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In-Depth Information
question, especially as the boundary layers are much
more important in the laboratory than in the atmosphere.
Furthermore, atmospheres do not have the Stewartson
layers to contendwith and thereforeshould not be affected
by the Prandtl number in the way as described in Section
3.5. However, through developing some understanding of
how the Prandtl number affects the annulus flow, it is
possible to disentangle the laboratory-specific dynamics
from those that are relevant to atmospheric or oceanic
dynamics and to the atmospheric modeler.
Appendix B: LOW-ORDER MODEL
OF BOUNDARY LAYER FEEDBACK
The model to couple the one-dimensional boundary
layer model with a single zonal baroclinic wave mode with
zonal mode
m
and radial mode
n
= 1 was formulated to
simulate the behavior for annulus parameters given by
γ
h
,
the horizontal aspect ratio,
π(a
+
b)/(b
−
a)
−
γ
v
,
the vertical aspect ratio,
d/(b
a)
Acknowledgments.
The author wishes to thank in
particular Raymond Hide, Peter Read, Patrice Klein,
Christoph Egbers, and Thomas von Larcher for many
inspiring discussions on the baroclinic annulus and the
concept of vacillation.
n
,
the zonal wave number
Pr
,
the Prandtl number, equation (3.16)
Ta
,
the Taylor number, equation (3.1)
,
the thermal Rossby number, equation (3.2)
β
=
df /dy
,
the
β
effect
Appendix A: MORALS CODE SETUP
Defining equivalences between the thermally driven
annulus and the two-layer system by using the thermal
wind as the definition for both, the Rossby number of the
two-layer system, and the shear forcing, we can associate
the two-layer terms on the left-hand side with thermal
annulus parameters on the right-hand side as
Ro =
θ/
2,
U
d
=
θ/
2,
U
s
=0,
F
=8
/θ
,
r
=
4
/
γ
v
Ta
1
/
4
2
/θ
.
The two-dimensional (2D) solver of MORALS was
set up for axisymmetric flow integration with a grid res-
olution of 24—24 and 32—32 without any appreciable
difference, where the grid was stretched using a hyperbolic
tangent function. As a result, no further grid refinement
was carried out.
The dimensions of the annulus were an inner radius
a
= 2.5cm, outer radius
b
= 8cm, and depth
d
=14cm,
and the temperatures were 18
◦
C at the inner wall and 22
◦
C
at the outer wall.
The fluid properties at the reference temperature
T
0
=22
◦
C were a density
ρ
0
= 1.043g
/
cm
3
, kinematic vis-
cosity of
ν
0
=0.0162g
/
cm
3
, and thermal diffusivity
κ
0
calculated to set the Prandtl number as follows:
(B.1)
Furthermore, the thickness of the
E
1
/
3
Stewartson layer
can be described through the vertical aspect ratio of the
annulus and the Taylor number as
δ
=
γ
−
1
/
2
Pr
0.1
0.5
0.7
1
1.4
2
κ
0
0.162
0.0324
0.0231
0.0162
0.01157
0.00810
Ta
−
1
/
6
.
(B.2)
Pr
7 10 13 26 50 100
κ
0
0.00231 0.00162 0.00129 0.000623 0.000324 0.000162
v
Using the correspondences, thermal annulus conditions
can be converted to those of the minimal baroclinic two-
layer model analyzed by
Lovegrove et al.
[2002] as
The variation of the fluid properties with temperature
was a volume expansion coefficient for the fluid of
α
=
3.07
k
1
=2
mπ/γ
h
,
K
2
=
k
1
+
π
2
,
s
=
r
1+
r
Ro
K
2
,
d
=
rK
2
/
K
2
+2
F
+
r
2
Ro
K
2
,
b
=
r π
2
/
π
2
+2
F
+
r
2
Ro
π
2
,
β
s
=
β/K
2
10
−
4
K
−
1
and quadratic approximations as
ρ
=
ρ
0
1
×
T
0
)
2
,
10
−
6
(T
−
α (T
−
T
0
)
−
7.83
×
−
ν
=
ν
0
1
10
−
2
(T
−
×
−
−
2.79
T
0
)
6.73
T
0
)
2
,
10
−
4
(
T
×
−
κ
=
κ
0
1
T
0
)
.
U
s
k
1
,
10
−
3
(T
−
2.33
×
−
−
