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@
y
ıB
z
@
z
ıB
y
sin
C
@
x
ıB
y
@
y
ıB
x
cos
D
i!
V
A
.E
x
sin
C
E
z
cos /;
(6.73)
i!
V
A
@
z
ıB
x
@
x
ıB
z
D
E
y
;
(6.74)
where V
A
D
c="
1=2
?
is the Alfvén velocity. These equations should be supplemented
by the Faraday's law given by Eq. (
4.2
), where
B
should be replaced by ı
B
.
The neutral atmosphere .
d<
z
<0/ is considered as an insulator, and the solid
Earth .
z
<
d/ as a uniform conductor with a constant conductivity
g
.Ifthe
displacement current in both media is disregarded, the electromagnetic perturbations
are described by Eqs. (
5.27
) and (
5.28
).
6.4.3
Transfer Matrices
Solution of the above problem with proper boundary conditions relates the magnetic
perturbations on the ground surface with the forcing functions, i.e., the amplitudes
of the MHD waves and of the wind-driven ionospheric current. We seek for the
solution of the problem in the form of spatiotemporal Fourier transform. This
implies that all quantities vary as exp.i
k
R
i!t/, where
k
is horizontal wave
vector, and
R
D
.x;y/.Letı
b
.
k
;!;
z
/ and ı
e
.
k
;!;
z
/ be Fourier transforms
of the magnetic and electric field variations, respectively. Let ı
b
.m/
.
k
;!/ be the
spectral amplitude of the incident Alfvén and FMS waves in the ionosphere, while
ı
I
.
w
/
.
k
;!/ stands for the height-integrated wind-driven ionospheric current. This
latter value denotes a Fourier transform of the functions I
.
w
x
, I
.
w
y
and I
.
w
z
given
by Eqs. (
6.70
) and (
6.72
), respectively. In consequence of linearity of both Maxwell
equations and boundary conditions, the spectral densities of the ionospheric and
atmospheric fields are coupled in a linear fashion through the transfer matrices
M
.
w
/
.
k
;!;
z
/ and
M
.m/
.
k
;!;
z
/
ı
b
.
k
;!;
z
/
D
M
.
w
/
.
k
;!;
z
/
ı
I
.
w
/
.
k
;!/
C
M
.m/
.
k
;!;
z
/
ı
b
.m/
.
k
;!/: (6.75)
We now omit the detailed derivation of the transfer matrices. The interested
reader is referred to the paper by Surkov and Hayakawa (
2007
,
2008
) for details.
If the vertical ambient magnetic field is assumed then an analytical solution of the
problem can be found for arbitrary value of
k
. As the magnetic field
B
0
is vertically
downward one should therefore substitute
D
=2 in the basic equations. In such
a case the set of Eqs. (
6.73
), (
6.74
), and (
4.2
) can be split into two independent sets,
which describe the shear Alfvén and FMS waves propagating in the magnetospheric
plasma. Similarly, the set of Eqs. (
5.27
) and (
5.28
) for the atmosphere and the
ground can be split into two independent sets, which describe the TM and TE modes
in the atmosphere. These two modes are coupled through boundary conditions at
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