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Coupled Waves Phase Mixing
At k y
= 0 interaction takes place between Alfven and FMS-waves. From (4.43)
and (4.45) we obtain
b ( x )
B 0
1
k A ( x )
∂x
ξ x ( x )=
,
k 2
b ( x )
B 0
ik y
k A ( x )
ξ y ( x )=
.
k 2
Then (4.44) becomes
k A ( x )
k y
k 2
||
d
dx
db ( x )
dx
1
k A ( x )
+
b ( x ) = 0
(4.52)
k 2
||
k A ( x )
k 2
or
+ k A ( x )
b ( x )=0 .
d 2 b ( x )
dx 2
dk A /dx
k A ( x )
db ( x )
dx
k y
k 2
(4.53)
k 2
In order to study the coupling between these two modes, let us consider
first the problem of wave incidence on a stratified plasma medium. Suppose
Alfven velocity c A ( x ) increases monotonically from c 1 at x
→−∞
to c 2 at
x
, with Alfven wave number k A ( x ) decreasing from k 1 to k 2 (Fig. 4.3,
top left panel). Let the incident wave amplitude be b 0 . Introduce reflection
and transmission coecients R and T . Then at x
+
, the wave magnetic
field b is the sum of the incident ( i ) and reflected ( r ) waves
→−∞
b = b ( i )
+ b ( r )
,
where
=exp
x + ik y y + ik z b 0 ,
b ( i )
iωt + ik (1)
x
= R exp
ik (1 x x + ik y y + ik z b 0
b ( r )
iωt
and at x
+
= T exp
x + ik y y + ik z b 0
b ( t )
iωt + ik (2)
x
= k 1 , 2
with k (1 , 2)
k 2
.
The problems of oblique incidence of FMS-waves on a layer and oblique
incidence of an electromagnetic wave of H -polarization on a plane-layered
medium are similar ([11], [12]). The electric field E in the H -polarized wave
is within the plane of incidence, while the magnetic field is perpendicular to
k y
x
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