Rings in the Gulf of Mexico and Stochastic Resonance
Benjamín Martínez-López, Jorge Zavala-Hidalgo and Carlos Gay García
Center for Atmospherc Sciences, National Autonomous University of Mexico, Ciudad Universitaria, D.F., Mexico
Keywords: Gulf of Mexico, Ring Shedding, Reduced Gravity Model, Seasonal Forcing, Stochastic Resonance.
Abstract: In this work, we used a nonlinear, reduced gravity model of the Gulf of Mexico to study the effect of a
seasonal variation of the reduced gravity parameter on ring-shedding behaviour. When small amplitudes of
the seasonal variation are used, the distributions of ring-shedding periods are bi-modal. When the amplitude
of the seasonal variation is large enough, the ring-shedding events shift to a regime with a constant, yearly
period. If the seasonal amplitude of the reduce gravity parameter is small but a noise term is included, then a
yearly regime is obtained, suggesting that stochastic resonance could play a role in the ring-shedding
process taking place in the Gulf of Mexico.
1 INTRODUCTION
Anticyclonic rings generated by meandering of
intense boundary current systems are long-lived,
intense near-surface features that dominate the
oceanic mesoscale in different regions of the World
Ocean. They substantially contribute to determine
the water mass characteristics as well as the upper-
ocean circulation patterns in these regions and, due
to their characteristic self-induced, westward
propagation, they often also play an important role
in the transfer of chemical and biological properties
across frontal zones (Olson, 1991). The large surface
temperature anomalies as well as the large surface
horizontal velocity shears associated with these rings
may profoundly influence human activities. In the
Gulf of Mexico (Fig. 1), for instance, the passage of
“warm-core” rings detached from the Loop Current
is able to disturb oil extraction activities, while it is
demonstrated that hurricanes may be intensified by
their interaction with the warm ring water (see
Halliwell et al., 2011 and references therein).
Predicting the onset and evolution of ring
shedding in the Gulf of Mexico (GOM) may
substantially contribute to the understanding of the
subtle dynamics involved in the local oceanic
phenomena and also to the reduction of the impact
on human activities caused, directly or indirectly, by
these rings. Observations show a nearly bi-modal
distribution (the most evident peaks existing around
6 months and 9 to 11 months). Current full-fledged
ocean numerical models can explain some of the
observed ring-shedding variability but fail in
simulating observed periods (Murphy et al., 1999);
(UWelsh and Inoue, 2000); (Romanou et al., 2004).
Simple models, on the other hand are only able
to reproduce an almost constant period (Hulburt and
Thompson, 1980); (Sturges et al., 1993); (Oey et al.,
2003), which was called the “natural” period of the
Gulf by Hulbert and Thompson (10-11 months). In
part, this deficiency of existing numerical model is
undoubtedly the result of inaccuracies induced in the
simulated dynamics by the imposed boundary
conditions, which unavoidably, tend to introduce in
the system an exaggeratedly strong yearly signal.
The discrepancy between observations and
simulations, however, may be used to gain a deeper
understanding of the subtle dynamics governing the
process of ring shedding in the Gulf of Mexico. In
fact, it results that as the strength of the yearly signal
imposed in realistic model simulations decreases, the
occurrence of yearly, ring-shedding events also
diminishes. This behaviour may indicate that, in
numerical models, a synchronization mechanism
exists, which is able to shift a “natural” ring-
shedding period toward a yearly one.
Considering that the seasonal cycle of sea
surface temperature (SST) in the Gulf of Mexico is a
natural forcing on the wide spectrum of physical
processes taking place there then emerges an
attractive possibility: stochastic resonance. If the
imposed forcing by the SST is strong enough to
drive the system, we can expect that ring-shedding
variability will contain spectral energy in the yearly
534
Martínez-López B., Zavala-Hidalgo J. and Gay García C..
Rings in the Gulf of Mexico and Stochastic Resonance.
DOI: 10.5220/0004165805340538
In Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (MSCCEC-2012), pages
534-538
ISBN: 978-989-8565-20-4
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

frequen
c
most of
forcing
m
signal i
n
b
y the a
s
In other
weak,
y
capable
sheddin
g
term co
u
with a y
e
Our
numeric
a
in the
G
model s
i
the infl
o
b
oundar
i
capable
Thomps
o
reduced
seasonal
can ex
p
b
ehavio
r
stochast
i
includin
g
b
uoyanc
2 M
O
A redu
c
simulate
Mexico
cycle u
p
assume
d
b
y usin
g
density
o
b
eing c
approxi
m
evolves
w
The
d
and a p
o
Atlantic
p
ropaga
t
Caribbe
a
their po
s
On the
located
s
interest,
p
lace n
e
reflectio
n
signific
a
GOM.
In e
a
c
y, but this i
s
the time. N
m
echanism, t
h
n
the forcing
c
s
sistance of n
o
words: if i
n
y
early signal
of inducing
g
process, th
i
u
ld be able t
o
e
arly period.
conjecture
m
a
l models. T
o
G
ulf of Mexic
i
mulating an
i
o
w/outflow o
f
i
es (see Fig.
of simulatin
g
o
n, 1980). In
gravity mod
e
ly forced an
d
p
lain the o
b
r
. Additionall
y
i
c resonance
i
g
a noise t
e
y ter
m
.
O
DEL SI
M
c
ed gravity
m
the edd
y
-sh
e
and to study
p
on it. The
d
constant in
s
g
a linear th
e
o
f the upper
l
onsistent wi
t
m
ation) a re
d
w
ith the time
d
omain cover
o
rtion of the
G
(see Fig.
t
ion of an
o
a
n Sea thro
u
s
sible effect
other hand,
s
ufficiently f
a
thus assuri
n
e
ar the open
b
n
of some
a
ntly with t
h
a
ch model
e
s
not the cas
ow, if we i
n
h
en we can e
x
c
an be amplif
i
o
ise (Gamma
i
n
the Gulf o
f
in the forci
n
a yearly pe
r
i
s yearly for
c
o
produce rin
g
m
ay be test
e
o
this purpose
,
o a nonlinea
r
i
dealized Loo
p
f
nea
r
-surface
1). This is t
h
g
ring sheddi
this study, it
e
l, where the
d
the dissipati
o
b
served peri
y
, we explore
i
n the ring-sh
e
e
rm in the
s
M
ULATI
O
m
odel (1.5 l
a
e
dding proce
s
the influenc
upper-ocea
n
s
pace but var
i
e
rmodynamic
l
ayer (to mak
e
t
h the plan
e
d
uced g
r
avity
is obtained.
s the GOM, t
h
G
ulf Stream
1). This c
h
o
malies fro
m
u
gh the Yuc
a
on the ring-
s
the eastern
o
a
r from our
r
n
g that the
a
b
oundary, an
d
waves, do
e
h
e internal
d
e
xperiment, t
h
e
, or at leas
t
n
clude noise
a
x
pect that a
w
i
ed and opti
m
i
toni et al., 2
0
f
Mexico exi
s
n
g, which is
r
iod in the
r
c
ing plus a
n
g
-shedding e
v
e
d using si
m
,
we impleme
r
, reduced-gr
a
p
Current thr
o
water throug
h
h
e simplest
m
n
g (Hurlburt
i
s proposed
t
h
buoyancy ter
m
o
n term is va
r
o
d of shed
d
the
p
ossibili
t
e
dding proce
s
s
easonally fo
O
NS
a
yers) is use
d
s
s in the Gu
l
e
of the seas
n
temperatur
e
i
es in time.
T
equation fo
r
e
sure that w
e
e
tary geostro
parameter
w
h
e Caribbean
area in the
N
h
oice allows
m
the we
s
a
tan Channel
s
hedding pro
c
o
pen bounda
r
r
egion of pri
m
a
djustment ta
k
d
the unavoi
d
e
s not inte
r
d
ynamics of
h
e equations
t
not
as a
w
eak
m
ized
0
00).
s
ts a
not
r
ing-
n
oisy
v
ents
m
ple
e
nted
a
vi
t
y
o
ugh
h its
m
odel
and
h
at a
m
is
r
ie
d
,
ding
t
y of
s
s by
o
rced
d
to
l
f of
s
onal
e
is
T
hus,
r
the
e
are
o
phic
w
hich
Sea,
N
orth
the
s
tern
and
c
ess.
r
y is
m
ary
k
ing
d
able
r
fere
the
are
int
e
a f
o
the
p
er
f
ho
r
of
ex
p
(g’
)
g’
p
ad
d
Fig
u
are
flo
w
hei
g
Up
p
hei
g
3
Wi
t
she
mo
d
an
d
fol
l
of
a
e
grated nume
r
o
rward Euler
active lay
e
f
ormed usin
g
izontal grid
w
1/4 degree
p
erimen
t
s, a
c
)
is used. In t
h
p
arameter is
c
d
ed to the sea
s
u
re 1: Domain
two eastern o
p
w
induce
d
b
y
g
ht. These inp
u
p
er panel: hei
g
g
ht anomalies
a
RESUL
T
t
hout season
a
dding perio
d
d
el geometr
y
d
g’ value (Fi
g
l
owing ones i
n
a
ring of the
L
r
ically for a
pe
scheme and
a
er
. All mo
d
g
a time step
w
ith 135 x 15
3
(see Fig. 1
c
onstant redu
c
h
e second set
,
c
onsidered. F
s
onally evolvi
n
used in our
m
p
en boundarie
s
a meridional
u
t and ouput
f
g
ht anomalies
a
t
day 500.
T
S
a
l forcing, a
s
d
is observe
d
y
, upper laye
r
g
. 2). A red d
o
n
dicate the ti
m
L
oop Current.
e
riod of 20 y
e
a
n Arakawa
C
del experi
m
of 90 seco
n
3
p
oints and
). In a fir
s
u
ced gravity
p
,
a seasonall
y
F
inally, a noi
s
n
g g’
p
arame
t
m
odel simulati
o
s
, which are f
o
gradient of u
p
f
lows are time
a
t day 50. Lo
w
s
ingle peak i
n
d
, which de
p
r
thickness,
d
o
t in this figu
r
m
ing of the d
e
e
ars using
C
grid for
m
ents are
n
ds and a
a
spacing
st
set of
p
arameter
evolving
s
y term is
t
e
r
.
o
ns. There
o
rced by a
p
per layer
invariant.
w
er panel:
n
the ring
p
ends on
d
iffusion,
r
e and the
e
tachment
RingsintheGulfofMexicoandStochasticResonance
535

Figure 2
:
height a
n
estimate
t
using a
period o
f
with zer
o
panel).
Whe
n
increase
d
characte
r
cruciall
y
the am
p
clearly
d
modal d
i
the obse
r
In t
h
signal,
emerges
N
ow
annual s
p
resent.
similar
t
between
simulate
:
Histograms
a
n
omalies over
t
he timing of r
i
diffusion coe
ff
f
seven month
s
o
diffusion pe
r
n
the amplit
u
d
, a more c
o
r
ized by a
y
depends on
p
litude is lar
g
d
ominates, bu
t
i
stributions a
p
r
ved one.
h
e case of l
a
a quite sim
p
: constant, ye
a
, we try to a
n
s
ignal is wea
k
This kind of
t
o that associ
a
atmosphere
them is b
y
a
nd time seri
e
the ring-shed
d
i
ng shedding e
v
ff
icient of 100
0
s
is obtained (
u
r
iod doubling
u
de of the
v
o
mplex beha
v
bi-modal di
s
the amplitu
d
g
e enough,
t
t
when it is w
e
p
pear, some
o
a
rge am
p
litu
d
p
le ring sh
e
a
rly ring-she
d
n
swer what
h
k
but high fre
forcing coul
d
a
ted with tur
b
and ocean.
A
y
using a n
o
s of area-ave
r
d
ing region us
e
v
ents (red dots
)
0
m²/s, a co
n
u
ppe
r
p
anel),
w
is observed (l
o
v
ariation of
g
v
iour is obse
r
s
tribution,
w
d
e (Fig 4).
W
t
he annual si
e
aker differe
n
o
f which rese
m
d
e of the a
n
e
dding beha
v
d
ding even
t
s.
h
appens whe
n
q
uency forci
n
d
be, fo
r
exa
m
b
ulent heat fl
u
A
simple wa
y
o
ise term, w
r
aged
e
d to
)
. By
n
stant
w
hile
owe
r
g
’ is
r
ved,
w
hich
W
hen
i
gnal
n
t bi-
m
ble
n
nual
v
iour
n
the
n
g is
m
ple,
u
xes
a
y to
w
hich
rep
r
an
d
Fig
u
sea
s
use
d
wit
h
Me
x
r
esents heat
e
d
the lower at
m
u
re 3: Ring-s
h
s
onal variation
d
to obtain a se
h
the seasonal
x
ico. Zero diff
u
e
xchange bet
w
m
osphere.
h
edding distrib
u
in g’ is includ
e
a
sonal variatio
n
cycle of SS
T
u
sion is used.
w
een the oce
a
b
utions obtaine
e
d. The phase
c
n
of density in
T
in the centr
a
a
n surface
d when a
c
onstant is
agreement
a
l Gulf of
SIMULTECH2012-2ndInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
Applications
536

Figure 4: Similar to Fig 3, but with a larger amplitude for
the seasonal variation of g’.
In this case, we used for the stochastic forcing a
series of random numbers with zero mean and
standard deviation of 0.015. Additionally, we
explore the effect of noise magnitude on the ring-
shedding process using a factor of two and three in
the noise term (Fig. 5).
Fig. 6 shows the ring-shedding period
distribution resulting of the seasonal variation of g’.
The amplitude of the seasonal forcing is not able to
produce yearly ring shedding, because it is not large
enough (upper panel). The inclusion of a small noise
term modifies the distribution of ring-shedding
periods but it is not able to induce ring-shedding
events with a yearly period. If the noise amplitude is
increased three times, then the noise term is able to
change this behaviour, inducing a dominance of
ring-shedding events with a period close to the
yearly one.
Figure 5: Stochastic forcing. The white curve represents
the seasonal variation of g’. In the green curve, a noise
term with zero mean and standard deviation of 0.015 is
added. This forcing is multiplied by a factor 2 and 3 in the
red and blue curves, respectively.
4 CONCLUSIONS
In the case of constant g’, it is shown that large
values of diffusion lead to ring-shedding events with
a constant period of several months, while with zero
diffusion the period is doubled.
By considering a seasonal variation of g’, a more
realistic ring-shedding distribution is obtained,
which is characterized by a bi-modal distribution
that crucially depends on the amplitude. If the
annual signal of g’ has large amplitude, a quite
simple ring shedding behaviour is observed: a
constant, yearly ring-shedding event.
If the amplitude of g’ is not large enough to
induce yearly variability in the ring-shedding
process, such behaviour can be obtained by
including stochastic forcing with a proper intensity.
Thus, it has been shown that stochastic resonance
could play a role in the ring-shedding process taking
place in the Gulf of Mexico.
0 5 10 15 20
-0.01
0
0.01
0.02
0.03
0.04
Years
g´
RingsintheGulfofMexicoandStochasticResonance
537

Figure 6: Ring-shedding period distributions using
seasonal forcing (upper panel) and seasonal forcing plus
noise. See the text for details.
REFERENCES
Gammaitoni L., Hanggi P., Jung P. and Marchesoni F.,
1998: Stochastic resonance. Rev Modern Phys 70:
223–287.
Halliwell, G. R., L. K. Shay, J. K. Brewster, W. J. Teague,
2011: Evaluation and sensitivity analysis of an ocean
model response to hurricane ivan. Mon. Wea. Rev.,
139, 921–945.
Hurlburt, H. E., and J. D. Thompson, 1980: A numerical
study of Loop Current intrusion and eddy shedding. J.
Phys. Oceanogr., 10, 1611–1651.
Murphy, S. J., H. E. Hurlburt, and J. J. O’Brien, 1999: The
connectivity of eddy variability in the Caribbean Sea,
the Gulf of Mexico, and the Atlantic Ocean. J.
Geophys. Res., 104, 1431–1453.
Oey, L.-Y., Lee, H.-C, and Schmitz, W.J. Jr., 2003.
Effects of Winds and Caribbean Eddies on the
Frequency of Loop Current Eddy Shedding: A
Numerical Model Study. J. Geophy. Res., 108(C10),
doi:10.1029/2002JC001698.
Olson, D. B., 1991: Rings in the ocean. Annu. Rev. Earth
Planet. Sci., 19, 283–311.
Romanou, A., E. P. Chassignet, and W. Sturges, 2004:
The Gulf of Mexico circulation within a high-
resolution numerical simulation of the North Atlantic
Ocean. J. Geophys. Res., 109.C01003,
doi:10.1029/2003JC001770.
Sturges, W., J. C. Evans, W. Holland and S. Welsh, 1993.
Separation of warm-core rings in the Gulf of Mexico,
J. Phys. Oceanogr., 23, 250–268.
Welsh, S. E and M. Inoue, 2000. LC rings and the deep
circulation in the Gulf of Mexico. J. Geophys. Res.
105:16,951-16,959.
SIMULTECH2012-2ndInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
Applications
538