Geoscience Reference
In-Depth Information
Now, the wave will grow if more plasma moves into a region of high density
than leaves that region. Consider the ion motion expressed by (4.38e). The real
part of the equation corresponds to the ion velocity in phase with the wave.
The
E
x
, which is also in phase and thus
is included in the real part of the expression. The imaginary part gives the out-
of-phase motion at the spatial positions where
(
+
ik
δφ)
term is just the electric field,
δ
E
x
both vanish and
thus the part of the ion motion that causes the wave to grow or decay. The
two vectors, pressure gradient force and inertial force, are shown in Fig. 4.32a.
For growth, the ion inertial force must be larger than the pressure term so that
the plasma moves horizontally from low density to high density. This in turn
requires that
δ
n
/
n
and
δ
k
k
B
T
i
/
M
(δ
ω
r
δ
V
i
x
>
n
/
n
)
(4.45)
In other words, the ion inertial force must be greater than the pressure gradient
force that is trying to smooth out the density enhancement by diffusion. Using
(4.38d) to eliminate
δ
V
i
x
, equation (4.45) can also be written as
>
k
B
T
i
/
M
k
2
ω
r
ω/
Near marginal stability,
γ
ω
r
,so
ω
ω
r
, and using the dispersion relation
(4.40a) for
ω
r
/
k
we have the requirement that
+
0
) >
k
B
T
i
/
M
1
/
2
V
D
/(
1
(4.46)
This is almost but not quite the required threshold condition for growth. The
exact result,
V
D
> (
C
s
, came out of the detailed determinant analysis,
since the electron pressure term adds to the ion pressure via the ambipolar diffu-
sion effect. Ion inertia is thus the destabilizing factor in the two-stream instability,
whereas ambipolar diffusion causes damping.
Zero-order temperatures in the E region are usually such that
T
i
=
1
+
0
)
T
n
=
T
. Then, if the wave process behaves isothermally, the ion acoustic speed is
C
s
T
e
=
M
. Farley and Providakes (1989) noted that the threshold speed
of short-scale E-region irregularities should be evaluated more carefully, based
on their observations of high latitude E-region irregularities moving with phase
speeds that were clearly faster than the isothermal ion-acoustic speed and much
closer to a speed associated with adiabatic electrons:
=
2
k
B
T
/
=
γ
i
T
i
+
γ
e
T
e
M
C
s
where
γ
i
,
e
are the specific heats at constant volume for the ion and electron gases.
For low-frequency waves, cooling will act quickly enough that both gases behave
isotropically and
1. At high frequencies, the behavior of one or both
gases may become adiabatic with a corresponding
γ
i
=
γ
e
=
3, which will increase
the phase velocity. However, since wave phase velocities approach
V
D
in the
γ
=
5
/
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