Geoscience Reference
In-Depth Information
Guckenheimer and Holmes
[1983]; e.g., case II
is the time
reverse of case II.
The results indicate that case Ib
applies for all mode
interaction points for all Prandtl numbers greater than 4.
This case corresponds to the “simple”case 1 of
Kuznetsov
[2004]. The results of
Lewis and Nagata
[2003], in which
a fluid of Prandtl number 7.1 is studied, fall into this
category; the details of the dynamics of all high Prandtl
number are thus qualitatively the same. This is consistent
with the experimental results reported in [
Fowlis and Hide
,
1965;
Pfeffer and Fowlis
, 1968].
This is an important case because it indicates the mech-
anism by which hysteresis of rotating waves occurs in the
annulus. In this case, all normal form coefficients are neg-
ative and
A
is also negative. The negative
a
and
d
indicate
that supercritical Hopf bifurcations occur at
μ
1
=0and
μ
2
= 0, respectively; i.e., there exist periodic orbits corre-
sponding to fixed points 2 and 3 for
μ
1
>
0and
μ
2
>
0,
respectively. Fixed point 4 exists in the wedge defined by
μ
2
>
μ
2
μ
2
=-
c
μ
1
ρ
2
ρ
1
μ
2
=-
μ
1/
b
s
μ
1
Figure 2.5.
Two-parameter bifurcation diagram for case Ib
(reproduced following [
Lewis and Nagata
, 2003]). The dia-
gram is displayed using the real parts of the critical eigenvalues
μ
1
,
μ
2
as the bifurcation parameters. The regions of different
character are separated by solid lines. In each region, the corre-
sponding phase portrait is drawn, where the phase portraits are
presented in (
ρ
1
,
ρ
2
) coordinates. Fixed points at the origin cor-
respond to the steady axisymmetric solutions of the full model
equations; fixed points along the horizontal (vertical) axis cor-
respond to wave number
m
1
(
m
2
) rotating waves, while fixed
points for which
ρ
1
>
0and
ρ
2
>
0 correspond to mixed-mode
flows for the model equations.
cμ
1
. Linear stability analy-
sis reveals that fixed point 2 is stable for
μ
2
<
−
μ
1
/b
and
μ
2
<
−
cμ
1
and unstable elsewhere (where it exists) and 3 is stable for
μ
2
>
−
μ
1
/b
and unstable elsewhere. Thus, both fixed
points 2 and 3 are stable inside the wedge, and thus, there
is bistability of the corresponding wave solutions. Fixed
point 4 is always unstable, and thus, the corresponding
mixed-mode flow would not be observable in experiment.
The corresponding two-parameter bifurcation dia-
gram with corresponding phase portraits is presented in
Figure 2.5. It is presented with
μ
1
and
μ
2
as parameters
with the understanding that these depend directly on the
physical parameters
and
T
and thus on the nondimen-
sional parameters, the thermal Rossby number
−
Stable periodic orbit (wave)
Unstable periodic orbit (wave)
Unstable mixed-mode
Stable steady solution
Unstable steady solution
Bifurcation point
R
, and the
Taylor number
, with which most regime diagrams are
presented. This figure and similar ones that follow should
be interpreted as follows: The line
μ
j
= 0 corresponds to
the neutral stability curve of wave number
m
j
,
j
=1,2.
Thus, the four quadrants in Figure 2.5 correspond to the
four regions created by the intersection of two neutral sta-
bility curves. The relative location of the curves in the
T
T
U
s
-
R
plane corresponding to the lines
μ
2
=
−
μ
1
/b
and
μ
2
=
cμ
1
are indicated in Figure 2.5, but the exact form
they take depends on the values of
b
and
c
and on how
μ
1
and
μ
2
depend on
and
T
.
The hysteresis between the wave solutions may occur on
a one-parameter path through parameter space such as
the one parameterized by
s
that is indicated in Figure 2.5.
See Figure 2.6 for the bifurcation diagram that would be
observed along the path
s
. Thus, it would be expected
that the transition from the rotating wave of dominant
wave number
m
1
to one of wave number
m
2
would occur
at different parameter values depending on whether you
followed a path of increasing or decreasing
s
.
−
Figure 2.6.
One-parameter bifurcation diagram that may be
observed along the path parameterized by
s
that is indicated
in the two-parameter bifurcation diagram for case Ib
plotted in
Figure 2.5.
For all mode interaction points, as the Prandtl number
is decreased, there is a transition from case Ib
to case
Ia
, equivalently from simple case 1 to simple case 2 of
Kuznetsov
[2004]. Case Ia
is another important case; the
change in case occurs via a change in sign of
A
while all
other coefficients remain of the same sign. The change in
