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Turning now to the continuity equation (4.8a), we have
(∂
n
/∂
t
)
+
V
x
(∂
n
/∂
x
)
+
V
z
(∂
n
/∂
z
)
=
0
The two velocity components may be obtained from (4.4) if we remember that
the electric field
δ
E
is of first order,
κ
i
is large, and the pressure-driven velocity
does not contribute to
V
x
is of first order, only the zero-order
V
x
contributes to the second term—hence,
V
x
=
·∇
n
. Since
∂
n
/∂
Mg
/
eB
=
Q
/
e
. In the third term,
∂
z
is of zero order due to the vertical density gradient in the plasma. We must
then include the first-order vertical velocity given by
V
z
=
δ
n
/∂
E
x
/
B
. The linearized
continuity equation is therefore
∂
n
/∂
t
+
(
Q
/
e
)(∂
n
/∂
x
)
−
(
1
/
B
)(∂φ/∂
x
)(∂
n
/∂
z
)
=
0
(4.12)
Using the plane wave solutions, (4.11b) and (4.12) may be written
n
0
k
2
P
−
ikQ
δ
n
+
δφ
=
0
(4.13a)
(
i
ω
−
ikQ
/
e
)δ
n
+
(
ik
/
B
) (∂
n
0
/∂
z
) δφ
=
0
(4.13b)
, and may be solved by
setting the determinant of coefficients equal to zero. This yields the dispersion
relation
These are two equations in two unknowns,
δ
n
and
δφ
i
g
/ν
in
(
ω
=
(
kQ
/
e
)
−
1
/
n
0
)(∂
n
0
/∂
z
)
(4.14)
The real part of
ω, ω
r
, shows that the plane waves propagate eastward with
phase velocity
V
given by
φ
V
φ
=
ω
r
/
k
=
Q
/
e
=
Mg
/
eB
(4.15a)
For an atomic oxygen plasma at the equator
V
φ
≈
6cm
/
s, which is quite
ω
small. The imaginary part of
is
ω
i
=−
g
/ν
in
[
(
1
/
n
0
)(∂
n
0
/∂
z
)
(4.15b)
]
When
∂
n
0
/∂
z
is positive (corresponding here to the density gradient antipar-
allel to
g
),
ω
i
is negative and
e
i
ω
t
e
i
ω
r
t
e
γ
t
=
where
γ
is positive and thus yields a growing solution. The parameter
γ
is the
growth rate of the instability and is given by
γ
=
g
/
L
ν
in
(4.16)
]
−
1
).
=[
(
/
n
0
)∂
n
0
/∂
where
L
is the gradient scale length (
L
1
z
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