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The use of such an analytical approximation makes it possible to express the
IAR eigenfunctions in terms of Bessel functions and to obtain simple analytical
expressions for the eigenfrequencies and the IAR damping/growth rates. In order to
make our consideration as transparent as possible we, however, choose a simplified
approximation (e.g., Pokhotelov et al.
2001
), which describes the Alfvén velocity in
terms of a piece-wise function, so that V
A
D
V
AI
V
A0
within the IAR .0<
z
<L/
and V
A
D
V
AM
V
A0
1
in the outer magnetosphere .
z
>L/, where V
AI
and
V
AM
are constant quantities and L denotes the characteristic width of the resonance
cavity (IAR). Notice that V
AI
is typically much smaller than Alfvén speed V
AM
in
outer magnetosphere. The model altitude profile of the Alfvén velocity used in this
study is shown in Fig.
5.4
with dotted line.
The vertical
z
axis is positive parallel to
B
0
, while the x and y axes are parallel
to the plane boundaries of ionospheric layers. The region above the E-layer is
supposed to be the area consisting solely of cold collisionless plasma. The plasma
dielectric permittivity tensor,
, in this region is assumed to be diagonal with
components given by Eqs. (
2.18
) and (
2.19
).
"
5.1.2
Fourier Transform of Maxwell Equations
In what follows all perturbed quantities are considered to vary as exp.
i!t/, where
! is the frequency. Let ı
B
be the small perturbation of the geomagnetic field
B
0
,so
that ıB
B
0
. The Maxwell equation (
2.17
) is reduced then
i!
c
2
"E:
r
ı
B
D
(5.2)
We are thus left with the set of Maxwell equations (
5.2
) and (
1.2
), where
B
should
be replaced by ı
B
. This set should be supplemented by Eq. (
2.18
) for the tensor of
dielectric permittivity of plasma.
In what follows the
z
axis is positive parallel to the constant and homogeneous
magnetic field
B
0
. Since the medium is assumed to be uniform and infinite in the
direction perpendicular to the unperturbed magnetic field
B
0
, it is customary to
seek for the solution of Maxwell equations in the form of Fourier transform over
perpendicular coordinates x and y, for example,
Z
Z
ı
B
.!;
;
z
/
D
exp.ik
x
x
C
ik
y
y/
b
.!;
k
?
;
z
/dk
x
dk
y
;
(5.3)
1
1
.x;y/ and
k
?
D
k
x
;k
y
are the position vector and the wave vector,
correspondingly, both perpendicular to the unperturbed magnetic field
B
0
.The
inverse Fourier transform is given by
where
D
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