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that it might, from a practical point of view, be as well an
infinite-dimensional system or a nondeterministic system.
To distinguish these two cases, the attractor dimen-
sion reconstructed from experimental or numerical data
can be used as a guide. So far, chaotic modulated
amplitude vacillation and similarly complex forms of
amplitude vacillation have always appeared to follow
fairly low-dimensional dynamics when their Grassberger-
Procaccia dimension was estimated, as demonstrated for
the thermally driven annulus by
GuckenheimerandBuzyna
[1983],
Read et al.
[1992],
Früh and Read
[1997], and
Sitte and Egbers
[2000]. From all studies, it is clear that
amplitude vacillation is a key candidate to explore a
number of standard bifurcation scenarios through sec-
ondary Hopf bifurcations, period-doubling cascades, and
intermittency-type bifurcation, to name but a few. The
majority of the evidence points to a picture whereby the
chaotic flow is in some way the result of global mode
instabilities or through attractor crises arising from two
coexisting attractors associated with different zonal global
wave modes. Either type of transition was always found
to lead to strictly low-dimensional behavior where the
Grassberger-Procaccia dimension tended to be less than 4.
Ultimately, the chaotic flow is normally terminated by
steady wave flow again, usually of a lower wave number
if Pr
>
1 and a higher wave number if Pr
<
1.
Another possible progression is to a structural vacilla-
tion. The evidence, however, points to an understanding
that AV and SV are fundamentally different types of flow
and that a transition from AV to SV occurs more by acci-
dent than through a systematic transition. This is because
the transition is only found at high Prandtl numbers where
AV is so ubiquitous in the regular wave regime that it is
virtually the only possible regular wave flow on which an
SV can develop, whereas SV does also develop on a steady
wave through an as-yet poorly understood mechanism.
The transition to turbulence, on the other hand, seems
to be closely linked to either structural vacillation through
the emergence of possibly localized flow structures, as the
Taylor number is increased, or the progressive emergence
of higher wave modes as the rotation rate is increased
(simultaneous increase of Ta and
). A reevaluation of
the dimension estimates for structural vacillation from
Guckenheimer and Buzyna
[1983] by
Pfeffer et al.
[1997]
supports the proposition that structural vacillation repre-
sentsa secondary instability ontop of theremaining stable
baroclinic wave, which gradually gains predominance as
the flow becomes turbulent. Their argument is based on
the observation that, at the onset of SV, the dimension
estimates suggest a Grassberger-Procaccia dimension of
1.6, a number that persists as “the answer.” At the same
time a second scaling region develops at smaller scales in
phase space. That second scaling region suggests a dimen-
sion of between 7 and 10, with an estimated dimension of
full geostrophic turbulence of 11. These observations are
consistent with those of
Sitte and Egbers
[2000] and
Früh
andRead
[1997], who observed two scaling ranges for their
weak structural vacillation, one suggesting a dimension
of 1.3, the other 4.5. Complementary dimension estimates
for the integrated total heat flux measured simultaneously
with the temperature measurements in the same experi-
ment gave inconsistent results with a suggested dimension
of 5.8. Usually the heat transfer dimension
D
Q
would be
related to that from the temperature data,
D
T
,as
D
Q
=
D
T
−
1 since the total heat transfer does not resolve
the spatial structure within the flow. To overcome the
difficulties presented by dimension estimates and spu-
rious Lyapunov exponents,
Pfeffer et al.
[1997] used
Lorenz analog diagrams to visualize the degree of chaos
by presenting the phase space distance between subse-
quent states to show how the apparently stable global
flow structure is being broken up by spatially separated
fluctuations.
3.8. CONCLUSIONS
This review of amplitude vacillation has attempted to
introduce a range of methods to investigate this phe-
nomenon from careful experimentation in a thermally or
a mechanically driven apparatus complemented by high-
resolution CFD as well as targeted low-order models. By
combining the findings from the various approaches, it has
been possible to build up a fairly comprehensive picture of
the processes leading to and involved in amplitude vacil-
lation. The main processes remain nonlinear wave-wave
interactions and wave-mean flow interactions but also
feedback mechanisms between the fluid interior and the
boundary layers.
This survey has reiterated the fact that the baroclinic
annulus and the two-layer experiment are key fluid exper-
iments to investigate a rich variety of nonlinear dynamics,
including chaotic flow and geostrophic turbulence. The
success, but also the challenges in modeling the observed
flows successfully in CFD models, makes this system a
good candidate for model development and validation.
For straightforward code validation it is possible to find
relatively simple flows that are (or should be!) easy to
model, and for model development there is the option to
model slightly more complex flows that either involve a
higher resolution or combine new processes such as grav-
ity waves. It is also possible to push the experimental
conditions to truly complex flows that are likely to remain
a serious challenge to computational modeling.
From a practical point of view, a frequently asked
question is how this experiment can possibly help to
understand real atmospheric flows, let alone help to pre-
dict weather and climate more accurately. This is a valid
