Geoscience Reference
In-Depth Information
When
γ
is positive, the waves will grow, and thus the requirement for instability is
2
r
k
2
C
s
ω
>
or
V
D
> (
1
+
0
)
C
s
This is the origin of the term
two-stream
because the electrons must drift through
the ions at a speed exceeding the sound speed to generate the waves.
We can now use these mathematical results to gain insight into the physical
instability mechanisms involved. First, from the electron continuity equation
we have
=
ω/
V
D
δ
δ
V
e
x
k
−
n
/
n
(4.41)
But using the two electron momentum equations [(4.38b) and (4.38c)] we also
must have
ν
e
e
(
)
−
ikk
B
T
e
/
m
(δ
)
e
δ
V
e
x
=−
+
ν
ike
δφ/
m
n
/
n
(4.42)
If we study the properties of the wave near marginal stability where
γ
=
0 (or
, we can replace
ω/
V
D
by
equivalently assume
ω
r
γ)
k
−
(ω
r
/
k
−
V
D
)
, which
in turn becomes
−
(
0
/
[1
+
0
]
)
V
D
, using (4.38a). Setting (4.41) and (4.42)
equal and using this result yield
ν
e
e
(
)
=
ν
e
ikk
B
T
e
/
m
e
e
e
+
ν
ike
δφ/
m
+
ν
+
0
V
D
/(
1
+
0
)
]
(δ
n
/
n
)
Now if we note that
δ
E
x
=
ik
δφ
and
V
D
=
E
z
0
/
B
, where
E
z
0
is the zero-order
vertical electric field, this may be written
m
e
m
e
ν
e
ikk
B
T
e
/
e
e
(
+
0
)
ν
e
e
/
+
ν
δ
E
x
=
+
ν
+
0
E
z
0
/
B
1
δ
n
/
n
and finally,
0
e
[
eE
z
0
2
e
2
δ
E
x
/
E
z
0
=
+
ν
ν
e
e
(
1
+
0
)
]
+
ikk
B
T
e
/
δ
n
/
n
(4.43)
n
is wave-
length dependent. For long wavelengths, the second term in the bracket is negli-
gible and
This expression shows that the relationship between
δ
E
x
and
δ
n
/
n
are either in phase or 180
◦
out of phase, depending on the
sign of
E
z
0
. Using the good approximation that
δ
E
x
and
δ
e
ν
e
and
0
=
ν
e
ν
i
/
e
i
we have for this case
δ
E
x
E
z
0
=
ν
i
δ
n
n
(4.44)
i
(
1
+
0
)
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