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et al.
[1981] suggested the route through the harmonic
(Figure 3.8b).
zonal mode is not usually the mode to which a flow
would eventually equilibrate. The final steady wave regime
would usually be dominated by a lower zonal wave num-
ber [
Hart
, 1973, 1981;
Pedlosky
, 1981]. For this reason,
there is an amplitude dependence of the growth rates of
the waves such that a higher mode can grow preferably
during a stage of strong zonal flow but that this mode
suffers stronger damping and reduced growth compared
to a longer wave when the zonal flow is reduced through
the original growth of the shorter wave, leading to an
alternation of which mode received more energy from the
zonal flow.
3.4.5. Wave Interaction Scenarios in Low-Order
Models
Numerous studies have investigated the onset of vac-
illation in a range of low-order models, each of them
isolating a few or even a single possible route by which
a steady, equilibrated wave starts to develop a vacilla-
tion. The most basic of them, for example, demonstrated
that the wave-mean flow interaction between a single
zonal wave and the mean flow is able to render a finite-
amplitude steady wave unstable to vacillation if the forc-
ing as quantified by the Froude number is large enough
or if the dissipation parameter
r
is small enough. As
a comprehensive review of the earlier two-layer mod-
els by
l2)/2.Klein
[1990] has shown, the “interesting” behavior
of vacillating and chaotic flows in the simplest mod-
els with a single unstable wave of wave number
(k
,
l)
was mostly found at an intermediate balance of forcing
and dissipation, as quantified by
r/
1
/
2
=
O(
1
)
,where
=
F
3.5. PRANDTL NUMBER EFFECTS
3.5.1. Observations
As discussed by
Lewis
[2013], the Prandtl number as
the ratio of the kinematic viscosity over the thermal
diffusivity,
Pr =
ν
(k
2
+
l
2
)/
2.
l2)/2.Klein
[1990] also found that the inclu-
sion of more wave modes into the models tends to stabilize
the flow but it does not fundamentally alter the types of
flows observed.
Früh
[1996] analyzed the various possible wave inter-
action scenarios in a set of low-order models where the
included wave modes were carefully chosen to allow or
suppress specific wave-wave interaction routes based on
the selection criteria (equation (3.12)). For the analysis,
the sideband phase-locking function in equation (3.14)
was adapted to the triad resonance condition in equa-
tion (3.13) to define a triad phase-locking function. In the
full model, they observed a sequence of bifurcations that,
at least superficially, resembled the types of transitions
found in the experiment, from a steady wave through an
amplitude vacillation to some forms of chaotic modulated
amplitude vacillations, all of which involved substantial
energy transfer between the different zonal wave modes,
and finally to fairly irregular flow within the constraints
of the dimensions of the system. All the flows involv-
ing more than one zonal mode showed clear resonant
triad interactions, where the strength depended on the
relative mode amplitudes. Removing specific triads from
these models resulted only in moderate changes of the
observed flows, which suggested that the dynamics would
make use of preferred triads if they are available but that
they could make use of alternative routes for energy trans-
fer. The results changed more substantially if all triads
were removed and only wave-mean flow interactions were
retained. In that case, the preferred route was through
a competition between different zonal modes. This can
be understood through the fact that the most unstable
κ
,
(3.16)
−
affects the first transition, from axisymmetric flow to reg-
ular waves. We have also already mentioned in Section 3.1
that the Prandtl number affects the transition to ampli-
tude vacillation strongly in a way that can be summarised
as follows: bifurcation to AV from a steady wave on
decrease of
if Pr
1, none or very little AV in water,
onset of AV on increase of
if Pr
10, and amplitude
vacillation prevalent if Pr
10. One key characteristic
defined largely by the Prandtl number alone is the relative
thickness of the momentum and temperature boundary
layers.
3.5.2. Possible Role of Boundary Layers
As the low-order models are, so far, all based on the
quasi-geostrophic equations that do not solve explicitly
the heat equations or the boundary layers, they rely on
capturing the effect of boundary layers implicitly through
Ekman suction from horizontal boundary layers and hor-
izontal diffusion from vertical boundary layers. How-
ever, the relative thickness of the thermal and velocity
boundary layers affects the relative contribution to the
heat transport through the boundary layers and through
the fluid interior, respectively. A linear analysis by
Bar-
cilon and Pedlosky
[1967] identified that two parameters,
namely the Ekman number and the product of the thermal
Rossby number and Prandtl number, organize the rela-
tive contribution from different boundary layers into three
scenarios,
