Geoscience Reference
In-Depth Information
The shallow-water parameter
α
is usually small in
the ocean. For typical isolated islands such as Madeira
[
Caldeira et al.
, 2002;
Caldeira and Sangra
, 2012], Gran
canaria[
Aristegui et al.
, 1994;
Sangra et al.
, 2005], Hawaï
[
Calil et al.
, 2008;
Chavanne et al.
, 2010], and even
the small Aoga Shima [
Hasegawa et al.
, 2004], we get
α
= 0.05-0.005. If the vortices generated downstream of
the island keep such small values of
α
, they will satisfy
the hydrostatic balance at the first order of approximation.
The layer thickness ratio
δ
controls the dynamic interac-
tions between the surface and the bottom layer. In the
open ocean, this parameter is also small,
δ
= 0.1-0.02,
the deep bottom layer acts as a neutral layer with no
motion, and the surface flow is not affected by the bottom
dissipation induced by the seafloor roughness.
In order to quantify the impact of Earth rotation on
the oceanic flow, we generally introduce a Rossby num-
ber. Taking into account the upstream current velocity
U
0
, the characteristic island radius
R
, and the local Cori-
olis parameter
f
, we can easily build the island Rossby
number:
(a)
D
U
0
h
(b)
U
0
h
H
Ro
I
=
U
0
fR
However, a more accurate description of the vortex
dynamics in the wake will be given by the vortex Rossby
number:
Figure 14.1.
Oceanic current impinging on (a) shallow-water
and (b) deep-water island configuration.
Ro =
V
max
fr
max
where
V
max
is the maximum velocity of the vortex and
r
max
the corresponding vortex radius. While the island and the
vortex Rossby numbers have generally the same order of
magnitude, the relative core eddy vorticity
ζ
0
/f
could be
much larger. Indeed, an intense anticyclone was detected
in the lee of Oahu island [
Chavanne et al.
, 2010] with a
finite vortex Rossby number Ro
deep oceanic layer is much larger than the upper thermo-
cline (
H
h
), the influence of the bottom drag could be
neglected, as discussed by [
Tomczak
, 1988]. When the hor-
izontal dissipation is weak, the (large) Reynolds number is
not a crucial parameter anymore and the combined effects
of rotation and stratification govern the wake pattern.
We restrict this review to the cases of deep ocean wake
(Figure 14.1b) where an upper surface current encounters
an isolated island or archipelago.
0.45 and a higher rel-
ative core vorticity
ζ
0
/f
1.5. For such large values
the corresponding vortex will satisfy the cyclogeostrophic
balance where the sum of the centrifugal and the Coriolis
forces balance the pressure gradient. On the other hand,
when a weak current encounters a large island, the cor-
responding Rossby number Ro
I
will be small and both
the wake flow and the eddies will satisfy the standard
geostrophic balance.
The vertical density gradient is quantified by the Brunt-
Vaisala frequency
N(z)
(where
N
2
=
−
14.2. DIMENSIONAL ANALYSIS AND
DYNAMICAL PARAMETERS
In order to classify the various dynamical regimes of the
idealized island wake, we first need to identify the main
dimensionless parameters which control the flow. Assum-
ing a symmetrical island shape with an effective diameter
D
=2
R
and an upstream surface current of thickness
h
(we consider here that the current has roughly the same
height as the upper thermocline) above a deep oceanic
layer of depth
H
, we get two geometric parameters:
α
=
h
g∂
z
ρ/ρ
), and the
relative strength of the oceanic stratification in compari-
son with the Earth rotation is given by the stratification
parameter
S
=
N/f
. This parameter is large for the
upper thermocline (
S
−
100 according to Table 14.1) and
induces a strong stratification regime for the deep-water
wake. Another way to account for the vertical stratifica-
tion is to introduce the Rossby deformation radius asso-
ciated with the first baroclinic mode
R
d
. If we consider
h
h
+
H
.
R
,
δ
=
