Geoscience Reference
In-Depth Information
Fig. 1.15
Electric,
E
,
magnetic, ı
B
?
, perturbations
and mass velocity,
V
?
,
caused by the shear Alfvén
wave propagation. Magnetic
field lines disturbed by
linearly polarized Alfvén
waves are shown with
wavy
lines
.
V
A
denotes the group
velocity of the Alfvén wave
V
A
z
V
↥
d
B
↥
y
ʸ
B
0
k
↥
k
E
x
and taking into account that ı
D
0,
B
0
ı
B
?
D
0 and
k
V
?
D
0, yields
!
0
0
V
?
C
.
k
B
0
/ı
B
?
D
0;
(1.56)
.
k
B
0
/
V
?
C
!ı
B
?
D
0:
(1.57)
Here we have taken into account only projection of the equations on the vertical
axes
z
.ThesetofEqs.(
1.56
)-(
1.57
) has a nontrivial solution for
V
?
and ı
B
?
only
if the determinant of the set equals zero. Hence we get the following dispersion
relation
!.
0
0
/
1=2
D˙
k
B
0
:
(1.58)
The phase velocity of the wave is thus given by
!=k
D˙
V
A
cos ;
(1.59)
where is the angle included between the vectors
k
and
B
0
, and V
A
denotes the
Alfvén speed
B
0
.
0
0
/
1=2
:
V
A
D
(1.60)
The group velocity
V
A
is pointed along
B
0
d!
d
k
D˙
B
0
.
0
0
/
1=2
:
V
A
D
(1.61)
This wave mode has been termed the shear Alfvén wave because the mass
velocity of the conducting fluid/plasma is perpendicular to both the group and phase
velocities of the wave. Moreover, once the shear Alfvén wave propagates in the
medium, the mass density remains unchanged; that is, the material can be considered
as incompressible.
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