Geoscience Reference
In-Depth Information
matrices from these Helmholtz and Poisson operators are
performed once during a preprocessing step.
three-dimensional solutions were previously validated by
Randriamampianina et al.
[1997] for a liquid-filled cavity
(Pr=13.07) with respect to the detailed results reported
by
Hignett et al.
[1985] from a combined laboratory
and numerical study. Comparisons have been carried out
between our computations and their measurements for a
regular steady three-wave flow (characterized by a domi-
nant azimuthal wave number
m
= 3). Very close agreement
has been obtained for the qualitative structure of the flow
pattern and for the quantitative comparison of the radial
variation of the azimuthal velocity at different heights.
Particular attention has been paid to the grid effect on
the solution, which has served as a basis for subsequent
studies.
16.2.5. Computational Details
For the transition from the upper symmetric regime to
the regular waves, the initial conditions corresponded to
the steady axisymmetric solution at each azimuthal node
to which a random perturbation was added to the tem-
perature field in azimuth. Subsequently, the strategy con-
sisted of progressively increasing the rotation rate without
adding any further perturbations for the computation of
the following successive three-dimensional solutions.
The length of time for each specific computed solu-
tion strongly depends on the fluid considered, e.g., on
the Prandtl number, due to the very different temporal
behaviors involved. Some values of the wave drift were
reported for air by
Randriamampianinaetal.
[2006]. Theo-
retically, only one drift period of the large-scale baroclinic
waves is necessary to have a complete analysis of the flow.
But close to a bifurcation, corresponding to significant
changes on the flow structure and temporal behavior, sev-
eral drift periods have been computed until the final state
is reached. This was the case during the computations
of the different bifurcations occurring at the transition
between wave numbers
m
=2and
m
= 3 for air, in par-
ticular during the bifurcation between the quasi-periodic
regimes QP2 and QP3 characterized by two and three
incommensurate frequencies, respectively. On the other
hand, the presence of small-scale inertia gravity waves in
the cases of liquids required much longer drift periods
of the baroclinic waves and higher resolutions to ensure
grid independency of the solution than for air, therefore
increasing the length of integration time to be simulated.
The different meshes with the corresponding time step
used are reported hereafter for each specific case treated.
For Pr = 16 with
T
=2Kat
= 0.5125 rad/s, about
7.62
16.3. RESULTS
Hide
[1958] and
Fowlis and Hide
[1965], from their pio-
neering experimental investigations of baroclinic instabil-
ity using liquids as working fluids, have delineated three
main classes of flow regimes: axisymmetric regimes, reg-
ular waves, and irregular waves or geostrophic turbu-
lence (see also
Hide and Mason
[1975]). The regular wave
regimes are composed of the steady waves, denoted
S
,
and the vacillation regimes subdivided into amplitude vac-
illation, AV or MAV (modulated amplitude vacillation),
and structural vacillation, SV. The steady waves are deter-
mined by a dominant azimuthal wave number
m
in space
and characterized by periodic oscillations in time induced
by the uniform angular drift of the waves with constant
amplitude. The amplitude vacillation regimes are defined
by periodic (AV), quasi-periodic, or chaotic (MAV) tem-
poral behavior of the amplitude of the dominant wave
number (for a detailed analysis of the amplitude vacilla-
tion phenomenon, see Chapter 3 in this topic). The SV
regime, an intermediate step before the transition toward
geostrophic turbulence, is characterized by a spatiotem-
poral chaos but still with a well-defined dominant wave
number, as shown by the experimental evidence of
Früh
and Read
[1997] (see also
Read et al.
[2008]).
Three specific fluids have been considered in the present
study, air, a water-glycerol mixture, and water, in order to
get insight into the important role played by the Prandtl
number on the spatiotemporal characteristics of the baro-
clinic instability. Indeed, the Prandtl number Pr is a
parameter of particular interest, also in the context of
other convection problems.
Fein and Pfeffer
[1976], who
carried out a careful survey of the main flow regimes
in a thermally driven annulus using mercury, water, or
silicon oils, found significant differences in the onset of
baroclinic instability in the region of the so-called lower
symmetric transition at low Taylor number, where viscous
diffusion and thermal diffusion are expected to play a
major role. Some substantial differences in the onset of
10
−
5
CPU seconds per time step and per mode on
the supercomputer
NEC
×
SX
5 (IDRIS, Orsay, France)
were necessary to compute the different scales occurring
simultaneously within the cavity once the transient was
removed (see hereafter the corresponding time step and
mesh used). The transient is assumed to be finished when
a clear temporal behavior can be identified from the time
evolution of one dependent variable taken at a fixed mon-
itoring point and when its random behavior disappears.
The transient is also associated with the flow structure
observed.
−
16.2.6. Validation
The numerical tools have been completely devel-
oped by the team [
Chaouche et al.
, 1990;
Hugues and
Randriamampianina
, 1998;
Raspo et al.
, 2002]. The
