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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