Geoscience Reference
In-Depth Information
(a)
(b)
10
2
IW
5
E
4
MIW
5
3-4
10
0
B
BI
4
10
1
MRW
1-2
AX
5
5
8
5
6
5
7
1-3
8
RK
4-8
4-8
KH
or
H
AX
Outcropping
4
RK
5
Stable
KH
or
H
1-2
5
6-7
3-4
BI
1-3
5
10
0
KH
IW
10
-1
AX
Baroclinic
instability
4
5
3
2
1
10
-1
AX
10
-2
10
-1
10
-2
10
-1
10
0
10
1
Bu
=
√
νΩ
H
∆Ω
d
Figure 11.3.
Regime diagrams of the different instabilities. (a) in (Bu,
d
)-space with the gray lines obtained for two-layer immiscible
fluids [
Williams et al.
, 2005] separating axisymmetric flow (AX), mixed regular waves (MRWs), and mixed irregular waves (MIWs),
and gray shaded areas representing regimes of Kelvin-Helmholtz or Hölmböe instability (KH or H), RK instability, and baroclinic
instability (BI) (see Figure 11.2) found for approximately the same density difference
g
in miscible fluids. The black dashed lines
delimit the region of KH and Hölmböe instability, and the gray dashed line baroclinic instability according to quasi-geostrophic
theory [
Hart
, 1972]; (b) in (Ro, Bu) space with the gray lines representing the theory of
Gula et al.
[2009a] for inviscid flows.
Adapted from
Flór et al.
[2011].
The required resolution in this case should be estimated
from the Batchelor scale for scalar diffusion, defined as
L
B
=
L
K
×
of the Burger number and dissipation number
d
.The
evolution of the instability in parameter space changes
from Kelvin-Helmholtz and RK instability to regular
baroclinic instability and eventually the irregular change
between different baroclinic modes known as the chaotic
regime [
Flór et al.
, 2011]. These figures are obtained for
increasing background rotation (i.e. for increasing inverse
Burger number 1
/
Bu, and increasing
d
) and fixed disk
rotation. With respect to results of
Williams et al.
[2005],
Flór et al.
[2011] showed RK and Hölmböe instability,
both of which will be discussed in more detail in the
sections below. When neglecting viscous effects, the full
regime diagram reduces to a two-dimensional one as
function of Ro and Bu, shown in Figure 11.3b.
Also amplitude vacillation has been observed [e.g.,
Pedlosky
, 1987]. Typically, the oscillation occurs at the
limit between RK and baroclinic instability. The flow
oscillates in time between the axisymmetric state and the
unstable baroclinic state. These periodic changes in wave
amplitude are related to the transfer of energy between
the wavefield and the zonal flow and have been observed
before in two-layer immiscible fluid flows [
Hart
, 1976,
1979;
Read et al.
, 1992;
Früh and Read
, 1997] but for
smaller Bu numbers, i.e, in the baroclinically unstable
regime. In Figure 11.4a the very contrasted crests and
troughs are separated by approximately 35 table rotation
periods; the oscillation in mode 2 obtained from these data
is represented in Figure 11.4b. Even though there is some
noise and the amplitude of the mode gradually decreases,
Sc
−
1
/
2
[
Batchelor
, 1959;
BuchandDahm
, 1996;
Rahmani
, 2011]. But since singularities due to the rela-
tively high Schmidt number and coarse grid in the present
simulations occurred far from the relatively smooth inter-
face, the interfacial dynamics were found to be well cap-
tured for a resolution rather estimated with
L
K
(grid step
around 1-5 times
L
K
).
Given the numerical constraints and purpose of our
study on rather small-scale phenomena, we thus found
a compromise of a
N
θ
700
grid with a Schmidt number of 700 and a Reynolds num-
ber of 3500 for the study of the secondary circulation.
For the 3D simulation of the flow we choose a grid of
1201
×
N
r
×
N
z
=1
×
700
×
257 with a spatial resolution of 1-5 mm asso-
ciated with Schmidt number of 100 and Reynolds number
of 10,000.
Depending on the Reynolds number of the simulation
and the associated value of the effective viscosity, the
time corresponding to the spin-up took 2-40 dimension-
less units.
×
301
×
11.3. FLOW EVOLUTION OF OBSERVED
INSTABILITIES
Figure 11.3a summarizes the different instability
regimes of the interface for a cyclonic forcing and
constant density difference as a function of the inverse
