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a second-order finite difference approximation in space
and in time. The approach was implemented on staggered
grids over a regular mesh for the pressure-temperature and
the velocity components. An explicit second-order leap-
frog scheme was employed to discretize both the space
and time derivatives. However the integration domain
was restricted to a sector, only admitting the dominant
wave and its harmonics. The cavity was filled with water,
assumed to satisfy the Boussinesq approximation, with the
density variation applied to the gravitational acceleration
[
Williams
, 1971].
Then
James et al.
[1981] carried out a combined labora-
tory and numerical study of the steady baroclinic waves.
Similar to the model proposed by
Williams
[1969], stag-
gered grids were used with a second-order finite difference
formula, but without any arbitrary truncation of the full
spectrum of the waves. A hyperbolic tangent transfor-
mation was introduced to stretch the mesh toward the
boundaries in the radial and axial directions while keep-
ing uniform distribution in the azimuthal direction with
Fourier series. To avoid severe constraint on the time
step stability condition associated with these small grid
sizes within the boundary layers, a Dufort-Frankel scheme
was used for the diffusion terms, taking into account
eventual variations of the viscosity, unlike the formula-
tion of
Williams
[1969]. Using a water-glycerol mixture
as working fluid, properties were assumed variable, with
quadratic and linear dependencies with the temperature
respectively for the density and for the kinematic viscos-
ity and the thermal diffusivity. However, the authors did
not achieve direct comparison of results between exper-
iment and numerical simulation under the same exter-
nal conditions, partially inferred to the restricted coarse
resolutions used, due to the existing computer capacity
constraints.
Hignett et al.
[1985] continued these studies and
obtained the same wave flow structure under identical
conditions for the laboratory experiment and numerical
simulation by using a different combination of water and
glycerol than
James et al.
[1981]. They introduced density
variations in centrifugal acceleration, contrary to
James
etal.
[1981]. They put forward the intransitivity of the flow
for steady wave state, corresponding to the coexistence of
different stable wave structures under the same external
conditions. They also mentioned the occurrence of hys-
teresis cycles during the transition from axisymmetric to
nonaxisymmetric solutions, as already observed by
Hide
[1958] with an open upper free-surface configuration.
A sophisticated version,
MORALS (Met Office/Oxford
Rotating Annulus Laboratory Simulation)
, derived from
the numerical tool proposed by
James et al.
[1981], was
implemented at the University of Oxford, UK (AOPP),
for the investigations of a wide spectrum of applications
devoted to geophysical fluid dynamics.
The choice of high-resolution spectral technique in the
present study stems from its ability to accurately pre-
dict the thresholds of the different bifurcations occurring
during time-dependent flow regimes, resulting from its
global character, in contrast with local finite difference
discretization [
Gottlieb and Orszag
, 1977;
Canuto et al.
,
1987]. In particular, the accuracy of spectral techniques
was discussed in detail by
Pulicani et al.
[1990] (see also
Randriamampianina etal.
[1990]) during the simulation of
oscillatory convection at low Prandtl number. Moreover,
the approach is well suited for the simulation of rotat-
ing flows in enclosures, where the boundary layer is three
dimensional from its inception.
16.2.2. Governing Equations
The physical model, the so-called baroclinic cavity, con-
sists of an annular domain of inner radius
a
, outer radius
b
, and height
d
rotating around its vertical axis of symme-
try. The cavity is filled with a liquid defined by a Prandtl
number Pr and is submitted to a temperature difference
T
=
T
b
−
T
a
between the inner, cold, and outer, hot,
cylinders closed by horizontal insulating rigid endplates.
One specific configuration involving an open upper free
surface is also considered.
In the meridional plane, the dimensional space vari-
ables
(r
∗
,
z
∗
)
∈[
a
,
b
]×[
0,
d
]
have been normalized into
[−
]×[−
]
the square
, a prerequisite for the use
of Chebyshev polynomials (where the asterisk denotes
dimensional variables):
1,1
1,1
2
r
∗
b
z
=
2
z
∗
d
b
+
a
b
a
−
−
r
=
a
,
1.
−
−
The fluid is assumed to satisfy the Boussinesq approxima-
tion [
Zeytounian
, 2003] with constant properties except for
the density when applied to the Coriolis, centrifugal, and
gravitational accelerations, where
ρ
∗
=
ρ
0
[
α(T
∗
−
,
where
α
is the coefficient of thermal expansion and
T
0
is a
reference temperature
T
0
=
(T
b
+
T
a
)/
2. However, it was
found that the contribution of density variation with the
Coriolis term
ρ
0
αT
∗
e
z
×
1
−
T
0
)
]
V
∗
was negligible compared to
the centrifugal and gravitational ones. Moreover, for the
imposed external conditions, the variations with temper-
ature of viscosity and thermal diffusivity remain small,
keeping the value of Prandtl number almost constant
(at least below the margins of error from measurements).
The reference scales are the velocity
U
∗
=
gαT/
2
and
the time
t
∗
=
(
2
)
−
1
, and the nondimensional normal-
ized temperature is 2
(T
∗
−
T
0
)/T
[
Randriamampianina
et al.
, 2006].
Depending on the type of solution sought, axisymmet-
ric or nonaxisymmetric, two different approaches are con-
sidered independently for the governing equations of the
flow dynamics. In the first case, a vorticity stream function
