Geoscience Reference
In-Depth Information
(a)
(b)
0
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1. 5
1. 0
0.5
0.0
-0.5
0
-200
-200
C4
-400
-400
-600
-600
A5
A4
-800
-800
-1000
-1000
C4
-1200
-1200
C3
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
-4.5
-5.0
-1400
-1400
-1600
-1600
-1800
-1800
A4
A3
-2000
-2000
C3
C2
-2200
-2200
250
500
750
1000
1250
1500
1750
2000
2250
250
500
750
1000
1250
mm
1500
1750
2000
2250
mm
Figure 14.8.
Surface vorticity field of an intense submesoscale wake within a thin (
h
c
= 6.7 cm) and stratified (
N/f
= 13) upper
layer at (a)
t
=0and(b)
t
2
T
0
when Ro
I
= 2.8, Bu = 48, and Re = 12, 500 [
Lazar et al.
, 2013b]. The color panel quantifies
the relative vorticity amplitude
5
<ζ/f <
5 in blue (red) for anticyclonic (cyclonic) vorticity. The experimental configuration
corresponds to figure 2 (c). For color detail, please see color plate section.
−
the vertical dissipation (i.e., the Ekman number
E
k
), and
the stratification is quantified at the submesoscale by the
Burger number Bu.
a Rankine vortex will become unstable when the vortex
Rossby number Ro =
V
max
/(fr
max
)
is larger than Ro
≥
0.5 and a Gaussian vortex when Ro
0.31. However,
for a stratified and viscous vortex the Rayleigh criterion
becomes less relevant. Close to the marginal stability limit,
where the growth rates are strongly controlled by the ver-
tical dissipation, the linear analysis of
Lazar et al.
[2013a]
reveals that the instability is not sensitive to the velocity
or the vorticity profiles. An analytical marginal stability
limit, which depends only on three dimensionless param-
eters (Ro, Bu,
E
k
), was derived for anticyclonic Rankine
vortex [
Lazar et al.
, 2013a]. The latter become linearly
unstable to inertial modes if, in addition to the inviscid
citerion (1), corresponding here to
Ro >
0.5, the following
equation is satisfied:
≥
14.5.2. Inertial-Centrifugal Instability of Shallow
Stratified Anticyclones
A global stability analysis of the stratified and rotat-
ing wake flow is hard to achieve, especially due to the
rapid evolution of the detached vortices in the near wake.
However, the vortex street usually reaches a quasi-steady
state in the far wake. If we neglect the vortex-vortex
interactions, the stability of single and circular vortices
could provide a first estimate of the threshold of inertial-
centrifugal instability in the far wake.
For inviscid and circular vortices, the generalized
Rayleigh criterion [
Kloosterziel and VanHeijst
, 1991;
Mutabazi et al.
, 1992] is a sufficient condition that all
anticyclonic vortex columns are unstable to inertial-
centrifugal perturbations if somewhere in the flow we get
χ(r)
=
ζ(r)
+
f
2
V(r)
r
8
3
Ro
2
√
2Ro
1
E
k
≥
|
a
0
|
3
Bu
1
7
,
(14.2)
−
where
= 2.3381 is the first zero of the Airy func-
tion. The above marginal stability equation appears to be
relevant for a wide variety of other vortices (parabolic,
conical, Gaussian) and could then be used to build a “first
guess” stability diagram for inertial-centrifugal destabi-
lization of viscous and stratified anticyclones. Assuming
that the vortex Rossby number Ro of the far wake vortices
scales as the island Rossby number Ro
I
in the turbulent
ocean, we could extend such stability diagram to the wake
flow.
|
a
0
|
+
f
<
0,
(14.1)
where
V(r)
is the azimuthal velocity profile and
ζ(r)
=
∂
r
V(r)
+
V/r
is the relative vorticity. The stratification or
the shallow-water ratio will not change this criterion if the
dissipation is negligible. According to this widely used cri-
terion, the stability of circular anticyclones depends cru-
cially on their velocity or vorticity profiles. For instance,
