Geoscience Reference
In-Depth Information
Sometimes matrix
T
Σ
is useful:
T
=
T
Σ
1
+
R
.
b
(
g
)
=
T
Σ
b
τ
,
(7.89)
Matrices
R
and
T
could be found after cumbersome but simple enough
algebra from the equalities (7.71)-(7.78) for
k
2
= 0. However, it is more
simple to proceed from (7.7)-(7.10) and the boundary condition (7.50).
Without considering the Hall conductivity and for
k
2
= 0, equations (7.7)-
(7.10) become
∂
∂x
3
−
ik
1
cot
I
b
1
=
κ
s
ik
0
E
2
,
(7.90)
∂
∂x
3
−
ik
1
cot
I
E
2
=
−
ik
0
b
1
,
(7.91)
for the FMS-waves and
κ
2
A
∂b
2
∂x
3
=
−
ik
0
E
1
,
(7.92)
∂E
1
∂x
3
=
ik
0
b
2
,
(7.93)
for Alfven waves. Here
κ
s
=
k
0
ε
m
−
k
1
1
/
2
,
Re
κ
S
>
0
and
,
A
=
k
0
ε
1
/
m
=
ω/c
A
.
The admittance matrix in the boundary condition (7.50) for
k
2
= 0 has the
form
κ
A
=
k
A
/
|
sin
I
|
⎛
⎝
⎞
⎠
Y
sin
I
ζ
4
ζ
3
−
X
−
Y
I
=
.
(7.94)
ζ
1
ζ
2
−
X
sin
2
I
Y
sin
I
−
Components
b
2
and
E
1
(Alfven waves) and components
b
1
and
E
2
(FMS-
waves) from (7.90)-(7.93) are given by
b
1
=
e
ik
1
x
3
cot
I
e
−iκ
s
x
3
+
R
SS
e
iκ
s
x
3
b
(
i
)
+
R
SA
e
iκ
s
x
3
b
(
i
2
,
(7.95)
1
κ
s
e
ik
1
x
3
cot
I
e
−iκ
s
x
3
R
SS
e
iκ
s
x
3
b
(
i
)
R
SA
e
iκ
s
x
3
b
(
i
2
,
k
0
E
2
=
−
−
(7.96)
1
+
e
−iκ
A
x
3
+
R
AA
e
iκ
A
x
3
b
(
i
)
b
2
=
R
AS
e
iκ
A
x
3
b
(
i
)
,
(7.97)
1
2
R
AS
e
iκ
A
x
3
b
(
i
1
+
e
−iκ
A
x
3
R
AA
e
iκ
A
x
3
b
(
i
2
,
k
0
κ
A
E
1
=
−
−
−
(7.98)
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