Geoscience Reference
In-Depth Information
Table 11.1.
Dimensions and parameter regimes of experiments on the instability of baroclinic fronts and the parameter ranges
considered.
Reference
γ
L
(cm)
Bu
Ro
E
k
d
10
−7
,3
10
−3
Hart
[1972, 1973, 1976,
0.15, 0.039 0.36 or 0.67 12.5
0.02-0.1
0.02-0.25 2.5
×
×
0.1-0.7
10
−5
1979, 1980, 1981, 1985]
3
×
10
−7
-1
10
−5
0.1-0.7
King
[1979]
0.01-0.02
2.19
5.7
0.04-0.125
0.035-0.1 1
×
×
10
−3
-1
10
−2
Carrigan
[1978]
0.027
0.67
7.62
0.0033-0.033 0.017-0.5 1
×
×
—
10
−6
-1
10
−4
Lovegrove et al.
[1999, 2000] 0.006
2
6.25
0.06-0.25
0.1-10
1
×
×
0.004-0.03
10
−6
-3
10
−3
Williams et al.
[2003, 2004a, 0.006
2
6.25
0.03-333
0.1-100
1
×
×
0.0003-0.03
2004b, 2005, 2008, 2010]
Flór et al.
[2011]
10
−5
-7
10
−4
0.006
0.18
75
0.02-5
0.06 -1.5 7
×
×
0.01-0.1
Note: Flór et al.
[2011] considered miscible salt-stratified fluids.
baroclinic instability. In this chapter, we focus on fronts
that do not intersect with the top or bottom boundary.
We consider ageostrophic flows and discuss the recently
investigated Rossby-Kelvin (RK) instability first discov-
ered by
Sakai
[1989] and aspects of the frontal instability
that are related to the thickness of the interface.
Sakai
[1989] considered ageostrophic modeling of baroclinic
instability and, following
HayashiandYoung
[1987], inves-
tigated the resonance of different type of waves, such as
Kelvin, Rossby, and Poincaré waves. The baroclinic insta-
bility was interpreted as a resonance (a coupling of the
phase speeds) between two Rossby waves, one in the upper
and one in the lower layer. The RK instability is a conse-
quence of the resonance between a Rossby wave in one
layer and Kelvin or Poincaré wave in the other layer.
Though gravity waves move faster than Rossby waves,
resonance occurs when the two wave frequencies match
due to the Doppler shift. Recently, this approach has
been continued for rectangular and annular geometries
respectively by
Gula et al.
[2009a, 2009b],
Gula and Zeitlin
[2010], and
Gulaetal.
[2010] showing especially significant
growth rates for the resonance of a Rossby wave in the
upper layer and a coastal Kelvin wave in the lower layer.
Sakai
[1989] used the term Rossby-Kelvin for all vorticity-
gravity wave resonances, but here we will reduce the term
“Rossby-Kelvin instability” to this particular resonance.
These results were compared to experimental results by
Flór et al.
[2011].
The choice of a fluid interface between two miscible
fluid layers is in between the limits of a very thin immis-
cible fluid interface [
Williams et al.
, 2005] and, at the
other end, a linear density stratification. For two-layer
fluids, it has been shown in numerical simulations [
Gula
et al.
, 2009a] that the RK instability occurs only for
thin interfaces. For interfaces with a thickness beyond
a certain threshold, i.e.
δ
ρ
/(
2
H)>
0.16, where 2
H
is the
fluid depth and
δ
ρ
the interface thickness (see Section
11.3.2), the growth rate of ageostrophic modes was found
to reduce to zero. Though we are not aware of simulations
on the RK instability in linearly stratified fluids, in consid-
ering a very thick interface as an approximation of a linear
stratification, one may expect also a zero growth rate
there. In general, relatively thin density interfaces allow
for a richer variety of shear instabilities, including Kelvin-
Helmholtz and Hölmböe instability (see Section 11.3.3).
Small-scale waves at the immiscible fluid interface in
the baroclinic unstable regime [
Lovegrove et al.
, 2000,
Williamsetal.
, 2003, 2004a, 2004b, 2005] were interpreted
as the spontaneous emission of inertia-gravity waves. In
these experiments the optical activity of the two immis-
cible fluids gave access to very accurate measurements
of interfacial perturbations. A constraint was that the
reduced gravity was fixed by the limited type of optical flu-
ids that is available (see Table 11.1). In a larger setup and
filled with a salt-stratified two-layer fluid [
Flóretal.
, 2011],
interfacial waves due to surface tension effects were elimi-
nated, and, as mentioned, access to a larger range of scales
of motion was achieved. Hölmböe instability was in part
considered responsible for similar type of small waves, a
mechanism that could also be efficient in the immiscible
fluid experiments mentioned above [
Flór et al.
, 2011].
In this chapter, we report numerical results for a two-
layer stratification with a smooth interface that have
earlier been presented by
Scolan
[2011], and experimen-
tal results by
Flór et al.
[2011]. We discuss these inter-
facial waves next to the different instabilities observed
in the parameter space set by Burger number, Rossby,
and Ekman or the dissipation number, defined in the
next section. In doing so, we consider experimental and
numerical results of the setup depicted in Figure 11.1.
The governing equations and pertinent nondimensional
numbers of this experimental system are presented in
Section 11.2.1, followed by a brief description of the
numerical approach in Section 11.2.2. The instability
regime diagrams and the baroclinic instability including
the recently reported observations of the RK instabil-
ity are presented in Section 11.3.1. In Section 11.3.2 the
secondary vertical circulation is investigated numerically.
