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diagram showing the scenario for the transition from the
upper symmetric regime to a steady wave and subsequent
amplitude vacillations for the
m
= 2 flows. The, albeit
narrow, hysteresis at the transition between the axisym-
metric flow and the regular steady wave suggests a sub-
critical Hopf bifurcation, resulting from a zonal symmetry
breaking. By using a weakly nonlinear stability analysis in
the same air-filled configuration,
Lewis
[2010] confirmed
the subcritical Hopf bifurcation observed during our
numerical study. While
Hide and Mason
[1978] reported
that such an hysteretic behavior was observed only if
the upper boundary was a free surface,
Koschmieder and
White
[1981] presented evidence for the possibility of
small hysteresis in their experimental study of a water-
filled cavity. On the other hand,
Castrejón-Pita and Read
[2007], using air as working fluid in their experiments,
mentioned the occurrence of the so-called
weak waves
,
characterized by
>
c
, prior to the onset of fully devel-
oped regular waves, but did not observe any hysteresis
cycle. During this study, we did not find any hint of such
weak waves.
By progressively increasing the rotation rate, the transi-
tion sequence from the upper symmetric US flow through
all observed two-wave flows follows clear steps of increas-
ing complexity before bifurcating to three-wave steady
flow (the number before the letter denotes the dominant
azimuthal wave number): US
Read et al.
[1998], in analogy with a noise-induced crisis
in a multistable system (see also
von Larcher and Egbers
[2005]). Crisis is characterized by a sudden change in
the flow dynamics and temporal behavior [see
Grebogi
et al.
, 1983]. A further increase in rotation rate up to
Ta = 2.3
10
5
from this chaotic solution leads abruptly
to the steady 3S regime, also due to a crisis. The tempo-
ral behaviors of all the AV and MAV solutions have been
confirmed by the calculations of the corresponding largest
Lyapunov exponent, reported in Figure 16.2. The quasi-
periodic solutions have a largest Lyapunov exponent
which cannot be distinguished from zero within the mar-
gin of error, while chaotic solutions are characterized by
a positive Lyapunov exponent: 2.2
×
10
5
<
Ta
<
2.3
10
5
×
×
[
Randriamampianina et al.
, 2006].
16.3.2. Liquid-Filled Cavity: Pr = 16
The details of the system, the fluid properties, and the
governing parameters are summarized in Table 16.1. The
configuration corresponds to one experimental rig used
at the University of Oxford, UK [
Wordsworth
, 2009]. It
consists of an annular domain of inner radius
a
= 4.5 cm,
outer radius
b
= 15 cm, and height
d
= 26 cm. The cavity is
filled with a liquid described by a Prandtl number Pr= 16
and is submitted to a temperature difference
T
=2 K
between the inner, cold, and outer, hot, cylinders closed
by horizontal insulating rigid endplates. Four values of
the rotation rate have been considered, covering different
flow regimes of baroclinic waves. For
= 0.25,0.35, and
0.5125 rad/s, a mesh of
N
→
2S
(
P
)
→
2AV
(
QP2
)
→
2MAV
(
QP3
)
3S
(
P
)
, as illustrated in
Figure 16.1 showing the mean amplitude and the enve-
lope of the vacillation. Here, P stands for periodic, QP
for quasi-periodic (QP2 is characterized by two incom-
mensurate frequencies, and QP3 by three frequencies)
and NP for aperiodic regime. Steady wave solution 2S is
obtained for 1.8
→
2MAV
(
NP
)
→
256
was used in the radial, axial, and azimuthal directions,
respectively, with a dimensionless time step
δt
= 0.0125.
For the rotation rate value
= 1.25 rad/s, a refined resolu-
tion in the radial and azimuthal directions was necessary,
N
×
M
×
K
= 128
×
150
×
10
5
10
5
. The first
×
≤
Ta
≤
2.05
×
10
5
, likely occurs via a sec-
ondary Hopf bifurcation from an oscillatory flow, also
known as a Neimark-Sacker bifurcation through a tem-
poral symmetry breaking. It is characterized by a second
frequency (QP2) resulting from the periodic oscillations of
the amplitude, in addition to the wave drift observed dur-
ing the regular steady flow. A further increase in rotation
rate brings a third frequency coming from the modula-
tion of the amplitude oscillations in the 2MAV regime.
This corresponds to a continuation of “the quasi-periodic
route to chaos” described by
Newhouse et al.
[1978], but
the nature of the initial solution as a quasi-periodic 2MAV
with three incommensurate frequencies was unusual. As
shown by
Newhouse et al.
[1978], generic three-frequency
flows are expected to be chaotic rather than periodic. To
our knowledge, no previous example of such a flow has
been reported from either numerical or experimental stud-
ies of baroclinic waves. The final type of flow dominated
by
m
= 2 was a chaotic 2MAV regime that can be induced
by a crisis as discussed in a similar baroclinic cavity by
2AV regime, at Ta = 2.1
×
×
M
×
K
= 150
×
150
×
320, with a dimensionless time
step
δt
= 0.00625.
The values of the control parameters used in the numer-
ical simulation of the flow at these four rotation rates are
represented in Figure 16.3 together with the experimental
cases considered by
Wordsworth
[2009] in a (Ta,
) regime
diagram. The slight difference between the measurements
and the computations along the traverse corresponding to
the temperature differenceT
T
= 2K in Figure 16.3 results
from the change operated on the outer radius of the exper-
imental setup when drawing the diagram (
b
exp
= 14.3 cm
instead of the value
b
= 15 cm used in the simulations;
see Table 16.1). However, no significant differences were
observed on the nature of the flow regime between mea-
surements and computed solutions at identical control
parameter values. Thus, in agreement with experimen-
tal investigations, the first value at
= 0.25 rad/s, corre-
sponding to
(
,Ta
)
=
(
2.3475,2.95
10
6
)
, yields a weak
wave flow, while for the two others, at
values of 0.35
×
