Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 295

The solution of these equations should match with that of Eqs. ( 7.63 )-( 7.65 ) through
standard boundary conditions at z D 0, as shown in Appendix G. As it follows from
equations, the solution of the problem for the atmosphere can be written as follows:

1
q .qk R p/ C s .sk R p/ exp . k R z C i /;

k R V 0
m .k R C p/

2

ıB x D

ıB z D iıB x ;ıB y D 0; (7.67)

where Rep>0and

1 D .iqB 0x B 0 z / ! 2

1

i!
m

C l

;

! 2
C t

1

2s.sB 0 z iB 0x /
.1 C s 2 /

i!
m

2 D : (7.68)

The electric field components are given by

( !p m iqk R C l ) exp . k R z C i /;

E x D iB 0y V 0 C l
C R

2
1 C s 2

qk R ! m iC l

iC R
C l E x ;E y D iqC R ıB x :

E z D (7.69)

As is seen from Eqs. ( 7.67 )-( 7.69 ), the amplitudes of the electromagnetic
perturbations decrease with altitude. A typical vertical scale of the field attenuation
in the atmosphere is of the order of k R , that is about Rayleigh wavelength.
If the vector of the Earth's magnetic field is parallel to the vertical plane in which
the medium particles move; that is, if B 0y D 0, then only three components ıB x ,
ıB z and E y are nonzero. Moreover, the density of electric charges is equal to zero
everywhere. This implies that the field is a vortex one in nature. By contrast, if
B 0y ¤ 0, then there is the vertical component of electric current j z D E z , which
leads to the generation of uncompensated electric charges in the conducting half-
space and on its surface.
The magnitude of the electromagnetic field variations in the region z <0
decreases with depth due to energy absorption depending on the relation between
seismic wavelength, R D 2=k R , and skin-layer depth in the conducting media,
ı D . 0 !=2/ 1=2 . It follows from Eqs. ( 7.115 )-( 7.117 ) that if ı 2 k R 1
which is valid under the requirement ! C R = m , then the typical scale of the
electromagnetic field damping is about seismic wavelength R . In such a case
the amplitudes of the magnetic field perturbations on the ground surface can be
estimated as follows: ıB x ıB z V 0 B 0 = . k R m / ıB y V 0 B 0 =C R while the
electric field estimates are E y C R ıB x E x E z V 0 B 0 . Not surprisingly, the