Geoscience Reference
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1. Two coordinates (say
x
1
,x
2
) set the field line of the external magnetic
field
B
0
and the third (
x
3
) sets the location of a point on this field-line.
2. Boundary surfaces where boundary conditions are set must coincide with
some coordinate surfaces.
Usually, there is no orthogonal coordinate system satisfying these two con-
ditions and field-lines are non-orthogonal to the boundary surfaces. In the
Earth's magnetosphere, the geomagnetic field-lines can be considered as or-
thogonal to the lower ionosphere boundary only in the polar regions, while in
the middle and low latitudes geomagnetic field inclination
I
=
π/
2mustbe
taken into account. In this situation, it is reasonable to abandon an orthogonal
system of coordinates in favour of a non-orthogonal one which results only in
an insignificant complication in writing plasma oscillation equations.
We use an orthogonal curvilinear coordinate system related to the geom-
etry of the background magnetic field
B
0
(
r
). Assuming that
B
0
(
r
)canbe
expressed through a scalar potential
Ψ
(
r
):
B
0
=
−
∇
Ψ
, we introduce the
following coordinates:
y
1
=
Φ
1
(
x, y, z
)
,
2
=
Φ
2
(
x, y, z
)
,
3
=
Ψ
(
x, y, z
)
,
(6.27)
are Cartesian coordinates. The coordinate
y
1
marks magnetic
shells (for example,
y
1
may be proportional to the magnetic flux inside the
corresponding shell), coordinate
y
2
specifies a field line on a chosen shell, and
y
3
{
x, y, z
}
where
is a coordinate along a field-line. In the axially symmetric case
y
2
=
ϕ
,
where
ϕ
is an azimuthal coordinate. The element length is
d
s
2
=
h
1
(d
y
1
)
2
+
h
2
(d
y
2
)
2
+
h
3
(d
y
3
)
2
,
where
h
1
,
h
2
,and
h
3
are Lame coecients.
In this system, the line of intersection of the surfaces
Φ
1
(
x, y, z
) = const
and
Φ
2
(
x, y, z
) = const with the equipotential surface
S
3
(
y
3
=
Ψ
(
x, y, z
)=
const) are the lines of curvature on the surface
S
3
. These are the lines tangents
to which in each point of
S
3
belong to one of two orthogonal planes of the
principal normal sections. The Lame coecients of this coordinate system are
h
1
=
∇
Φ
1
−
1
,
2
=
∇
Φ
2
−
1
,
3
=
∇
Ψ
3
−
1
.
Thus, the orthonormal local basis is
i
1
=
h
1
∇
Φ
1
,
Φ
2
,
i
2
=
h
2
∇
i
3
=
h
3
∇
Ψ.
Denote by
S
a
the interface between the ionosphere and the atmosphere
and by
S
m
the magnetopause surface. Write the equation of the surface
S
a
in
the form
y
3
=
Γ
N
(
S
)
(
y
1
,y
2
)
,
(6.28)
where index
N
(
S
) denotes the Northern (Southern) Hemisphere. Set the ex-
ternal boundary of the magnetosphere by
y
1
=const
.
(6.29)
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