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Z
Z
1
.2/
2
b
.!;
k
?
;
z
/
D
exp.
ik
x
x
ik
y
y/ı
B
.!;
;
z
/dxdy:
(5.4)
1
1
The same representation is true for the electric field
E
. In the text we use the big
letters to represent the functions of spatial variable
, while the small letters are
used to represent the functions of the perpendicular wave vector
k
?
, that is, the
Fourier transform of the electromagnetic field. Applying a Fourier transform (
5.4
)
to Maxwell equations (
5.2
) and (
1.2
) yields
i!
c
2
"e
i .
k
?
b
/
CO
z
@
z
b
D
;
(5.5)
i .
k
?
e
/
CO
z
@
z
e
D
i!
b
;
(5.6)
where
b
.!;
k
?
;
z
/ and
e
.!;
k
?
;
z
/ denote the Fourier transforms of the electromag-
netic variations and
O
z
D
B
0
=B
0
is the unit vector parallel to
B
0
.
5.1.3
Three- and Two-Potential Representation
of Plasma Waves
As it follows from the analysis in Sect.
2.2
, the electromagnetic perturbation in
plasma can be split into the shear Alfvén and compressional/FMS modes. As shown
in Appendix C, the electromagnetic field can be presented by scalar, dž, and vector,
A
, potentials or by three scalar potentials dž, A, and ‰. Particularly applying Fourier
transforms to Eqs. (
5.78
) and (
5.79
) we come to the following field representation
through the potentials
b
D
i
k
?
@
z
‰
C
i .
k
?
O
z
/A
CO
z
k
2
‰;
(5.7)
?
and
e
D
i
k
?
dž
.
k
?
O
z
/!‰
CO
z
.i!A
@
z
dž/:
(5.8)
In the ULF/ELF range when !
!
pe
, the absolute value of the parallel
component (
2.19
) of the plasma dielectric permittivity is much greater than unity
and thus can be assumed to be infinite whereas the total field-aligned Alfvén current
which includes the conduction and displacement currents, must be finite as it follows
from Eqs. (
5.82
) and (
5.86
). On account of equation j
z
D
i!"
0
"
k
e
z
, one can
conclude that the field-aligned electric component e
z
tends to zero in this frequency
range. Hence it follows that the potential A is coupled to dž via i!A
D
@
z
dž and we
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