Environmental Engineering Reference
In-Depth Information
5.4
Nonlinear Phenomena in Plasmas
5.4.1
The Lighthill Criterion
Plasma instabilities cause the development of certain types of oscillations. The sub-
sequent evolution of these oscillations is determined by interactions between the
oscillations and the plasma itself. The strength of these interactions depends on
the wave amplitude. Hence, if a plasma instability develops, we deal with nonlin-
ear plasma phenomena. But the role of nonlinear plasma processes is not restrict-
ed to the development of instabilities [61]. Nonlinear processes affect the nature
of wave propagation in a plasma. In fact, interaction between elementary waves
acts through the interaction of these waves with the plasma. Therefore, nonlinear
phenomena determine both the profile and the character of the propagation of all
waves in a plasma except those of low intensity. Along with this, nonlinear phe-
nomena create a new plasma state [54, 62-64].
To study the development of perturbations in a plasma, we examine a perturba-
tion in the form of a one-dimensional wave packet consisting of waves encompass-
ing a narrow range of wave numbers (
Δ
k
k
). The perturbation amplitude at
point
x
is
X
a
(
x
,
t
)
D
a
(
k
)exp(
ikx
i
ω
t
) ,
(5.102)
k
where
a
(
k
) is the amplitude of the wave with wave number
k
.Wetakeintoaccount
the wave dispersion, expressible as [65]
@
ω
(
k
0
)
@
@
2
ω
(
k
0
)
k
0
)
2
ω
(
k
)
D
ω
(
k
0
)
C
(
k
k
0
)
C
(
k
k
@
k
2
2
@
v
g
(
k
0
)
1
k
0
)
2
.
D
ω
C
v
g
(
k
k
0
)
C
(
k
(5.103)
0
@
k
Here
k
0
is the mean wave number of the wave packet and
v
g
is the group velocity of
the wave. Since wave number values are restricted to an interval of width
k
near
k
0
, then according to (5.102) the perturbation is initially concentrated in a spatial
region of extent
Δ
k
. As the wave packet evolves, it diverges owing to the
different group velocities of individual waves. The initial wave packet, which has a
size of the order of 1/
Δ
x
1/
Δ
k
)
1
.
That is, the wave dispersion usually leads to increasing spatial extension of the wave
as a function of time.
The interaction of waves of different
k
values influences the behavior of this
process. In particular, it can even lead to compression of the waves. To assess this
possibility, we take into account the dependence
Δ
k
, diverges on a timescale given by
τ
(
Δ
k
2
)
1
(
@
v
g
/
@
E
2
ω
D
ω
α
(5.104)
0
