is valid if ! =."" 0 /
10 7 -10 8 Hz. This means that in the seismic frequency range (0.1-10 Hz) the
displacement current is much smaller than the conduction one.
Considering seismic waves travelling through ground-based stations, we note
that in practice the sensors of acoustic and electromagnetic fields are situated in
the reference frame, K 0 , moving with the mass velocity V of the ground. As a first
step we start with consideration of the electromagnetic fields in an earth-fixed or
motionless coordinate system, K. In such a case Maxwell equation ( 1.1 ) is reduced
to the form given by Eq. ( 1.9 ).
Let ı B be the perturbation of the Earth's/external magnetic field B 0 , so that
B D B 0 C ı B . Usually the GMPs caused by the seismic waves are as much as
1-100 nT, that is much smaller than the typical value of the undisturbed geomagnetic
field B 0 50T. We consider therefore only weak perturbations, i.e., j ı B jj B 0 j .
Hence the last term on the right-hand sides of Eq. ( 1.9 ) can be reduced to the form
V B D V . B 0 C ı B / V B 0 . So, the basic equations of the problem are
given by

r ı B D 0 . E C V B 0 /; (7.7)

r E D @ t ı B : (7.8)

To find the variations of magnetic induction ı B we first divide Eq. ( 7.7 )by and
then take the cross product of the operator r with both sides of Eq. ( 7.7 ). Using

r ..1=/ r ı B / Dr .1=/ . r ı B / C .1=/ r . r B /