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In-Depth Information
Uncoupled Waves Phase Mixing
We first consider the case
k
y
= 0. Then the coupled equations (4.43)-(4.45)
split into two non-interacting subsystems
=
B
0
k
A
−
ξ
x
,
db
dx
k
2
(4.46)
b
B
0
dξ
x
dx
=
−
(4.47)
for the FMS-wave and
k
A
(
x
)
ξ
y
=0
.
k
2
−
(4.48)
for the Alfven wave.
Uncoupled equation (4.48), despite its extreme simplicity, describes an
important phenomenon arising in Alfven wave propagation - the so-called
phase mixing, essential for the comprehension of wave processes in space and
laboratory inhomogeneous plasma. The essence of the effect linked to Alfven
wave guiding in inhomogeneous plasma is as follows. Alfven waves propagate
independently along individual field-lines, at its phase velocity
c
A
(
x
)oneach
line. The phase difference between the Alfven waves propagating on close
field-lines increases proportionally to distance
z
.
This results in the growth of the transverse wavenumber
k
x
in the direc-
tion of the inhomogeneity and in the decrease of the transverse scale. The
scale diminishes until either the dissipative effects (e.g. viscosity and electric
conductivity) lead to wave damping, or the transversal dispersion results in
the transformation of large-scale Alfven oscillations into small-scale modes.
Both dissipation and dispersion increase with
k
x
.
Heating by an Alfven wave
dissipating due to phase mixing is often adduced for explaining the heating of
space and laboratory plasma. For instance, solar coronal heating can be con-
nected with phase mixing in an Alfven wave arising due to subphotospheric
convection and propagating along the magnetic field of the coronal hole.
In the simple case of non-interacting Alfven and FMS-waves phase mixing
can be obtained from (4.48) in the space-time domain
∂
2
ξ
y
∂z
2
−
∂
2
ξ
y
∂t
2
1
c
2
A
=0
.
(4.49)
Suppose that a source of Alfven waves is turned on in plane
z
= 0 at time
t
= 0. If the plasma density is independent of
y
, then the displacement
ξ
y
in
the wave traveling towards positive
z
can be written as
ξ
y
(
x, y, t
)=
ξ
y
0
(
x
)
f
+
(
z
−
c
A
t
)
,
where the function
f
+
is determined by the time dependence of a signal in
the source. For harmonic perturbations
ξ
y
(
x, y, t
)
∝
exp[
i
(
−
ωt
+
Φ
0
(
x
)+
k
A
(
x
)
z
)]
,
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