Geoscience Reference
In-Depth Information
A simple way to describe the dispersion relation for
ω
ω
a
is
2
b
k
y
ω
2
ω
=
(6.8)
k
y
+
k
z
+
(
2
1
/
2
H
)
(Note that many authors use
k
for
k
y
and
m
for
k
z
.) The reader is asked to show
that the product of the vertical phase and group velocities,
,
is negative. Thus, in the important case of upward energy propagation, when
d
(ω/
k
z
)(
d
ω/
dk
z
)
ω/
dk
z
is positive, the vertical phase velocity is downward. This downward
phase propagation is evident in Fig. 6.3 and, as we shall see, explains why the
plasma layers in Fig. 5.23a and c all move downward, at least, on average.
It is crucial to note that in (6.8)
is the intrinsic frequency, the wave frequency
in the neutral wind reference frame. The Doppler formula relates
ω
ω
,
ω
to
ω
=
ω
+
k
·
u
ω
is the frequency in the frame in which
u
is measured (usually the earth
frame). When the horizontal wave phase velocity in the earth frame matches
the wind velocity,
where
0 and the wave simply vanishes. This is called a critical
layer. Such a layer cannot happen for the sound wave branch, since the wind
would have to be supersonic. But IGWwaves are much slower, and critical layers
happen all the time. For example, consider an isotropic wave source in the lower
atmosphere propagating up into an atmospheric jet. As long as
ω
=
u
peak
,a
critical layer must be found for waves nearly parallel to the jet. Waves in the
other direction are simply upshifted and go right through. (This is illustrated in
Fig. 7.5 in the next chapter.) Waves propagating into the critical layer add their
momentum to the layer when they are absorbed. This accelerates the jet even
more. A classic example is the quasi-biennial oscillation in which a critical layer
moves downward as gravity waves are absorbed in the stratosphere. The process
requires almost two years to complete a cycle of zonal wind reversal at a given
height (Holton, 1979).
If we ignore the curvature of the earth but allow the wave frequency to
approach
f
ω/
k
<
, then the rotation of the earth cannot be ignored. The
frequency
f
is called the inertial frequency and is equal to 2
=
2
sin
θ
τ
F
is
the time it takes for the plane of oscillation of a Foucault pendulum to return
to its initial plane at
t
π/τ
F
, where
=
0. At the latitude of Arecibo this period is 52 hours.
Such long-period waves are not easy to study, but they have been observed in
the stratosphere (Cornish and Larsen, 1989; Cho, 1995). Another observation
that seems to be due to such a wave at 35
◦
latitude in this frequency range is
shown in Fig. 6.6. Here a long-duration Leonid meteor trail was photographed
82 seconds after the meteor struck the atmos
p
here (Kelley et al., 2003c
).
The
trail formed into a helix, and the mean wind,
u
, the perturbation wind,
u
, and
the vertical wave number (
k
z
) could be measured (Drummond et al., 2001). The
ambient temperature profile was determined using a lidar located at the same site
δ
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