Geoscience Reference
In-Depth Information
geomagnetic dipole field in the spherical coordinates
{
r, θ, ϕ
}
with the polar
cos
θ/r
2
. One can introduce the
axes directed along the dipole moment is
Ψ
∝
dipole coordinates
{
ν, µ, ϕ
}
connected with the spherical coordinates as
sin
2
θ
r
µ
=
cos
θ
r
2
ν
=
−
,
ϕ
=
ϕ,
.
(6.33)
A field line lying within a magnetic shell
ν
= const, crosses the equatorial
plane
θ
=
π/
2at
r
=
r
e
=
LR
E
,
where
L
=
−
1
/
(
νR
E
) is the McIllwain
parameter.
Basic vectors of the dipole coordinate system (
e
ν
,e
ϕ
,e
µ
) expressed in
terms of orthonormal basis (
i
r
,
i
θ
,
i
ϕ
) of the spherical coordinate are
e
ν
=
sin
2
θ
r
2
sin 2
θ
r
2
1
r
sin
θ
i
ϕ
,
e
µ
=
2cos
θ
r
3
sin
θ
r
3
i
θ
,
e
ϕ
=
i
r
−
−
i
r
−
i
θ
.
(6.34)
|
−
1
yields the Lame coe
cients for the
Substitution of (6.34) into
h
n
=
|
e
n
dipole system:
r
2
sin
θ
(1 + 3 cos
2
θ
)
1
/
2
,
h
ν
=
ϕ
=
r
sin
θ,
r
3
(1 + 3 cos
2
θ
)
1
/
2
.
h
µ
=
h
ν
h
ϕ
=
(6.35)
Suppose that the ionosphere-atmosphere interface
S
a
is a sphere of the
radius
R
I
≈
R
E
, where
R
E
is the Earth's radius. Then the equation for
S
a
is
r
=
R
E
and in the dipole coordinates it becomes
R
−
E
(1 +
νR
E
)
1
/
2
,
where plus (minus) corresponds to the northern (southern) ionosphere, re-
spectively.
For the coecients
K
1
and
K
2
from (6.30) at
d
=2weget
µ
=
Γ
N,S
(
ν
)=
±
and
K
2
=
R
E
(1
νR
E
)
−
1
/
2
.
K
1
=0
,
−
(6.36)
And for new coordinates
ξ
=
x
1
,η
=
x
2
,ζ
=
x
3
we obtain
ζ
=
µR
E
(1 +
νR
E
)
−
1
/
2
,
ξ
=
ν,
η
=
ϕ,
(6.37)
and from (6.37)
µ
=
ζR
−
2
E
(1 +
νR
E
)
1
/
2
.
ν
=
ξ,
ϕ
=
η,
In the coordinates
ξ, η, ζ
, the region
Ω
between two magnetic shells
L
1
and
L
2
, is transformed into a parallelepiped
Π
:(
L
−
1
1
L
−
2
,
0
−
<ξ<
−
≤
η<
2
π,
1
<ζ<
1)
.
The boundary
S
a
is transformed at these coordinates
into rectangles
S
+
and
S
−
for the northern and southern ionospheres. The
components of the metrical tensor for this coordinate system are given in
Appendix 6.A.
−
Search WWH ::
Custom Search