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ρ
1
and
ρ
2
of the top and bottom layers are then adjusted
to obtain a small or large Rossby radius
R
d
in comparison
with the cylinder diameter. A more realistic stratification
is obtained when the thin upper layer is linearly strat-
ified (Figure 14.2c). In such case, we consider that the
motion of the cylinder transfers momentum mainly over
the heigth of the submerged cylinder
h
c
and the defor-
mation radius associated with the first baroclinic mode
is defined by
R
d
=
Nh
c
/f
. Theses idealized salt strat-
ifications are needed to study a wide range of Burger
numbers from large mesoscale wake, Bu = 0.1, to small
submesoscale wake, Bu = 70.
In the laboratory, the molecular viscosity controls both
the horizontal and the vertical dissipation, and unlike the
oceanic case, the Ekman and Reynolds numbers are not
independent parameters. Sometimes it could be more con-
venient to use 1
/E
k
instead of the Ekman number because
it evolves as the Reynolds number. Weak dissipation will
correspond to a large value of 1
/E
k
according to
1
E
k
the two-dimensional nonrotating flow in the limit of large
Reynolds number, for which the Strouhal number,
S
t
=
f
s
D/U
0
=
D/L
e
0.24, at large Reynolds num-
bers [
Wen and Lin
, 2001]. Both experimental [
Boyer and
Kmetz
, 1983;
Stegner et al.
, 2005;
Teinturier et al.
, 2010;
Lazar et al.
, 2013b] and numerical studies [
Heywood et al
,
1996;
Perret et al.
, 2006a] confirm this result in a deep-
water configuration when the bottom friction is negligible.
Hence, a quasi-geostrophic wake will have the same pat-
tern as the standard nonrotating Karman street even if the
separation and the vortex shedding occur at higher critical
Reynolds numbers [
Boyer and Kmetz
, 1983;
Boyer et al.
,
1984].
0.2
−
14.4.2. Large-Scale Geostrophic Wake
When the Rossby and the Burger numbers are both
small, the flow is expectedto follow the frontal geostrophic
regime [
Cushman-Roisin
, 1986]. The isopycnal deviations
scale as the ratio of the Rossby over the Burger number,
λ
=Ro
/
Bu, and when
λ
is finite, the evolution of cyclonic
and anticyclonic eddies, confined in the upper surface
layer, should differ strongly. Other studies on the stability
of isolated eddies have shown that, beyond the quasi-
geostrophic regime, anticyclones tend to be more stable
and coherent than their cyclonic counterparts [
Arai and
Yamagata
, 1994;
Stegner and Dritschel
, 2000;
Baey and
Carton
, 2002;
Graves et al.
, 2006]. Hence, when the mean
island radius is two or three times larger than the local
deformation radius, we could expect a significant cyclone-
anticyclone asymmetry of the vortex wake. Indeed, for
some extreme cases, coherent cyclones do not emerge at
all, and only an anticyclonic vortex street appears several
diameters behind the circular island [
Perret et al.
, 2006b].
For these experiments, where the drifting cylinder was
confined in a thin (
δ
=
fh
2
ν
=
Re
Ro
α
2
,
where
α
=
h/R
is the shallow-water ratio parameter. If
the later is small, 1
/E
k
could be much smaller than Re
and the vertical dissipation will become more important
than the horizontal one. In small facilities, typically 1 m
diameter turntables, both Reynolds (Re = 10
2
10
3
)and
−
Ekman (1
/E
k
=10
2
10
3
) numbers correspond to viscous
laminar dissipation. On a large-scale turntable, such as
the 13 m diameter Coriolis platform, the Reynolds num-
ber may reach turbulent values up to 10
4
−
10
5
while the
Ekman number will generally stay in the viscous range
(1
/E
k
=10
2
−
10
3
) when the flow is confined in a shallow
upper layer (Figure 14.2c).
−
0.1) and light (but non-stratified)
surface layer, both dye visualization (Figure 14.3) and
vorticity measurements (Figure 14.4) show this suprising
behavior of large-scale geostrophic wake. This asymme-
try was first explained by the linear stability analysis of
parallel wake flows in the framework of rotating shallow-
water equations [
Perret et al.
, 2006a;
Perret et al.
, 2011].
In the frontal regime, the most unstable mode is fully
localized in the anticyclonic shear region. Hence, the anti-
cyclonic perturbations, leading to large-scale anticyclones,
have the fastest growth rates. Besides, a spatiotemporal
analysis of the wake flow behind the cylinder reveals a
change in the nature of the instability: For a large-scale
cylinder the wake flow is convectively unstable [
Perret
et al.
, 2006a] while the quasi-geostrophic wake and the
classical Karman street is absolutely unstable [
Pier
, 2002;
Chomaz
, 2005]. Since the near wake flow is nearly par-
allel in the large-scale (frontal) regime, see Figures 14.3
and 14.4, this unstable wake would behave like a noise
14.4.MESOSCALE VORTEX WAKE
In this section, we consider large-scale wakes where the
Burger number Bu is smaller than or equal to unity. In
other words, the characteristic radius of the eddies,
r
max
,
within the Karman street will be larger than or equal to
the first baroclinic radius
R
d
.
14.4.1. Quasi-Geostrophic Wake
The quasi-geostrophic regime corresponds to small
Rossby number Ro
I
1 while the Burger number
remains close to unity, Bu
1. For this range of parame-
ter, the cyclonic and anticyclonic eddies which are periodi-
cally shed behind a cylindrical obstacle are identical in size
and intensity as for the classical Karman street. The dis-
tance
L
e
between two vortices having the same sign, pos-
itive for cyclonic and negative for anticyclonic, is roughly
equal to five cylinder diameters. This is in agreement with
