Environmental Engineering Reference
In-Depth Information
(
E
0
) and wave fields. Since the wave parameters are proportional to exp(
i
k
r
i
ω
t
),
we have
E
z
D
E
0
ik
z
'
. The plasma conductivity is proportional to the electron
C
Σ
0
D
Σ
N
e
/
N
0
), where
number density, so
0
is the plasma conduc-
tivity,
N
0
is the electron number density in the absence of perturbations, and
Σ
D
Σ
0
(1
C
Σ
0
Σ
0
and
N
e
are the corresponding perturbed parameters. From the above expressions,
the condition for conservation of the current density in the field direction is
' C
Σ
0
E
0
Σ
0
ik
z
D
0,
or
N
e
N
0
E
0
ik
z
' C
D
0 .
(5.152)
The continuity equation (4.1) for electrons is
2
N
e
@
@
N
e
@
D
a
@
C
div(
N
e
w
)
D
0,
t
z
2
where
D
a
is the ambipolar diffusion coefficient for the plasma. We have ignored the
diffusive flux of electrons perpendicular to the magnetic field, because the magnet-
ic field strength is high and this flux is small. Assuming a harmonic dependence
of the perturbed parameters on time and spatial coordinates, we can rewrite this
equation to first order as
w
x
@
N
0
@
k
z
D
a
)
N
e
C
i
ω
C
D
(
0 ,
(5.153)
x
where the
x
-axis is in the direction of the maximum gradient of the equilibrium
number density.
Equations (5.151) and (5.152) give
cE
y
H
0
c
ik
y
'
H
0
k
y
k
z
N
e
N
0
cE
0
H
0
w
x
D
D
D
.
Substituting this into dispersion relation (5.153), we obtain
k
y
k
z
cE
0
H
0
L
,
k
z
D
a
i
ω
D
C
(5.154)
where 1/
L
0) will develop if
the ratio
k
y
/
k
z
has an appropriate sign and value [78]. The instability has a thresh-
old with respect to the electric field. The magnetic field must be high enough to
meet the conditions described above.
Equation (5.154) shows that the optimum condition for this instability is such
that the cyclotron frequency for ions is of the order of the collision frequency of ions
with gas particles. The threshold for this instability is connected to the diffusion
of charged particles. If the diffusion rate is sufficiently high, perturbations of the
plasma are damped by the diffusion.
D
d
(ln
N
0
)/
dx
. One can see that instability (Im
ω
<
