Geoscience Reference
In-Depth Information
following numerical values V
A
.0:5-1/
10
3
km/s and L
10
3
km we obtain
the rough estimate f
n
.0:5-0:25/n (in Hz), where n
D
1;2;:::. Thus, the IAR
highlights eigenfrequencies of several Hz that lie below the Schumann resonances.
To study with rigorous formulation of the problem of eigen oscillations one
should examine the dispersion relation of the IAR, that is the dependence of !
D
!.k
?
/. In the framework of the model developed above, the IAR eigenfrequencies
are defined by the zeros of the factor q in the denominator of Eq. (
5.32
). Hence
taking q
D
0, we get
.is
C
x
0
Ǜ
P
/.LJ
1
C
Ǜ
P
/
C
x
0
Ǜ
2
H
D
0:
(5.36)
This equation defines an implicit dependence of eigenfrequencies on k
?
.
5.2.2
Shear Alfvén and FMS Modes for the Case of Zero
Hall Conductance
We start our analysis with the simplified case of small Hall conductance †
H
.
In the first approximation we omit the terms proportional to Ǜ
2
H
in Eq. (
5.33
)
for F
0
and Eq. (
5.34
)forq whereas the terms linear in Ǜ
H
can be kept
(Pokhotelov et al.
2001
). The general dispersion relation (
5.36
) in this case
decouples into two branches/modes. The first one corresponds to the roots of
the following equation:
LJ
1
C
Ǜ
P
D
0:
(5.37)
The dispersion relation (
5.37
) does not depend on k
?
and correspond to the shear
Alfvén mode. Indeed, on account of Eq. (
5.21
)forLJ
1
,Eq.(
5.37
) can be rewritten as
1
C
1
.1
C
Ǜ
P
/
.1
Ǜ
P
/
:
exp .2ix
0
/
D
(5.38)
Decomposing the dimensionless frequency x
0
in Eq. (
5.38
) into its real and imagi-
nary parts, i.e., x
0
D
C
i, one finds
1
C
1
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
1
C
Ǜ
P
1
Ǜ
P
exp .2i
2/
D
exp .2i#
Ǜ
C
2in/;
(5.39)
where n is an integer, n
D
1;2;:::and
2
arg
.1
C
Ǜ
P
/
1
#
Ǜ
D
.1
Ǜ
P
/
:
(5.40)
At the nighttime conditions the Pedersen conductivity is small so that Ǜ
P
<1.It
follows from Eq. (
5.40
) that #
Ǜ
D
0 when 0
Ǜ
P
<1and #
Ǜ
D
=2 when Ǜ
P
>1.
Search WWH ::
Custom Search