Geoscience Reference
In-Depth Information
3.1.2. Transition to Amplitude Vacillation
the same for the onset of amplitude vacillation in direct
numerical simulations of a low-Pr fluid on decreasing
.
The intermediate case was covered by
Sitte and Egbers
[2000], who were able to show that both existed, a Hopf
bifurcation from a steady wave 2 to a 2AV on decreas-
ing
toward the
m
= 3 region and a Hopf bifurcation
from a steady wave 3 on increasing
toward the
m
=2
region. The region between these two bifurcations showed
secondary bifurcations to chaotic modulated vacillations,
each involving both modes,
m
=2 and
m
= 3, but domi-
nated by their respective original mode. While the hys-
teresis in the transition between modes involving only
steady waves and amplitude vacillation is substantial, the
transition between the 2-dominated and the 3-dominated
chaotic flows seen by
Sitte and Egbers
[2000] had little
hysteresis and is more gradual, similar to the transition
between complex amplitude vacillation flow observed by
Früh and Read
[1997]. It appears that the modulated vac-
illations always involve activity in other modes, especially
the sidebands of the dominant mode, and thereby facili-
tate the transition from one dominant mode to the next
lower or higher mode.
In the corresponding two-layer experiment,
Hart
[1972]
observed that amplitude vacillation emerged from a steady
wave when the driving of the lid and consequently the
Rossby number were decreased. If the vertical velocity
shear, either driven by the lid or through the thermal wind
balance, is taken as the “forcing” of the system, then the
Rossby number defined by the lid rotation takes the equiv-
alent role as the thermal Rossby number defined by the
thermal wind. In that case, the observed transition from a
steady wave to the amplitude vacillation in the two-layer
system corresponds to the thermally driven annulus filled
with a low-Prandtl-number fluid.
Amplitude vacillations tend to develop from their cor-
responding steady wave flow through a supercritical bifur-
cation as precursor to a mode transition to a different
wave number. While the occurrence of structural vacilla-
tion appears to be determined more by dissipation, the
onset of amplitude vacillation and mode transitions are
more determined by the thermal forcing. Other factors
known to affect the occurrence of amplitude vacillation
are the tank geometry and the fluid's Prandtl number.
An impressionistic synthesis of the various experimental
reports by, in particular,
Hide et al.
[1977],
Jonas
[1981],
Hignett
[1985],
Pfeffer et al.
[1973],
Pfeffer et al.
[1980b],
Buzyna et al.
[1984],
Sitte and Egbers
[2000], and
von
Larcher and Egbers
[2005] suggests the following gener-
alization:
Vacillations seem to be more prevalent in a wider
anddeeperannulusilledwithahigherPrandtlnumberluid
.
The fact that vacillation appears more easily in a wider
gap could be a different phrasing of another observation,
namely that amplitude vacillation tends to be seen more
when the baroclinic wave has a relatively low wave number.
Hide and Mason
[1970] showed that only a finite range of
wave number can be observed, given by the ratio of the
zonal wave length to the gap width,
α
=
mπ(a
+
b)
(b
as 0.25
α
≤
m
≤
0.75
α
.
(3.3)
−
a)
The key difference between the regime diagram for a
low-Prandtl-number fluid (Pr
1 in Figure 3.3a) and for a
high-Prandtl-number fluid (10
80 in Figure 3.3b) is
the relative position of the vacillating regime. For a lower
Prandtl number, a steady wave can develop an amplitude
vacillation as the thermal forcing is reduced, prior to a
transition to a flow with a higher wave number, while the
onset of amplitude vacillation in a fluid with a higher
Prandtl number is usually found when the thermal forcing
and stratification are increased. While there is no exper-
imental evidence for a systematic trend in low-Prandtl-
number fluids, it has been observed in many experiments
that vacillation is rare in water but becomes more widely
observed at higher Prandtl numbers, to a degree where
steady waves become rare as the Prandtl number reaches
values in excess of 40. For example, the water-filled annu-
lus of
von Larcher and Egbers
[2005] only exhibited flow
resembling amplitude vacillation in the region between
the
m
=2 and
m
= 3 dominated range in the narrow-gap
annulus, whereas the annulus filled with a silicone fluid of
Pfeffer et al.
[1980b] appears to show always vacillating
flows in the regular wave range.
Bifurcation studies in a high-Prandtl-number fluid by
Read et al.
[1992] have suggested that the onset of vacilla-
tion on increasing
is consistent with a Hopf bifurcation,
and similarly
Randriamampianina et al.
[2006] showed
Pr
3.2.MECHANICS OF AMPLITUDE VACILLATION
In this section, some possible processes resulting in
amplitude vacillation will be presented. As all observa-
tions suggest that AV is a global modulation of a finite-
amplitude steady wave mode and that the steady wave
originates from a global mode instability of a zonal flow,
the processes are usually expressed in terms of energy
transfer between modes and the underlying zonal flow. In
this framework, the energy is described in such terms as
zonal kinetic energy
,
zonal potential energy
,
eddy kinetic
energy
,and
eddy potential energy
[
Lorenz
, 1955]. Fol-
lowing
Hart
[1976], the possible transfer routes can be
illustrated as in Figure 3.4. In the mechanically driven
two-layer system, the lid injects kinetic energy into the
two fluids, which are then converted to a distortion of
the interface, while in the thermally driven annulus, the
imposed horizontal temperature difference sets up the
sloped isotherms and the vertical shear flow. In both
