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where
B
0
.x/
Œ
0
.x/
1=2
;
V
A
.x/
D
(6.33)
is the Alfvén velocity. Despite the fact that the wave equations (
6.31
) and (
6.32
)are
independent, both the MHD modes can be coupled through the boundary conditions
at the E region of the ionosphere. So we need to consider the effect of the boundary
conditions on the spectrum of normal oscillations.
6.2.2
FLR Eigenfrequencies
For all frequencies of interest here the E region of the ionosphere can be considered
as a thin conductive layer with integral Pedersen and Hall conductivities. The
boundary condition (
5.26
)attheE layer relates the jump of horizontal magnetic
field across this layer to the horizontal electric field. We recall that the interaction
between the magnetospheric MHD waves and the ionosphere depends on value of
the dimensionless parameters Ǜ
P
and Ǜ
H
, which are equal to the ratio of height-
integrated Pedersen †
P
and Hall †
H
conductivities to the Alfvén wave parallel
conductance of the magnetosphere †
w
D
.
0
V
A
/
1
. As would be expected,
considering the FLRs due to the shear Alfvén waves propagation, the ionospheric
Pedersen conductivity plays an important role in closing of the field-aligned Alfvén
currents in the ionosphere, that is in the closing of the field lines perpendicular
to
B
. In this picture the Hall conductivity in the ionosphere is of minor importance
and in the first approximation it can be ignored (e.g., Krylov and Fedorov
1976
;
Krylov and Lifshitz
1984
). In this approach the wave perturbations coming from the
magnetosphere cannot penetrate through the conducting ionosphere so that we can
neglect the variations of the magnetic field below the ionosphere. In this way the
boundary condition (
5.26
) at the ionosphere reduces to
ıB
y
D˙
0
†
P
E
x
; and
ıB
x
D˙
0
†
P
E
y
;
(6.34)
where the sign plus on the right-hand side of Eq. (
6.34
) corresponds to the northern
ionosphere .
z
D
l
1
/ and the sign minus corresponds to the southern ionosphere
.
z
D
0/. Furthermore, †
P
stands for the height-integrated Pedersen conductivity
of the northern ionosphere while †
P
denotes the same value for the southern
ionosphere. Combining Eqs. (
6.27
), (
6.28
), and (
6.34
) we finally obtain
@
z
E
?
D˙
i!
0
†
P
E
?
;
(6.35)
where
E
?
D
E
x
;E
y
and
z
D
0 or l
1
.
We choose first to study the free Alfvén oscillations at J
x
D
0. In such a case
the solution of Eq. (
6.31
) can be written
E
x
D
C
1
exp .i!
z
=V
A
/
C
C
2
exp .
i!
z
=V
A
/;
(6.36)
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