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to estimate the width
δ
of FLR, and three resonant frequencies
f
A
,f
+
,f
−
of the field-lines crossing the meridian between two stations (
x
=
x
c
)andat
some points to the north and to the south (
x
=
x
−
,x
=
x
+
).
Inverse Problem of MHD-Sounding
We choose a class of functions
C
ρ
for the field-aligned plasma density distri-
bution in the form
ρ
=
ρ
0
(
L
)
LR
E
r
q
,
where
ρ
0
(
L
) is the plasma density at the top of the field line with the McIllwain
L
parameter,
r
is the distance along the field line from the Earth's center to
a point on the field line
L
,
LR
E
is the distance from the Earth's center to the
top of this field-line. Assume
n
=
ρ/m
p
,
where
m
p
is the proton mass. For a
multicomponent plasma
n
=
α
n
α
m
α
m
p
,
(11.8)
where
n
α
and
m
α
are the concentration and mass of
α
-ions. Let the distrib-
ution of the plasma concentration in the magnetospheric equatorial plane be
a power function of
L
:
n
0
=
aL
−s
,
0
(
L
)=
m
p
n
0
.
(11.9)
Consider only the inner magnetosphere where the dipole model of the external
magnetic field is valid and rather a narrow
L
range with a negligible variation
of the parameters
a
and
s
.
Below, two variants of the hydromagnetic diagnostics are treated. In the
first method
q,
and
n
0
are obtained from frequencies of two or more FLR-
harmonics at one magnetic shell. In the second one, the parameters
a
and
s
are found from the FLR-harmonic measured at two or more
L
-shells (11.9).
FLR-resonance frequencies for a dipole magnetic field and high integral
ionospheric conductivity are found from the boundary problem (6.79), (6.80).
New dependent variables are chosen. Let
e
2
=
i
L
2
R
E
c
B
eq
2
√
πm
p
n
0
L
4
R
E
,
e
1
=
y
1
,
Ωy
2
,
Ω
(
n
0
,L
)=
where
Ω
(
n
0
,L
) is frequency scale (6.93),
B
eq
is the magnetic field on the
Earth's equator. The boundary problem (6.79), (6.80) for the field-aligned
plasma density distribution (11.8) becomes
dy
1
dw
w
2
)
6
−q
y
2
,
−
ω
(1
−
=
dy
2
dw
=
ωy
1
,
(11.10)
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