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⎛
⎞
p
1
K
2
p
1
p
2
K
2
p
1
K
2
h
1
+
h
3
h
3
h
3
−
⎝
⎠
γ
2
K
2
p
1
p
2
K
2
p
2
K
2
h
3
h
2
+
h
3
h
3
(
g
ik
)=
−
,
(6.A.1)
h
3
K
2
p
1
K
2
p
2
K
2
h
3
h
3
−
−
⎛
⎝
⎞
⎠
1
h
1
p
1
p
2
K
2
p
1
h
1
h
3
g
ik
=
p
1
p
2
K
2
1
h
2
p
2
h
2
h
3
,
(6.A.2)
p
1
h
1
p
2
h
2
K
2
h
3
+
p
1
h
1
+
p
2
h
2
where
p
α
=
∂x
(3)
/∂y
(
α
)
=
∂K
1
/∂y
(
α
)
+
y
(3)
∂K
2
/∂y
(
α
)
. The metric tensors of
the non-orthogonal dipole coordinates
ξ, η, ζ
, obtained from (6.36), (6.A.1)
and (6.A.2), are
g
11
=
h
ν
+
R
E
h
µ
4
r
4
cos
2
θ
22
=
h
−
2
R
E
sin
2
θ/r
)
2
,
,
ϕ
(1
−
g
33
=
h
µ
(1
R
E
sin
2
θ/r
)
/R
E
,
−
h
µ
2
r
2
R
E
cos
θ
g
13
=
g
31=
R
E
sin
2
θ/r
)
1
/
2
,
(6.A.3)
(1
−
g
11
=
h
−
2
ν
22
=
h
−
2
ϕ
,
,
g
33
=
1
h
µ
+
R
E
h
µ
4
r
4
cos
2
θ
R
E
R
E
sin
2
θ/r
,
R
E
sin
2
θ/r
)
2
(1
−
1
−
R
E
cos
θ
g
13
=
g
31
=
−
R
E
sin
2
θ/r
)
3
/
2
.
(6.A.4)
2
r
2
h
ν
(1
−
References
1. Allan, W., S. P. White, and E. M. Poulter, pulse-excited hydromagnetic cavity
and field resonances in the magnetosphere,
Planet. Space. Sci.
,
34
, 371, 1986.
2. Allan, W. and D. R. McDiarmid, Magnetospheric cavity modes: numerical model
of a possible case,
J. Geophys. Res.
,
94
, 309, 1989.
3. Angerami, J. J. and J. O. Thomas, The distribution of ions and electrons in the
Earth's exosphere,
J. Geophys. Res.
,
69
, 4537, 1964.
4. Bhatacharjee, A., C. A. Kletzing, Z. W. Ma, C. S. Ng, N. F. Otani, and X.
Wang, Fourfield model for dispersive fieldline resonances: Effects of coupling
between shear Alfven and slow modes,
Geophys. Res. Lett.
,
26
, 3281, 1999.
5. Bilitza, D., IRI 2000,
Radio Science
,
36
, 261, 2001.
6. Boyce, W. E. and R. C. DiPrima,
Elementary Differential Equations and Bound-
ary Value Problems
, 7th edn., John Wiley & Sons, Inc., New York, 2001.
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