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aspect ratio 0.18 in each layer and Rossby number between
0.1 and 1, we have
γ
2
Ro between 0.003 and 0.03, which
ensures that (a) the vertical acceleration can be neglected,
(b) the hydrostatic balance is verified, and (c) the shallow-
water approximation is valid. This condition is not always
met in former experiments with larger aspect ratios (see
Table 11.1).
Both models consider a jump in density with sepa-
rated equations for each layer coupled via pressure and,
if included, viscous effects. The dynamics of miscible
interfaces are however not captured, and especially inter-
actions between density and vorticity at the interface with
shear instabilities are not included. Below we consider
experiments and direct numerical simulations (DNSs)
built on the full Boussinesq Navier-Stokes equations
(11.2) in the annular configuration (see Section 11.2.2).
the dye visualizations, was of the order of 3 mm for a
tank of 1 m radius and allowed to measure the presence
of small-scale waves.
The lack of information about the flow characteristics
such as the local density gradient and three-dimensional
velocity field motivated us to obtain additional flow infor-
mation from DNSs for a series of almost identical, or at
least comparable, initial values as in the experiments.
11.2.2. Numerical Approach
The numerical simulations were performed using the
DNS code of
Verzicco et al.
[1997]. The governing equa-
tions are the Boussinesq equations, including Corio-
lis and centrifugal forces in the annular configuration.
Equations are written in cylindrical coordinates (
v
r
,
v
θ
,
v
z
)
and discretized on a staggered mesh by central second-
order accurate finite difference approximation. Details
of the numerical scheme are described by
Verzicco and
Orlandi
[1996] and
Verzicco and Camussi
[1997]. The ini-
tial (nondimensional) density profiles are
ρ
2
=1and
ρ
1
= 0 to which a random value is added as a perturba-
tion. At
t
= 0, the velocity field (
v
r
,
v
θ
,
v
z
) is at rest. At the
solid walls, the no-slip condition is used, except for the
top boundary,
z
=2
H
, where the rotating disk imposes
an azimuthal velocity
v
θ
=
.
This code has been tested for different initial conditions.
The resolution of the grid amounted in most 3D simula-
tions
N
θ
×
11.2.1. Experimental Modeling of Fronts
Abaroclinicfrontinthermalwindbalanceiscreatedina
two-layer rotating fluid with the shear across the interface
driven by a rotating lid at the fluid surface. For the study
of baroclinic instability, this forcing is preferable because
the density difference is set from the beginning, and flow
measurements of the velocity field and the density field
are better accessible. The mixing of the miscible interface
amounted to a very small percentage of the volume of the
layers and therefore has a negligible influence on the buoy-
ancy frequency at the interface. The flow dimensions of
the experimental setup are indicated in Figure 11.1 and the
parameter ranges are indicated in Table 11.1. In order to
investigate the frontal evolution for the Bu number ranges
indicated in Table 11.1, the disk rotation
was set con-
stant for a very slowly increasing background rotation
(t)
such that the values of the Ro, Ek, and Bu numbers
changed correspondingly. This same method had initially
been tested successfully by, e.g.,
Williams et al.
[2005], and
the comparison with the same experiment for fixed back-
ground rotation assured that Ekman pumping effects due
to spin-up did not affect the flow dynamics and instability.
The density difference across the interface,
ρ
=
ρ
1
−
97. Tests
with 2 times higher resolution grids as well as with 2 times
lower resolution grids showed essentially the same results
but with less detail.
In DNS, the full equations are solved, but care must be
taken with the evolution at small scales. To assure stability,
the grid size must be of the same order of magnitude as
the Kolmogorov scale, i.e. the scale at which viscosity dissi-
pates energy,
L
K
=Re
−
3
/
4
N
r
×
N
z
= 257
×
97
×
257 or 97
×
97
×
l
, with
l
the integral scale and
Re the Reynolds number. If the grid scale is larger than
L
K
, energy may accumulate artificially at this length scale
without being dissipated, leading to a numerical instabil-
ity. In the laboratory experiments, the Reynolds number
based on the velocity of the rotated disk is of
O
(50,000).
Since this Reynolds number would require too long calcu-
lationtimes,asmallerReynoldshasbeenusedthatisbased
on a larger viscosity and a grid size that is adapted to the
small-scale flow features of interest, as discussed below.
To reproduce the dynamics of the front in coherence
with the laboratory experiments (Sc = 700), the Schmidt
number is necessarily large and requires a high resolu-
tion. For small Schmidt numbers, the density interface was
found to diffuse too rapidly, i.e., before the front becomes
unstable. For high Sc numbers, scalar diffusion is much
weaker than viscous diffusion so that its typical length
scale is also much smaller than the Kolmogorov scale.
×
ρ
2
,
and reduced gravity,
g
=
gρ /
ρ
(with
g
the gravity con-
stant), were kept constant and close or equal to the values
by
Williams et al.
[2005] (see Table 11.1).
To visualize the flow, the upper fluid layer was dyed
with fluorescein dye and illuminated by a horizontal laser
sheet at middepth such that the inclined interface of the
front was visible in the top-view images. The wavelengths,
phase speeds, and growths of the instabilities are obtained
from spatiotemporal sequences of the top-view frontal
evolution, also known as Hövmüller diagrams. In addi-
tion, PIV (Particle Image Velocimetry) measurements
from tracer particles gave access to typical velocity fields
in each layer. The resolution in the experiments, especially
¯
