Geoscience Reference
In-Depth Information
Its solution requires rather complicated computations to be carried out for a
wide range of wavenumbers
k
τ
. However, these equations can be simplified in
the case of a 'thin' ionosphere. This approach holds when the 'optical' thick-
ness of the ionosphere for the transversal Alfven waves is small, and the skin
depth scale
d
P
=
c/
(2
πωσ
P
)
1
/
2
is larger than the thickness
l
I
of the highly
conductive ionosphere region. We shall restrict ourselves to the consideration
of oscillations with periods
T
≥
10 s, for which the
d
P
l
I
.
Two limiting cases must be distinguished here:
•
Small horizontal wavenumbers
k
τ
, when
k
τ
l
I
1.
Then the electric field variation in the ionosphere is a small constant with
an accuracy of an order of
d
P
/l
I
and
k
τ
l
I
.
•
Large
k
τ
. In this case
k
τ
l
I
1. It is seen from (7.7)-(7.10) that the electric
field can vary significantly at the ionospheric thickness. But at large
k
τ
,
wave mode coupling is weak, FMS-waves scarcely affect the Alfven waves
and it is possible to use the thin ionosphere model for the Alfven waves
in this case as well. The precise condition for the applicability of this
approximation will be given in Section 7.7. The numerical calculations have
shown that the application area of these two simple limiting cases overlap
and the interaction of the MHD-waves with the ionosphere is described
within these simple approximations.
Large Horizontal Scales
Let the scales
k
τ
l
1. Outside the equatorial region, integrating (7.7)-(7.10)
with a condition
k
τ
l
I
1 along the field-lines, yields
=
4
π
{
E
τ
}
=0
,
{
b
τ
}
c
I
×
z
,
I
=
Σ
E
,
(7.47)
b
τ
=
b
1
b
2
,
E
τ
=
E
1
E
2
,
Σ
=
Σ
11
,
Σ
12
(7.48)
Σ
21
Σ
22
Σ
P
sin
2
I
,
Σ
H
sin
I
,
Σ
11
=
Σ
21
=
−
Σ
12
=
−
Σ
22
=
Σ
P
.
(7.49)
Here
A
−
is the discontinuity of '
A
' on the ionosphere;
Σ
P
and
Σ
H
are the integral Pedersen and Hall conductivities, respectively, given by
{
A
}
=
A
+
−
l
I
l
I
Σ
P
=
σ
P
(
z
)
dz,
Σ
H
=
σ
H
(
z
)
dz.
0
0
Then the boundary conditions at the ionosphere for the tangential com-
ponents of the magnetic field
b
τ
and electric field
E
τ
become
b
τ
(+0) =
Y
I
E
τ
(+0)
,
(7.50)
where
Y
I
=
Y
(+0) is the admittance matrix above the ionosphere.
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