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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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