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In-Depth Information
and
Kuznetsov
[2004]. Here we present a short summary
in the context of the present application.
The last two equations of the normal form, (2.18) and
(2.19), indicate that to first order the variables
θ
1
and
θ
2
are
simply linear functions of time
t
and therefore represent a
constant rotation. Thus, we need only consider the depen-
dence on the radial variables
ρ
1
and
ρ
2
, i.e., the dynamics
can be found from considering only (2.16) and (2.17).
For the reduced system (2.16)-(2.17), the following fixed
points exist when the quantities inside the square root
signs are positive:
1.
ρ
1
=
ρ
2
=0
the dynamics are determined. For further discussion of
pseudo-arc length continuation see Section 2.5. The nor-
mal form coefficients have previously been calculated for
two different annuli with fluids of different Prandtl num-
ber:
Lewis and Nagata
[2003] compared to experiments
of
Fein
[1973], who used water as the working fluid, and
Lewis
[2010] compared to numerical experiments of
Ran-
driamampianina et al.
[2006], in which the fluid parameters
are taken to be those of air. Here we connect and extend
these results by varying the kinematic viscosity
ν
and ther-
mal diffusivity
κ
such that for high Prandtl number they
correspond to the values for a gycerol-water mixture as
in the experiments of
Hignett et al.
[1985], for moderate
Prandtl number the values correspond to those of water
[
Fein
, 1973], for near unity Prandtl number they corre-
spond to the values of air [
Castrejon-Pita and Read
, 2007],
while at low Prandtl number they correspond to mercury
[
Fein and Pfeffer
, 1976]. We do not consider variations
in the coefficient of thermal expansion
α
of the various
fluids, because a change in
α
can be unambiguously com-
pensated for with a change in
T
, as they both only
appear in the thermal Rossby number. Thus, although a
change in
α
will change the specific value of
T
at which
the transition occurs, the change in
α
would have no effect
on the critical values of the nondimensional parameters,
nor would it affect the predicted dynamics.
The parameters for the geometry of the annulus are
taken to be
2.
ρ
2
=0and
ρ
1
=
ρ
p
=
μ
1
−
a
3.
ρ
1
=0and
ρ
2
=
ρ
q
=
μ
2
−
d
=
−
dμ
1
+
bμ
2
A
4.
ρ
1
=
ρ
(T)
and
1
=
cμ
1
−
aμ
2
ρ
2
=
ρ
(T)
2
A
Fixed point 1 corresponds to the trivial fixed point of
the full normal form equations (2.16)-(2.19) and corre-
sponds to the steady axisymmetric solution of the model
equations. For fixed point 2, because we have
ρ
2
=0,
the constant rotation due to the
θ
1
dependence implies
that there exists a periodic orbit when
ρ
1
=
ρ
p
>
0. This
periodic orbit corresponds to a wave number
m
1
rotat-
ing wave of the full model equations. Likewise fixed point
3 corresponds to a rotating wave of wave number
m
2
.
Fixed point 4 has
ρ
1
>
0and
ρ
2
>
0, and thus, the
θ
1
and
θ
2
dependence implies that the corresponding orbit
of (2.16)-(2.19) lies on a 2-torus. If
ω
1
and
ω
2
are not
rationally related, then the orbit is quasi-periodic on the
torus. This is the case for all bifurcation points consid-
ered here. Fixed point 4 corresponds to a mixed-mode
flow for the full model equations. A straightforward anal-
ysis can determine the dependence of the stability of the
fixed points on the coefficients [
Guckenheimer andHolmes
,
1983;
Kuznetsov
, 2004]. The correspondence between the
fixed points 1-4 and the fluid flows described here holds
for the entire discussion below, i.e. we determine how
the stability of these flows, and the extent in parameter
space in which they exist, change as the Prandtl number is
varied.
r
a
=3.48cm,
r
b
= 6.02 cm,
D
=10cm.
These values are similar to those used by
Randria-
mampianina et al.
[2006] and
Pfeffer and Fowlis
[1968]. We
also fix
10
−
3
K
−
1
,
g
= 980 cm
/
s
2
.
α
= 3.30
·
Results for three mode-interaction points are presented
in Figures 2.2-2.4, which display the values of the nor-
mal form coefficients
a
,
b
,
c
,and
d
and of
A
=
ad
bc
as a function of Prandtl number Pr. Other mode interac-
tion points exist but for only some values of the Prandtl
number; these are not included here. Also, due to spatial
resonance, the
(m
1
,
m
2
)
=
(
2,1
)
mode interaction point
involves a different, more complicated analysis and is a
topic of future research. The regions of qualitatively dif-
ferent behavior are separated with vertical lines, and the
corresponding dynamics are labeled according to the cases
identified by
Guckenheimer and Holmes
[1983], where the
borders of these regions correspond to a change in sign
of one of the coefficients
a
,
b
,
c
,and
d
or of
A
.For
a table which relates the signs of the coefficients to the
given dynamic region, see the work of
Guckenheimer and
Holmes
[1983]. The prime in the case labels indicates that
it is the “time-reverse” case of the corresponding case of
−
2.4.1. Results Over a Range of Prandtl Numbers
In this section, we investigate the effects of Prandtl num-
ber on the dynamics that are observed close to the mode
interaction points. Specifically, as the Prandtl number is
varied, we use a pseudo-arc length continuation method
[see, e.g.,
Govaerts
[2000]] to track the location of the mode
interaction points in parameter space, and at each loca-
tion, we compute the normal form coefficients from which
