Geoscience Reference
In-Depth Information
free surface are undesirable and are usually neglected. In
order to justify it, the experimental setup must be designed
to have
β
η
7.3. LARGE-SCALE EXPERIMENTS
O
(10 M)
The very first attempts to perform experiments with a
rotating tank were rather modest, as the pioneering work
by
Taylor
[1921]. During the 1950s and 1960s the study of
geophysical flows by means of laboratory models showed
a trend toward the construction of large-scale apparatus.
Part of the story of these developments is narrated by
von Arx
[1957], who described the 4 m diameter rotat-
ing tank constructed at the Woods Hole Oceanographic
Institution. In 1960 the Coriolis platform, a 13 m diame-
ter device, was built by the University of Grenoble and a
group of French science agencies in Grenoble, France. The
original purpose was to model tidal motions in English
Channel (La Manche) affected by the rotation of Earth.
The platform was dismantled in 2010 and has recently
been rebuilt with numerous technological improvements.
Another well-known large-scale facility is the 5 m diame-
ter platform at the Norwegian University of Science and
Technology in Trondheim, Norway.
With a large-scale platform it is possible to generate
flows with very low Rossby and Ekman numbers, as
required for cases where rotation effects are fundamen-
tal. For modeling fluid phenomena over variable topog-
raphy, a large-scale tank is very convenient because a
topographic feature can be constructed with great detail
given the range of typical depths (around
H
β
t
. In some experiments, a parabolic bot-
tom plate was introduced for a specific
value having
the same shape as the free surface in order to eliminate
any topographic effects associated with the free surface
deformation [e.g.,
Trieling et al.
, 2010].
7.2.5. Shallow Flows
Flows in shallow fluid layers, with the layer depth
H
significantly smaller than the horizontal scales
L
,
i.e.,
δ
=
H/L
1, are anisotropic because of the differ-
ence in scales: On the basis of the continuity equation
it is commonly argued that the horizontal and verti-
cal velocity components
U
and
W
, respectively, scale as
W
U
. In the ideal case of a stress-free bottom
and upper surface, one would have
w
=0and
∂/∂z
=0,
so that the vorticity vector is
ω
=
(
0,0,
ω)
and the motion
is purely 2D and governed by
∂ω
∂t
+
J(ω
,
ψ)
=
ν
∼
Uδ
2
ω
,
∇
(7.22)
with
ψ
the stream function.
The presence of a no-slip bottom is usually taken into
account by adopting a parabolic (Poiseuille) profile in the
z
direction. This quasi-2D (planar) flow is then assumed
to be governed by
0.5 to 1 m).
Another advantage is that the Ekman time scale, propor-
tional to
H
, can be relatively large in comparison with
smaller rotating tanks, which makes possible to perform
long experiments without the influence of Ekman damp-
ing (
T
E
can be as long as 30 min, whereas in a medium-size
tank it is typically 3-5 min).
∼
∂ω
∂t
+
J(ω
,
ψ)
=
ν
2
ω
∇
−
λω
(7.23)
with
λ
=
ν(π/
2
H)
2
the Rayleigh friction coefficient [see,
e.g.,
Clercx and van Heijst
, 2009]. In recent studies by
Akkermansetal.
[2008a, 2008b],
Ciesliketal.
[2009b], and
Kamp
[2012], however, it was found that significant ver-
tical motions may occur in shallow-layer flows even for
δ
7.3.1. The
β
-Drift of Monopolar Vortices
0 and that a correlation exists between the vertical
motion and the rotation/strain of the primary horizontal
flow field.
Experimentally, shallow flows have been studied in lab-
oratory configurations of varying sizes. Large-scale exper-
iments on turbulent mixing layers, for example, have been
carried out in tanks of horizontal dimensions ranging
from 1 m to tens of meters [see, e.g.,
Jirka and Uijtte-
waal
, 2004]. Such flows are usually characterized by larger
Re values, and hence are turbulent. Shallow flow exper-
iments in much smaller geometries (with a water depth
H
→
On a planetary
β
-plane in the Northern Hemisphere,
cyclonic vortices move northwestward, while anticyclones
drift in the southwestern direction [see, e.g.,
McWilliams
and Flierl
, 1979]. This effect is due to the redistribution of
relativevorticityinthevortexcore,asluidcolumnschange
latitude while conserving potential vorticity. As a result,
vortices preserve their quasi-circular shape, superimposed
by a dipolar component causing the vortex motion.
Laboratory experiments of planetary flows cannot sim-
ulate this phenomenon directly, because the first-order
effects of the curvature of Earth are not present in a uni-
formly rotating fluid tank. However, the planetary
β
-plane
can be simulated by taking advantage of the dynamical
equivalence (7.20) shown above: Using a uniform weak
sloping topography over the length of a rotating tank, a
nearly uniform topographic
β
t
is obtained over the whole
domain. Then, the shallow part corresponds to the north
and the deep part to the south. The experimental
β
t
value
50 cm) have
been performed by a number of researchers [e.g.,
Tabel-
ing et al.
, 1991;
Xia et al.
, 2009;
Figueroa et al.
, 2009],
with the purpose of studying the characteristics of 2D
turbulence. The motion in these experiments is typically
generated by electromagnetic forcing, as will be described
in Section 7.5.
≈
10 mm and a typical horizontal size
L
≈
