Geoscience Reference
In-Depth Information
coecient
C
0
:
⎛
⎝
⎞
⎠
−
1
d/
2
d/
2
cm
2
2
ω
r
(
z
0
)
g
22
0
g
11
0
√
g
0
g
2
0
e
1
d
x
3
d
d
x
3
d
x
3
.
C
0
=
i
e
1
e
2
(6.73)
−d/
2
−d/
2
Here the small terms relate to Hall conductivity are neglected. These terms are
small if the length of the field line in the ionospheric
E
-layer is small in com-
parison with the length of the field line between the conjugated ionospheres.
It is seen from (6.59) with taking into account (6.70), (6.71) and (6.73) that
near the resonance magnetic shell
x
1
=
x
0
the distributions of the components
b
2
(
x
1
)and
E
1
(
x
1
) are the resonance curves with the half-width
δ
.Iftosuppose
that an external source is given with a real frequency
ω
, then the boundary
problem (6.68), (6.69) determines the corresponding magnetic shell
x
0
=Re
z
0
and the resonance half-width
δ
=Im
z
0
.At
δ
=0
,
from the boundary problem
(6.68), (6.69) we can find resonance frequencies
ω
and decrement
γ
.
In (6.68) the boundary conditions (6.69) correspond to one field line and
it is not necessary to write it in the non-orthogonal coordinates. Return to
the initial orthogonal coordinate system. From the change of variables (6.30)
we have
∂
∂x
3
1
K
2
∂
∂y
3
=
and (6.68) reduces to
⎛
⎞
ik
0
h
1
h
2
1
h
3
d
d
y
3
e
1
(
y
3
)
e
2
(
y
3
)
= 0
−
⎝
⎠
(6.74)
1
h
3
d
d
y
3
ik
0
h
2
h
1
−
ε
⊥
with
e
1
= 0 at the field line pierce point between ionosphere and atmosphere.
The obtained formulae yield a key to the principal features of Alfven reso-
nance oscillations. In the next sections of this chapter, numerical methods for
the realization of these theoretical results and their application to the Earth's
magnetosphere are given.
6.4 FLR in the Dipole Geomagnetic Field
Before we turn to a closer examination of the wave-field structure in the
vicinity of a resonant line in the dipole field, we shall describe the simplified
models of the magnetospheric plasma distributions and the ionosphere which
are used in numerical calculations.
Model Magnetospheric Plasma Distribution
The models of a plasma distribution in the Earth's magnetosphere are
based on equations of hydrostatic equilibrium with taking into account the
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