Geoscience Reference
In-Depth Information
∂T
∂t
Nondimensionalization is usually not performed, essen-
tially because little benefit is gained by doing so. Unlike in
some other classical fluid dynamics problems, the intro-
duction of dimensionless parameters does not signifi-
cantly reduce the dimension of the parameter space. In
particular, there are a total of seven dimensional parame-
ters:
T
b
−
2
T
=
κ
∇
−
(
u
·∇
)T
,
(2.5)
∇·
u
= 0,
(2.6)
where
u
(r
,
ϕ
,
z
,
t)
=
u(r
,
ϕ
,
z
,
t)
e
r
+
v(r
,
ϕ
,
z
,
t)
e
ϕ
+
w(r
,
ϕ
,
z
,
t)
e
z
is the fluid velocity vector that has been decomposed
into components that lie along the standard cylindrical
polar basis vectors
e
r
,
e
ϕ
,and
e
z
;
=
e
z
is the rota-
tion vector;
=
R
,and
D
.Many
choices of nondimensionalization are possible; however,
in all cases, the resulting equations depend on no less than
six dimensionless parameters (see Section 2.1).
T
a
≡
T
,
,
ν
,
κ
,
α
,
r
b
−
r
a
≡
|
|
is the rate of rotation about the
axis of the annulus;
p
is the pressure deviation from
p
0
=
ρ
0
g(D
z)
+
ρ
0
2
r
2
/
2;
ν
is the kinematic viscosity;
κ
is the coefficient of thermal diffusivity;
g
is the gravita-
tional acceleration;
T
0
is the reference temperature; and
∇
−
2.3. TRANSITION CURVE
is the usual gradient operator in cylindrical coordinates.
The domain is
Most of the numerical studies of the annulus use
a numerical experimentation approach. This essentially
consists of performing a series of numerical integrations
of the governing equations from a specific initial state for
a small set of parameter values; this is similar to the pro-
cedure used in many laboratory experiments. This can be
a computationally efficient method for determining the
basic features of the flow in various regions in the param-
eter space, including approximations of the extent of the
various flow regimes. However, such methods have diffi-
culty pinpointing the boundaries between the various flow
regimes and in uncovering certain details of the regimes.
For instance, regions in which two flows are stable and
the mechanism by which these bistable flows are generated
cannot be determined. Here we implement an alterna-
tive approach that uses numerical linear stability analysis
[
Lewis and Nagata
, 2003;
Lewis
, 2010]. This not only pro-
vides a method for precisely determining the location of
the transition curve but also is the first step in a weakly
nonlinear analysis (center manifold reduction and normal
forms) that can provide important information about the
flow transitions. Although computationally more chal-
lenging than numerical experimentation, the analysis sup-
plies complementary information regarding the primary
transition. We will describe the nonlinear analysis in the
next section, while in this section we will provide some
details of the linear stability analysis. The methods and
presentation follow the work of
Lewis and Nagata
[2003],
where more details can be found.
In the linear stability analysis, we seek the parameter
values at which the steady axisymmetric solution (basic
state) loses stability to small (baroclinic) perturbations.
These locations in parameter space will correspond to the
primary transition, i.e., the axisymmetric-to-wave transi-
tion. We find the steady axisymmetric solution by look-
ing for solutions of equations (2.4)-(2.6) subject to the
boundary conditions (2.7) and (2.8) in the form
u
=
u
(
0
)
(r
,
z)
,
p
=
p
(
0
)
(r
,
z)
,
T
=
T
(
0
)
(r
,
z)
,
r
a
≤
r
≤
r
b
,
0
≤
ϕ<
2
π
,
0
≤
z
≤
D
.
The boundary conditions are chosen to mimic the
ideal experimental apparatus. In particular, for experi-
ments using a rigid, flat top and bottom of the annu-
lus, the boundary conditions on the fluid velocity are
taken to be nonslip at all boundaries. The differential
heating is imposed through the boundary conditions on
the temperature; that is, the inner wall of the annulus
is held fixed at
T
=
T
a
and the outer wall at
T
=
T
b
.
We take
T
=
T
b
−
T
a
>
0, as in most studies of the
annulus, although some studies have considered
T <
0
[
Koschmieder and White
, 1981]. It is assumed that the top
and bottom of the annulus are perfect thermal insulators.
Thus, the boundary conditions are
u
=
0
on
r
=
r
a
,
r
b
and
z
=0,
D
,
(2.7)
T
=
T
a
on
r
=
r
a
,
T
=
T
b
on
r
=
r
b
,
∂T
∂z
=0 on
z
=0,
D
,
(2.8)
with 2
π
periodicity in
ϕ
for
u
,
T
,and
p
.
For the situation of a free upper surface instead of a
rigid flat top,
Williams
[1971] and
Miller and Butler
[1991]
assumed stress-free boundary conditions at
z
=
D
.The
differences in boundary conditions induce some small but
interesting differences in the observations. In particular, a
hysteretic primary transition is seen at parameter values
for which this did not occur in the rigid-lid case. We do
not investigate this further here.
The effects due to centrifugal buoyancy are included
via the term
2
rα (T
T
0
)
e
r
. These effects are generally
included in models because they are not insignificant at
higher rotation rates and are necessary to obtain a quanti-
tative comparison with experiments. See
Lewis and Nagata
[2004] for a discussion of this term's quantitative effects on
the primary transition.
−
i.e., solutions that are independent of time
t
and the
azimuthal variable
ϕ
. The axisymmetric solution also
