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 s x 3 b ( i )
+ R SA e 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 s x 3 b ( i )
R SA e s x 3 b ( i 2 ,
k 0
E 2 =
(7.96)
1
+ e −iκ A x 3 + R AA e A x 3 b ( i )
b 2 = R AS e A x 3 b ( i )
,
(7.97)
1
2
R AS e A x 3 b ( i 1 + e −iκ A x 3
R AA e A x 3 b ( i 2 ,
k 0
κ A
E 1 =
(7.98)
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