Environmental Engineering Reference
In-Depth Information
the electron velocity is
v
0
C
v
e
,where
v
0
is the electron velocity due to the initial
plasma wave and
v
e
is the electron velocity due to the small-amplitude plasma wave.
The current density due to the small-amplitude plasma wave is then
j
0
D
N
i
)(
C
v
e
)
eN
e
v
e
eN
i
v
0
,
e
(
N
e
C
C
eN
e
v
0
D
v
0
E
0
/
j
0
where we ignore second-order terms. The Maxwell equation
@
@
t
C
4
π
D
0
can now be rewritten as
@
E
0
@
eN
i
v
0
eN
e
v
e
4
π
4
π
D
0.
t
The electron equation of motion is
m
e
d
v
e
dt
eE
0
.
D
Eliminating
E
0
fromtheseequations,weobtain
N
i
N
e
ω
@
2
v
e
2
p
v
e
C
2
p
C
ω
D
0
(5.125)
v
0
@
t
2
as the equation
for the elec
tron velocity due to the small-amplitude plasma wave.
Here
D
p
4
N
e
e
2
/
m
e
is the frequency of plasma oscillations ignoring the
thermal motion of electrons. One can see that if we do not take into account the
interaction between the small-amplitude plasma wave and the initial plasma os-
cillation and the ion acoustic wave (that is, if we ignore the last term in (5.125)),
then the assumptions used here give the small-amplitude plasma wave frequency
as being equal to the plasma frequency.
Let us represent the solutions of (5.124) and (5.125) in the form
ω
π
p
e
t
),
N
i
D
u
0
cos(
k
0
x
ω
0
t
),
D
a
cos(
k
e
x
ω
D
bN
0
cos(
k
i
x
ω
i
t
),
v
0
v
e
where
a
and
b
are slowly varying oscillation amplitudes,
e
and
k
e
are the frequen-
cy and the wave number of the small-amplitude plasma wave,
ω
i
and
k
i
are the
frequency and the wave number of the ion sound, and
N
0
is the equilibrium num-
ber density of the charged particles. (We assume that
N
e
ω
N
0
in (5.125).) Since
the oscillation amplitudes vary slowly, the temporal and spatial dependencies are
identical, so we find from (5.124) and (5.125) that
D
ω
D
ω
C
ω
I
k
0
D
k
e
C
k
i
,
(5.126)
0
e
i
as in (5.121). This condition is similar to that for parametric resonance of coupled
oscillators, and therefore the instability that we analyze is termed a parametric in-
stability. In addition, because of the nature of this process, this instability is called
modulation instability. This instability has a universal character and is suitable both
for plasma oscillations and for electromagnetic waves [75, 76].
Taking into account a slow variation of the oscillation amplitudes and condi-
tion (5.126), we find from (5.124) and (5.125) [35]
@
a
b
ω
e
u
0
4
@
b
@
a
m
e
k
i
u
0
4
m
i
D
D
,
@
t
t
ω
i
