Environmental Engineering Reference
In-Depth Information
can be rewritten for electrons as
@
v
e
@
C
v
e
@
v
e
@
m
e
N
@
1
p
e
@
eE
m
e
D
C
C
0 ,
(5.123)
t
x
x
where
p
e
NT
e
is the the electron gas pressure,
T
e
is the electron temperature,
and
N
e
is the electron number density. After averaging over fast oscillations, we
find the first term in this equation is zero. We write the electron velocity as
v
e
D
D
C
v
e
,where
v
e
is the electron velocity due to the small-amplitude plasma wave.
Then we have
v
0
v
e
@
v
e
@
1
2
@
v
e
1
2
@
@
@
@
D
D
x
(
C
v
e
)
2
D
x
(
v
0
v
e
),
v
0
x
@
x
where the bar denotes averaging over fast oscillations. We assume that
T
e
const,
corresponding to a high rate of energy exchange in the ion acoustic wave. During
the ion motion, the electron number density has a relaxation time for maintaining
the quasineutrality of the plasma. Therefore, after averaging, we find the deviation
of the electron number density from equilibrium is the same as that for ions, and
the third term in the Euler equation (4.6) becomes
D
N
i
@
m
e
N
@
1
p
e
@
m
e
N
0
@
T
e
D
(
N
N
0
).
x
x
The Euler equation after averaging is transformed into
x
v
0
v
e
C
N
i
@
@
@
m
e
N
0
@
T
e
eE
m
e
D
C
0.
x
Substituting the electric field derived from this equation into (5.122), we obtain
2
N
i
@
2
N
i
@
x
2
v
0
v
e
D
2
@
m
i
@
T
e
m
e
N
0
m
i
@
0 .
(5.124)
t
2
x
2
@
If we ignore the last term in (5.124) and assume harmonic dependence of the
ion density on time and spatia
l coord
inates, we obtain dispersion relation (5.62) for
ion sound:
D
p
T
/
m
i
. To take into account interaction between ion
sound and plasma oscillations, we have to trace the motion of electrons in the field
of the small-amplitude plasma wave. To do this, we shall use the Maxwell equation
for the electric field of the small-amplitude wave, assuming the magnetic field to
be zero. This gives
ω
D
c
s
k
,
c
s
0. Here
E
0
is the electric field of the wave and
j
0
is the electric current density generated by it.
For simplicity, we shall ignore thermal motion of the electrons, since it has only
a small effect on the oscillation frequency. Therefore, when writing the expression
for the current density, we can ignore the variation of the electron number density
due to the electron pressure of the plasma wave. We assume the electron number
density to have the form
N
e
E
0
/
j
0
D
@
@
t
C
4
π
N
i
,where
N
e
includes the equilibrium electron
number density and its variation under the action of the initial plasma wave and
N
i
is the variation of the ion number density owing to the motion of ions. Accordingly,
C
