Geoscience Reference
In-Depth Information
Equations for Alfven and FMS-Waves
Let us write Maxwell's equations in a curvilinear magnetic field in a coordi-
nate system
x
1
,x
2
,x
3
{
}
, which is generally not-orthogonal. The contravariant
components of
∇
×
are easily expressed through the covariant components.
For example,
∇
×
of the electric field is given by
e
1
e
2
e
3
∂
1
∂
2
∂
3
E
1
E
2
E
3
1
g
1
/
2
∂
∂x
k
,
∇
×
E
=
,
k
=
(6.38)
where
g
= det(
g
ik
).
For the perfect longitudinal conductivity,
σ
→∞
, the component of the
electric field parallel to the geomagnetic field
E
3
=
E
e
3
vanishes and
E
=
E
⊥
=
E
1
e
1
+
E
2
e
2
. The other components may be written as a 5-component
1-column matrix
·
U
=(
b
2
,E
1
,b
1
,b
3
,E
2
)
tr
,
where
E
k
,b
k
are the covariant components of the electric and magnetic fields;
the tr marks the transpose operation,
A
tr
is transposed to
A
.
Combining Maxwell's equations with Ohm's law and using (6.38), we ob-
tain the equation for
U
:
A
1
∂
x
2
+
C
U
=0
,
∂
∂
x
1
+
A
2
∂
(6.39)
where
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
000 0 0
000 0 0
000 0 0
000 0
00000
00010
00000
01000
00000
A
1
=
,
A
2
=
,
−
1
−
000
10
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
−
g
22
00
g
23
0
0
ε
11
00
ε
12
00
g
11
0
1000
−
10000
00001
00000
00100
∂x
3
+
ik
0
√
g
∂
g
13
0
,
C
=
g
32
0
g
31
g
33
0
ε
21
00
ε
22
0
ε
11
=
ε
⊥
g
11
, ε
22
=
ε
⊥
g
22
,
ε
12
=
ε
⊥
g
12
+
ε
g
33
/g,
ε
g
33
/g.
ε
21
=
ε
⊥
g
21
−
In the magnetosphere
ε
⊥
=
ε
m
and in the ionosphere
ε
⊥
=
i
4
πσ
P
/ω
.
Respectively,
ε
= 0 in the magnetosphere and
ε
=
i
4
πσ
H
/ω
in the
ionosphere.
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