Geoscience Reference
In-Depth Information
The spectra of toroidal and poloidal oscillations can be found from the solution
of 1D-eigenvalue problems (6.47) and (6.50).
1) The toroidal mode
s
=0;
S
+
ω
2
c
2
A
∂
∂s
h
2
h
1
∂E
1
∂s
h
2
h
1
E
1
=0
,
∂E
1
∂s
=0
,
(11.1)
2) The poloidal mode
s
=0;
S
+
ω
2
c
2
A
∂
∂s
h
1
h
2
∂E
2
∂s
h
1
h
2
E
2
=0
,
∂E
2
∂s
=0
,
(11.2)
where
h
1
and
h
2
are Lame coecients of the coordinate system (6.27) related
to the geomagnetic field, the field-aligned coordinate
s
is a distance along the
field-line,
S
is the length of the field line between two conjugate ionospheres.
Let
L
be a McIllwain parameter,
λ
is a geomagnetic longitude of the field
line footpoint on the ground, then the eigenvalues of boundary problems (11.1)
and (11.2) form two discrete sets continuously dependent on
L
and
λ
:
ω
n
(
L, λ
)and
ω
n
(
L, λ
)
.
Each field line is characterized by two sets of resonance frequencies com-
pletely determined by the field-aligned plasma density distribution and field
line geometry. Field-aligned distribution of plasma density can be unambigu-
ously determined from the spectra of these boundary problems (11.1) and
(11.2). The problem of the hydromagnetic diagnostics of the magnetosphere
is reduced to the experimental determination of FLR-frequencies and solution
of the inverse problems (11.1)-(11.2) ([7], [9], [18], [20]).
The usefulness of the inverse problem hydromagnetic diagnostics is limited
by the instability of the solution, i.e., its sensitivity to small disturbances of
the initial data. Such disturbances always exist because it is impossible to get
a complete FLR-spectrum from measured geomagnetic pulsations. Not only
because of measurement inaccuracy but also because of the variability of the
magnetosphere itself as plasma density changes during observation time. To
make the inverse problem stable, it is necessary to select a class of functions
C
ρ
for the sought plasma density distributions
ρ
(
s
). Then the distribution
found from the inverse problem will have a bounded confidence band
.
The bandwidth will be narrower for a narrower class of functions
C
ρ
and for
more accurate and full initial data. Such a technique can be used if the class
of functions
C
ρ
can be determined from the information about the plasma
density distribution obtained with other experimental methods.
Summarizing, the plan for the solution of the problem of hydromagnetic
diagnostics is as follows:
{
ρ
(
s
)
}
1. Find FLR-frequencies from measured ULF-fields.
2. Select class of functions
C
ρ
for the plasma density distribution from inde-
pendent experimental data.
Search WWH ::
Custom Search