Genesis, evolution, and apocalypse of Loop Current rings
GGenesis, evolution, and apocalypse of Loop Current rings
F. Andrade-Canto, a) D. Karrasch, b) and F. J. Beron-Vera c) Instituto de Investigaciones Oceanológicas, Universidad Autónoma de Baja California, Ensenda, Baja California,México Technische Universität München, Zentrum Mathematik, Garching bei München,Germany 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 Mexicoby applying three objective (i.e., observer-independent) coherent Lagrangian vortex detection methods onvelocities derived from satellite altimetry measurements of sea-surface height (SSH). The methods revealmaterial 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 withprecision. We find that the stretching- and diffusion-withstanding assessments produce consistent results,which show large discrepancies with Eulerian assessments that identify vortices with regions instantaneouslyfilled with streamlines of the SSH field. The even-rotation assessment, which is vorticity-based, is found tobe 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 Currentitself and the anticyclonic (counterclockwise) mesoscale(100–200-km radius) vortices, so-called rings, shed fromit, strongly influences the circulation, thermodynamics,and biogeochemistry of the Gulf of Mexico (GoM). As important long-range carriers, westward-propagatingLoop Current rings (LCRs) provide a potential mech-anism for the remote connectivity between the GoM’swestern basin and the Caribbean Sea.
In par-ticular, bringing warm Caribbean Sea water within, theheat content of the LCRs is believed to be as significantas for LCRs to promote the intensification of tropical cy-clones (hurricanes). On the other hand, as regions ofstrong flow shear, LCRs may be capable of producingstructural damage on offshore oil drilling rigs. For allthese reasons, LCRs are routinely monitored. LCRs leave footprints in satellite altimetric sea-surfaceheight (SSH) maps, so sharp that the routine detec-tion of LCRs consists in the identification of regionsfilled with closed streamlines of the SSH field assum-ing a geostrophic balance, a practice widely followed inoceanography. However, this eddy detection approachuses instantaneous Eulerian information to reach long-term conclusions about fluid (i.e., Lagrangian) transport,which are invariably surrounded by uncertainty due tothe unsteady nature of the underlying flow. At the heartof the issue with Eulerian eddy diagnostics of this type istheir dependence on the observer viewpoint: they give a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] different results for observers that rotate and translaterelative to one another. The issue is most easily graspedby bringing up, one more time as we believe is central yetwidely overlooked, the example first discussed by Haller and thereafter by others. Consider the exact so-lution to the Navier–Stokes equation in two spatial di-mensions: v ( x, t ) = ( x sin 4 t + x (2 + cos 4 t ) , x (cos 4 t − − x sin 4 t ) , where x = ( x , x ) ∈ R denotes posi-tion and t ∈ R is time. The flow streamlines are closedat 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 ( x , x ) (cid:55)→ (¯ x , ¯ x ) = ( x cos 2 t − x sin 2 t, x sin 2 t + x cos 2 t ) , un-der which v ( x, t ) (cid:55)→ (¯ x , ¯ x ) ≡ ¯ v (¯ x ) . In other words, thede-facto oceanographic eddy detection diagnostic mis-classifies the flow as vortex-like. The ¯ x -frame is spe-cial inasmuch the flow in this frame is steady, and thusflow streamlines and fluid trajectories coincide. Henceshort-term exposition pictures of the velocity field by theobserver in the ¯ x -frame determine the long-term fate offluid particles. The only additional observation to havein mind to fully determine the Lagrangian motion is thatthe observer in the ¯ x -frame rotates (at angular speed ).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. Thus one can neverbe sure which observer gives the right answer when thede-facto oceanographic eddy detection diagnostic isapplied. As a consequence, neither false positives norfalse negatives can be ruled out, and 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. Thiswill be done in line with recent but growing work thatmakes systematic use of geometric tools from nonlinear a r X i v : . [ phy s i c s . a o - ph ] S e p dynamics to frame vortices objectively. Wewill specifically apply three methods which define coher-ent Lagrangian vortex boundaries as material loops that(i) defy stretching, (ii) resist diffusion, and (iii)whose elements rotate evenly, respectively. The fluidenclosed by such loops can be transported for long dis-tances without noticeable dispersion. The rest of the paper is organized as follows. In thenext section we briefly review the formal definition ofeach of the above coherent Lagrangian vortex notions.In Sec. III we present a technology that enables framingvortex genesis and apocalypse robustly and with preci-sion. Section IV presents the data (satellite altimetry) onwhich 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 introducesthe databases which deliver Eulerian assessments of the“birth” and “decease” dates of LCRs, which are used forreference. The results of our study are presented in Sec-tion V. Finally, concluding remarks are offered in SectionVI.
II. COHERENT LAGRANGIAN RING DETECTION
Consider F tt : x (cid:55)→ x ( t ; x , t ) , (1) the flow map resulting from integrating a two-dimensional incompressible velocity field, namely, v ( x, t ) = ∇ ⊥ ψ ( x, t ) ( x ∈ R and t ∈ R as stated above),where ψ denotes sea-surface height. If the pressure gra-dient force is exclusively due to changes in the SSH field,the latter is given by g − f ψ ( x, t ) , where g is gravity and f is the Coriolis parameter, assuming a quasigeostrophicbalance. A. Null-geodesic (NG) rings
Following Haller and Beron-Vera we aim to iden-tify fluid regions enclosed by exceptional material loopsthat defy the typical exponential stretching experienced bygeneric material loops in turbulent flows . This is achievedby detecting loops with small annular neighborhoods ex-hibiting no leading-order variation in averaged materialstretching.These considerations lead to a variational problemwhose solutions are loops such that any of their subsetsare stretched by the same factor λ > under advectionby the flow from t to t + T for some T . The time- t positions of such uniformly λ -stretching material loopsturn out to be limit cycles of one of the following twobidirectional vector or line fields: η ± λ ( x ) := (cid:115) λ ( x ) − λ λ ( x ) − λ ( x ) ξ ( x ) ± (cid:115) λ − λ ( x ) λ ( x ) − λ ( x ) ξ ( x ) , (2)where λ ( x ) < λ < λ ( x ) . Here, { λ i ( x ) } and { ξ i ( x ) } satisfying < λ ( x ) ≡ λ ( x ) < , (cid:104) ξ i ( x ) , ξ j ( x ) (cid:105) = δ ij , (3) i, j = 1 , , are eigenvalues and (orientationless) nor-malized eigenvectors, respectively, of the Cauchy–Green(strain) tensor, C t + Tt ( x ) := D F t + Tt ( x ) (cid:62) D F t + Tt ( x ) . (4)The tensor field C t + Tt ( x ) objectively measures materialdeformation over the time interval [ t , t + T ] . Limit cy-cles of (2) or λ -loops either grow or shrink under changesin λ , forming smooth annular regions of non-intersectingloops. The outermost member of such a band of materialloops is observed physically as the boundary of a coher-ent Lagrangian ring . The λ -loops can also be interpretedas so-called null-geodesics of the indefinite tensor field C t + Tt ( x ) − λ Id , 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 . In this case one aims at identify-ing fluid regions that defy diffusive transport across theirboundaries . Note that by flow invariance, any fluid re-gion has vanishing advective transport across its bound-ary. In turbulent flows, however, a generic fluid regionhas massive diffusive leakage through its boundary, whichcorrelates with the typical exponential stretching of thelatter.A technical challenge is that the diffusive flux of a vir-tual diffusive tracer through a material surface over a fi-nite time interval [ t , t + 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 show that the diffusive fluxthrough a material surface can be determined by the gra-dient of the tracer at the initial time instance and a ten-sor field T that can be interpreted as the time average ofthe diffusion tensor field along a fluid trajectory. In thecase of isotropic diffusion, this reduces to the average ofinverse Cauchy–Green tensors, T ( x ) := 1 T (cid:90) t + Tt (cid:0) C t + tt ( x ) (cid:1) − d t. (5)Searching for material loops with small annular neigh-borhoods exhibiting no leading-order variation in thevanishing-diffusivity approximation of diffusive transportleads to a variational problem whose solutions are limitcycles of (2), where now λ i and ξ i are, respectively, eigen-values and eigenvectors of the time-averaged Cauchy–Green tensor ¯ C t + Tt ( x ) := 1 T (cid:90) t + Tt C t + tt ( x ) d t. (6)This simple tensor structure assumes isotropic diffusionand an incompressible fluid flow. We refer to vorticesobtained by this methodology as diffusion-barrier (or DB ) rings . Due to the mathematical similarity to thegeodesic ring approach, we may use the same computa-tional method as for NG rings, simply by replacing C t + Tt by ¯ C t + Tt . C. Rotationally-coherent (RC) rings
In our analysis, we also employ a third methodology,which was developed by Haller et al. . It puts less em-phasis on specific properties of the boundary (like stretch-ing or diffusive flux) of coherent vortices, but highlightsthat coherent vortices are often associated with concen-trated regions of high vorticity . Defining vortices in termsof vorticity has a long tradition, but in unsteady fluidflows it comes with a number of drawbacks, one of whichis the lack of objectivity. In Haller et al. , the au-thors overcome these challenges by showing that the La-grangian averaged vorticity deviation (or
LAVD ) field
LAVD t + Tt ( x ) := (cid:90) t + Tt (cid:12)(cid:12) ω (cid:0) F tt ( x ) , t (cid:1) − ¯ ω ( t ) (cid:12)(cid:12) d t, (7)is an objective scalar field. Here, ω ( x, t ) is the vorticityof the fluid velocity at position x and time t , and ¯ ω ( t ) is the vorticity at time t averaged over the tracked fluidbulk. In this framework, vortex centers are identifiedas maxima of the LAVD field, and vortex boundaries asoutermost convex LAVD-level curves surrounding LAVDmaxima. Because loops are composed of fluid elementsthat complete the same total material rotation relativeto 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. In con-trast to the two previously described methods, the LAVDapproach therefore does not address vortex boundariesdirectly (say, via a variational approach), but deducesthem 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 ofview. Since there is no generally agreed definition of theconcept of a coherent vortex, we need to employ severalproposed 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 toeliminate a couple of potentially biasing issues. First, asstated above, we include several Lagrangian methodolo-gies in our study. Second, we want to avoid potentialsensitivities due to implementation details (such as al-gorithm or parameter choices). Recall that Lagrangianapproaches choose not only an initial time instance t ,but also a flow horizon T . A naive approach to the de-termination of the decease of a coherent vortex wouldbe to simply take the maximum of t + T for whicha Lagrangian method detects a coherent vortex, where t and T are taken from a range of reasonable values.While this approach yields a definite answer, it may betotally inconsistent with other computations run for dif-ferent choices of t and T . For instance, if a Lagrangiancomputation detects a coherent vortex over the time in-terval [ t , t + T ] , it should also detect a vortex over thetime interval [ t + δt, ( t + δt ) + ( T − δt )] = [ t + δt, t + T ] for small | δt | , if t + T was 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 simulationson a temporal double grid as follows. We roll the initialtime instance t over a time window roughly coveringthe time interval of vortex existence, which we seek todetermine. For each t , we progress T in 30-day stepsas long as the Lagrangian method successfully detects acoherent vortex. Thus, we obtain for each t a life ex-pectancy T max ( t ) , which is the maximum T for which aLagrangian simulation starting at t successfully detecteda coherent vortex.Ideally, we would like to see the following T max ( t ) pattern. Assume a coherent Lagrangian vortex breaksdown on day 200, counted from day 0. Then for t = 0 the longest successful vortex detection should yield a T max (0) = 180 d. Similarly, for t = 5 , , , weshould get a T max = 180 d. From t = 25 on, however,we should start seeing T max dropping down to days,because for T = 180 days, the Lagrangian flow horizonreaches beyond the vortex breakdown. As a consequence,we would like to see a wedge-shaped T max ( t ) distribu-tion, which would indicate that all Lagrangian coherenceassessments predict the breakdown consistently, thoughslightly smeared out regarding the exact date. If encoun-tered, such a consistent prediction of breakdown wouldarguably remove the possibility of degenerate results. Tosummarize, in an ideal case, a Lagrangian simulation ofthe lifespan of a coherent vortex would therefore startwith a large T max -value, which consistently decreases as t progresses forward in time.It turns out that in many cases such wedge-shaped T max ( t ) -patterns can be indeed observed, sometimeswith 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 givendaily on a 0.25 ◦ -resolution longitude–latitude grid. Thisrepresents an absolute dynamic topography, i.e., the sumof 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. The SSHanomaly is referenced to a 20-yr (1993–2012) mean, ob-tained from the combined processing of data collected byaltimeters on the constellation of available satellites. Computationally, we detect NG and DB ringsfrom the altimetry-derived flow by the method de-vised in Karrasch, Huhn, and Haller and recentlyextended for large-scale computations in Karraschand Schilling, as implemented in the package CoherentStructures.jl . It is written in the modernprogramming language
Julia , and is freely availablefrom https://github.com/CoherentStructures/CoherentStructures.jl . In turn, RC ring detection,computationally much more straightforward, was imple-mented in
MATLAB R (cid:13) as described in Beron-Vera et al. (a software tool, not employed here, is freely distributedfrom https://github.com/LCSETH/Lagrangian-Averaged-Vorticity-Deviation-LAVD ). The spacingof the grid of initial trajectory positions in all cases isset to 0.1 km as in earlier Lagrangian coherence analysesinvolving altimetry data. Trajectory integration iscarried out using adaptive time-stepping schemes andinvolves cubic interpolation of the velocity field data.NG and DB rings are sought with stretching parameter( λ ) ranging over the interval λ ∈ [1 ± . . Recallthat λ = 1 NG-vortices reassume their arc length at t + T . When the flow is incompressible (as is thecase of the altimetry-derived flow) such λ = 1 vorticesstand out as the most coherent of all as their boundariesresist stretching while preserving the area they enclose.Following Haller et al. the convexity deficiency is setto − for the RC ring extractions.As our interest is in LCRs, we concentrate on the timeintervals on which these were identified by Horizon Ma-rine, Inc. as part of the EddyWatch R (cid:13) program. Thisprogram identifies LCRs as regions instantaneously filledwith altimetric SSH streamlines. The
EddyWatch R (cid:13) program has been naming LCRs and reporting their birthand decease dates since 1984. Our analysis is restricted tothe period 2001–2013, long enough to robustly test the-oretical expectations and for the results to be useful inapplications such as ocean circulation model validation. Alternative assessments of the genesis and apocalypseof LCRs are obtained from the AVISO+
Mesoscale EddyTrajectory Atlas Product, which is also computed fromthe Eulerian footprints left by the eddies on the globalaltimetric SSH field. V. RESULTS
We begin by testing our expectation that Lagrangianlife expectancy ( T max ) should decrease with increasingscreening time ( t ), exhibiting a wedge shape. We do thisby focusing on LCR Kraken , so named by
EddyWatch R (cid:13) and recently subjected to a Lagrangian coherence study. In that study the authors characterized
Kraken as an NGring using altimetry data. Furthermore, they presentedsupport for their characterization by analyzing indepen-dent data, namely, satellite-derived color (Chl concen-tration) and trajectories from satellite-tracked driftingbuoys. This rules out the possibility that LCR
Kraken isan 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 framingthe genesis and apocalypse of the ring with precision wasbeyond the scope of their work.The top panel of Fig. 1 shows T max ( t ) for Kraken based on NG (red), DB (solid black), and RC (dashedblack) coherence assessments. First note that the NGand DB assessments are largely consistent, producing awide-base T max ( t ) wedge with height decreasing with in-creasing t in addition to a short, less well-defined wedgeprior to it. This becomes very evident when compared tothe RC assessment, which produces intermittent wedge-like T max ( t ) on various short t -intervals. We have ob-served that this intermittency is typical, rather than ex-ceptional, for the RC assessment. Thus we consider NGcoherent Lagrangian vortex detection, which in generalproduces 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 verticaldashed line) is our estimate of the birth date of LCR Kraken , t = t = t marking the leftmost end of the T max ( t ) wedge withthe longest base (highlighted). Our first decease dateestimate (d1) is given by the birth date plus its life ex-pectancy, set by the height of the wedge or 12/Jan/2014 − = 239 d. The second decease date esti-mate (d2) is given by the t marking the rightmost end ofthe 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 atits rightmost end, namely, 21/Apr/2014 − = 59 d.The bottom panel of the Fig. 1 shows, in orange, LCR Kraken on its estimated birth date (b), and its advected
FIG. 1. (top panel) For Gulf of Mexico’s Loop Current ring (LCR)
Kraken , life expectancy as a function of screening timeaccording null-geodesic (NG), diffusion-barrier (DB), and rotationally-coherent (RC) Lagrangian vortex assessments. Indicatedare the birth date of the ring (b), and three decease date estimates (d1, d2, and d3); cf. text for details. Birth and decease datesaccording to
EddyWatch R (cid:13) 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 dateestimated are (shown in cyan) the NG vortex extractedon the second decease date estimate and its advected im-age on the third decease date estimate. Note that on thefirst and second decease date estimates
Kraken does notshow any noticeable signs of outward filamentation. Onthe third decease date estimate most of the original fluidmass enclosed by the ring boundary exhibits a coherentaspect. Evidently, the first and second decease date esti-mates are too conservative, so it is reasonable to take thethird one as the most meaningful decease date estimateof the three. We will refer to it as the decease date.Indicated by open and filled triangles in the abscissaof the T max ( t ) plot in the top panel of the Fig. 1 are theEulerian assessment of birth and decease dates of Kraken by EddyWatch R (cid:13) , and AVISO+ , respectively.
EddyWatch R (cid:13) 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 isneeded.The results from such an analysis are presented inFig. 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 < , aroundthe Kraken ’s decease date. The top panel of the fig- ure shows (now) | T max | as a function of screening time t . Note that the NG and DB coherence assessmentsproduce single wide-base | T max | ( t ) wedges with heightdecreasing with decreasing t , nearly indistinguishablefrom one another. As in the forward-time analysis, theRC assessment shows intermittent wedge-like | T max | ( t ) on various short t -intervals, suggesting a much shorterlife expectancy than observed in reality. Thus we turnour attention to the NG (or DB) assessment. Thisproduces a backward-time birth date estimate on t = t = AVISO+ and
EddyWatch R (cid:13) , assessments,respectively, which are instantaneous, i.e., they do notdepend on the time direction on which they are made.The backward-time estimate of Kraken ’s decease datecan be taken to represent a forward-time conception dateestimate for the ring. This is quite evident from the in-spection of the bottom panel of Fig. 2, which shows (inorange) LCR
Kraken as extracted from backward-time
FIG. 2. As in Fig. 1, but for assessments made in backward time.FIG. 3. (left panel) LCR
Kraken on birth date overlaid inorange on the forward-advected image of the ring extractedfrom backward-time computation on 08/Apr/2013 (thirdbackward-time decease estimate). (right panel) Backward-advected image of the fluid region indicated in cyan in theleft panel. computation on the backward-time birth date estimate,and images thereof under the backward-time flow on thethree backward-time decease date estimates. On the lasttwo decease date estimates, these are shown overlaid onthe ring extracted from backward-time computation onthe second backward-time decease date estimate and itsbackward-advected image on the third backward-time de-cease date estimate, which represents, as noted above, aconception date for
Kraken .Indeed, the fluid region indicated in cyan contains atall times the fluid region indicated in orange. Thus theorange fluid is composed of the same fluid as the cyanfluid. Furthermore, the cyan fluid, which can be tracedback into the Caribbean Sea, ends up forming the fluidthat 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 computationon its birth date overlaid on the forward-advected imageof the cyan fluid. In the right panel we show a backward-advected image of the cyan fluid that reveals its origin inthe Caribbean Sea. The supplementary material includesan animation (Mov. 1) illustrating the full life cycle ofLCR
Kraken .We note that the need of introducing the conceptiondate estimate could have been anticipated from the in-spection of the forward-time assessment. Note the shortwedge-like T ( t ) before the long-base wedge in Fig. 1 em-ployed in assessing genesis and apocalypse. In a way thepresence of that short wedge-like T ( t ) was already insin-uating that coherence was building sometime before thering was declared born. Similar disconnected wedge-like T ( t ) patterns may be observed past the main wedge, ascan be seen in Fig. 4, which shows the same as Fig. 1 butfor LCR Yankee . These wedge-like patterns, however, arenot signs of the ring’s “resurrection,” but actually corre-spond to vortex structures in general unrelated or onlypartly 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 ( t ) wedge. Let us concentrate attention on the earliest of thetwo short wedges. We infer a forward-time birth dateis 30/Jun/2007, and two forward-time decease dates on18/Sep/2017 and 17/Oct/2017. The bottom panels ofthe figure show how these characterize the life cycle of avortex, newly formed and composed only in part of LCR Yankee ’s fluid. This is evident by comparing the positionof the vortex on birth date and first decease date esti-
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 (cid:13) and AVISO+ nonobjective Eulerian vortex assessmentsfail to frame this, largely overestimating the decease dateof
Yankee .We compile in Table I the objective Lagrangian es-timates of conception, birth, and decease dates for allLCRs during the altimetry era. An entry of table leftblank means that the ring could not be classified as coher-ent. The objective estimates are compared with nonob-jective Eulerian estimates by
EddyWatch R (cid:13) and AVISO+ ,with the former only providing the month of the yearwhen birth and decease take place. Note the tendency ofthe Eulerian assessments to overestimate the birth anddecease dates of the rings. Indeed, the Eulerian assess-ments cannot distinguish between conception and birth.They typically keep track of vortex-like structures pastthe decease date of the rings, which, present around thatdate, are not formed by the fluid mass contained by therings. Moreover, the Eulerian assessments classify as co-herent, and even name, rings that turn out not to beso. The supplementary material includes two anima-tions supporting these conclusions for features classifiedas LCRs
Quick (Mov. 2) and
Sargassum (Mov. 3) by the
EddyWatch R (cid:13) and AVISO+ nonobjective Eulerian assess-ments.We conclude by highlighting the disparities betweenthe 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 timeover 2001–2011, the difference (in d) between NG andE (dots), and NG and A (circles) assessments of birth(left) and decease (right) dates. The differences can bequite large (up to 1 yr!) with Eulerian assessments, whichin general underestimate the birth dates of the rings andoverestimate their decease dates.
VI. CONCLUSIONS
We have carried out an objective (i.e., observer-independent) Lagrangian assessment of the life cycle ofthe Loop Current rings (LCRs) in Gulf of Mexico de-tected from satellite altimetry. Three objective methodsof coherent Lagrangian vortex detection were consideredhere. These reveal material vortices with boundariesthat defy stretching or diffusion, and whose elementsrotate evenly. A modest technology advance was per-formed which enabled framing vortex genesis and apoc-alypse with robustness and precision. We found thatthe stretching- and diffusion-defying assessments produceconsistent results. These in general showed large discrep-ancies with Eulerian assessments which identify vorticeswith regions instantaneously filled with streamlines of theSSH 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-
FIG. 5. As in Fig. 4, with a focus on the highlighted piece of the T max ( t ) plot.Birth date Decease dateRing Conception date Objective EddyWatch R (cid:13) AVISO+
Objective
EddyWatch R (cid:13) AVISO+
Nansen
Odesa
Pelagic
Quick
Sargasum
Titanic
Ulises
Extreeme
Yankee
Zorro
Albert
Cameron
Darwin
Ekman
Hadal
Icarus
Jumbo
Kraken rotation assessment, which is vorticity-based, was found to be quite unstable, suggesting life expectancies much
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 (cid:13) , (resp., AVISO+ ) assessments. shorter than those produced by all other assessments.The inconsistency found adds to the list of known issuesof LAVD-based vortex statistics, 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 ofthe features classified as LCRs
Quick and
Sargassum , re-spectively, by the
EddyWatch R (cid:13) and AVISO+ nonobjectiveEulerian 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 Latinoamericanode 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 theCIGoM project.
AIP PUBLISHING DATA SHARING POLICY
The gridded multimission altimeter products were pro-duced by SSALTO/DUACS and distributed by
AVISO+ ( ), with supportfrom CNES. The Mesoscale Eddy Trajectory Atlas Prod-uct was produced by SSALTO/DUACS and distributedby AVISO+ ( ) withsupport from CNES, in collaboration with Oregon StateUniversity with support from NASA. The EddyWatch R (cid:13) data are available from Horizon Marine, Inc.’s website at . Andrade-Canto, F., Sheinbaum, J., and Sansón, L. Z., “A La-grangian approach to the Loop Current eddy separation,” Nonlin.Processes Geophys. , 85–96 (2013). Beron-Vera, F. J., Hadjighasem, A., Xia, Q., Olascoaga, M. J.,and Haller, G., “Coherent Lagrangian swirls among submesoscalemotions,” Proc. Natl. Acad. Sci. U.S.A. , 18251–18256 (2019). Beron-Vera, F. J., Olascaoaga, M. J., Wang, Y., Triñanes, J., andPérez-Brunius, P., “Enduring Lagrangian coherence of a LoopCurrent ring assessed using independent observations,” ScientificReports , 11275 (2018). Beron-Vera, F. J., Wang, Y., Olascoaga, M. J., Goni, G. J.,and Haller, G., “Objective detection of oceanic eddies and theAgulhas leakage,” J. Phys. Oceanogr. , 1426–1438 (2013). Chelton, D. B., Schlax, M. G., and Samelson, R. M., “Globalobservations of nonlinear mesoscale eddies,” Prog. Oceanogr. ,167–216 (2011). Donohue, K., Watts, D., Hamilton, P., Leben, R., and Kennelly,M., “Loop current eddy formation and baroclinic instability,” Dy-namics of Atmospheres and Oceans , 195–216 (2016), the LoopCurrent Dynamics Experiment. Forristall, G. Z., Schaudt, K. J., and Cooper, C. K., “Evolutionand kinematics of a Loop Current eddy in the Gulf of Mexicoduring 1985,” J. Geophys. Res. , 2173–2184 (1992). Hadjighasem, A., Farazmand, M., Blazevski, D., Froyland, G.,and Haller, G., “A critical comparison of Lagrangian methodsfor coherent structure detection,” Chaos , 053104 (2017). Haller, G., “An objective definition of a vortex,” J. Fluid Mech. , 1–26 (2005). Haller, G., “Lagrangian coherent structures,” Ann. Rev. FluidMech. , 137–162 (2015). Haller, G., “Climate, black holes and vorticity: How on Earthare they related?” SIAM News , 1–2 (2016). Haller, G.and Beron-Vera, F. J., “Geodesic theory of transportbarriers in two-dimensional flows,” Physica D , 1680–1702(2012). Haller, G.and Beron-Vera, F. J., “Coherent Lagrangian vortices:The black holes of turbulence,” J. Fluid Mech. , R4 (2013). Haller, G.and Beron-Vera, F. J., “Addendum to ‘Coherent La-grangian vortices: The black holes of turbulence’,” J. Fluid Mech. , R3 (2014). Haller, G., Hadjighasem, A., Farazmand, M., and Huhn, F.,“Defining coherent vortices objectively from the vorticity,” J.Fluid Mech. , 136–173 (2016). Haller, G., Karrasch, D., and Kogelbauer, F., “Material barriersto diffusive and stochastic transport,” Proceedings of the Na-tional Academy of Sciences , 9074–9079 (2018). Haller, G., Karrasch, D., and Kogelbauer, F., “Barriers to thetransport of diffusive scalars in compressible flows,” SIAM Jour-nal on Applied Dynamical Systems , 85–123 (2020). Kantha, L., “Empirical models of the loop current eddy detach-ment/separation time in the gulf of mexico,” Journal of Wa-terway, Port, Coastal, and Ocean Engineering , 04014001(2014). Karrasch, D., Huhn, F., and Haller, G., “Automated detection ofcoherent Lagrangian vortices in two-dimensional unsteady flows,”Proc. Royal Soc. A , 20140639 (2014). Karrasch, D.and Schilling, N., “Fast and robust computation ofcoherent lagrangian vortices on very large two-dimensional do-mains,” The SMAI journal of computational mathematics , 101–124 (2020). Kuznetsov, L., Toner, M., Kirwan, A. D., Jones, C. K. R. T.,Kantha, L. H., and Choi, J., “The Loop Current and adjacentrings delineated by Lagrangian analysis of the near-surface flow,”J. Mar. Res. , 405–429 (2002). Le Traon, P. Y., Nadal, F., and Ducet, N., “An improved map-ping method of multisatellite altimeter data,” J. Atmos. OceanicTechnol. , 522–534 (1998). Leben, R. R., “Altimeter-derived loop current metrics,” in
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