Oceanic Island Wake Flows in the Laboratory - Modeling Atmospheric and Oceanic Flows - page 261

Table 14.1. Characteristic scales and dynamical parameters for various oceanic wakes.

Islands U 0 ( cm / s ) R (km) h (m) S = N/f Ro I Bu

Madeira 20-30 10-25 ~100 ~350 0.1-0.3 2-6

Gran Canaria 40 25 ~100 — ~0.2 —

Hawaï (Oahu) 20-35 15-20 ~100 150-200 0.2-0.4 1-3

Aldabra 40-60 30 ~150 ~150 0.6-0.8 ~0.7

Aoga Shima 100-150 2 ~100 ~200 6-10 ~100

a purely linear stratification in the upper layer, the latter
is defined by R d = Nh/f = Sh . We could then define a
dimensionless Burger number from the relative island size:
Bu = R d
R

tude smaller than the horizontal one κ h and the Ekman
and Reynold numbers are therefore two independent
parameters.

2

14.3. IDEALIZED LABORATORY SETUPS FOR
OCEANIC ISLAND CONFIGURATION

For a geostrophic and circular vortex the Burger number is
equal to the ratio of the kinetic over the potential energy.
Hence, for mesoscale flows, where the typical horizontal
scale R is equal to or larger than the Rossby radius R d
(Bu

We should first recognize that due to strong experi-
mental constraints the laboratory experiments are always
highly simplified in comparison with oceanic island flows.
The turbulent coastal boundary layer strongly differs from
the viscous boundary layer obtained in the laboratory
and the interaction between an oceanic current and a com-
plex island coast could hardly be reproduced on a rotating
turntable. While the experimental forcing and the bound-
ary layer detachment in the near wake may differ from
the ocean, the global wake pattern (geometric structure,
distance between eddies, cyclone-anticyclone asymetries)
and its dynamics (vortex interactions, secondary insta-
bilities) will be close to the oceanic ones if the geometric
and dynamic similarities are satisfied. In other words, the
values of the main dimensionless numbers obtained in
the laboratory should be comparable to the oceanic ones.
Among the geometric numbers, the similarity of the
shallow-water parameter α is probably the most difficult
to achieve. Indeed, to generate a Karman street which con-
tains several eddies, the typical size of the rotating tank
should be at least 10 D ,where D is the typical island diam-
eter. Hence, even on the world's largest turntable, the 13 m
diameter Coriolis platform (http://coriolis.legi.grenoble-
inp.fr/?lang=en) 1 , the diameter D =2 R should not exceed
1 m. For a layer thickness h of a few centimeters the
shallow-water parameter α = h/R can hardly go below
α

1 ), there is a significant amount of potential energy
available due to the isopycnal deviations within the ther-
mocline. On the other hand, for the submesoscale flow,
which corresponds to R < R d (Bu

≤

1), the energy is

mainly kinetic.
The Reynolds number is the most widely used param-
eter to quantify the dissipation. However, as far as we
consider the nonisotropic turbulent ocean, it becomes a
challenge to estimate an effective turbulent viscosity (i.e.,
the mean eddy diffusivity). This means that turbulent dif-
fusivity depends on the spatial and temporal scales used
for the averaging. Moreover, due to the vertical stratifi-
cation, there is a net distinction between the horizontal
diffusivity κ h and vertical diffusivity κ z . For an isolated
island in deep water we can use a horizontal Reynolds
number:

Re = U 0 D
ν t
where ν t is the effective eddy (turbulent) viscosity. The
distribution of tracers such as sulfur hexafluoride (SF6)
was recently used to estimate the open-ocean eddy diffu-
sivity [ Ledwell and Watson , 1998], which varies from 1 to
10m 2 / s for horizontal scales of tens of kilometers. Hence,
if we assume ν t

0.1, which is still one order of magnitude larger than
the oceanic values, α

10 − 2 . In the pioneering experi-
ments on rotating wakes [ Boyer and Kmetz , 1983; Boyer
et al. , 1984; Tarbouriech and Renouard , 1996] the shallow-
water constraint was not strictly satisfied with α = 0.6-6.
Smaller values (see Table 14.2) were only reached with
a two-layer configuration [ Perret et al. , 2006b; Te i ntur i er
et al. , 2010; Stegner et al. , 2012] where the upper layer

κ h , the horizontal Reynolds number will
typically range from 10 2 to 10 4 for the above-mentionned
islands (Table 14.1). Taking into account the rotation, we
introduce the Ekman number
E k = κ z
fh 2

to quantify the vertical dissipation of shallow and coher-
ent structures within the upper thermocline. The vertical
diffusivity κ z in the ocean is several orders of magni-

1 LEGI-Coriolis,

Grenoble, France. http://coriolis.legi.grenoble-

inp.fr/?lang=en