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Genesis, evolution, and apocalypse of Loop Current rings

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We carry out assessments of the life cycle of Loop Current vortices, so-called rings, in the Gulf of Mexico by applying three objective (i.e., observer-independent) coherent Lagrangian vortex detection methods on velocities derived from satellite altimetry measurements of sea-surface height (SSH). The methods reveal material vortices with boundaries that withstand stretching or diffusion, or whose fluid elements rotate evenly. This involved a technology advance that enables framing vortex genesis and apocalypse robustly and with precision. We find that the stretching- and diffusion-withstanding assessments produce consistent results, which show large discrepancies with Eulerian assessments that identify vortices with regions instantaneously filled with streamlines of the SSH field. The even-rotation assessment, which is vorticity-based, is found to be quite unstable, suggesting life expectancies much shorter than those produced by all other assessments.
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Genesis, evolution, and apocalypse of Loop Current rings
F. Andrade-Canto,1, a) D. Karrasch,2, b) and F. J. Beron-Vera3 , c)
1)Instituto de Investigaciones Oceanológicas, Universidad Autónoma de Baja California, Ensenda, Baja California,
México
2)Technische Universität München, Zentrum Mathematik, Garching bei München,
Germany
3)Department of Atmospheric Sciences, Rosenstiel School of Marine and Atmospheric Science, University of Miami,
Miami, Florida, USA
(Dated: 22 September 2020)
We carry out assessments of the life cycle of Loop Current vortices, so-called rings, in the Gulf of Mexico
by applying three objective (i.e., observer-independent) coherent Lagrangian vortex detection methods on
velocities derived from satellite altimetry measurements of sea-surface height (SSH). The methods reveal
material vortices with boundaries that withstand stretching or diffusion, or whose fluid elements rotate evenly.
This involved a technology advance that enables framing vortex genesis and apocalypse robustly and with
precision. We find that the stretching- and diffusion-withstanding assessments produce consistent results,
which show large discrepancies with Eulerian assessments that identify vortices with regions instantaneously
filled with streamlines of the SSH field. The even-rotation assessment, which is vorticity-based, is found to
be quite unstable, suggesting life expectancies much shorter than those produced by all other assessments.
PACS numbers: 02.50.Ga; 47.27.De; 92.10.Fj
I. INTRODUCTION
The Loop Current System, namely, the Loop Current
itself and the anticyclonic (counterclockwise) mesoscale
(100–200-km radius) vortices, so-called rings, shed from
it, strongly influences the circulation, thermodynamics,
and biogeochemistry of the Gulf of Mexico (GoM).34
As important long-range carriers, westward-propagating
Loop Current rings (LCRs) provide a potential mech-
anism for the remote connectivity between the GoM’s
western basin and the Caribbean Sea.1,3,6,7,21,24,25 In par-
ticular, bringing warm Caribbean Sea water within, the
heat content of the LCRs is believed to be as significant
as for LCRs to promote the intensification of tropical cy-
clones (hurricanes).32 On the other hand, as regions of
strong flow shear, LCRs may be capable of producing
structural damage on offshore oil drilling rigs.18 For all
these reasons, LCRs are routinely monitored.33
LCRs leave footprints in satellite altimetric sea-surface
height (SSH) maps,23 so sharp that the routine detec-
tion of LCRs consists in the identification of regions
filled with closed streamlines of the SSH field assum-
ing a geostrophic balance, a practice widely followed in
oceanography.5However, this eddy detection approach
uses instantaneous Eulerian information to reach long-
term conclusions about fluid (i.e., Lagrangian) transport,
which are invariably surrounded by uncertainty due to
the unsteady nature of the underlying flow. At the heart
of the issue with Eulerian eddy diagnostics of this type is
their dependence on the observer viewpoint:29 they give
a)Electronic mail: andcanfer@gmail.com
b)Electronic mail: karrasch@ma.tum.de
c)Electronic mail: fberon@miami.edu
different results for observers that rotate and translate
relative to one another. The issue is most easily grasped
by bringing up, one more time as we believe is central yet
widely overlooked, the example first discussed by Haller9
and thereafter by others.4,10,13,15 Consider the exact so-
lution to the Navier–Stokes equation in two spatial di-
mensions: v(x, t)=(x1sin 4t+x2(2 + cos 4t), x1(cos 4t
2) x2sin 4t), where x= (x1, x2)R2denotes posi-
tion and tRis time. The flow streamlines are closed
at all times suggesting an elliptic structure (i.e., a vor-
tex). However, this flow actually hides a rotating saddle
(pure deformation), as it follows by making (x1, x2)7→
x1,¯x2) = (x1cos 2tx2sin 2t, x1sin 2t+x2cos 2t), un-
der which v(x, t)7→ x2,¯x1)¯v(¯x). In other words, the
de-facto oceanographic eddy detection diagnostic5mis-
classifies the flow as vortex-like. The ¯x-frame is spe-
cial inasmuch the flow in this frame is steady, and thus
flow streamlines and fluid trajectories coincide. Hence
short-term exposition pictures of the velocity field by the
observer in the ¯x-frame determine the long-term fate of
fluid particles. The only additional observation to have
in mind to fully determine the Lagrangian motion is that
the observer in the ¯x-frame rotates (at angular speed 2).
This tells us that the flow under consideration is not actu-
ally unsteady as there is a frame (¯x) in which it is steady.
In a truly unsteady flow there is no such distinguished ob-
server for whom the flow is steady.26 Thus one can never
be sure which observer gives the right answer when the
de-facto oceanographic eddy detection diagnostic5,23 is
applied. As a consequence, neither false positives nor
false negatives can be ruled out,4and thus the signifi-
cance of life expectancy estimates is unclear.
Our goal here is to carry out objective (i.e, observer-
independent) assessments of the life cycle of LCRs. This
will be done in line with recent but growing work that
makes systematic use of geometric tools from nonlinear
arXiv:2009.09050v1 [physics.ao-ph] 18 Sep 2020
2
dynamics to frame vortices objectively.2,4,8,10–17,19,31 We
will specifically apply three methods which define coher-
ent Lagrangian vortex boundaries as material loops that
(i) defy stretching,13,14 (ii) resist diffusion,16 and (iii)
whose elements rotate evenly,15 respectively. The fluid
enclosed by such loops can be transported for long dis-
tances without noticeable dispersion.36,37
The rest of the paper is organized as follows. In the
next section we briefly review the formal definition of
each of the above coherent Lagrangian vortex notions.
In Sec. III we present a technology that enables framing
vortex genesis and apocalypse robustly and with preci-
sion. Section IV presents the data (satellite altimetry) on
which our assessments of the life cycle of LCRs are ap-
plied. It also presents numerical details of the implemen-
tation of the vortex detection methods, and introduces
the databases which deliver Eulerian assessments of the
“birth” and “decease” dates of LCRs, which are used for
reference. The results of our study are presented in Sec-
tion V. Finally, concluding remarks are offered in Section
VI.
II. COHERENT LAGRANGIAN RING DETECTION
Consider
Ft
t0:x07→ x(t;x0, t0),(1)
the flow map resulting from integrating a two-
dimensional incompressible velocity field, namely,
v(x, t) = ψ(x, t)(xR2and tRas stated above),
where ψdenotes sea-surface height. If the pressure gra-
dient force is exclusively due to changes in the SSH field,
the latter is given by g1(x, t), where gis gravity and
fis the Coriolis parameter, assuming a quasigeostrophic
balance.
A. Null-geodesic (NG) rings
Following Haller and Beron-Vera 13 ,14 we aim to iden-
tify fluid regions enclosed by exceptional material loops
that defy the typical exponential stretching experienced by
generic material loops in turbulent flows. This is achieved
by detecting loops with small annular neighborhoods ex-
hibiting no leading-order variation in averaged material
stretching.
These considerations lead to a variational problem
whose solutions are loops such that any of their subsets
are stretched by the same factor λ > 0under advection
by the flow from t0to t0+Tfor some T. The time-t0
positions of such uniformly λ-stretching material loops
turn out to be limit cycles of one of the following two
bidirectional vector or line fields:
η±
λ(x0):=sλ2(x0)λ2
λ2(x0)λ1(x0)ξ1(x0)±sλ2λ1(x0)
λ2(x0)λ1(x0)ξ2(x0),(2)
where λ1(x0)< λ2< λ2(x0). Here, {λi(x0)}and
{ξi(x0)}satisfying
0< λ1(x0)1
λ2(x0)<1,hξi(x0), ξj(x0)i=δij ,(3)
i, j = 1,2, are eigenvalues and (orientationless) nor-
malized eigenvectors, respectively, of the Cauchy–Green
(strain) tensor,
Ct0+T
t0(x0):= DFt0+T
t0(x0)>DFt0+T
t0(x0).(4)
The tensor field Ct0+T
t0(x0)objectively measures material
deformation over the time interval [t0, t0+T]. Limit cy-
cles of (2) or λ-loops either grow or shrink under changes
in λ, forming smooth annular regions of non-intersecting
loops. The outermost member of such a band of material
loops is observed physically as the boundary of a coher-
ent Lagrangian ring. The λ-loops can also be interpreted
as so-called null-geodesics of the indefinite tensor field
Ct0+T
t0(x0)λId, which is why we also refer to them as
null-geodesic (or NG)rings.
B. Diffusion-barrier (DB) rings
Another recent approach to coherent vortices in geo-
physical flows has been put forward in Haller, Karrasch,
and Kogelbauer 16 . In this case one aims at identify-
ing fluid regions that defy diffusive transport across their
boundaries. Note that by flow invariance, any fluid re-
gion has vanishing advective transport across its bound-
ary. In turbulent flows, however, a generic fluid region
has massive diffusive leakage through its boundary, which
correlates with the typical exponential stretching of the
latter.
A technical challenge is that the diffusive flux of a vir-
tual diffusive tracer through a material surface over a fi-
nite time interval [t0, t0+T]depends on the concrete evo-
lution of the scalar under the advection–diffusion equa-
tion. In the limit of vanishing diffusion, however, Haller,
Karrasch, and Kogelbauer 16 show that the diffusive flux
through a material surface can be determined by the gra-
dient of the tracer at the initial time instance and a ten-
sor field Tthat can be interpreted as the time average of
the diffusion tensor field along a fluid trajectory. In the
3
case of isotropic diffusion, this reduces to the average of
inverse Cauchy–Green tensors,
T(x0):=1
TZt0+T
t0Ct0+t
t0(x0)1dt. (5)
Searching for material loops with small annular neigh-
borhoods exhibiting no leading-order variation in the
vanishing-diffusivity approximation of diffusive transport
leads to a variational problem whose solutions are limit
cycles of (2), where now λiand ξiare, respectively, eigen-
values and eigenvectors of the time-averaged Cauchy–
Green tensor
¯
Ct0+T
t0(x0):=1
TZt0+T
t0
Ct0+t
t0(x0) dt. (6)
This simple tensor structure assumes isotropic diffusion
and an incompressible fluid flow. We refer to vortices
obtained by this methodology as diffusion-barrier (or
DB)rings. Due to the mathematical similarity to the
geodesic ring approach, we may use the same computa-
tional method as for NG rings, simply by replacing Ct0+T
t0
by ¯
Ct0+T
t0.
C. Rotationally-coherent (RC) rings
In our analysis, we also employ a third methodology,
which was developed by Haller et al. 15. It puts less em-
phasis on specific properties of the boundary (like stretch-
ing or diffusive flux) of coherent vortices, but highlights
that coherent vortices are often associated with concen-
trated regions of high vorticity. Defining vortices in terms
of vorticity has a long tradition,27,38 but in unsteady fluid
flows it comes with a number of drawbacks, one of which
is the lack of objectivity.10 In Haller et al. 15, the au-
thors overcome these challenges by showing that the La-
grangian averaged vorticity deviation (or LAVD) field
LAVDt0+T
t0(x0):=Zt0+T
t0ωFt
t0(x0), t¯ω(t)dt, (7)
is an objective scalar field. Here, ω(x, t)is the vorticity
of the fluid velocity at position xand time t, and ¯ω(t)
is the vorticity at time taveraged over the tracked fluid
bulk. In this framework, vortex centers are identified
as maxima of the LAVD field, and vortex boundaries as
outermost convex LAVD-level curves surrounding LAVD
maxima. Because loops are composed of fluid elements
that complete the same total material rotation relative
to the mean material rotation of the whole fluid mass,
we will refer to the vortices as rotationally-coherent (or
RC )rings. In practice, the convexity requirement is re-
laxed, using a “tolerable” convexity deficiency.15 In con-
trast to the two previously described methods, the LAVD
approach therefore does not address vortex boundaries
directly (say, via a variational approach), but deduces
them as level-set features of the objective LAVD field.
III. GENESIS AND APOCALYPSE
Our main goal is to study genesis, evolution, and apoc-
alypse of LCRs from an objective, Lagrangian point of
view. Since there is no generally agreed definition of the
concept of a coherent vortex, we need to employ several
proposed methods to rule out the possibility that the re-
sults are biased by the specific choice of method.
To determine the “birth” or the “decease” of a coher-
ent Lagrangian vortex in a robust fashion, we need to
eliminate a couple of potentially biasing issues. First, as
stated above, we include several Lagrangian methodolo-
gies in our study. Second, we want to avoid potential
sensitivities due to implementation details (such as al-
gorithm or parameter choices). Recall that Lagrangian
approaches choose not only an initial time instance t0,
but also a flow horizon T. A naive approach to the de-
termination of the decease of a coherent vortex would
be to simply take the maximum of t0+Tfor which
a Lagrangian method detects a coherent vortex, where
t0and Tare taken from a range of reasonable values.
While this approach yields a definite answer, it may be
totally inconsistent with other computations run for dif-
ferent choices of t0and T. For instance, if a Lagrangian
computation detects a coherent vortex over the time in-
terval [t0, t0+T], it should also detect a vortex over the
time interval [t0+δt, (t0+δt)+(Tδt)] = [t0+δt, t0+T]
for small |δt|, if t0+Twas really the date of breakdown.
In order to make our predictions statistically more ro-
bust and prove internal consistency, we employ the fol-
lowing approach. First, we run Lagrangian simulations
on a temporal double grid as follows. We roll the initial
time instance t0over a time window roughly covering
the time interval of vortex existence, which we seek to
determine. For each t0, we progress Tin 30-day steps
as long as the Lagrangian method successfully detects a
coherent vortex. Thus, we obtain for each t0alife ex-
pectancy Tmax (t0), which is the maximum Tfor which a
Lagrangian simulation starting at t0successfully detected
a coherent vortex.
Ideally, we would like to see the following Tmax(t0)
pattern. Assume a coherent Lagrangian vortex breaks
down on day 200, counted from day 0. Then for t0= 0
the longest successful vortex detection should yield a
Tmax(0) = 180 d. Similarly, for t0= 5,10,15,20 we
should get a Tmax = 180 d. From t0= 25 on, however,
we should start seeing Tmax dropping down to 150 days,
because for T= 180 days, the Lagrangian flow horizon
reaches beyond the vortex breakdown. As a consequence,
we would like to see a wedge-shaped Tmax(t0)distribu-
tion, which would indicate that all Lagrangian coherence
assessments predict the breakdown consistently, though
slightly smeared out regarding the exact date. If encoun-
tered, such a consistent prediction of breakdown would
arguably remove the possibility of degenerate results. To
summarize, in an ideal case, a Lagrangian simulation of
the lifespan of a coherent vortex would therefore start
with a large Tmax-value, which consistently decreases as
4
t0progresses forward in time.
It turns out that in many cases such wedge-shaped
Tmax(t0)-patterns can be indeed observed, sometimes
with astonishing clarity, given the finitely resolved ve-
locity fields and the complexity of the Lagrangian calcu-
lation and vortex detection algorithms.
IV. DATA AND NUMERICAL IMPLEMENTATION
The SSH field from which the flow is derived is given
daily on a 0.25-resolution longitude–latitude grid. This
represents an absolute dynamic topography, i.e., the sum
of a (steady) mean dynamic topography and the (tran-
sient) altimetric SSH anomaly. The mean dynamic to-
pography is constructed from satellite altimetry data,
in-situ measurements, and a geoid model.30 The SSH
anomaly is referenced to a 20-yr (1993–2012) mean, ob-
tained from the combined processing of data collected by
altimeters on the constellation of available satellites.22
Computationally, we detect NG and DB rings
from the altimetry-derived flow by the method de-
vised in Karrasch, Huhn, and Haller 19 and recently
extended for large-scale computations in Karrasch
and Schilling,20 as implemented in the package
CoherentStructures.jl. It is written in the modern
programming language Julia, and is freely available
from https://github.com/CoherentStructures/
CoherentStructures.jl. In turn, RC ring detection,
computationally much more straightforward, was imple-
mented in MATLAB R
as described in Beron-Vera et al. 2
(a software tool, not employed here, is freely distributed
from https://github.com/LCSETH/Lagrangian-
Averaged-Vorticity-Deviation-LAVD). The spacing
of the grid of initial trajectory positions in all cases is
set to 0.1 km as in earlier Lagrangian coherence analyses
involving altimetry data.3,4,28 Trajectory integration is
carried out using adaptive time-stepping schemes and
involves cubic interpolation of the velocity field data.
NG and DB rings are sought with stretching parameter
(λ) ranging over the interval λ[1 ±0.5]. Recall
that λ= 1 NG-vortices reassume their arc length at
t0+T.13 When the flow is incompressible (as is the
case of the altimetry-derived flow) such λ= 1 vortices
stand out as the most coherent of all as their boundaries
resist stretching while preserving the area they enclose.
Following Haller et al. 15 the convexity deficiency is set
to 103for the RC ring extractions.
As our interest is in LCRs, we concentrate on the time
intervals on which these were identified by Horizon Ma-
rine, Inc. as part of the EddyWatch R
program. This
program identifies LCRs as regions instantaneously filled
with altimetric SSH streamlines.23 The EddyWatch R
program has been naming LCRs and reporting their birth
and decease dates since 1984. Our analysis is restricted to
the period 2001–2013, long enough to robustly test the-
oretical expectations and for the results to be useful in
applications such as ocean circulation model validation.
Alternative assessments of the genesis and apocalypse
of LCRs are obtained from the AVISO+ Mesoscale Eddy
Trajectory Atlas Product, which is also computed from
the Eulerian footprints left by the eddies on the global
altimetric SSH field.5
V. RESULTS
We begin by testing our expectation that Lagrangian
life expectancy (Tmax ) should decrease with increasing
screening time (t0), exhibiting a wedge shape. We do this
by focusing on LCR Kraken, so named by EddyWatch R
and recently subjected to a Lagrangian coherence study.3
In that study the authors characterized Kraken as an NG
ring using altimetry data. Furthermore, they presented
support for their characterization by analyzing indepen-
dent data, namely, satellite-derived color (Chl concen-
tration) and trajectories from satellite-tracked drifting
buoys. This rules out the possibility that LCR Kraken is
an artifact of the satellite altimetric dataset, thereby con-
stituting a solid benchmark for testing our expectation.
The authors of the aforementioned study estimated a La-
grangian lifetime for Kraken of about 200 d, but framing
the genesis and apocalypse of the ring with precision was
beyond the scope of their work.
The top panel of Fig. 1 shows Tmax(t0)for Kraken
based on NG (red), DB (solid black), and RC (dashed
black) coherence assessments. First note that the NG
and DB assessments are largely consistent, producing a
wide-base Tmax(t0)wedge with height decreasing with in-
creasing t0in addition to a short, less well-defined wedge
prior to it. This becomes very evident when compared to
the RC assessment, which produces intermittent wedge-
like Tmax(t0)on various short t0-intervals. We have ob-
served that this intermittency is typical, rather than ex-
ceptional, for the RC assessment. Thus we consider NG
coherent Lagrangian vortex detection, which in general
produces nearly identical results as DB vortex detection,
in the genesis and apocalypse assessments that follow.
Indicated in the top panel of Fig. 1 (with a vertical
dashed line) is our estimate of the birth date of LCR
Kraken,t0=18/May/2013, and three estimates of its
decease date, to wit, t0=12/Jan/2014, 21/Feb/2014,
and 21/Apr/2014. The birth date corresponds to the
t0marking the leftmost end of the Tmax(t0)wedge with
the longest base (highlighted). Our first decease date
estimate (d1) is given by the birth date plus its life ex-
pectancy, set by the height of the wedge or 12/Jan/2014
18/May/2013 = 239 d. The second decease date esti-
mate (d2) is given by the t0marking the rightmost end of
the wedge, which is 40-d longer than its life expectancy.
Our third decease date estimate (d3) is given by the sec-
ond decease date estimate plus the height of the wedge at
its rightmost end, namely, 21/Apr/2014 21/Feb/2014
= 59 d.
The bottom panel of the Fig. 1 shows, in orange, LCR
Kraken on its estimated birth date (b), and its advected
5
FIG. 1. (top panel) For Gulf of Mexico’s Loop Current ring (LCR) Kraken, life expectancy as a function of screening time
according null-geodesic (NG), diffusion-barrier (DB), and rotationally-coherent (RC) Lagrangian vortex assessments. Indicated
are the birth date of the ring (b), and three decease date estimates (d1, d2, and d3); cf. text for details. Birth and decease dates
according to EddyWatch R
and AVISO+ Eulerian vortex assessments are indicated with open and filled triangles, respectively.
(bottom panel) Based on the NG assessment, LCR Kraken on birth date and the three decease date estimates.
image under the altimetry-derived flow on the first (d1),
second (d2), and third (d3) decease date estimates. Over-
laid on the later on the second and third decease date
estimated are (shown in cyan) the NG vortex extracted
on the second decease date estimate and its advected im-
age on the third decease date estimate. Note that on the
first and second decease date estimates Kraken does not
show any noticeable signs of outward filamentation. On
the third decease date estimate most of the original fluid
mass enclosed by the ring boundary exhibits a coherent
aspect. Evidently, the first and second decease date esti-
mates are too conservative, so it is reasonable to take the
third one as the most meaningful decease date estimate
of the three. We will refer to it as the decease date.
Indicated by open and filled triangles in the abscissa
of the Tmax(t0)plot in the top panel of the Fig. 1 are the
Eulerian assessment of birth and decease dates of Kraken
by EddyWatch R
, and AVISO+, respectively. EddyWatch R
overestimates the decease date by about 180 d, while
AVISO+, underestimates it somewhat, by 19 d. To eval-
uate the performance of Eulerian vortex detection in as-
sessing the birth date of Kraken an additional analysis is
needed.
The results from such an analysis are presented in
Fig. 2, which shows the same as in Fig. 2 but as ob-
tained from applying all the Lagrangian vortex detec-
tion methods backward in time, i.e., with T < 0, around
the Kraken’s decease date. The top panel of the fig-
ure shows (now) |Tmax|as a function of screening time
t0. Note that the NG and DB coherence assessments
produce single wide-base |Tmax |(t0)wedges with height
decreasing with decreasing t0, nearly indistinguishable
from one another. As in the forward-time analysis, the
RC assessment shows intermittent wedge-like |Tmax|(t0)
on various short t0-intervals, suggesting a much shorter
life expectancy than observed in reality. Thus we turn
our attention to the NG (or DB) assessment. This
produces a backward-time birth date estimate on t0=
02/Apr/2014, and three backward-time decease date es-
timates on 06/Apr/203, 08/Apr/2013, and 10/Mar/2013.
In forward time, 02/Apr/2014 represents a decease date
estimate, which is only 19-d earlier than the decease
date obtained above from forward-time computation.
The largest discrepancy between forward- and backward-
time assessments are seen for the birth date. Follow-
ing the forward-time computation reasoning above, the
backward-time computations sets it 296 d earlier, on t0=
10/Mar/2013. This lies 21 and about 30 d later and ear-
lier than to the AVISO+ and EddyWatch R
, assessments,
respectively, which are instantaneous, i.e., they do not
depend on the time direction on which they are made.
The backward-time estimate of Kraken ’s decease date
can be taken to represent a forward-time conception date
estimate for the ring. This is quite evident from the in-
spection of the bottom panel of Fig. 2, which shows (in
orange) LCR Kraken as extracted from backward-time
6
FIG. 2. As in Fig. 1, but for assessments made in backward time.
FIG. 3. (left panel) LCR Kraken on birth date overlaid in
orange on the forward-advected image of the ring extracted
from backward-time computation on 08/Apr/2013 (third
backward-time decease estimate). (right panel) Backward-
advected image of the fluid region indicated in cyan in the
left panel.
computation on the backward-time birth date estimate,
and images thereof under the backward-time flow on the
three backward-time decease date estimates. On the last
two decease date estimates, these are shown overlaid on
the ring extracted from backward-time computation on
the second backward-time decease date estimate and its
backward-advected image on the third backward-time de-
cease date estimate, which represents, as noted above, a
conception date for Kraken.
Indeed, the fluid region indicated in cyan contains at
all times the fluid region indicated in orange. Thus the
orange fluid is composed of the same fluid as the cyan
fluid. Furthermore, the cyan fluid, which can be traced
back into the Caribbean Sea, ends up forming the fluid
that forms Kraken on its (forward-time) birth date. This
is illustrated in left panel of Fig. 3, which shows Kraken
(in orange) as obtained from forward-time computation
on its birth date overlaid on the forward-advected image
of the cyan fluid. In the right panel we show a backward-
advected image of the cyan fluid that reveals its origin in
the Caribbean Sea. The supplementary material includes
an animation (Mov. 1) illustrating the full life cycle of
LCR Kraken.
We note that the need of introducing the conception
date estimate could have been anticipated from the in-
spection of the forward-time assessment. Note the short
wedge-like T(t0)before the long-base wedge in Fig. 1 em-
ployed in assessing genesis and apocalypse. In a way the
presence of that short wedge-like T(t0)was already insin-
uating that coherence was building sometime before the
ring was declared born. Similar disconnected wedge-like
T(t0)patterns may be observed past the main wedge, as
can be seen in Fig. 4, which shows the same as Fig. 1 but
for LCR Yankee. These wedge-like patterns, however, are
not signs of the ring’s “resurrection,” but actually corre-
spond to vortex structures in general unrelated or only
partly related to the ring in question.
We illustrate the above in Fig. 5. Note the appear-
ance of two short wedge-like patterns past the main T(t0)
wedge. Let us concentrate attention on the earliest of the
two short wedges. We infer a forward-time birth date
is 30/Jun/2007, and two forward-time decease dates on
18/Sep/2017 and 17/Oct/2017. The bottom panels of
the figure show how these characterize the life cycle of a
vortex, newly formed and composed only in part of LCR
Yankee’s fluid. This is evident by comparing the position
of the vortex on birth date and first decease date esti-
7
FIG. 4. As in Fig. 1, but for LCR Yankee.
mates and their forward-advected images with those of
Yankee as revealed on 13/Sep/2006. The EddyWatch R
and AVISO+ nonobjective Eulerian vortex assessments
fail to frame this, largely overestimating the decease date
of Yankee.
We compile in Table I the objective Lagrangian es-
timates of conception, birth, and decease dates for all
LCRs during the altimetry era. An entry of table left
blank means that the ring could not be classified as coher-
ent. The objective estimates are compared with nonob-
jective Eulerian estimates by EddyWatch R
and AVISO+,
with the former only providing the month of the year
when birth and decease take place. Note the tendency of
the Eulerian assessments to overestimate the birth and
decease dates of the rings. Indeed, the Eulerian assess-
ments cannot distinguish between conception and birth.
They typically keep track of vortex-like structures past
the decease date of the rings, which, present around that
date, are not formed by the fluid mass contained by the
rings. Moreover, the Eulerian assessments classify as co-
herent, and even name, rings that turn out not to be
so. The supplementary material includes two anima-
tions supporting these conclusions for features classified
as LCRs Quick (Mov. 2) and Sargassum (Mov. 3) by the
EddyWatch R
and AVISO+ nonobjective Eulerian assess-
ments.
We conclude by highlighting the disparities between
the objective Lagrangian and nonobjective Eulerian as-
sessments of the genesis and apocalypse of LCR rings
in Fig. 6. The figure presents, as a function of time
over 2001–2011, the difference (in d) between NG and
E (dots), and NG and A (circles) assessments of birth
(left) and decease (right) dates. The differences can be
quite large (up to 1 yr!) with Eulerian assessments, which
in general underestimate the birth dates of the rings and
overestimate their decease dates.
VI. CONCLUSIONS
We have carried out an objective (i.e., observer-
independent) Lagrangian assessment of the life cycle of
the Loop Current rings (LCRs) in Gulf of Mexico de-
tected from satellite altimetry. Three objective methods
of coherent Lagrangian vortex detection were considered
here. These reveal material vortices with boundaries
that defy stretching or diffusion, and whose elements
rotate evenly. A modest technology advance was per-
formed which enabled framing vortex genesis and apoc-
alypse with robustness and precision. We found that
the stretching- and diffusion-defying assessments produce
consistent results. These in general showed large discrep-
ancies with Eulerian assessments which identify vortices
with regions instantaneously filled with streamlines of the
SSH field. The Eulerian assessments were found inca-
pable to distinguish conception from birth of the rings.
They also tended to track past their decease dates vortex-
like features unrelated to the rings in question. The even-
8
FIG. 5. As in Fig. 4, with a focus on the highlighted piece of the Tmax(t0)plot.
Birth date Decease date
Ring Conception date Ob jective EddyWatch R
AVISO+ Objective EddyWatch R
AVISO+
Nansen 27/02/01 13/03/01 04/01 18/03/00 31/07/01 12/01 04/01/02
Odesa 02/07/01 31/07/01 09/01 23/03/01 09/10/01 12/01 05/11/01
Pelagic 12/01 07/09/01 05/02 17/02/02
Quick 03/02 19/02/02 04/03 05/05/03
Sargasum 05/03 30/03/02 12/03 16/01/04
Titanic 10/11/03 09/12/03 10/03 01/08/03 21/06/04 10/04 13/11/04
Ulises 22/11/04 11/12/04 05/04 07/12/03 15/05/05 09/05 07/10/05
Extreeme 03/06 13/01/06 09/06 13/11/06
Yankee 04/09/06 13/09/06 07/06 26/04/06 31/05/07 01/08 03/01/08
Zorro 10/03/07 24/03/07 04/07 26/08/06 16/08/07 08/07 17/08/07
Albert 02/11/07 01/12/07 11/07 21/03/07 05/03/08 05/08 23/04/08
Cameron 20/06/08 20/06/08 07/08 15/06/08 10/02/09 05/09 18/06/09
Darwin 29/01/09 02/02/09 12/08 13/02/08 25/10/09 11/09 10/11/09
Ekman 11/04/09 09/07/09 07/09 04/02/09 20/05/10 03/11 28/08/10
Hadal 22/06/11 11/07/11 08/11 23/11/10 23/12/11 03/12 26/12/11
Icarus 08/10/11 22/10/11 11/11 19/07/11 11/09/12 02/13 04/10/12
Jumbo 28/04/12 28/04/12 06/12 18/04/12 21/08/12 02/13 10/11/12
Kraken 10/03/13 08/04/13 04/13 17/02/13 02/04/14 10/14 10/04/14
TABLE I. Objective Lagrangian estimates of conception, birth, and decease dates of Loop Current rings in the Gulf of Mexico
identified from satellite altimetry over 2001–2013 along with nonobjective Eulerian estimates of birth and decease dates.
rotation assessment, which is vorticity-based, was found to be quite unstable, suggesting life expectancies much
9
FIG. 6. As a function of time, difference (in d) between objec-
tive Lagrangian and nonobjective Eulerian estimates of birth
(left panel) and death (right panel) dates. Dots (resp., circles)
involve EddyWatch R
, (resp., AVISO+) assessments.
shorter than those produced by all other assessments.
The inconsistency found adds to the list of known issues
of LAVD-based vortex statistics,35 including high sensi-
tivity with respect to the choice of computational param-
eter values. Our results can find value in drawing unam-
biguous evaluations of material transport and should rep-
resent a solid metric for ocean circulation model bench-
marking.
SUPPLEMENTARY MATERIAL
The supplementary material contains three anima-
tions. Movie 1 illustrates the complete life cycle of LCR
Kraken as assessed objectively using NG-ring detection.
Movies 2 and 3 show sequences of advected images of
the features classified as LCRs Quick and Sargassum, re-
spectively, by the EddyWatch R
and AVISO+ nonobjective
Eulerian assessments.
AUTHOR’S CONTRIBUTIONS
All authors contributed equally to this work.
ACKNOWLEDGMENTS
This work was initiated during the “Escuela interdisci-
plinaria de transporte en fluidos geofísicos: de los remoli-
nos oceánicos a los agujeros negros,” Facultad de Cien-
cias Exactas y Naturales, Universidad de Buenos Aires,
5–16/Dec/2016. Support from Centro Latinoamericano
de Formación Interdisciplinaria is sincerely appreciated.
This work was supported by CONACyT–SENER (Mex-
ico) under Grant No. 201441 (FAC, FJBV) as part
of the Consorcio de Investigación del Golfo de Méx-
ico (CIGoM). FAC thanks CICESE (Mexico) for allow-
ing him to use their computer facilities throughout the
CIGoM project.
AIP PUBLISHING DATA SHARING POLICY
The gridded multimission altimeter products were pro-
duced by SSALTO/DUACS and distributed by AVISO+
(https://www.aviso.altimetry.fr/), with support
from CNES. The Mesoscale Eddy Trajectory Atlas Prod-
uct was produced by SSALTO/DUACS and distributed
by AVISO+ (https://www.aviso.altimetry.fr/) with
support from CNES, in collaboration with Oregon State
University with support from NASA. The EddyWatch R
data are available from Horizon Marine, Inc.’s website at
https://www.horizonmarine.com/.
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