Geoscience Reference
In-Depth Information
Wave-vector components are determined by covariant components
k
1
,k
2
,
k
A
3
and
k
1
,k
2
,k
S
3
. Transformation of the covariant components of oblique
coordinates to Cartesian coordinates has the form
k
1
sin
I
k
αx
=
−
+
k
α
3
cos
I,
k
αy
=
k
2
,
k
αz
=
−
k
α
3
sin
I,
(7.69)
where
α
=
A
or
α
=
S
.
The transformation inverse to (7.69), applied to components
E
or
b
, per-
mits the polarization vector to be found in oblique coordinates. For instance,
for
b
we get:
b
1
=
−
b
x
sin
I
−
b
z
cos
I,
b
2
=
b
y
,
b
z
sin
I
.
b
3
=
−
(7.70)
Using transformation (7.70), we find:
Alfven Waves
=
E
(
i,r
)
=
,
k
(
i,r
)
−
Ax
sin
I
k
(
i,r
)
Ay
c
A
k
(
i,r
)
A
E
(
i,r
)
Aτ
A
1
E
(
i,r
)
A
2
∓
=
b
(
i,r
)
=
k
(
i,r
)
,
Ay
sin
I
k
(
i,r
)
Ax
1
k
(
i,r
)
A
b
(
i,r
)
Aτ
A
1
b
(
i,r
)
A
2
E
(
i,r
)
A
3
(
i,r
)
A
3
=0
,
=0
.
(7.71)
FMS-Waves
⎛
⎝
−
⎞
⎠
,
=
E
(
i,r
)
=
k
(
i,r
)
Sy
sin
I
k
0
k
(
i,r
)
z
E
(
i,r
)
Sτ
S
1
E
(
i,r
)
S
2
k
(
i,r
)
Sx
k
(
i,r
)
S
−
⎛
⎞
=
b
(
i,r
)
=
sin
I
+
k
Sx
+
k
Sy
k
Sz
k
(
i,r
)
Sx
−
cos
I
1
k
(
i,r
)
S
⎝
⎠
b
(
i,r
)
Sτ
S
1
b
(
i,r
)
S
2
,
k
Sy
k
Sx
+
k
Sy
k
(
i,r
)
Sz
b
Sz
sin
I
1
sin
I
E
(
i,r
)
S
3
(
i,r
)
S
3
=0
,
=
−
=
.
(7.72)
k
(
i,r
)
S
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