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where
V
0
⊥
is the magnitude of the zero-order drift parallel to
∇
n
. Using (10.5),
which can be written
γ
=
(δ
E
k
/
BL
)(δ
n
/
n
)
and substituting
δ
E
k
from (10.8)
yields
√
2
3
E
||
γ
=
cc
1
/
2
BL
(σ
P
/σ
0
)
and
the zero-order parallel drift velocity difference of the ions and electrons, which
may be written
The usual practice is to compare these two growth rates in terms of
V
0
⊥
V
||
=
σ
0
E
||
/
ne
Then
1
/
2
|
γ
EB
|
|
γ
cc
|
3
√
2
V
0
⊥
B
(σ
P
σ
0
)
=
V
ne
||
This result makes it easy to compare the two effects. First, we note that if both
processes are unstable, (10.9) will show which one is more important. Second, if
E
0
⊥
is destabilizing, instability is still pre-
dicted if (10.9) is less than one. To estimate the magnitude of
J
is in the stable configuration while
J
||
required to over-
come, say, a stabilizing 10mV/m perpendicular electric field (
V
0
⊥
=
||
400m/s), we
A/m
2
. This is a sizable
current but is not out of the question for the auroral zone.
Using the same expressions for
10
4
cm
−
3
to find
J
use
V
||
=
j
||
/
ne
and set
n
=
5
×
||
≥
7
μ
1
/
2
σ
P
and
σ
0
as used previously, (
σ
P
σ
0
)
=
1
/
2
and Eq. (10.10) becomes,
(
e
ν
i
/
i
ν
e
)
e
ν
i
i
ν
e
1
/
2
|
γ
EB
|
|
γ
cc
|
3
√
2
V
0
⊥
V
=
(10.9)
||
Finally, once again, we note that the quantity in parentheses is approximately
equal to unity in the F region, and we have
γ
EB
/γ
cc
=
2
V
0
⊥
/
V
||
which is identical to the result usually quoted. A complete analysis (Ossakow
and Chaturvedi, 1979; Vickrey et al., 1980) yields the following expression for
the local growth rate of the current convective instability including the possible
existence of
E
0
⊥
:
)
(
−
||
θ
max
1
/
2
γ
cc
=
(
−
1
/
L
E
0
⊥
/
B
)(ν
in
/
i
)
+
V
max
(
i
/ν
in
+
e
/ν
ei
) θ
+
ν
in
i
k
2
⊥
ν
ei
e
i
1
i
max
(10.10)
k
2
||
C
s
ν
in
C
s
2
in
2
2
−
−
+
ν
/
(ν
ei
ν
in
/
i
)
+
θ
e
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