Geoscience Reference
In-Depth Information
0.6
(a)
(b)
5.0
3.00
0.5
C
2
C
2
D
0.4
C
B
A
-5.0
-3.0
0.3
-6.0
4.8
-3.0
3.1
C
1
C
3
(c)
(d)
0.2
2.5
3.0 3.5
Taylor number, 10
6
4.0
5.0
3.00
Figure 1.12.
Bifurcation diagram showing the maxima of
m
=2
temperature amplitudes at successive values of
C
2
C
2
T
keeping
constant at
1.75. Adapted from
Young and Read
[2008].
Copyright 2008 with permission from Elsevier.
-5.0
-3.0
where
σ
,
r
a
,and
b
are constants,
X
is related to
A(τ )
,
Y
is related to
V(τ )
and
Z
-6.0
4.0
-3.0
3.0
C
1
C
3
F(A
,
V)
[e.g.,
Brindley and
Moroz
, 1980;
Gibbon and McGuinness
, 1980;
Pedlosky and
Frenzen
, 1980;
Klein
, 1990], which is the famous set of
equations that can result in the Lorenz attractor [
Lorenz
,
1963a]. In the presence of a “planetary vorticity gradient”
or
β
effect, the wave-zonal flow interaction problem may
reduce to a set analogous to the
complex
Lorenz equations
[e.g.,
Gibbon and McGuinness
, 1980;
Fowler et al.
, 1982;
Lovegrove et al.
, 2001, 2002]:
dX
dτ
∼
Figure 1.13.
Phase portraits [(a) and (c)] and Poincaré sections
[(b) and (d)] obtained from measurements of temperature in
a rotating annulus showing a transition from amplitude vac-
illation (top) to a chaotic “modulated amplitude vacillation”
(bottom). Adapted from
Read et al.
[1992] with permission.
the observed “chaos” [
Read et al.
, 1992;
Früh and Read
,
1997]. The transition may be illustrated in reconstructed
phase portraits derived from time series of temperature
measurements, for example. Examples are illustrated in
Figure 1.13 from the experiments reported by
Read et al.
[1992]. The other main route may be via the period-
doubling sequence found in model simulations by
Young
and Read
[2008] although, as mentioned above, this route
has so far proved elusive in real experiments.
=
σ(Y
−
X)
,
(1.18)
dY
dτ
=
XZ
+
r
a
X
−
aY
,
(1.19)
dZ
dτ
=
1
2
(X
∗
Y
+
XY
∗
)
−
bZ
,
(1.20)
where
X
and
Y
are now complex variables and
r
a
and
a
are complex parameters. The onset of chaos in these
models as parameters are smoothly varied is characterized
by a particular sequence of transitions typically involv-
ing either a sudden “snap-through” bifurcation from an
initially steady wave as dissipation is reduced or a period-
doubling cascade from an “amplitude vacillation”state as
dissipation is increased [
Klein
, 1990].
In the thermal annulus, the situation seems more com-
plicated, with the possibility of at least two distinct routes
to chaotic behavior. In one case, periodic AV gives way
to an azimuthally asymmetric, chaotically modulated vac-
illation in which two or more adjacent wave numbers
occur in irregular competition. The final “chaotic” state
appears to comprise at least three independent frequen-
cies together with a “noisy” component associated with
1.3.6. Structural Vacillation and Transition
to Geostrophic Turbulence
“Structural vacillation” (also known as “shape” or
“tilted-trough vacillation” or SV [
White and Koschmieder
,
1981;
Buzyna et al.
, 1984]) occurs as the irregular flow
transition is approached and, in its purest expression, is
characterized by a nearly periodic, horizontal tilting of
the radial axes of wave peaks and troughs [
Weng et al.
,
1986;
Weng and Barcilon
, 1987]. However, in practice it
takes many different forms, depending upon a variety of
factors, including how close the dominant wave number
m
may be to Hide's maximum stable wave number
m
max
[cf. equation (1.13)]. This becomes more pronounced as
