Geoscience Reference
In-Depth Information
need one result of the detailed linear analysis made by Ossakow and Chaturvedi
(1979). They linearized the continuity equation, the momentum equation, and
∇·
J
=
0 and showed that, for maximum growth,
−
1
1
/
2
i
k
ν
i
i
ν
i
+
e
||
θ
=
k
ν
e
⊥
σ
P
=
ne
2
i
and
σ
0
=
ne
2
ν
e
for the zero-order conduc-
2
Substituting
ν
i
/
M
/
m
tivities in this expression yields
σ
P
σ
0
1
−
1
/
2
σ
P
2
1
/
2
+
i
ν
e
e
ν
i
=
θ
=
σ
0
where in the last step we use the fact that for typical F-region conditions
i
ν
e
/
e
ν
i
≈
θ
=
1. Substituting this result into (10.7) and using the fact that
sin
θ
θ
=
and cos
1,
E
k
=
δσ
0
/σ
0
√
2
E
2
1
/
δ
||
/
(σ
P
/σ
0
)
3
and, finally, because
σ
0
∝
n
, we have for maximum growth
1
/
2
δ
E
k
=
(
.
)(δ
/
)
||
(σ
P
/σ
0
)
0
47
n
n
E
(10.8)
This electric field is very nearly perpendicular to
B
, since
k
⊥
>>
k
and, refer-
||
ring to Fig. 10.15, we see that it is such that
δ
E
k
is in the
a
x
direction when
ˆ
δ
n
/
n
>
0. The perturbation
E
×
B
drift in this phase of the wave is thus equal to
+
(δ
a
y
, which means that a high-density region moves down the gradient
to lower-density regions and therefore grows in relative amplitude. Therefore,
the plasma is unstable.
In this same configuration, suppose a background perpendicular zero-order
electric field
E
0
⊥
=
E
k
/
B
)
ˆ
E
0
⊥
ˆ
a
x
existed in addition to the zero-order parallel electric
field
E
0
||
B
instability theory presented in Chapters 4 and 5
shows that the growth rate is given by
. A review of the
E
×
γ
EB
=−
E
0
⊥
/
BL
=+
V
0
y
/
L
That is, if the perpendicular electric field is in the
+ˆ
a
x
direction, the system is
stable, while if it is in the
a
x
direction, instability occurs.
We nowmay compare the growth rate of the pure current convective instability
to the classical
E
−ˆ
×
B
process. The latter we write in the form
γ
EB
=
E
0
⊥
/
BL
=
V
0
⊥
/
L
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