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exactly. Since the plasma behaves as an incompressible fluid, the density variation
with time at a fixed point in space is given by the advective derivative
∂
n
/∂
t
=−
δ
V
·∇
n
(4.18)
where
n
is the plasma density and
V
the perturbation plasma velocity. Here we
again use a right-hand coordinate systemwith
x
positive in the eastward direction
and
y
positive toward the north. For perfect spatial resonance the perturbation
neutral velocity in the plasma reference frame is time independent (the wave
frequency is Doppler shifted to zero) and (4.18) becomes
δ
w
0
e
−
ik
x
x
+
ik
z
z
∂
n
/∂
t
=−
(∂
n
/∂
z
)
(4.19)
For an initial density profile of the form
n
0
e
z
/
L
the solution to (4.19) is
n
0
e
z
/
L
e
γ(
x
,
z
)
t
(
,
,
)
=
n
x
z
t
(4.20a)
where
R
e
−
ik
x
x
+
ik
z
z
γ(
x
,
z
)
=
−
(
w
0
/
L
)
(4.20b)
where
R
means “take the real part.” These equations describe a spatial pat-
tern of rising and falling density contours in the plasma reference frame. For
w
0
=
is 10
−
4
s
−
1
. This “perfect”
spatial resonance theory gives a perturbation of 5% in 550 s. Note, however,
that the perturbation plasma ion velocity due to the presence of the wave
can
never exceed
the amplitude of the wave-induced neutral velocity. This con-
straint severely limits the altitude modulation of the F layer due to a pure grav-
ity wave-driven process. In Klostermeyer's (1978) nonlinear approach to this
problem, an anomalous diffusion due to plasma microinstabilities was mod-
eled, and the saturation amplitude of the density perturbation for perfect spatial
resonance was found to be at most about one order of magnitude (e.g., see
his Fig. 4.1). This corresponds to an uplift of about 3 plasma scale heights or
about 50 km. It is now clear that much larger perturbations occur (e.g., see
Fig. 4.1) and a pure gravity wave theory explaining CEIS height modulation is not
tenable.
This simple model can be extended to address the question of how “resonant”
a wave must be to create a significant effect. Kelley et al. (1981) showed that
the horizontal phase velocity of the gravity wave only has to be within only
about 100m
2m
/
s and
L
=
20 km, the peak value of
γ
s of the horizontal plasma drift velocity to produce a 5% seed-
ing effect in one-half of a wave period. This is not a very severe constraint
and makes it very plausible that gravity waves can be responsible for seed-
ing spread F. Huang and Kelley (1996) performed nonlinear simulations (see
Section 4.3.2) with and without spatial resonance and verified that it is not
necessary.
/
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