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low frequency limit, when !<
j
k
?
j
V
AM
, the real part of the function
M
becomes
negative that means that the FMS mode exponentially decreases with altitude in the
region
z
>L.
Inside the resonance cavity .0<
z
<L/ the solutions of the wave equa-
tions (
5.12
) and (
5.13
) must include both the upward and downward propagating
waves, which arise due to the wave reflection from the lower .
z
D
0/ and upper
boundaries .
z
D
L/ of the IAR. These solutions should be matched at this boundary
under the requirement of continuity of the magnetic field components and transverse
electric field
e
?
at
z
D
L . Whence there follows the requirement of continuity of
the potentials dž and ‰ and their derivatives, @
z
dž and @
z
‰,at
z
D
L. The solution
of the problem derived in Appendix D can be written as
dž
D
dž.0/
cos
!
z
:
V
AI
C
iLJ
1
sin
!
z
(5.18)
V
AI
‰
D
‰.0/
cosh
I
z
:
L
C
LJ
2
sinh
I
z
(5.19)
L
Here we made use of the following abbreviations
I
D
k
2
L
2
x
0
(5.20)
?
1
C
.1
/ exp .2ix
0
/
1
C
C
.1
/ exp .2ix
0
/
;
LJ
1
D
(5.21)
I
C
M
.
I
M
/ exp .2
I
/
I
C
M
C
.
I
M
/ exp .2
I
/
:
LJ
2
D
(5.22)
where
D
V
AI
=V
AM
1 is the ratio of the Alfvén velocity in the IAR to that in
the magnetosphere.
5.1.5
Boundary Conditions at the
E
Layer of the Ionosphere
At altitudes of the E-layer the electron gyrofrequency, !
H
, is much greater than the
sum,
e
, of electron-ion and electron-neutral collision frequencies. In contrast, the
ion gyrofrequency,
H
, is much smaller than the sum,
i
, of ion-electron and ion-
neutral collision frequencies. This means that inside the E-layer the collisions play
a crucial role in the behavior of ions whereas the electrons are guided by the Earth's
magnetic field lines since they are magnetized. The tensor of plasma conductivity
by Eq. (
2.5
) describes this kind of the anisotropy in the plasma conductivity which
occurs in the E-layer of the ionosphere. Furthermore, in the ULF range the tensor
components are nearly independent of the wave frequency, !, because the value of
! is much smaller than all the collision frequencies and gyrofrequencies entering the
tensor. In this case the components of the tensor can be expressed through Pedersen,
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