Geoscience Reference
In-Depth Information
1.3.5. Vacillating Waves and Wave-Zonal Flow
Interactions
controversial. However, other forms of nonlinear interfer-
ence vacillation, for example, involving the superposition
of two modes with differing (but adjacent) azimuthal wave
numbers but similar radial and vertical structures [
Ohlsen
and Hart
, 1989b], may also manifest themselves as peri-
odic modulations of baroclinic eddy variance while also
modulating the azimuthal mean flow through nonlinear
triad interactions of harmonics with the mean zonal flow.
Although the basic amplitude vacillation (AV) regime is
typically a quasi-periodic flow characterized by two inde-
pendentfrequenciesassociated with(a)theazimuthal drift
of a monochromatic wave number pattern and (b) its peri-
odic modulation in amplitude, transitions to more chaotic
states have also been observed, still apparently within the
“regular”wave regime. These include transition sequences
via period-doubling bifurcations to chaotic amplitude
vacillations [
Hart
, 1985, 1986] and routes involving more
complex transitions directly to chaos from doubly peri-
odic “modulated” amplitude vacillation [
Farmer et al.
,
1982;
Read et al.
, 1992;
Früh and Read
, 1997]. The result-
ing flows were apparently consistent with the interaction
of a relatively small number of spatial modes but dif-
fered in their basic azimuthal symmetry properties. Period
doubling was observed as a typical route to chaos in the
two-layer, open cylinder experiments of
Hart
[1985] and
Ohlsen and Hart
[1989a]. This does not seem to be typi-
cal for thermal annulus experiments, however, which tend
to be dominated by higher wave number baroclinic modes
(
m
Baroclinic waves in the regular wave regime may be
either steady (apart from a slow drift) or “vacillating”
(i.e., with a periodic or nearly periodic time dependence;
see Chapter 3 for a more detailed discussion). Labora-
tory observations of “amplitude vacillation” indicate that
(for fluids with Pr
1) it occurs close to the upper sta-
bility threshold of its wave number
m
, around where a
transition from
m
to
m
−
1 is observed, at moderate-high
Taylor number. At lower values of Pr, however, it appears
that this transition sequence is reversed, with transitions,
for example using air as the working fluid (with Pr
0.7), from steady waves into the vacillating regime as
is
increased
[e.g.,
Randriamampianinaetal.
, 2006;
Castrejón-
Pita and Read
, 2007]; see also Chapter 16 in this volume.
The “vacillating”state then comprises the periodic modu-
lation of both the amplitude and drift frequency of the
wave on a time scale
T
100 “days”. Figure 1.9
shows a sequence of streak images taken from a typical
m
= 3 flow undergoing an amplitude vacillation cycle at
a level around 0.8
d
above the annulus base. This clearly
shows the wave amplitude growing, reaching a maximum
in amplitude in Figure 1.9d and then decaying before the
cycle repeats. The wave is modulated in both amplitude
and drift rate (phase speed) during the cycle, indicating a
nonlinear interaction with the background zonal flow.
More detailed diagnostics show periodic variations in
total heat transport and in potential energy exchanges
between the wave and zonal flow [see
Pfeffer et al.
,
1980;
Hignett et al.
, 1985]. In particular variations in the
slope of the azimuthal mean isotherms (see Figure 1.10)
clearly show modulations in the potential energy stored
in the azimuthal mean flow. The zonal flow structure (see
Figure 1.10a and 1.10b) is seen to oscillate between a
single jet pair at minimum wave amplitude and two double
jets at maximum amplitude.
In this regard, it appears that nonlinear interactions
between the dominant wave and the azimuthal mean flow
are critically important for the phenomenon of amplitude
vacillation. In practice, however, it may not be easy to
distinguish this behavior from interference arising from
a quasi-linear superposition of two wave components
with the same azimuthal wave number and differing ver-
tical structure and drift frequencies
ω
1
and
ω
2
. Apparent
“vacillation” then takes place at the difference frequency
of the two components
∼
10
−
3). In the latter, the more typical route involves the
development of a third period through the emergence of
an additional wave mode that is not harmonically related
to the initial dominant wave [
Read et al.
, 1992;
Früh and
Read
, 1997].
However,
YoungandRead
[2008] did observe a sequence
of period doublings from a wave number
m
= 2AV flow
in a set of numerical simulations which led to chaotic
states consistent with the endpoint of a period-doubling
cascade over limited regions of parameter space.
Figure 1.11 shows two examples of such flows, illustrat-
ing chaotic ((a), (b)) and period 3 ((c), (d)) vacillations. In
this regime, the amplitude modulations vary in strength,
alternating between two intensities in the period 2 state
with successive doublings as
was increased until chaotic
vacillation ensued. The whole sequence shows a sequence
of bifurcations as successive period doublings lead to
chaotic behavior followed by an indication of period
3 “periodic windows”. Figure 1.12 illustrates such a
sequence at even higher values of
T
. If the two compo-
nents cross-interact with the zonal flow, effects such as
phase locking and zonal flow modulation may occur,
reproducing several aspects of the observed flows. Some
observers claim to have identified this mechanism in mea-
surements in the laboratory [
Lindzen et al.
, 1982], though
the more general relevance of this mechanism remains
|
ω
1
−
ω
2
|
showing the maxi-
mum and minimum wave amplitudes of the equilibrated
baroclinic flow at various values of
T
T
while keeping
at a fixed value of
1.75. This shows clear evidence
of a series of period-doubling transitions with chaotic
regions near points A, B, and D interspersed with regions
characterized by quasi-periodic vacillations.
