Environmental Engineering Reference
In-Depth Information
The dispersion relation for plasma oscillations (5.57) has the form
1
!
C
γ
˝
v
x
˛
k
2
E
2
2
2
p
2
ω
D
ω
(5.113)
16
π
N
0
T
when (5.112) is taken into account, where
p
is the plasma frequency in the ab-
sence of fields. In (5.113) we used (1.12) for the plasma frequency.
We can write (5.113) in a more convenient
for
m. Take the electr
ic
field strength
ofthewaveintheform
E
ω
E
0
/2. Using
v
T
/
m
e
for
electrons and introducing the Debye-Hückel radius
r
D
according to (1.7), we rep-
resent (5.113) in the form
D
E
0
cos
ω
t
,so
E
2
D
x
D
p
1
r
D
k
2
.
E
0
2
2
ω
D
ω
N
0
T
C
2
γ
(5.114)
32
π
The first term on the right-hand side of this expression is considerably larger than
the other two terms.
One can see that dispersion relation (5.114) satisfies the Lighthill criterion (5.106)
becauseinthiscasewehave
α
@
v
g
@
γω
p
D
<
0.
k
32
π
m
e
N
0
Thus, nonlinear Langmuir oscillations can form a soliton. Dispersion relation
(5.114) shows that if the energy density of plasma oscillations is high enough
(so that the second term of (5.114) is larger than the third one), the oscillations
cannot exist far from the soliton. The oscillations create a potential well in the
plasma and are enclosed in this well. They can propagate in the plasma together
with the well and occupy a restricted spatial region. The size of the potential well
(or the soliton size) decreases with increase of the energy density of the plasma
oscillations. Because
r
D
k
1, the solitons are formed when the energy density
of the oscillations is small compared with the specific thermal energy of charged
particles of the plasma. Thus, this analysis demonstrates the tendency to form
solitons from long-wave plasma oscillations. However, the analysis used does not
allow one to study the evolution of oscillations at large amplitudes. Nevertheless,
this exhibits the conditions for the existence of the Langmuir soliton that were
realized experimentally [68-72] and that can exist in a solar plasma [73].
5.4.5
Nonlinear Ion Sound
We now analyze nonlinear ion sound when the nonlinearity is large [74]. For this
purpose we use the Euler equation (4.6), the continuity equation (4.1), and the Pois-
son equation (1.4), which, in the linear approach, leads to dispersion relation (5.62)
for ion sound. When we use these equations without any linear approximation,
they can be written as
@
v
i
@
2
C
v
i
@
v
i
@
m
i
@'
e
@
N
i
@
C
@
(
N
i
v
i
)
@
@
'
x
C
D
D
D
e
(
N
e
N
i
) . (5.115)
0,
0,
4
π
t
@
x
t
x
@
x
2
