Geoscience Reference
In-Depth Information
The matrix
R
(0)
and
T
(0)
in dimensionless variables may be written
⎛
2
k
A
−
2
Y
k
A
−
⎞
k
2
k
2
⎝
⎠
1+
sign
I
∆
(0)
SK
∆
(0)
A
∆
(0)
SK
R
(0)
=
,
(7.124)
2
Y k
A
2
k
A
Y
2
R
(0)
−
AA
+
∆
(0
A
2
∆
(0)
A
∆
(0)
SK
sin
I
∆
(0)
SK
|
sin
I
|
⎛
⎞
Y
sign
I
∆
(0)
A
T
(0)
SS
T
(0)
SS
⎝
⎠
T
(0)
=
,
(7.125)
k
0
Y
∆
(0)
S
T
(0)
AA
T
(0)
AA
−
sin
I
where
−
√
ε
m
|
2
ik
0
ε
a
∆
(0)
A
∆
(0)
S
∆
(0)
AA
=
X
sin
I
|
R
(0)
T
(0)
,
AA
=
SK
|
sin
I
|
,
(7.126)
X
+
√
ε
m
|
sin
I
|
k
sinh
k
2
T
(0)
SK
cosh
k
(
k
0
h
)
−
1
Z
(
m
)
i
tanh
k/k
,
SS
=
−
(7.127)
∆
(0)
−
g
τ
K
+
k
A
−
k
2
X
|
−
√
ε
m
,
SK
=
k
0
h
ζ
4
∆
(0)
A
∆
(0)
=
−
ζ
3
−
(7.128)
|
sin
I
where
τ
K
=
τ
D
+
k
0
hY
2
X
,
τ
D
=
k
0
hX.
Equations (7.124), (7.125) were obtained for the right-hand half-plane on
the physical sheet (Re
k>
0). In the left-hand half-plane, it is necessary to
replace
k
by
|
k
|
. It can be seen that
∆
SK
1
.
This allows a simplified expression to be written immediately for the trans-
formation coecient of an incident Alfven wave into the reflected Alfven wave:
−
√
ε
m
|
AA
=
X
sin
I
|
R
AA
=
R
(0)
.
X
+
√
ε
m
|
(7.129)
sin
I
|
In (7.129), the neglected term is of the order of
≈
2
√
ε
m
sin
I
∆
SK
k
0
hY
2
X
+
√
ε
m
2
2
k
A
1
.
1
,
√
ε
m
=
c/c
A
It is taken into account here that
X
X, Y
.
Equation (7.129) at
I
=
π/
2 transforms into the known expression for
R
AA
([20], [21]).
∼
Y
,
∆
SK
∼
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