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(a)
(b)
(c)
(d)
14 cm
5
10
21
8
4
6
3
4
2
20
z
z
z
2
z
1
0
0
-2
-1
19
-4
-2
-6
-3
-8
-4
-10
0
49.75
55.25
cm
r
r
r
r
Figure 1.17.
Cross sections in the
(r
,
z)
plane of (a) azimuthal velocity (mm/s), (b) temperature (
◦
C), ( c) radial gradient of QG
potential vorticity (s
−1
) (computed by
Bell and White
[1988]), and (d) meridional stream function (cm
2
/s
1
) in the axisymmetric
regime of a rotating annulus subject to differential heating at the sidewalls. (Adapted from
Read
[2003] with permission.)
apparently as efficient as possible as measured by the
nondimensional Nusselt (
largely confined to the Ekman layers. This is clearly seen in
the temperature and stream function fields of the rotating,
axisymmetric flow illustrated in Figure 1.17a and 1.17d.
Meanwhile, the azimuthal flow assumes a form where
geostrophic balance applies except in the Ekman layers,
within which viscous accelerations dominate.
Since the radial flow becomes confined to Ekman lay-
ers when
N
) or Péclet (Pe) numbers
Total heat transport
Heat transport by conduction alone
,
N
=
(1.23)
Advective heat transport
Heat transport by conduction alone
.
Pe =
(1.24)
= 0, the efficiency of advective heat trans-
port is governed primarily by the mass transport, which
can be accommodated within an Ekman layer. For a
given geostrophic zonal flow in the interior, the advec-
tive Ekman transport is proportional to the depth of
the Ekman layer (i.e., proportional to
V
When the system is rotated, Coriolis accelerations begin
to influence the circulation, deflecting horizontal radial
motion into the azimuthal direction. The axisymmetric
low at low values of
is therefore similar to the non-
rotating case, except (a) an azimuthal component of flow
is induced, producing jets antisymmetric about middepth
(for identical upper and lower boundary conditions), pro-
grade at the top (where radial flow is inward) and retro-
grade below (where radial flow is outward), and (b) radial
low becomes largely confined to Ekman layers adjacent to
the horizontal boundaries (for
sufficiently large). When
Coriolis accelerations dominate the interior flow, any
O
(1)
radial flow has to be geostrophic, requiring an azimuthal
pressure gradient. Such a gradient cannot occur in an
axisymmetric circulation unless a rigid meridional barrier
is present, so radial flow strong enough to carry signifi-
cant amounts of heat energy across the annulus becomes
1
/
2
,where
E
is the Ekman number and
V
the azimuthal velocity
scale; see below). Since
E
is inversely proportional to
,
the efficiency of advective heat transport must decrease
with
given a constant
T
,
V
is proportional to
−
1
, and Ekman transport is proportional to
−
1
E
1
/
2
∝
−
3
/
2
. Thus, as
increases, advective heat transport
must decrease. Such a decrease is clearly apparent in
Figure 1.18, which shows a compilation of both labora-
tory measurements of Nusselt number and from numeri-
cal simulations [
Read
, 2003;
Pérez
, 2006]. The variation of
Nusselt number
E
N
in a pure axisymmetric flow in which
baroclinic waves are suppressed is indicated by diamonds,
which broadly confirm the decay as
−
3
/
2
, though note
