Geoscience Reference
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theory, and by Buneman (1963), using the Navier-Stokes equation. They have
shown that the plasma is unstable for waves propagating in a cone of angle
φ
C
s
. For smaller drift
velocities the plasma still can be unstable, provided there is a plasma density
gradient of zero order oriented in the right direction relative to the electric field
driving the electrons. This instability was first studied by Simon (1963) and Hoh
(1963) for laboratory plasmas and is termed the gradient drift instability. Many
features of the type 2 radar echoes are explained by this instability.
We first investigate the linear theory for the two-stream instability. The linear
theory is important because two-stream waves have been observed to move with
a phase speed of the order of the ion-acoustic speed (which is not necessarily
isothermal and which matches the phase speed at the instability linear threshold),
no matter what exact nonlinear mechanism has produced them. We will start
with the isothermal linear theory. Consider the zero-order condition in which
the electrons stream east with a velocity
V
D
ˆ
about the plasma drift velocity such that
V
D
cos
φ>
a
x
driven by a vertically downward
zero-order polarization electric field
a
z
(see Chapter 3). This corresponds to
nighttime conditions. The ions are assumed to be at rest to zero order due to the
high collision frequency (
−
E
z
0
ˆ
ν
in
i
). For the moment we ignore any zero-order
plasma gradient. Perturbations in density, electrostatic potential, and velocity
are of the form
ne
i
(ω
t
−
kx
)
,
e
i
(ω
t
−
kx
)
, and
V
e
i
(ω
t
−
kx
)
, and we have taken
k
δ
δφ
δ
to be in the
a
y
. The
linearized electron continuity equation, ignoring production and loss, which are
assumed to be in equilibrium in the zero-order equations, is
a
x
direction (
k
ˆ
=
k
a
x
)
ˆ
, which is perpendicular to
B
=
B
ˆ
i
ωδ
n
+
V
D
(
−
ik
δ
n
)
−
ikn
0
δ
V
ex
=
0
(4.37)
where the second term comes from the (
V
V
).
In this expression the common factor
e
i
(ω
t
−
kx
)
has been factored out. Notice that
since
k
is horizontal and there is no vertical gradient, the vertical perturbation
drift does not change the electron density. Equation (4.37) can be rewritten as
·∇
n
) term and the third from
n
(
∇·
V
ex
=
ω/
V
D
(δ
δ
−
/
)
k
n
n
(4.38a)
where the subscript zero has been dropped from the mean density
n
0
. Notice that
we have already assumed quasi-neutrality by using
δ
n
as a variable rather than
δ
n
e
. In the electron momentum equation the term
md
V
e
/
dt
is dropped because
of the small electron mass, leaving
)
−
k
B
T
e
/
m
(
∇
0
=
(
−
e
/
m
)(
E
+
V
e
×
B
n
/
n
)
−
ν
e
V
e
where again we ignore gravity and where
ν
e
is the electron-neutral collision
frequency. We simplify
ν
i
in the E region. The linearized
version of this in component form is, in the
z
direction,
ν
en
and
ν
in
to
ν
e
and
0
=−
e
δ
V
e
x
−
ν
e
δ
V
e
z
(4.38b)
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