Geoscience Reference
In-Depth Information
Perkins instability growth rate
λ
F
(Equation 6.24), and the isolated
E
s
LI growth
rate
λ
E
s
:
λ
EFCL
=
λ
E
s
(
R
F
+
2
R
0
)
+
λ
F
(
R
E
s
+
2
R
0
)
2
[
R
E
s
+
R
F
+
2
R
0
]
2
2
R
0
)
+
λ
F
R
E
s
+
2
R
0
λ
E
s
λ
F
2
R
0
R
E
s
+
λ
E
s
(
R
F
+
±
2
R
E
s
+
2
R
0
−
2
R
0
(6.31)
R
F
+
R
F
+
HE
ρ
i
PE
θ
cos
θ
cos
I
λ
E
s
=
k
s
u
e
sin
(6.32)
θ
is the azimuthal angle of the phase
fronts with the north-south direction in magnetic field-aligned coordinates (45
◦
is northwest to southeast phase fronts; see Cosgrove and Tsunoda 2003), and
u
e
k
s
is the zonal wind shear at the
E
s
L altitude. Equations (6.32) and (6.31) are
valid whenever
where
I
is the magnetic field dip angle,
(
E
u
E
s
×
B
+
)
·ˆ
e
=
0, where
u
E
s
is the wind velocity at the
E
s
L
altitude, and
e
is a unit vector toward the east. When this relation is not satisfied,
(6.30) may still provide approximations to the growth rates. The reader should
refer to Cosgrove and Tsunoda (2004a).
Equations (6.31), (6.32), and (6.24), together with the definitions in Fig. 6.41,
allow study of the growth rates for structure in the
E
s
and F layers for various
amounts of electrical coupling (values of
R
0
). For the highly coupled case (
R
0
=
0), which occurs for long wavelengths under conditions such that the relative
velocity of the
E
s
and F layers is zero, equation (6.31) gives the two growth
rates:
ˆ
⎧
⎨
=
λ
E
s
R
F
+
λ
F
R
E
s
R
F
+
R
E
s
λ
EFCL
|
R
0
=
0
=
(6.33)
⎩
0
Equation (6.33) shows only a single unstable mode for which the isolated growth
rates combine according to the internal impedances of the
E
s
- and F-layer
sources, in the same way as the mapped polarization electric field. For the uncou-
pled case (
R
0
infinity), which occurs for short wavelengths, equation (6.31)
gives the growth rates
=
λ
E
s
λ
F
λ
EFCL
|
R
0
→∞
=
(6.34)
In this case there are two unstable modes: the growth rate of the Perkins
instability, and the growth rate of the
E
s
layer instability.
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