Geoscience Reference
In-Depth Information
(a)
(b)
0.12
0.12
0.1
m
4
−
m
3
ω
4
−
ω
3
0.1
m
3
−
m
2
2ω
3
− ω
4
ω
3
−
ω
2
0.08
0.08
0.06
m
6
−
m
4
0.06
m
2
0.04
0.04
ω
2
m
3
−
m
2
ω
3
−
ω
2
0.02
0.02
n
= 1
n
= 1
n
= 2
n
n
= 2
m
4
−
m
3
= 3
ω
4
−
ω
3
n
= 3
0
0
0
1
2
3
4
5
6
0
1
2
3
4
5
6
7
8
m
m
Figure 3.8.
Two possible routes of energy transfer to the sidebands, either (a) involving two triads coupled by the common long
wave with
m
= 1 or (b) through the harmonic of the dominant mode,
m
=2
m
.
triads depends on the wave amplitudes and resonance
conditions.
3.4.4. Wave Interactions in Experiments and CFD
Hide et al.
[1977] developed a method to quantify the
degree to which sidebands interact from spatially resolved
temperature measurements in the thermally driven rotat-
ing annulus. A Fourier analysis gave the amplitude
A
and
phases
φ
of the zonal modes (but not resolving radial
modes). Integrating equation (3.14) suggests
3.4.3. Higher-Order Wave Interactions
If no triad is fully resonant, higher-order interaction
scenarios can affect the baroclinic wave. One classic exam-
ple of this is the Benjamin-Feir instability [
Benjamin and
Feir
, 1967] where a monochromatic surface wave with
wave vector
k
in a channel develops a slow modulation
through the rise of a long wave of wave vector
δ
k
and
“sideband”waves with wave vectors
k
2
φ
m
−
φ
m
−
1
−
φ
m
+1
≈
const,
which lead to the definition of a sideband phase-locking
function
δ
k
. As with the res-
onance condition for the resonant triads (equation (3.13)),
a condition for the sideband instability can be written as
[
Zakharov
, 1968]
±
m
=2
φ
m
−
φ
m
−
1
−
φ
m
+1
,
(3.15)
and
Hide et al.
[1977] observed that this phase-locking
function was indeed fluctuating around a constant value
of
m
≈
2
ω
k
−
ω
k
−
δk
−
ω
k
+
δk
= 0.
(3.14)
In the rotating annulus or two-layer experiments, the pos-
sible wave numbers are a discrete set,
k
=2
πm/L
,and
the longest possible wave is that with the wave number
m
= 1, i.e.,
δk
=2
π/L
. With this, we can propose
an illustration of how this sideband instability can occur
through a coupled set of triads; one option invokes this
long wave,
m
= 1, while the other possible route involves
the first harmonic of the main wave mode, as illustrated
in Figure 3.8.
The questions that arise from this framework are: Is any
particular set of possible nonlinear interaction the essen-
tial process to destabilize a steady baroclinic wave and
lead to amplitude vacillation? Do a range of interaction
possibilities allow for all or specific types of amplitude
vacillation.
π
for fully developed steady waves and ampli-
tude vacillations. This did not hold for irregular flow or
for flows with a noticeable structural vacillation. The fluc-
tuation around a constant value implies that the resonance
would only be nearly satisfied and that nonlinear interac-
tions couple the waves when they are strong enough, that
the waves start to drift apart when that coupling becomes
weaker as the main mode decays, and that they become
reentrained when the wave grows again in the vacillation
cycle. While the sideband phase locking confirmed the
presence of nonlinear wave interactions, it does not dis-
tinguish between the two possible interaction routes illus-
trated in Figure 3.8. A theoretical study by
Plumb
[1977]
suggested the route through the long wave (Figure 3.8a),
while an analysis of a numerical simulation by
James
