IOP PUBLISHING
JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL
doi:10.1088/1751-8113/46/25/254023
J. Phys. A: Math. Theor. 46 (2013) 254023 (21pp)
Ecological implications of eddy retention in the open
ocean: a Lagrangian approach
Francesco d’Ovidio 1 , Silvia De Monte 2,3,4 , Alice Della Penna 1,5 ,
Cedric Cotté 1,6 and Christophe Guinet 6
1 Laboratoire d’Océanographie et du Climat: Expérimentation et Approches Numériques, Institut
Pierre Simon Laplace, CNRS/UPMC/IRD/MNHN, Paris, France
2 École Normale Supérieure, Unité Mixte de Recherche 7625, Écologie et Évolution, 46 rue
d’Ulm, 75005 Paris, France
3 Université Pierre et Marie Curie-Paris 6, Unité Mixte de Recherche 7625, Écologie et
Évolution, CC 237-7 quai Saint Bernard, 75005 Paris, France
4 Centre National de la Recherche Scientifique, Unité Mixte de Recherche 7625, Écologie et
Évolution, 46 rue d’Ulm, 75005 Paris, France
5 Univ. Paris Diderot, Sorbonne Cité, Paris, France
6 Centre d’Études Biologiques de Chizé, Centre National de la Recherche Scientifique, 79360
Villiers en Bois, France
E-mail: [email protected]
Received 29 August 2012, in final form 13 December 2012
Published 4 June 2013
Online at stacks.iop.org/JPhysA/46/254023
Abstract
The repartition of tracers in the ocean’s upper layer on the scale of a few
tens of kilometres is largely determined by the horizontal transport induced
by surface currents. Here we consider surface currents detected from satellite
altimetry (Jason and Envisat missions) and we study how surface waters may
be trapped by mesoscale eddies through a semi-Lagrangian diagnostic which
combines the Lyapunov approach with Eulerian techniques. Such a diagnostic
identifies the regions of the ocean’s upper layer with different retention times
that appear to influence the behaviour of a tagged marine predator (an elephant
seal) along a foraging trip. The comparison between predator trajectory and
eddy retention time suggests that water trapping by mesoscale eddies, derived
from satellite altimetry, may be an important factor for monitoring hotspots of
trophic interactions in the open ocean.
This article is part of a special issue of Journal of Physics A: Mathematical and
Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to
applications’.
PACS numbers: 47.52.+j, 92.20.jm
(Some figures may appear in colour only in the online journal)
1751-8113/13/254023+21$33.00 © 2013 IOP Publishing Ltd
Printed in the UK & the USA
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1. Introduction
Remote sensing and in situ observations of the ocean and atmosphere show strong variability
in physical and biogeochemical properties at several spatiotemporal scales. This variability has
been associated with the presence in the velocity field of coherent structures that create regions
of retention (tracer reservoirs) or of mixing [73]. Several diagnostics have been proposed
to identify Eulerian and Lagrangian coherent structures and to detect their effect on the
distribution of an advected tracer field [9]. The Lyapunov exponent, in particular, has been
shown to be very effective in identifying regions of geophysical flow where the stretching rates
are maximal, which are in turn candidate locations of tracer frontal structures and transport
barriers.
The structuring effect of the velocity field over physical or biogeochemical tracers is
dramatically evident at the ocean surface on scales of tens to hundreds of kilometres, where
satellite images of surface chlorophyll concentration or sea surface temperature display strong
contrasts [44, 42] reminiscent of meteorological images of weather patterns. This regime,
sometimes referred to as the ocean weather, covers the so-called mesoscale and part of the
sub-mesoscale, so that in the following we will refer to it as the (sub-)mesoscale [40]. In this
regime, the ocean surface velocity field is dominated by the presence of coherent eddies [17],
i.e. axis symmetrical patches of strong vorticity with lifetimes ranging from several weeks to
more than one year and typical radii of tens/hundreds of kilometres. The core and the periphery
of these structures act in a very different way on the tracer’s redistribution. At the core of the
eddies, water parcels tend to recirculate and are only marginally mixed with intruding ambient
water. Therefore, under some conditions, eddy cores retain water parcels—and hence their
biotic or abiotic content—for timescales of weeks/months, and sometimes even longer [11, 53,
55, 38, 16, 15, 64, 45]. At their peripheries, where strain prevails over vorticity, eddies have
the inverse effect and enhance water mixing. In this case, tracer anomalies are redistributed by
the eddy field in the form of filaments and become eventually mixed with the ambient waters
on relatively fast timescales (days to weeks).
The different structuring role of eddy cores and eddy peripheries is well captured by
Lagrangian diagnostics such as finite-size or finite-time Lyapunov exponents, which analyze
properties of the velocity field along particle trajectories. Such diagnostics typically identify
lobular ridges of maximal stretching rates that delimit the eddy core. Depending on the intensity
and the temporal variability of the eddy, different situations can arise. In the limiting case of
no temporal variability, closed orbits are also material lines. In this case, Lyapunov-detected
transport barriers associated with eddies form a closed curve, so that the eddy core is not
permeable to lateral advection. When the velocity field changes in time, such closed barriers
break into interleaved spirals along which ambient water can be exchanged with the interior
of the eddy.
Idealized quasi-geostrophic models of oceanic mesoscale circulation have shown that
eddies can systematically trap and transport water parcels for timescales comparable to their
lifetime, shielding their content from dispersion [11]. In the real ocean, however, eddies
strongly interact with jets, topography and with each other. As a result, trapping of water
parcels is a more transient phenomenon than in idealized models and it typically depends on
a parcel being stirred by an ensemble of structures. Lagrangian and Eulerian analyses based
on multisatellite data [39, 13, 23, 15] indicate that 10–100 km wide lobular patches of trapped
tracers arise due to stirring by mesoscale currents, with lifetime shorter than the lifetime
of eddies (days/weeks compared to weeks/months and more). Primary producers—typically
phytoplankton, whose doubling time is of the order of a day—have been shown to be affected
by such patches, which may act as fluid dynamical niches [21].
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Patchiness in the ocean is known to affect not only primary producers, but also higher
trophic levels. It is indeed conjectured that mesoscale turbulence structures the distribution
of higher trophic levels from zooplankton and fish [32, 65, 61] up to top predators (e.g., [72,
26, 68, 18, 46]). However, grazers and predators need some time to grow and aggregate over
patches of high phytoplanktonic concentration. Therefore, in order to be found in association
with eddy cores, their distribution may require retention times longer than those sufficient
for confining primary producers. In particular, they should be mostly affected by water
that remained confined for a time comparable with or longer than the duration of a bloom
(about 30 days), which is the period during which phytoplanktonic biomass is accumulated.
The situation of isolated and strong mesoscale eddies, where such a long retention
condition may be met, is not typical in the ocean but does occur (e.g., [17]). This is the
case for instance of the eddies shed by energetic currents like the Gulf Stream and the
Antarctic Circumpolar Current (so-called Rings), or for eddies trapped by the topography like
the first Alboran gyre in the West Mediterranean and the Great Whirl off the Somali coast [58].
Indeed, these eddies have been found to be associated with spot-like anomalies of physical or
biogeochemical properties like sea surface temperature, chlorophyll and even with peculiar
phytoplanktonic community structures [6, 45, 21]. Due to their long retention times and their
observed structuring role on primary producers, it is natural to ask whether eddies with a long
retention time are also hotspots of trophic interactions.
In this paper, we aim at exploring the role of trapping by altimetry-derived mesoscale eddy
cores on trophic interactions. In order to achieve this, we start by quantifying retention times
from real data. Lagrangian diagnostics like the Lyapunov exponents can probe the permeability
of an eddy core, but they are not informative of its content. On the other hand, Eulerian
quantities like the vorticity identify the core of an eddy but not its retention time. Following
an approach which has been proposed on model systems [27] and generalizing the calculation
of Lehahn and co-authors [38], here we combine the Lagrangian and Eulerian approaches
and define a diagnostic able to estimate the retention time of mesoscale eddies observable
from satellite-derived surface currents. After discussing how the temporal variability affects
retention times, we consider the case of the energetic eddies present in the meanders of the
Antarctic Circumpolar Current. We locate eddies with different retention times, compute the
surface which they cover climatologically and compare them with the foraging behaviour
of a tagged elephant seal, which made a long foraging trip (thousands of kilometres) in a
region with eddies of contrasted characteristics. Remarkably, this predator halted and foraged
intensively in an eddy with large retention time (>30 days), but did not stop when crossing
other eddies with retention times of a few days.
Section 2 is dedicated to the introduction of the key observational and theoretical tools
used in the paper, in particular, the Okubo–Weiss (OW) parameter and the finite-size Lyapunov
exponents (FSLEs), and provide examples of the applications of such different diagnostics
to the ocean surface velocity field derived from satellite altimetry. Section 3 introduces the
retention parameter as a semi-Lagrangian diagnostic which integrates the information provided
by a Lagrangian approach and the OW parameter. Section 4 deals with the coupling of
marine ecosystems and of the physical variability at the sub-mesoscale, applying the retention
parameter to the analysis of a foraging trip of an elephant seal through the meanders of the
Antarctic Circumpolar Current East of Kerguelen Islands.
2. Surface currents and tracer distribution in the ocean
2.1. Satellite altimetry and the Lyapunov exponent calculation
Since the launch of ERS-1 and Topex-Poseidon altimetry satellites in the 1990s, information
on the oceanic surface circulation is almost globally accessible from remote sensing. The
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-2
[cm]
80
-46
60
[days ]
0.2
-46
0.15
40
-48
0.1
-48
0
-50
0.05
lat
lat
20
0
-50
-20
-52
-0.05
-52
-40
-60
-54
70
75
80
lon
85
90
95
-0.1
-0.15
-54
-80
70
75
80
70
75
80
lon
85
90
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90
95
-0.2
-1
[days ]
0.2
-46
-46
0.15
0.1
-50
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lat
-48
lat
-48
-50
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0.05
-54
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lon
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Figure 1. Four different analyses of the surface velocity field in a region around the Kerguelen
Islands. (a) Sea surface height (SSH) measured by satellite (AVISO, Kerguelen product) and
superimposed surface velocity field, obtained by assuming that the flow is quasi-geostrophic.
(b) Okubo–Weiss parameter computed on the velocity field of panel (a). Blue regions indicate
points where vorticity dominates over strain (e.g. the eddy cores), red regions the opposite. The
OW parameter is a Eulerian diagnostic, and its computation is based on the instantaneous flow.
(c) Backward finite-size Lyapunov exponents map: higher values of the dominant exponent indicate
higher stretching of water parcels, and identify potential locations of tracer fronts. (d) Distribution
of a virtual, passive tracer (red dots) advected by the (time-varying) flow of panel (a) after being
initialized with uniform concentration inside the region delimited by the blue lines. The tracer
distribution fronts are largely aligned with the FSLE ridges (black lines).
principle behind satellite altimetry is to measure the sea surface height (SSH, figure 1(a))
above a reference geoid and to compute geostrophic velocities by balancing the pressure
gradient (which depends on the gradient of the SSH) with the Coriolis force (which depends
on the horizontal velocity, the unknown to be retrieved). This is formally equivalent to the
definition of a time-varying two-dimensional Hamiltonian system, where the SSH corresponds
to the Hamiltonian. The spatiotemporal resolution of the SSH estimation—and hence of the
derived velocities—depends on the number of altimetry satellites in orbit at the same time and
on their orbit characteristics. The error depends on the precision of the on-board altimeters,
the orbit estimation, the precision at which the shape of the reference geoid is known and
the algorithms which process the satellite signals. More technical information on satellite
altimetry, including the characteristics of past, current and incoming altimetry constellation,
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can be found at the CNES/CLS AVISO website www.aviso.oceanobs.com [28]. Although
all these parameters constantly change, altimetry is considered reliable for the detection of
geostrophic circulation structures of about 70–100 km and lifetimes of at least a week—hence
for mesoscale studies [36]. Note that satellite altimeters are radar instruments and are therefore
unhindered by cloud coverage.
The availability of two-dimensional ocean surface velocity fields on an extended and
uninterrupted spatial and temporal domain (tens of years with almost a global coverage) has
opened the way to the application of several mathematical approaches originally developed for
the analysis of dynamical systems or idealized geophysical flows (see for instance the reviews
by Wiggins [74] and Samelson [62]). The main aim of these studies is the identification of
the key mesoscale properties and dynamical structures which control transport and mixing at
the ocean surface and which may link the current field to the biogeochemical dynamics of the
ocean.
We briefly review chronologically some of these previous results, focusing on the
Lyapunov exponent calculations applied to altimetry data. Following some previous
applications to atmospheric or model systems (e.g., [56]), probably the first calculation of
Lyapunov exponents from altimetry data has been made by Abraham and Bowen [1], who
estimated mean stirring rates and showed that the ridges of Lyapunov exponents were in
close agreement with fronts of sea surface temperature. In order to further explore the relation
between stirring and sea surface temperature, Abraham and Bowen also advected a tracer with
either a longitudinal or a latitudinal gradient, and applied to the ocean dynamics a formalism
which was proposed for chaotic advected chemical fields [49]. This approach was further
developed for the case of the Tasman sea [71] and then to the global oceans [70], where a strong
correlation was found between Lyapunov exponents and Eulerian quantities like the strain rate
and the eddy kinetic energy. A second regional study (East Mediterranean) [23] compared
Lyapunov exponents and Eulerian diagnostics at submesoscale-resolving resolution, showing
the superior ability of the former in detecting filaments and fronts associated with sea surface
temperature. The study also concluded that the correlation between Eulerian diagnostics and
Lyapunov exponents does not hold when basin-scale climatologies are computed by averaging
submesoscale-resolving maps. Transport barriers derived from altimetry by computing the
finite-size Lyapunov exponent have been compared to satellite-derived [39, 38] and modelled
[57] chlorophyll patches. These comparisons showed the structuring role of stirring on primary
production and in particular provided evidence of how the temporal variability of the mesoscale
velocity field can break closed orbits associated with mesoscale eddies and create submesoscale
spiralling patterns visible in satellite chlorophyll images. This difference between closed
Eulerian eddies and spiralling Lagrangian structures was later described also by Beron-Vera
et al [7] who compared ridges of Lyapunov exponents with the trajectory of one Lagrangian
drifter. A larger number of drifter trajectories and transport barriers were then compared by
Resplandy et al [58] who combined in situ, altimetry and a biogeochemical circulation model
to interpret the strong pCO2 variability observed by CARIOCA drifters, in terms of stirring
and transport barriers. The biogeochemical role of horizontal stirring was further investigated
by Calil et al [13] and Rossi et al [60, 59] where the Lyapunov analysis was used to link
the horizontal stirring dynamics to upwelling and primary production. In particular, Calil
et al showed that ridges of Lyapunov exponents can be associated with hotspots of primary
production in oligotrophic waters, by either creating frontal regions where nutrients can be
upwelled by secondary circulation or by structuring in filament patches of blooming waters.
In spite of the large number of case studies in which altimetry-derived Lyapunov ridges
have been shown to visually match fronts of physico-chemical or biological fields (with
some exceptions: see Nencioli et al [48] for dramatic errors in stirring patterns derived from
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altimetry along the coast), there are not many pieces of work where the robustness of the
altimetry-derived Lyapunov analysis has been assessed. Hernández-Carrasco et al [29] tested
the effects of noise and of the dynamics of unresolved scales, using data from a primitive
numerical model as a benchmark. Lacorata et al [33] already in 1996 compared Lyapunov
exponents computed from drifters and from model data showing agreement between the two
diagnostics. However, to our knowledge, the only systematic validation of Lyapunov-derived
frontal regions and in situ fronts detection has been done by Dèspres et al [20], who found that
the front spatiotemporal distribution of tracers like temperature or salinity can be predicted by
Lagrangian analyses of altimetry data only if information on the large-scale tracer distribution
is also known. The effect of noise on altimetry-based Lyapunov calculation has been also briefly
discussed by Cotté et al [18]. The spatial structure of the Lyapunov ridges was shown to be
highly robust to delta-correlated, high-frequency noise (which mimics small-scale diffusion),
and an analytical scaling relation integrating noise intensity in the value of the Lyapunov
exponents was proposed.
More recently, altimetry-derived Lyapunov exponent calculations have been used to
explore the ecological role of horizontal stirring in ecological processes. Tew Kai et al [31]
for the first time related Lyapunov ridges to the distribution of a top marine predator (Fregata
minor) of the Mozambique Channel. Using satellites to track the position of eight frigatebirds,
the authors provided evidence that these predators co-localized with Lyapunov ridges. Thus,
they argued that coherent Lagrangian structures may be detected by visual, olfactory or
wind cues. The co-localization between position of frigatebirds and Lyapunov ridges was
recently revisited by De Monte and co-authors [19] who used higher resolution (GPS),
behaviour-resolving tracking data, and showed that behavioural switches occur in the presence
of Lyapunov-detected transport fronts especially when they coincide with thermal fronts.
An important role of Lyapunov ridges for marine predators was further evidenced by Cotté
et al [18], who analysed the relation between Mediterranean whales and ocean physics in the
Mediterranean sea at various spatiotemporal scales. The authors found that whales were associated with Lyapunov ridges only during the summertime phytoplankton minimum, suggesting
that patchiness in food resources induced by stirring may be especially critical for marine
predators during oligotrophic conditions. In the same year, d’Ovidio et al [21] compared
remotely sensed dominant phytoplankton types and altimetry-derived transport structures.
This analysis showed that the phytoplanktonic landscape is organized in (sub-)mesoscale
patches of dominant types separated by physical fronts. These fronts, induced by horizontal
stirring, are identified from altimetry data through a Lyapunov analysis. The authors named
‘fluid-dynamical niches’ such water patches delimited by Lyapunov ridges and supporting
different dominant types. The concept of a water patch which maintains its integrity for several
weeks and longer has been also exploited by biogeochemical campaigns, which have tracked
and sampled such semi-closed systems, viewing them as megacosms. This approach has
allowed for instance the characterization of the biogeochemistry and ecology of an artificially
fertilized water patch from the beginning of the bloom to the export of organic material to the
deep ocean at the end of the bloom [69, 12].
2.2. Eulerian and Lagrangian diagnostics
We introduce here the formal definitions of the diagnostics that will be used in the following
sections, and that are possibly the most widespread tools for detecting the structures generated
in the ocean by horizontal transport. For a more extensive introduction to these and other
statistics of a velocity field, we refer to [9] and [23].
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Given a streamfunction ψ (x, y, t ) := g/ f H, where g is the acceleration due to gravity, f
is the Coriolis parameter which takes into account the Earth’s angular velocity and H is the
SSH with respect to a reference geoid, the geostrophic velocities are defined as
∂ψ
,
(1)
vx (x, y, t ) = −
∂y
∂ψ
,
(2)
∂x
where x, y and t are respectively longitude, latitude and time.
It is easy to check that such a velocity field has zero divergence for all the scale lengths
at which f (and g) can be considered constant.
The OW parameter compares the intensity of the strain to the vorticity:
vy (x, y, t ) =
OW(t, x, y) := s(t, x, y)2 − ω(t, x, y)2 ,
where s is the strain and ω is the vorticity.
The strain and vorticity are obtained from the derivative of the velocity field as
∂vy 2
∂vy
∂vx 2
∂vx
2
−
+
s =
+
,
∂x
∂y
∂x
∂y
(3)
(4)
∂vy
∂vx 2
ω =
.
(5)
−
∂x
∂y
It can be shown that the OW parameter is equal to the determinant of the Jacobian of the
velocity field multiplied by −4. When calculated at a stagnation point of the velocity field, it is
hence negative for an elliptic point (with two complex conjugated eigenvalues) and positive for
a hyperbolic point (with two real and opposite eigenvalues). The OW parameter can therefore
be seen as a way to extend the properties of a stagnation point to its neighbourhood.
The region of the plane (x, y) where OW(t, x, y) < 0, and hence recirculation prevails
over stretching, can be taken as an operative definition of an eddy core [17, 30, 47, 23].
The isolines of the OW parameter are used as a tool for identifying the Eulerian
eddy boundaries. The OW parameter cannot however account for the generation of
the complex filamentous structures induced by chaotic stirring on a tracer field, since these
structures depend on the temporal variability of the velocity field. In order to study these
filaments, it is necessary to move from the Eulerian framework to a Lagrangian one, using
statistics computed over particle trajectories, such as the Lyapunov exponent. This way,
one can reconstruct the advection history of a tracer field, integrating spatial and temporal
information along the trajectory of a material point. This approach has another advantage.
Tracer filaments produced by chaotic stirring and retrieved by Lagrangian techniques occur
typically at spatial scales smaller than the size of mesoscale eddies. Therefore, Lagrangian
techniques are a way to explore a regime which is not directly resolved by altimetry data.
We recall here some general concepts related to Lyapunov exponents and fluid dynamics
applications. More details can be found for instance in [37] and in the tutorial by S Shadden
(http://mmae.iit.edu/shadden/LCS-tutorial/).
The trajectory of a material point advected by the flow can be obtained by solving the
differential equations (1) for the boundary condition x(t0 ) = (x0 , y0 ), which provides the
position of the Lagrangian particle at time t0 . Given that δ0 represents the distance between
x(t0 ) and a nearby point and that δ represents the separation at a time t of the trajectories
originating at those points, Lyapunov exponents are formally defined [52] by the double limit
δ
1
(6)
λ = lim lim ln .
t→∞ δ0 →0 t
δ0
2
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This expression allows us to discriminate chaotic and non-chaotic domains of ergodic
dynamical systems. However, the limit is exactly determined only when the velocity field is
known at all times and all points in the plane. In order to apply the Lyapunov approach to a field
that, as derived from altimetry data, has a limited spatiotemporal resolution, the relaxation of
one of the two limits in the definition of the Lyapunov exponents is necessary.
One can either prescribe the integration time t leading to the finite-time Lyapunov exponent
(FTLE) or a spatial scale (the ratio δ/δ0 ) leading to the finite-size Lyapunov exponent (FSLE).
Both these quantities approach, in the limit of complete time-scale separation, the Lyapunov
exponent defined by equation (6). In this work, we will use the dominant FSLE which has
been introduced in [4, 3, 9]. (For more details, see appendix A in [24].)
Oseledec’s theorem states that if the flow is ergodic, then the limit in equation (6) is a
unique λ∗ irrespective of the perturbation applied to a point [51] and for almost all initial
conditions on the ergodic set. Such asymptotic value is positive for points belonging to chaotic
domains. For finite times and separations, however, spatial homogeneity no longer holds.
This turns out to be an advantage for geophysical applications, because the dominant FSLE
assumes different values for different positions in space, revealing regions with different
kinematic properties.
In the case of mesoscale turbulence, it is not difficult to choose an integration time
or particle separation for FTLE and FSLE such that most of the calculations occur in the linear
regime. There the finite-time and the finite-size calculations provide in practice a similar
outcome (compare for instance FTLE maps in [7] with FSLE maps in [39]). Although we do
not use this approach here, we also mention that another use of the FSLE is the exploration of
different dispersion regimes [4, 3, 9, 67].
The computation of the dominant FSLE can be performed either forward or backward
(when t < 0) in time. Here, we will focus on the dominant FSLE computed backward in time,
since it allows us to detect the barriers to transport in the flow.
2.3. Okubo–Weiss parameter and Lyapunov exponent applied to altimetry-derived velocities
An example of a snapshot of the mesoscale velocity field derived by satellite altimetry is
shown in figure 1(a) in terms of SSH (colour) and geostrophic velocity (arrows). The panel
depicts the Kerguelen region (Indian sector of the Southern Ocean). The Antarctic Circumpolar
Current—one of the most energetic currents of the world—is steered along its eastward flow by
the Kerguelen plateau and splits into several meandering branches, creating a very contrasted
mesoscale field. There is a low-energetic region on the wake of the plateau (70◦ –85◦ E)
followed eastward by strong and dynamically active eddies (85◦ –100◦ E). Mesoscale eddies
can be evidenced as closed streamlines with a radius of about 100 km. A map of the OW
parameter for this flow is shown in figure 1(b). This image depicts the eddy field (blue spots)
which is at the origin of horizontal stirring. In order to estimate the effect of these eddies over
any advected tracer, a Lagrangian analysis is necessary.
Figure 1(c) shows an example of FSLE calculation backward in time, where the map is
obtained by measuring the rate of separation of particles initialized in nearby positions. In
this example, the parameters for the computation of FSLEs are δ0 = 2.5 km and δ = 60 km.
This choice is related to the spatial scales of the system: since we are interested in mesoscale
structures, the final separation δ is chosen as a typical mesoscale structure’s dimension, whereas
δ0 is chosen close to the resolution of the grid [22]. The timescale t for going from a separation
of δ0 to δ is related to the mixing properties of different regions: typical values are about 1–4
weeks, corresponding to FSLEs in the range [0,0.14] days−1.
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An intuitive understanding of the physical meaning of FSLE maxima (ridges) can be
gathered by examining the advection of a passive tracer (figure 1(d)). A backward-in-time
large separation of particles initialized nearby means forward-in-time convergence of water
parcels that were located far apart from each other. If these water parcels had contrasted
properties, such as different tracer concentrations, the confluence dynamics would result in
the amplification of the tracer gradient, because the distance between the water parcels, and
the tracers that they transport, is reduced. Therefore, a ridge of large exponents can be found
in association with a biogeochemical front. When there is no temporal variability, the ridges
of the Lyapunov map coincide with the homoclinic and heteroclinic orbits surrounding elliptic
points, and thus provide a separation of the plane into elliptic or hyperbolic domains that is
consistent with what one would get by connecting hyperbolic points with streamlines.
2.4. Horizontal stirring of biogeochemical tracers and Lyapunov analysis
A powerful application of the Lyapunov exponent calculation in oceanography is the
identification of fronts of tracers advected by the mesoscale (tens to hundreds of kilometres)
velocity field.
Note however that lines of large Lyapunov exponents do not necessarily have a one-toone correspondence to the fronts of a tracer. Indeed, horizontal transport can only intensify
pre-existing heterogeneities, such as large-scale tracer gradients. For this reason, ridges of
Lyapunov exponents (which may be considered as transport or kinematic fronts) are typically
much more numerous than tracer fronts. There are also several factors which may affect the
effectiveness of the Lyapunov exponents calculation for the detection of tracer fronts, namely
(i) if the large-scale tracer gradient is oriented orthogonally to the stretching direction, then it
is not amplified; (ii) the tracer has an active dynamics so that initial contrasts in converging
waters are not preserved during advection; (iii) transport mechanisms different from stirring
by mesoscale turbulence are also active (e.g., upwelling and downwelling, convection); and
(iv) altimetry-derived geostrophic velocities do not perfectly represent horizontal stirring (e.g.,
in terms of resolution and in terms of missing ageostrophic components). These factors, which
are often all present at the same time, may induce one to think that the application of the
Lyapunov exponents is hopeless in any realistic case.
Nevertheless, several studies (see section 2.1) have shown empirically that the
aforementioned factors do not compromise the application of the Lyapunov exponent
calculation, although misplacements of the fronts of the order of 10–20 km can typically
arise.
A comparison between a numerically advected tracer and the FSLE field of figure 1(c)
is shown in figure 1(d). The tracer patch, initialized within a box, is structured in convoluted
filaments typical of chaotic advection. The contours of these filaments are almost perfectly
identified by ridges of FSLEs. As previously remarked, however, the Lyapunov ridges are
much more numerous than the tracer filament boundaries (which depend on the specific initial
conditions). Note that by visually inspecting the tightness of the spiralling ridges, it is not
difficult to guess which eddies may have longer lived cores; however, the Lyapunov exponent
map neither quantitatively provides this information nor indicates which points lie inside or
outside an eddy core.
Figure 2 shows an example of the matching between a real tracer, chlorophyll, and
FSLE ridges. Even if the tracer is not conservative (that is, its local concentration changes
in time) and the currents are actually affected by both ageostrophic dynamics (notably,
wind-induced Ekman transport) and unresolved geostrophic velocities, then the fact that
such a correspondence holds provides yet another empirical confirmation of the relevance
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[µg/l]
3
-46
1.5
-48
1
lat
0.75
-50
0.5
-52
0.2
-54
70
75
80
lon
85
90
95
0.1
Figure 2. Chlorophyll (Chl) distribution (from water-leaving radiance, SeaWiFS) and ridges of
FSLEs, in the same region as figure 1. The FSLE ridges show remarkable agreement with Chl
fronts, in spite of the fact that Chl is not a passive tracer. The possibility of matching Chl tracer
fronts with FSLE lines lies in the fact that the tracer evolves on a slower timescale than the velocity
field. It is worth noting that the actual distribution of Chl essentially depends of the initial location
of the phytoplankton bloom, that is, in the case illustrated here, localized on the Kerguelen plateau.
of Lagrangian approaches to the study of the oceanographic dynamics, at least at the
(sub-)mesoscale.
3. Time-varying velocity fields: estimating the timescale of eddy trapping
Let us first discuss the idealized case of a time-invariant streamfunction (this can be a good
approximation of a long-lived eddy trapped by the bathymetry). The stagnation points of
the velocity field (equilibria of equations (1)) play a structuring role for the circulation. In
particular, the invariant manifolds stemming from hyperbolic, saddle points act as transport
barriers and are thus able to constrain the displacement of particles advected by the surface
currents.
Imagine that a tracer is present in the water at concentration c(x, y), and that it is
conservative (the tracer is not produced or consumed) and passive (it does not affect the
velocity field). If diffusion is negligible, as when considering scales above a few kilometres in
the ocean, its dynamics is described by the advection equation
∂c
= −v · ∇c.
(7)
∂t
When the velocity field is independent of time, the tracer cannot cross a streamline
(line of constant ψ), since ∇c ⊥ v. On the other hand, the invariant manifolds (homoclinic
or heteroclinic orbits) of the saddle points are streamlines that connect equilibria, and can
enclose regions that contain elliptical stagnation points. These domains correspond to oceanic
eddies, within which the water circulates, and are separated by the free water that flows outside
their boundaries. It is easily shown that if the velocity field is time-independent, then the total
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J. Phys. A: Math. Theor. 46 (2013) 254023
F d’Ovidio et al
concentration C = c dω of a tracer within a region , whose contour ∂ is given by
streamlines, is conserved:
∂c
dC
=
dω = − ∇(vc) dω = −
(vc) · n ds = 0.
(8)
dt
∂t
∂
If applied to a homoclinic orbit, such as the contour of certain eddies in time-invariant fields,
this means that the tracer content of the region it contours is constant in time.
As soon as the field changes in time, however, the correspondence between streamlines
and particle trajectories does not hold any longer and particles can escape from eddy cores.
Still, the structure of a time-invariant field is maintained up to a certain approximation
(related to the observation time scale), so that the regions that have high retention times
(typically, eddy cores) and other regions (typically, eddy peripheries) where instead mixing is
dominant exist.
We follow here some previous works that have addressed the issue of distinguishing
retention zones with different mixing properties in idealized time-varying vector fields [35,
27, 34, 50, 66, 43]. We adopt a hybrid Lagrangian and Eulerian approach to give a definition
of the time during which particles remain trapped within eddies. In this section, we introduce
a retention parameter (RP) and discuss its relationship with the OW parameter and FSLEs. In
section 4, we will then propose an application to the case study of a foraging trip of a tagged
top marine predator.
With RP at a given time t0 and at a point (x0 , y0 ) we indicate the number of days a water
parcel at that given location has remained confined within an eddy core, that we define as a
region where the OW parameter is negative. This amounts to computing the OW parameter
OW(t, x, y), with t ∈ I = {t0 −tback , t0 }, along the trajectory (x(t ), y(t )) obtained by integrating
backwards in time for a time tback the velocity field with boundary values (t0 , x0 , y0 ), and finding
the time when OW changes sign, indicating that the particle has left the eddy core.
If we define te = max {t ∈ I|OW(t, x(t ), y(t )) = 0} the first singular value of the OW
parameter, then we can formally define the RP as
RP(t0 , x0 , y0 ) = t0 − te ,
or equivalently, and in analogy to [27], as
t0
RP(t0 , x0 , y0 ) =
dt.
(9)
(10)
te
Note that this definition is different from the ‘ellipticity time field’ proposed by Haller
[27], since the latter was based on a measure of the ellipticity properties along the trajectory,
and the integration takes place at all times that the trajectory spends in a region of given
topological properties. In our case, instead, we decided to base our definition on an Eulerian
statistics, the OW parameter, since this is what is most commonly used in oceanography
for identifying eddy cores (see section 2.4). Also, in the definition we propose here, we do
not allow a particle to leave and re-enter eddy cores. The rationale behind this choice is the
following: we assume that as soon as a water parcel gets in a hyperbolic region (i.e. outside
eddy cores), its content is stretched and mixed, so that its specific content is lost.
Figure 3 illustrates an example of the calculation of the RP (with a maximum integration
time tback = 60 days) computed for two positions in space. In both cases, the backtrajectories
have been obtained by backward integration in the same time interval, and plotted together
with the sign of the OW parameter calculated along them. Note that the blue segment (negative
OW) corresponds to the time in which the particle is circulating in an eddy core, that is, the
value of the RP.
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J. Phys. A: Math. Theor. 46 (2013) 254023
F d’Ovidio et al
Figure 3. Illustration of the rationale behind the definition of the retention parameter: such a
diagnostic is meant to quantify the amount of time a water parcel has remained in a confined region
of the ocean. Backtrajectories (tback = 60 days) originating from two points (red dots) situated
at time t0 in nearby eddies are represented together with the sign of the OW parameter calculated
along them (blue, negative; red, positive). The values of the retention parameters associated with
these two points are displayed, showing that the point with higher retention parameter is associated
with a trajectory spinning around an eddy core, whereas a smaller retention parameter is associated
with a water parcel that has been more recently trapped within an eddy.
3.1. Retention times of altimetry-detected mesoscale eddies
What are the retention times of oceanic mesoscale eddies? We address this question by
constructing some instantaneous and climatological maps of the RP for the Kerguelen region.
In order to evidence how properties of oceanic eddies are influenced by the temporal
variability of the velocity field, for any time t0 at which we compute a map, we artificially tune
the timescale of the currents derived from satellite observation. We rescale the time variable
so as to generate a family of fields v(t, t0 , ) = v(t0 − (t0 − t )) such that ∈ [0, 1] sets
the variability timescale: when it is zero, the velocity field is frozen at all times under the
condition observed at time t0 : v(t, t0 , 0) = v(t0 ) ∀t. All eddy cores have the same, maximal,
value of RP (equal to tback ), since the water within them recirculates indefinitely. If is unitary,
then the field has its natural temporal variability: v(t, t0 , 1) = v(t ).
Such time rescaling does not affect Eulerian measures at time t0 , which do not depend on
the state of the velocity field in the future or in the past. Lagrangian diagnostics, however, are
modified, since the trajectories are now computed over a field that is slowed down according
to the parameter .
Figures 4(a)–(c) display the dominant FSLE for an observed velocity field timed at
different values of .
As expected, the increased exchange of water between an eddy and its surroundings
reduces the retention time of most eddies identified by the Eulerian statistics. We find however
a surprising result. In some cases, the variability in time can generate retention regions that
were not present in the time-invariant case. A possibility may be an eddy at the end of its
lifetime. Backward trajectories for the time-invariant case only experience the weak vorticity
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J. Phys. A: Math. Theor. 46 (2013) 254023
−46
(a)
F d’Ovidio et al
(b)
FSLE
[days−1]
(c)
−48
0.1
0.05
latitude
−50
0
(e)
−46 (d)
RP
[days]
(f )
60
−48
40
20
−50
77
79
81
77
79
81
77
79
81
0
longitude
Figure 4. (Semi-)Lagrangian diagnostics of an altimetry-derived velocity field with artificially
tunable temporal variability, as explained in the text. (a)–(c) Dominant finite-size Lyapunov
exponent (δ0 = 1 km, δ = 40 km); (d)–( f ) retention parameter computed for tback = 60
days. From left to right the variability in time increases: (a), (d) = 0, (b), (e) = 0.5, (c), ( f )
= 1. In the last case, the temporal variability reflects the variation of the measured velocity field.
Slow variations of the velocity field do not substantially alter the landscape one would obtain for
a frozen field, where transport barriers contour recirculation regions. Time variability instead can
erase, or even generate (e.g. around 50.5◦ South, 78.5◦ East), retentive regions, such processes
being related to the opening and folding of transport barriers.
associated with this vanishing eddy and escape the eddy, hence resulting in a short retention
time; trajectories constructed in the time-dependent case however also experience the stronger
vorticity associated with the eddy at a younger life stage and therefore may recirculate in the
eddy for a longer time, thus resulting in a larger retention time.
The complex interplay of eddy core intensity and temporal variability is further confirmed
by a climatologic analysis (figure 5). The comparison of patterns of retention and patterns of
OW, averaged in time for one year, clearly shows that there is no clear association between
regions of largest negative OW and regions of strongest retention. Indeed, attempting to
produce a scatter plot does not provide any compact relation between the two quantities (not
shown).
Figure 5 suggests that retention times in excess of one and even two months—therefore
likely able to structure secondary and higher production, as discussed in the introduction—are
rare but possible. We quantify their occurrence by considering all the daily maps of retention
for one year in the Kerguelen region and constructing the cumulative probability of retention
times. We perform the calculation both for = 0 and = 1 and plot the results in figure 6.
This distribution indicates that cores older than one month cover about 3% of the ocean
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[d-2]
0.3
[d]
30
-46
-46
25
lat
20
15
-50
10
-52
-48
0.1
lat
-48
0.2
0
-50
-0.1
-52
5
-0.2
-54
70
-54
75
80
lon
85
90
0
95
70
75
80
lon
85
90
-0.3
95
Figure 5. Comparison between the retention parameter (left) and Okubo–Weiss parameter (right)
averaged over one year and computed on the same region of the ocean. The patterns do not show
a clear correlation between regions of largest negative Okubo–Weiss and highly retentive areas.
0
0
10
10
-1
Area covered
Area covered
10
10-1
10-2
-2
10 0
-3
10
20
30
retention > days
40
50
60
10 0
10
20
30
retention > days
40
50
60
Figure 6. Cumulative distribution of the retention parameter for no time variability of the velocity
field = 0 (left) and for natural time variability = 1. The importance of the time variability of
the velocity field for retention issues is underlined by the fact that in the case of natural temporal
variability eddy cores that are older than one month cover about 3% of the ocean surface around
the Kerguelen region, whereas in the case of a frozen velocity field, one gets approximately double
the value.
surface around the Kerguelen region. This value would approximately double if the temporal
variability of the velocity field were absent. Considering that an eddy core is about 100 km
wide and supposing that the eddies are randomly distributed, this means that a linear path
encounters on average one core with RP larger than 1 month every 3000 km.
4. Eddy retention and trophic interaction: an elephant seal case study
The observation of top marine predator displacements is an important tool of marine ecology.
High trophic levels indeed play the role of ‘integrators’ of the food chain, since their ecology is
altered by misfunctioning of the ecosystem [14]. If the underlying ecological processes, which
relate the predator behavioural response to changes in its environment, are known, one can
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[m]
0
-46
-1000
-48
lat
-2000
-50
-3000
-52
-4000
-54
70
75
80
lon
85
90
95
-5000
Figure 7. Trajectory of an elephant seal trip, starting from the Kerguelen Islands on 23 January
2009 and ending when the recording stopped on 14 March 2009, superimposed on the bathymetry
of the Southern Indian Ocean. The trajectory becomes more tortuous and the animal slows down
when it comes close to an eddy. These features of the azimuthal trajectory of elephant seals
is typically associated with an increase in capture events, indicating intensive foraging activity
[26, 54]. The three red dots indicate the position of the elephant seal at three dates two weeks apart
(1 February, 14 February and 1 March 2009).
hope to infer from physical features the modification in the predator habitat, and consequently
of its trophic function.
This however requires to clarify not only the statistical correlations with environmental
patterns at the large scale, such as for instance with sea surface temperature in migrating
animals [8], but also at the finer scale on which the animal perceives the physical features of
its habitat [19].
In this section, we provide an example of the use of previously discussed diagnostics of
horizontal stirring, the FSLE and the RP, for investigating a top predator’s behaviour at the
(sub)mesoscale.
4.1. How do elephant seals move in a turbulent field?
Elephant seal (Mirounga leonina) populations of the Kerguelen Islands (49◦ South, 69◦ East)
have been the object of long-term demographic monitoring that has been complemented in
recent years by satellite tracking at increasing temporal and spatial resolutions. Here, we
analyse the post-moulting trip of an adult female equipped with a ARGOS tag, who left the
Kerguelen Islands on January 2009. The animal position is measured with a resolution of 1 km
and 1/4 day, for a total duration of the recording of 79 days [25].
Elephant seals, and particularly females, forage in the open ocean, where they alternate
hundreds of metres deep diving in search for fish or squid, and short breathing intervals at
the surface (with a total period of about 20 min). After moulting on land, females leave their
colony to engage in solitary trips, lasting several months, where they typically head towards a
zone of strong mesoscale activity between the subtropical and the subpolar fronts (see figure 7
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J. Phys. A: Math. Theor. 46 (2013) 254023
F d’Ovidio et al
−48 (a)
−48 (b)
−48.5 (c)
−49
−49
−49.5
−50
−50
−50.5
OW
−2
[days ]
2
0
latitude
78
79
80
78
79
80
−2
90
−48 (d )
−48 (e)
−48.5 (f )
−49
−49
−49.5
−50
−50
−50.5
78
79
80
78
79
80
90
−48.5 (i )
−49
−49
−49.5
−50
−50
−50.5
80
78
79
80
FSLE
[days−1]
0.05
−48 (h)
79
92
0.1
−48 (g)
78
91
91
92
0
RP
[days]
50
90
91
92
0
longitude
Figure 8. Physical diagnostics derived by remote-sensing of sea surface height for three days
during the elephant seal trip (see figure 7 for their locations along the trajectory). The dot
indicates the position of the animal along the trajectory on a given day (specified in the panel title).
(a)–(c) Okubo–Weiss parameter: eddy cores are characterized by negative values. (d)–( f ) Finitesize Lyapunov exponents (δ0 = 1 km, δ = 40 km): larger values identify stronger transport barriers.
(g)–(i) Retention parameter: measures the number of days the trajectory stemming from a point
and computed backwards in time remains within the same eddy, that is larger for longer-lasting
structures.
for an overview of the trip analysed here) [5]. This region is the target for foraging trips not
only of elephant seals, but of many predator species [10].
The region displays highly variable physical structures, including eddies and filaments,
that we characterize here by both Eulerian diagnostics (OW) and by Lagrangian methods
(FSLE and RP).
On the timescale of the predator’s trip, these structures appear and vanish. At the
mesoscale, the time variability of the velocity field is evident irrespective of the method
one chooses for looking at the structures generated by transport. Figure 8 illustrates the three
aforementioned diagnostics at three different times during the elephant seal trip, two weeks
apart from each other. The red dot in each panel indicates the position of the animal at the time
when the diagnostic was computed. The tagged animal remained for more than two weeks
close to a mesoscale eddy that is clearly evidenced both by the OW parameter (figures 8(a)
and (b)) and the FSLEs (figures 8(d) and (e)).
The submesoscale fronts generated by the temporal variability of the turbulent field
are evidenced by the FSLEs, whereas the OW parameter only identifies the cores of the
recirculation regions.
In spite of the more accurate spatiotemporal resolution achievable by the use of Lagrangian
diagnostics, that exploit at best the spatiotemporal variability of the velocity field, the FSLE
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F d’Ovidio et al
map appears to bear more information than one would need, and may not be the key prompt
to understand the predator’s habitat use.
Indeed, the case study reported here shows that even if the elephant seal seems to
concentrate its foraging activity on a mesoscale eddy identifiable by the OW parameter,
and to follow the submesoscale structures evidenced by the FSLEs, nevertheless not all eddies
or all fronts are equivalent. For instance, figures 8(c), ( f ) and (i) display the animal crossing
an eddy, revealed by both the OW parameter and by the FSLEs, without halting.
We suggest here to use the RP introduced in section 2.2 as a tool to discriminate
between eddies that may or may not be interesting for foraging. Indeed, figures 8(g)–(i) are
consistent with the hypothesis that the elephant seal spends longer time in eddies with higher
retention.
The RP is of course not the only feature to be considered in evaluating the potential
productivity of oceanic mesoscale eddies. Indeed, one primary factor affecting the biomass
transfer to higher trophic levels is the origin of the water mass trapped within the eddy, and as
a consequence the type of ecosystem it is able to sustain [21].
It is however notable that the elephant seal halts where the RP has high values, returning
towards the highly retentive eddy after having moved towards a more permeable one, and then
following the filament evident in figure 8(h). These features stand out clearly when looking at
the retention time, but not in the FSLE and OW map.
5. Discussion and conclusions
The application of dynamical systems theory to the two-dimensional velocity field of oceanic
currents, obtained by reprocessing altimetric measures, revealed a powerful approach to
identify and elucidate some key mechanisms involved in (sub)mesoscale circulation and
environmental structuring. The finite-size Lyapunov exponent evidences the transport barriers
that control the horizontal exchange of water in and out of eddy cores. However, the
Lyapunov exponent does not provide information about the time that a water parcel has spent
inside an eddy. This information is of critical relevance for the ecology of the open ocean,
where patchiness in the distribution of the biomass along all the trophic chains is typically
found.
Here, we study the retention of oceanic mesoscale eddies by introducing a novel
diagnostic, the retention parameter (RP), that estimates the time a water parcel has been
recirculating inside an eddy core. Along the line of Lyapunov exponent computation, this
diagnostic is based on the evaluation of a property of the velocity field, the Okubo–Weiss (OW)
parameter, along the trajectory that emanates backwards in time from a given point. Negative
values of the OW parameter are classically used to identify eddy cores. The retention time of
an eddy is hence obtained by measuring along a particle trajectory the time of persistence of a
negative OW value. The RP characterizes eddy cores, whereas the analysis based on Lyapunov
exponents reveals a multitude of filaments, which in some cases enclose hotspots of retention.
Eddy cores with retention times relevant for structuring higher trophic levels (of one month
or larger) appear to be relatively rare: in the interfrontal region east of Kerguelen, they occupy
about 3% of the ocean surface.
If such regions have supported high primary production during the blooming season, it is
reasonable to think that high-retention regions may support higher trophic levels as well. Since
marine predator displacements are able to easily overcome the distance between eddies [8],
the active search of highly retentive regions should be rewarding in particular when resources
are rare and patchy (likely in post-bloom, oligotrophic conditions [18]). We have provided a
preliminary confirmation of this hypothesis by considering the case study of a foraging trip
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F d’Ovidio et al
performed by an elephant seal in the Austral summer. This animal crossed several eddies, but
showed a transition from straight movement to intensive searching only for an eddy with large
retention time.
If the horizontal current field had no temporal variability, then the closed orbits encircling
mesoscale eddies would be material lines and therefore all eddy cores would have infinite
retention times. The analysis of eddy retention that we have performed has led us naturally to
a formalism by which the temporal variability of the velocity field, which is at the origin of
finite retention time, can be introduced as a tunable perturbation over a time-invariant velocity
field. This approach may be useful for performing perturbative analyses of eddy permeability
on altimetry data and to better understand which terms in the temporal variability lead to the
breaking of eddy cores, that are isolated in time-invariant fields. Approaches like Melnikov
theory may be promising in this regard [63, 2].
Finally, here we focused on the impact of horizontal stirring in structuring the environment
that sustains marine ecosystems. Several other physical processes however exist which control
patchiness and dispersion of biotic tracers at various scales (e.g., [41]). A complete analysis of
retention processes would need to include these other mechanisms together with the horizontal
dynamics of eddy cores.
Acknowledgments
The altimeter products were produced by Ssalto/Duacs and distributed by Aviso with support
from Cnes. Chlorophyll images were obtained by the GLOBCOLOUR project. ADP thanks the
Fondation Bettencourt-Schueller (through the program Frontières du Vivant) and Fondazione
CRT (through the program Marco Polo) for their financial support. The elephant seal work
was conducted as part of the Institut Paul Emile Victor national research program (no.
109, H Weimerskirch and the observatory Mammifères Explorateurs du Milieu Océanique,
MEMO SOERE CTD 02). This work was carried out within the framework of the ANR
Blanc MYCTO-3D-MAP and ANR VMC 07: IPSOS-SEAL programs, OSTST program
ALTIMECO and CNES-TOSCA program (Élephants de mer océanographes). CG also thanks
the Total Foundation for financial support.
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