Geoscience Reference
In-Depth Information
6.6 Summary
In this chapter we treated both free and excited oscillations and the structure
of the MHD-waves in the axially symmetric magnetospheric models as well.
1. We found that spectrums of Alfven oscillations in 2D case can be reduced
to two 1D boundary problems:
•
determination of the toroidal oscillation spectrum from (6.51), (6.53)
and
determination of the poloidal oscillation spectrum from (6.52), (6.54).
2. We examined also the excitement of the oscillations by a harmonic source
of frequency
ω
close to the frequency
ω
(
T
j
(
L
r
) of the toroidal resonance
mode
j
of one of the FLR-shells with
L
=
L
r
.
It was shown that the spatial
distribution of the azimuthal magnetic component
b
ϕ
and of the electric
component
E
ν
orthogonal to the FLR-shellin the toroidal mode are Lorentz
resonance curves:
•
b
(
j
)
E
(
j
)
ϕ
ϕ
b
ϕ
≈
E
ν
≈
,
,
(
L
r
+
iδ
(
j
L
)
(
L
r
+
iδ
(
j
L
)
L
−
L
−
ϕ
and
E
(
j
)
ν
can be found from the longitudinal magnetic compo-
nent
b
(see (6.113), (6.114)).
3. The decrement
γ
(
T
)
j
where
b
(
j
)
(
L
) of the toroidal oscillations is connected with the
half-width
δ
(
j
)
L
and the spatial scale of the FLR-frequency
ω
(
T
)
j
:
d
ω
(
T
)
j
d
L
−
1
δ
(
j
)
L
γ
(
T
)
j
≈
.
L
=
L
r
4. Besides of toroidal modes, there are also poloidal modes. Contrary to
toroidal oscillations which manifest themselves in azimuthal magnetic com-
ponents and in electrical components orthogonal to the shell, poloidal os-
cillations of the shell are resonance oscillations of the magnetic component
orthogonal to the shell and of the azimuthal electric component.
5. Polarization splitting of the FLR-frequencies for the toroidal and poloidal
oscillations is determined by the difference of the two main curvatures of
the surface orthogonal to geomagnetic field-lines. In other words, the split
is determined by the difference of the convergency
/
divergency rate of the
field-lines within the meridional and equatorial plains.
Appendix A
The metric tensors
g
ik
and
g
ik
of the non-orthogonal system coordinates
x
1
,x
2
,x
3
{
}
are
Search WWH ::
Custom Search