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written as:
P
=
K 1 N
K 2 P
(6.22)
t
P = P 0 + . Then,
Consider a small perturbation,
K 2 P 0 + =−
K 2
=
K 1 N 0
t
since, in equilibrium,
P 0 /∂
t
=
0, the first two terms cancel, leaving
1
= γ =−
K 2
(6.23)
t
The growth rate is negative and the perturbation decays.
Perkins (1973) pointed this out but went on to show that if, in addition to the
equilibrium wind/electric field requirement, an eastward wind and/or northward
electric field is present, the system is unstable. The full growth rate is (Hamza,
1999; Garcia et al., 2000b):
E e cos I
BH
γ =
k e
k 2
cos I
BH E 0 ·
+
k
(6.24)
where E 0 =
B and E e is the eastward component. The real part of the
E
+
U
×
frequency is
B 2
ω r =
k
· (
E 0 ×
B
)/
(6.25)
only depends on E 0 , the electric field in the earth-
fixed frame, not the wind vector. The phase velocity is simply the E
Note that the real part of
ω
×
B velocity
component parallel to k
. Expressions can be derived for pure electric field or
pure wind-driven processes; following Garcia et al. (2000b), these are
k e k n
k 2
k n
k 2
1 E n
cos I
BH
γ E =
E e +
(6.26a)
k e k n U e
k 2
k n
k 2
1 U s sin I cos I
H
γ W =
+
(6.26b)
where subscripts e and n stand for east and north, respectively. The most unsta-
ble k vector for case (6.26a) lies halfway between E
and the eastward direction
E e this would be 22.5 north of east. For U s sin I
=
=
(Perkins, 1973). For E n
U e
the same condition applies. For the latter case, since
ω
=
0 these structures
r
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