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however, is dominated by thermal conduction so Pe
→
0
1
0.1
and
N
→
1.
-1/2
+1/2
1.4.2.3. Experimental Verification.
The axisymmetric
regimes discussed in Section 1.4.2.2 are capable (at least in
principle) of existing in real systems given an experimen-
tal system operating in an appropriate parameter range.
In practice, however, regimes (iv) - (vi) are not usually
obtainable because of the development of nonaxisym-
metric baroclinic waves within regime (iv) and beyond.
This is consistent with the notion that baroclinic waves
develop when Ekman layers begin to inhibit meridional
heat transport.
An exception was provided by
Hignett
[1982], who made
heat transport measurements in a rotating annulus with
parallel sloping upper and lower endwalls that sloped
strongly in the same sense as the isotherms. Such a con-
figuration tends to inhibit the development of baroclinic
waves (by constraining fluid trajectories away from the
“wedge of instability”; [
Hide and Mason
, 1975;
Mason
,
1975]). As a result, Hignett was able to show the effect of
almost the full range of behavior from zero to very strong
rotation on the total heat transport by axisymmetric flow
in a rotating annulus. His results are shown in Figure 1.20.
The dependence of
0.1
0.01
+1
Baroclinic waves
0.01
0.01
0.1
1
10
Boundary layer ratio
P
Figure 1.21.
Schematic dependence of zonal
(azimuthal)
velocity scale on
in a rotating annulus subject to internal
heating. Adapted from
Read
[1986] with permission.
P
1.4.3. Quantifying Baroclinic Eddy Transport
in a rotating annulus
subject to internal heating was also investigated by
Read
[1986], that also confirmed the above analysis provided the
definition of
N
and
V
on
P
When baroclinic waves are not suppressed, as shown
in Figure 1.18, heat transport evidently remains close to
its nonrotating value as a result of eddy-induced trans-
ports.
Read
[2003] suggested that the latter can be viewed
as adding to and enhancing the heat transport occurring
in the axisymmetric boundary layer circulation, and the
strength of this eddy-induced transport can therefore be
diagnosed directly from measurements or simulations of
total heat transport as the difference in Nusselt number
between that of the fully three-dimensional flow and
was modified appropriately to measure
heat transport efficiency in terms of the temperature con-
trast obtained with a given heat flux; the results are shown
in Figure 1.21. These clearly show the linear scaling of
V
with
N
P
in the weak rotation regime with a transition
1
/
2
as the moderate rotation regime is entered
while the Péclet number also begins to reduce toward a
P
−
1
/
2
dependence in the moderate rotation regime.
toward
P
N
obtained in a purely axisymmetric flow under the same
experimental conditions.
Read
[2003] diagnosed this from
a combination of numerical simulations of axisymmet-
ric flows and experimental measurements. The results are
shown in Figure 1.22, (a) as a function of both
and
boundary layer ratio
0.5
I
and (b) as a function of the
“supercriticality” of the flow defined with respect to a
supercritical rotational Froude number
P
0.1
A
F
s
,deinedas
0.05
F
−
F
0
m
=
1
1
0
m
F
s
=
−
.
(1.57)
Here
F
is defined as
F
=1
/
and
0
m
=1
/
F
0
m
B
0.01
represents the critical values of
and
for the onset of
baroclinic instability of azimuthal wave number
m
.
In this figure, the difference in Nusselt number repre-
sents an additional or “excess” Péclet number Pe
xs
due
to the presence of baroclinic waves. The effectiveness of
baroclinic eddy heat transport grows rapidly with
from
the first onset of baroclinic instability, rising to a value
F
0.01
0.05 0.1 0.5
Boundary layer ratio,
1
5
P
Figure 1.20.
Scaled measurements of total heat transport in the
axisymmetric regime of a rotating annulus as a function of
.
Adapted from
Hignett
[1982] by permission of Taylor & Francis
Ltd., http://www.tandf.co.uk/journals.
P
