oceanic turbulence and stochastic models from subsurface...

1884 VOLUME 34J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

q 2004 American Meteorological Society

Oceanic Turbulence and Stochastic Models from Subsurface Lagrangian Data for theNorthwest Atlantic Ocean

MILENA VENEZIANI

Division of Meteorology and Physical Oceanography, Rosenstiel School of Marine and Atmospheric Science, University of Miami,Miami, Florida

ANNALISA GRIFFA

Division of Meteorology and Physical Oceanography, Rosenstiel School of Marine and Atmospheric Science, University of Miami,Miami, Florida, and IOF/CNR, La Spezia, Italy

ANDY M. REYNOLDS

Silsoe Research Institute, Silsoe, Bedford, United Kingdom

ARTHUR J. MARIANO

Division of Meteorology and Physical Oceanography, Rosenstiel School of Marine and Atmospheric Science, University of Miami,Miami, Florida

(Manuscript received 9 May 2003, in final form 13 January 2004)

ABSTRACT

The historical dataset provided by 700-m acoustically tracked floats is analyzed in different regions of thenorthwestern Atlantic Ocean. The goal is to characterize the main properties of the mesoscale turbulence andto explore Lagrangian stochastic models capable of describing them. The data analysis is carried out mostly interms of Lagrangian velocity autocovariance and cross-covariance functions. In the Gulf Stream recirculationand extension regions, the autocovariances and cross covariances exhibit significant oscillatory patterns on timescales comparable to the Lagrangian decorrelation time scale. They are indicative of sub- and superdiffusivebehaviors in the mean spreading of water particles. The main result of the paper is that the properties of Lagrangiandata can be considered as a superposition of two different regimes associated with looping and nonloopingtrajectories and that both regimes can be parameterized using a simple first-order Lagrangian stochastic modelwith spin parameter V. The spin couples the zonal and meridional velocity components, reproducing the effectsof rotating coherent structures such as vortices and mesoscale eddies. It is considered as a random parameterwhose probability distribution is approximately bimodal, reflecting the distribution of loopers (finite V) andnonloopers (zero V). This simple model is found to be very effective in reproducing the statistical propertiesof the data.

1. Introduction

Lagrangian data provide direct information on oceancurrents in terms of velocity and transport. Extensivehistorical datasets are available today, both at and belowthe ocean surface [useful information and lists of ref-erences can be found online at the following Web sites:Drifting Buoy Data Assembly Center (Atlantic Ocean-ographic and Meteorological Laboratory, Miami, Flor-ida), http://www.aom1.noaa.gov/phod/dac/dac.html;Subsurface Float Data Center (Woods Hole, Massachu-setts), http://wfdac.whoi.edu], and their statistical anal-

Corresponding author address: Milena Veneziani, RSMAS/MPO4600 Rickenbacker Causeway, Miami, FL 33149.E-mail: [email protected]

yses have contributed significantly to improving ourknowledge of the ocean circulation (e.g., Patterson1985; Davis 1991; Owens 1991; Swenson and Niiler1996; Bauer et al. 1998; Lavender et al. 2000; Poulain2001; Fratantoni 2001; Bower et al. 2002). Lagrangianinstruments are particularly suitable for transport andmixing studies because they move approximately withthe ocean currents on (sub) mesoscale time scales (Davis1991; Riser 1982). In a number of previous investiga-tions (e.g., Freeland et al. 1975; Riser and Rossby 1983;Krauss and Boning 1987; Zhang et al. 2001; Bauer etal. 2002), mostly focused on the ocean surface, La-grangian data have been used to characterize transportproperties related to mesoscale turbulence and to testpossible parameterization methods. Turbulence param-

AUGUST 2004 1885V E N E Z I A N I E T A L .

eterizations are crucial for climate applications of oceangeneral circulation models (OGCMs), since the eddyfield is usually not resolved and the results rely heavilyupon the correct representation of the eddy effects onthe simulated large-scale circulation.

The simplest possible parameterization of turbulenttransport is through the commonly used ‘‘eddy diffu-sivity,’’ which neglects correlations between successivevelocity fluctuations. This approach cannot be used tomodel transport processes on time scales shorter thanthe integral Lagrangian time scale, and over which ve-locity fluctuations remain significantly correlated. Fur-thermore, the eddy-diffusivity approximation is notreadily extended to include the influence of non-Gauss-ian velocity statistics and particle inertia (e.g., Tennekesand Lumley 1972). An alternative class of parameteri-zation methods that are easily generalizable and have awide applicability is based on Markovian stochasticmodels describing the motion of single particles in tur-bulent flows (e.g., Risken 1989). These models havebeen widely used in applications in physics of the at-mosphere (e.g., Thomson 1987) and in oceanography(e.g., Griffa 1996), and within this context they are oftenreferred to as Lagrangian stochastic (LS) models (Saw-ford 1991). A hierarchy of LS models can be derivedby increasing the order of the joint Markovian variables,providing a means of reproducing increasingly morecomplex statistical properties.

Previous results on the analysis of surface Lagrangiandata (e.g., Krauss and Boning 1987; Poulain and Niiler1989; Figueroa and Olson 1989; Falco et al. 2000) sug-gest that, away from strong shear currents and from theequatorial regions, turbulent processes can be modeledby using a simple, one-dimensional, first-order LS mod-el. This implies that the two velocity components of thefluctuation field are independent (one dimensionality),and that the position x and the velocity u9 evolve jointlyas a Markovian process (first-order property). Lagrang-ian velocity autocovariance functions are exponential,with an e-folding scale equal to the decorrelation timescale TL, while diffusion processes are ballistic at shorttimes (i.e., the mean particle dispersion1 ^x2& is pro-portional to t2, for t K TL) and diffusive at longer times(^x2& } t, for t k TL). These models describe ‘‘simple’’situations of well-developed turbulence in the absenceof coherent structures.

Such a description does not apply when the eddydynamics is characterized by the presence of coherentstructures. This is the case, for instance, in the equatorialPacific (Bauer et al. 1998, 2002), where the flow isdominated by wave activity. Lagrangian data in this areagive rise to strongly oscillating autocovariance func-tions, and to an intermediate time regime of anomalousdiffusion, primarily subdiffusive, due to the trapping ofparticles in the wave field. Coherent structures are also

1 The angle brackets used here, and elsewhere in this paper, denotesome kind of ensemble average.

expected to play an important role at midlatitudes whereinstabilities of strong shear currents (e.g., the westernboundary currents extensions) lead to the formation ofintense vortices such as Gulf Stream rings.

Suitable parameterizations of turbulent processes inthe presence of coherent structures are still under debate.Berloff and McWilliams (2002), on the basis of nu-merical trajectories simulated in a double-gyre model,have suggested the use of one-dimensional, higher-orderLS models. A possible application of this kind of modelto the equatorial Pacific data is discussed in Bauer etal. (2002). Alternatively, Reynolds (2002a) has consid-ered the use of two-dimensional, first-order LS models.In this case, the coherent structure effects are modeledthrough the correlation between the eddy velocity com-ponents, which determines a rotational motion of thetrajectories. Other authors have suggested that quasi-geostrophic or two-dimensional turbulence might beconsidered as a suitable paradigm for the interpretationof oceanic turbulence (Elhmaidi et al. 1993; Provenzale1999; Reynolds 2002b), so that appropriate parameter-ization models should take into account non-Gaussian-ities and the existence of at least two different regimes,one associated with the strong coherent vortices and theother related to the more quiescent background (Pas-quero et al. 2001). Non-Gaussianities have indeed beenseen in the analysis of subsurface float data in the NorthAtlantic (Bracco et al. 2000), as well as in numericaltrajectories simulated by a high-resolution model (Brac-co et al. 2003). Furthermore, Richardson (1993) hasidentified two distinct classes of trajectories in the sub-surface float data. He has classified the floats as ‘‘loop-ers’’ and ‘‘nonloopers,’’ recognizing that many trajec-tories experienced marked looping behaviors due to thetrapping effects of strong vortices.

A definitive answer to which model and which phys-ical interpretation of mesoscale turbulence is the mostappropriate cannot be given at this stage because notenough evidence from the data is available. The needfor data analyses that specifically test the various hy-potheses provides motivation for the present work.

In this paper, the question of how to characterize andparameterize oceanic turbulence in the presence of co-herent structures is addressed by considering the his-torical dataset of subsurface floats at 700-m depth in thenorthwest Atlantic. The goal is twofold. On the one side,we aim at better understanding oceanic turbulence, and,by characterizing it in terms of stochastic models, weverify hypotheses on its degree of complexity and onthe nature of the coherent structures. For simplicity, wefocus on specific geographical subregions that are ‘‘qua-si homogeneous’’ in terms of dynamics and statistics,so that the applications can be considered in good ap-proximation as ‘‘local.’’ On the other side, the work willalso provide indications on which type of transport mod-el is most suitable for coarse-grain GCMs to simulatesubgrid-scale transport by unresolved (or partially re-solved) mesoscale eddies. Our results can be considered


as a first step in this direction. The actual use of LSmodels in OGCMs will involve ‘‘global’’ applications,where the spatial inhomogeneities will have to be in-cluded (Berloff and McWilliams 2002).

We concentrate on the regions of the Gulf Streamextension and recirculation because they are character-ized by the presence of strong vortices and rings asclearly illustrated by the large number of looping tra-jectories (Richardson 1993) and as seen in satellite im-ages (Brown et al. 1986). We focus on the statisticalproperties of the eddy turbulent field u9, estimated as aresidual from the mean flow U in each of the geograph-ical subregions mentioned above. After a preliminaryanalysis to verify that the properties of the eddy fieldare independent from the averaging process of the meanflow estimation, the u9 statistics are computed in termsof autocovariance and cross-covariance functions. Theanalyses are performed over the whole trajectory en-semble, and separately over the looper and nonloopersubsets. They are then used to identify suitable La-grangian stochastic models capable of describing thestatistics of observed covariability.

The paper is organized as follows. The first part pro-vides a background overview on LS models (section 2),while the second part is devoted to the Lagrangian dataanalysis. In section 3, information on the data and onthe methodology used to compute the mean flow is pro-vided. The results from the statistical analysis of theeddy field are shown in section 4, together with the testson Lagrangian stochastic model applicability. In section5, a discussion on the properties of the loopers andnonloopers populations is presented. A summary and abrief discussion on open questions and future devel-opments are provided in section 6.

2. Lagrangian stochastic models

Material transport in inhomogeneous, nonstationaryturbulence, typical of oceanic gyres, is most readily pre-dicted from the trajectories of particles simulated byLagrangian stochastic models (e.g., Griffa 1996). Thesimplest such model is the random walk or zeroth-orderLS model in which successive increments in the positionof a particle are taken to be uncorrelated and the particleposition is modeled as a Markovian process.

A time scale representative of the ‘‘energy contain-ing’’ scales of motions is introduced at first order. Inthese models, particle positions and velocities evolvejointly as a continuous Markovian process. This is jus-tifiable only when the Lagrangian acceleration autocor-relation function approaches a d function at the origin,corresponding to an uncorrelated component in the ac-celeration and hence to a Markov process (Sawford1991). At second order, particle positions, velocities,and accelerations evolve jointly as a Markovian process(Sawford 1991; Berloff and McWilliams 2002). Math-ematically, at least, this hierarchy of models can be ex-

tended to higher orders (Berloff and McWilliams 2002;Reynolds 2003a).

Notice that there is a correspondence between LSmodels, which are stochastic differential equations de-scribing particle motion in a turbulent flow, and Fokker–Planck (or Kolmogorov) equations, which are partialdifferential equations describing the particle probabilitydensity function (pdf ) in the appropriate space (e.g.,Risken 1989). For zeroth-order models, the appropriatespace is simply the physical space (x) and the Fokker–Planck equation corresponds to the commonly used ad-vection–diffusion equation. For first-order models, the(x, u9) space has to be considered, while for second-order models, the acceleration is also included (x, u9,A). In practical applications like the present one, the LSmodel formulation appears more natural than the Fok-ker–Planck equation approach because, aside from be-ing easier to implement, it provides a direct way to testthe model and compare it with Lagrangian observationsin terms of particle trajectories and autocorrelation func-tions.

First-order models are generally prescribed by

du9 5 a (u9, x, t)dt 1 b (u9, x, t)dj ,i i ij j (1)

where u9(t) is the Lagrangian turbulent velocity of aparticle (hereinafter the primes will be dropped for sim-plicity of notation), x(t) is the particle position along atrajectory, t is time, and dj is an incremental Weinerprocess with independent components, having 0 meanand variance dt.

One of the constraints that determine the precise formof the model is the well-mixed condition (Thomson1987). The well-mixed condition is equivalent to themost stringent criterion that has so far been establishedas distinguishing between well- and poorly formulatedmodels. Physically, it implies that a passive tracer uni-formly mixed over the full domain remains uniformlymixed at all times. Therefore, if, at some time t0, theprobability density distribution P of positions and ve-locities of passive tracers is proportional to PE, the Eu-lerian joint pdf of position and velocity (i.e., wellmixed), then, according to the well-mixed condition, Pmust remain proportional to PE at all later times t . t0.

For a given Eulerian flow, the deterministic terms ai

in the LS model can be constrained, but in general notdetermined uniquely, by the Eulerian statistics of theflow. This nonuniqueness can manifest itself as a ‘‘spin’’term, , that induces a mean rotation of the Lagrangiana9iturbulent velocity vector (Borgas et al. 1997; Sawford1999; Reynolds 2002a). Models having nonzero meanspin statistics are associated with spiraling particle tra-jectories, oscillatory velocity autocorrelation functions,and suppressed rates of turbulent dispersion for giventurbulent kinetic energies and turbulent kinetic energydissipation rates. These characteristics are the hallmarksof coherent flow structures such as eddies and vortices.The spin term therefore provides a means of incorpo-rating, into the modeling framework, the effects of en-


ergetic coherent structures seen in Lagrangian and sat-ellite data.

The simplest application of the well-mixed conditionis to isotropic, homogeneous, stationary, and incom-pressible turbulence with Gaussian velocity statistics,mean 0, and variance s 2. The simplest form of thecorresponding LS models arises when the spin term istaken to be linear in velocity, so that 5 e ijkV juk, wherea9ieijk is the antisymmetric unit tensor and V j are rotationalfrequencies. These models are given by

21du 5 2u T dt 1 e V u dt 1 b dj ,i i L ijk j k ij i (2)

where TL is the Lagrangian decorrelation time scale andbij 5 dij(2s 2/TL)1/2.

The parameters TL, V, and bij in Eq. (2) are consideredindependent from ui, so that the model is fully linear;that is, the stochastic increment is independent from thestate of the system. Nonlinearity has not been consideredhere because the correspondence between data and mod-el in our application does not support the employmentof a more complex, nonlinear formulation.

The Lagrangian velocities simulated through Eq. (2)are Gaussian with mean 0 and variance s 2 and, there-fore, exactly consistent with the prescribed Eulerian ve-locity statistics. For mesoscale ocean dynamics the two-dimensional form of this model can be considered(Reynolds 2002a), for which the horizontal velocitycomponents u and y are orthogonal to the axis of ro-tation, V3 (hereinafter simply referred to as V). Thespin term, which is absent in one-dimensional models(i.e., in models in which the velocity components evolveindependently from each other), allows for u and y tobe coupled during the entire flow evolution, and thecoupling is realized through the angular velocity V ofthe fluctuation field. This parameter is related to themean rotation ^ds& 5 ^u dy 2 y du& during a time in-crement dt, by (Sawford 1999)

^ds&V 5 , (3)

2dtEKE

where EKE is the eddy kinetic energy, equal to (^u2& 1^y2&)/2. The model represented by Eq. (2) has Lagrang-ian velocity autocorrelation and cross-correlation func-tions prescribed by

2t /TLR 5 R 5 e cos(Vt) anduu yy

2t /TLR 5 2R 5 e sin(Vt). (4)uy yu

The autocorrelations, Ruu and Ryy, have pronounced neg-ative lobes (i.e., the first negative lobe is more pro-nounced than the first positive lobe), and, as a conse-quence, transport is subdiffusive at intermediate times.This means that, for t ø TL, the associated mean particledispersion ^x2& is proportional to ta, with a , 1—thatis, it grows slower than in a typical diffusive process(for equal s 2 and TL) given by Taylor’s hypotheses (Tay-lor 1921).

A similar subdiffusive pattern in the velocity auto-

correlation functions can be produced by a one-dimen-sional, second-order model (Berloff and McWilliams2002). However, such a model cannot reproduce oscil-latory cross-correlation functions, Ruy and Ryu, of thekind represented in Eq. (4), because it does not takeinto account the correlation in time between orthogonalvelocity components. Furthermore, the one-dimensionalsecond-order model cannot produce spiraling trajecto-ries.

The two-dimensional model of Eq. (2) and its cor-responding statistics given by Eq. (4) are based on theassumption that the effects of eddy structures can berepresented by fixed parameters s 2, TL, and V. Recentnumerical simulations of particle trajectories (Reynolds2002a) and studies of dispersion processes in idealizedmodels of oceanic gyres (Berloff and McWilliams 2003)have suggested, however, that the particle diffusion canbe predicted more realistically by Lagrangian modelswhose parameters are random variables distributed ac-cording to a specified probability density function. Inparticular, Reynolds (2002a) found that an appropriatechoice of this probability distribution can reproducesubdiffusive and superdiffusive features in the Lagrang-ian autocorrelation functions.

In this paper, we investigate what kind of Lagrangianstochastic model and associated parameter distributionis most suitable to describe the observed float trajec-tories in the northwest Atlantic. The simplest distribu-tion capturing the coexistence of loopers and nonloopersis a bimodal distribution. A more general trimodalityhypothesis would be required if cyclonic and anticy-clonic loopers are to be distinguished.

3. Data and methods

a. The float dataset

The dataset analyzed in this paper is archived at theSubsurface Float Data Assembly Center (WFDAC) atWoods Hole. The data span the period from the late1970s to 1994 and are composed of a number of ex-periments whose details and related references are listedin Table 1.

Acoustically tracked isobaric floats in the northwest-ern Atlantic Ocean at the main thermocline level, thatis, at a nominal depth of 700 m, have been considered.We chose not to include the isopycnal float data becausewe did not intend to mix the effects of the isobaric andisopycnic observations within the statistical analyses.Besides, not enough isopycnal data were available fora possible comparison of statistical results from the twodatasets. Except for the isopycnical floats and the datafrom the Topographic Gulf Stream Experiment (TO-POGULF) (Ollitrault and Colin de Verdiere 2002a,b),the Lagrangian dataset analyzed in this paper is similarto that used for past investigations by a number of au-thors (e.g., Owens 1991; Richardson 1993; Rupolo etal. 1996; Bracco et al. 2000; LaCasce 2000).


TABLE 1. List of SOFAR/RAFOS float experiments in the northwest Atlantic from which the present statistical results were drawn.

Expt Reference

Abaco BasinEastern BasinGulf StreamGulf Stream RecirculationIberian Basin

Leaman and Vertes (1996)Spall et al. (1993)Schmitz et al. (1981)Owens (1984)Rees and Gmitrowicz (1989)

IFM Iberian BasinLDEMODENewfoundland BasinNorth Atlantic Tracer Release

Zenk et al. (1992)Rossby et al. (1986)Rossby et al. (1975)Schmitz (1985)Sundermeyer and Price (1998)

Northeast AtlanticPre-LDERingSite LSubduction 1991Topographic Gulf Stream

Klein and Mittelstaedt (1992)Riser and Rossby (1983)Cheney et al. (1976)Price et al. (1987)Sundermeyer and Price (1998)Ollitrault and Colin de Verdiere (2002a,b)

FIG. 1. (a) Spaghetti plot of the acoustically tracked isobaric floatsavailable in the northwestern Atlantic at 700-m depth. (b) Numberof daily observation per 18-square bin of the same dataset.

Float positions were mostly recorded daily. Whenevera different recording time step was used, the velocitieswere either interpolated or averaged to yield daily ob-servations. The ‘‘spaghetti’’ plot of the trajectories, to-gether with the number of observations per 18-squarebin is shown in Figs. 1a and 1b, respectively. The av-erage data density in the Gulf Stream extension is ø100days per bin, while it decreases to 50–60 days fartherdownstream. Higher concentrations, ranging between150 and 400 days per 18 3 18 box, are found southeastof the jet bifurcation and in the recirculation area south-west of the Gulf Stream.

b. Basin-scale statistics: Estimates of mean flow andeddy kinetic energy

The usual first task when analyzing Lagrangian datais to determine an accurate estimate of the mean flowU. The mean flow can be estimated in either an Eulerianor a Lagrangian framework. Following what has beenhistorically done (e.g., Davis 1991), and given that thenorthwest Atlantic can be characterized by regions withquasi-homogeneous statistical properties, we choose touse an Eulerian-based estimate for the mean flow (alsobecause the way to best estimate a Lagrangian-basedmean field remains an open question). The float veloc-ities are then averaged or interpolated to yield an Eu-lerian distribution in space and time of the mean cir-culation (‘‘pseudo-Eulerian’’ field). A commonly usedmethod is the ‘‘binning technique’’ (e.g., Poulain andNiiler 1989; Owens 1991), through which the float ve-locities are averaged over small spatial subregions (bins)and over a certain period of time. Alternative methodsfor computing the mean flow can be used, involvingobjective mapping (Davis 1998) or bicubic spline in-terpolation (Bauer et al. 1998). The ‘‘eddy’’ field isestimated as the residual fluctuation u with respect tothe estimated mean flow, and as such it includes dif-ferent kinds of fluctuation phenomena, like vortices due


FIG. 2. (a) Binned mean velocity field obtained by averaging the700-m float data over 18 3 18 bins. (b) Binned eddy kinetic energyfield and contours of the five subregions where detailed data analyseshave been carried out.

to flow instabilities, wave fluctuations, and possible sea-sonal-to-interannual variability.

In this work, the estimate of the mean flow was ac-complished through both the bin averaging and thespline interpolation technique. The first method was ap-plied at basin scale over the whole northwest Atlanticregion, while both methods have been applied and com-pared in the more detailed studies of the quasi-homo-geneous subregions (section 3c). The statistical resultsshown in section 4 are based on the spline-interpolatedmean field. In all cases, the average was performed overthe complete dataset in time, since there are not enoughdata to resolve nonstationarities at the seasonal or in-terannual level. This is of course a limitation of thepresent study, and many more data will be needed inthe future to significantly address nonstationary statis-tics.

The basin-scale investigation was carried out by firstchanging the bin size in order to test the sensitivity ofthe results. A 18 3 18 binning was finally chosen asgood tradeoff between the importance of resolving bothspatial shears of the mean flow and eddy scales on theorder of the internal Rossby radius of deformation, andthe necessity of keeping a high enough data density perbin to guarantee statistical significance of the results.The binned mean flow and the eddy kinetic energy(EKE) fields are presented in Figs. 2a and 2b, respec-tively, for bins with a number of independent measure-ments, n*, higher than 10. The value of n* was com-puted as nDt/2TL, where n is the total number of dailyobservations, Dt is the sampling interval and TL is as-sumed to be ø10 days (Riser and Rossby 1983; Owens1991).

The mean circulation (Fig. 2a) shows a strong sub-surface Gulf Stream system, with typical averaged ve-locities of 50 cm s21 beyond Cape Hatteras and 20–30cm s21 farther downstream. A North Atlantic Currentof 30–40 cm s21 is also present around 448N, 438W. Apronounced Gulf Stream recirculation is found between768 and 658W, together with recirculation gyres southof the stream, approximately centered at 368N, 628 and538W. These results are comparable to similar analysesperformed by Owens (1991).

Sources of errors for the mean flow estimates includemeasurements errors, inaccuracies associated with thefact that velocities are computed from the observed floatpositions, and sampling errors related to the finite num-ber of data available. The latter are by far the mostrelevant, and they can be quantified as (Riser and Ross-by 1983; Davis 1991)

2 1/2Er 5 (s /n*) ,U u (5)

where is equal to ^u2& (similarly, the subscript y is2s u

used for the meridional velocity component). The sam-pling error of the binned mean flow in Fig. 2a is higherin those regions of the ocean interior and eastern basinwhere velocities are weak and the data density is low.More quantitative error estimates will be given in sec-

tion 4 within the discussion of the regional analysis.Other error sources are the biases due to gradients inthe data spatial distribution and gradients in eddy dif-fusivities (Davis 1991; Freeland et al. 1975). They aretypically smaller than the sampling errors and are con-sidered negligible for the purposes of this work (Gar-raffo et al. 2001a).

The eddy kinetic energy distribution (Fig. 2b) showsthe highly energetic region of the Gulf Stream extension,with EKE ø 800 cm2 s22 around 388N, 658W. The restof the basin has much lower eddy variability, with typ-ical interior EKE ø 50 cm2 s22. The sampling error ofthe eddy kinetic energy field can be estimated as

1/24 4s 1 su yEr 5 . (6)EKE [ ]2(n* 2 1)


FIG. 3. Sample of (top), (middle) loopers and (bottom) nonloopersin the recirculation region RECW. Arrows along the trajectories areplotted every 5 days.

TABLE 2. Number of total Lagrangian data, nonlooping, and loopingtrajectory data in float days, for each region of the northwesternAtlantic Ocean. The loopers percentage is also in terms of days.

REG Tot No loop

Loop

Cycl Acycl Tot Loop (%)

RECWRECEGSWGSEAZ

22533797133911553554

19172738

979946

2134

127966283182

1195

209937727

225

3361059

360209

1420

1528271840

Our 18 3 18 binned EKE field has typical errors between20% and 30%.

c. Quasi-homogeneous regions

In Fig. 2b, superimposed on the distribution of eddykinetic energy, are the contours of five subregions wherewe have focused our study of the mesoscale turbulentfield. These areas are characterized by quasi-homoge-neous eddy kinetic energy levels, in order to identifyregions that are approximately statistically homoge-neous. The two areas named GSW and GSE (GulfStream west and east, respectively) are the most ener-getic, with eddy kinetic energy ranging between 300and 800 cm2 s22, and are located in the Gulf Streamextension between 698 and 448W. Regions RECW andRECE (Recirculation west and east) are the recirculationareas southwest and south of the jet axis. They arecharacterized by relatively weaker EKE levels (50–200cm2 s22). Last, region AZ (for the Azores Current) islocated southeast of the Gulf Stream bifurcation and haseddy energy lower than 50 cm2 s22.

In addition to the energy characterization, these re-gions also exhibit different dynamical regimes. GSWand GSE are dominated by the Gulf Stream jet, a highlycoherent and nonlinear structure that limits cross trans-port and mass exchanges. Float trajectories tend to beadvected out of the area within time scales shorter than30 days. The recirculation regions RECW and RECEare dominated by highly energetic eddies and GulfStream rings, because of the instability processes andnonlinear dynamics typical of this part of the ocean.Region AZ is also eddy dominated, but the characteristicenergy of the vortices is significantly lower than thatfound in the Gulf Stream recirculation area.

The direct effect of coherent structures, such as rings,submesoscale coherent vortices, and large-amplitudemeanders, is to produce two distinct categories of floattrajectories (Richardson 1993). The loopers are thequickly rotating floats that are trapped inside highlyenergetic eddies, while the nonloopers represent the restof trajectories, which experience little looping behav-iors, and are typically associated with the less energeticbackground flow. Richardson (1993) has introduced asimple criterion of defining a looper as a trajectory thatundergoes at least two consecutive loops in the samedirection. This same criterion has been adopted here asinitial identification method. Furthermore, in the fewcases in which a float stopped looping (i.e., when itstopped being influenced directly by the coherent struc-ture) the subsequent part of the trajectory has been con-sidered as nonlooper. A sample of looping and non-looping trajectories found in region RECW (GulfStream recirculation) is shown in Fig. 3.

A more quantitative method of looper identification,based on LS parameter estimation, has also been used(see section 5), in order to take into account loopingtrajectories that do not exhibit a clear spiraling behavior(because of temporary weakening of the vortex they areembedded in, or because they are influenced by strongbackground flows, e.g.).

The distribution of loopers and nonloopers resultingfrom this criterion, in number of float days, and for eachregion, is presented in Table 2. The highest percentageof looping floats is found in region AZ (southeast ofthe Gulf Stream bifurcation, slightly west of the Mid-Atlantic Ridge), where a number of trajectories wereembedded in fairly steady vortices. Some of the loopers


in GSW and GSE (Gulf Stream axis) are in meanders,and it is not clear whether they are merely followingthe jet and its local recirculation gyres or they weretrapped in Gulf Stream rings and were subsequentlyejected. The total number of float days in all the fivesubregions is 12 098. Out of these, 3384 days (28%)are from looping trajectories. The percentage is slightlyhigher than the 21% given by Richardson (1993), prob-ably because the more recent TOPOGULF experimentfloat data were not included in Richardson’s analysisand these floats contribute to the number of loopers inregion AZ.

d. Mean flow estimates in the quasi-homogeneousregions

In the subregions, the statistical analysis of the re-sidual velocity u has been carried out by computingautocovariance and cross-covariance functions (section4). Since u was determined by subtracting the meanflow U from the total Lagrangian velocities, an impor-tant prerequisite to the analysis is that the results areindependent of the averaging scales and of the methodused to compute U. This has been verified in a prelim-inary investigation, in which the two methods of binningand spline interpolation have been applied and com-pared in each region. Notice that, with respect to thebinning, the bicubic spline interpolation (Bauer et al.1998) has the advantage of providing a continuousspace-dependent mean flow, so that large-scale hori-zontal shears can be directly taken into account. Fur-thermore, Lagrangian statistics are expected to be eval-uated more accurately, since the separation betweenmean and eddy field is performed continuously for eachtrajectory point.

To test the robustness of the eddy statistics with re-spect to the U estimates, we have performed an exten-sive sensitivity investigation by varying the parametersthat characterize the spline-interpolated field (see ap-pendix A). It has been found that, for all the splineparameter choices, the eddy statistics tend to behaveasymptotically at high values of roughness r (see ap-pendix A for definitions), with the autocovariance andcross-covariance structures becoming independent fromthe specific value of r. A further comparison has beencarried out with the results obtained by using the binnedmean flow, showing that the same features of the eddyfield are reproduced. This suggests that the mean flowshear is correctly resolved, within the limits of the pres-ent float data coverage, and that the u statistics are ro-bust, independent from the specific estimates of U.

4. Lagrangian statistical analysis

In this section, the turbulent residual velocity u isanalyzed in the various subregions, computing auto-covariance and cross-covariance functions from theoverall Lagrangian dataset, and separately from the

looper and the nonlooper datasubsets. Both methods oflooper identification have been used, the Richardsoncriterion and the quantitative method described in sec-tion 5, although no noticeable differences in the resultshave been observed.

a. Gulf Stream recirculation region (RECW)

Region RECW (Fig. 2b) is located at the western edgeof the Gulf Stream recirculation and is dominated byan eastward recirculation mean flow of 2.7 6 0.9 cms21. The standard error is computed from Eq. (5) byusing the zonal variance and n* 5 187, obtained2s u

using an a posteriori choice of the Lagrangian time scaleequal to 6 days (see subsequent parameter estimationfor details). The overall eddy kinetic energy is 142.3 610.4 cm2 s22, where the sampling error is estimated fromEq. (6). The correspondent root-mean-square velocity,Vrms 5 (2EKE)1/2, is 16.7 cm s21, which indicates thatthe fluctuation field is significantly more energetic thanthe mean flow.

The majority of subsurface Lagrangian data presentin this region are from SOFAR floats released at 700-m nominal depth during the Site L Experiment (Priceet al. 1987), and in the Local Dynamics Experiment(LDE; Rossby et al. 1986). Most of the floats (80%)leave the region in less than 40 days, whereas some ofthem, typically the looping floats, remain trapped in thearea for 3–6 months. Anticyclonic looping trajectoriesdominate over the cyclonic ones in terms of days (Table2). They may be due to the presence of anticyclonicsubsurface warm lenses whose formation has not beencompletely clarified yet. Possible explanations involvethe separation of 188C water patches from the GulfStream (Brundage and Dugan 1986) and interactionswith the Corner Rise seamounts at 368N, 528W (Rich-ardson 1980).

The autocovariance and cross-covariance functionsfrom the overall data in RECW and from the separateddatasets of the nonloopers and anticyclonic loopingfloats are shown in Figs. 4 and 5, respectively. Theresults obtained from the cyclonic floats are too noisybecause of the small amount of cyclonic data and arenot displayed here. The 95% confidence limit (CL) isalso plotted. For the autocovariance case, this is esti-mated as 2s 2/(n*)1/2 (Priestley 1981). For the cross-covariance case, the 95% CL is computed as 2[( 2 2s su y

2 )/n*]1/2 (Sciremammano 1979), where is the2 2s suy uy

squared cross covariance at 0 time lag (^uy&).The overall autocovariance function (Fig. 4a) exhibits

a strong oscillation pattern, with a significant positivelobe that is more pronounced than the first negative lobe.This behavior is representative of ‘‘superdiffusion’’ pro-cesses because the associated eddy dispersion growsfaster than the typical diffusive spreading, at interme-diate times (^x2& } ta, with a . 1, for t ø TL). A similaroutcome was obtained by Berloff et al. (2002) in theirnumerical simulations of the mesoscale transport dis-


FIG. 4. Zonal (solid line) and meridional (dashed line) velocityautocovariance functions (cm2 s22) computed in the recirculation re-gion RECW from (a) the overall dataset and separately from (b) thenonloopers and (c) the anticyclonic looping trajectories. The dottedlines denote the 95% CL.

FIG. 5. Velocity cross-covariance functions (Ruy and Ry u, plottedas solid and dashed lines, respectively), computed in RECW from(a) the overall Lagrangian data and separately from (b) the nonloopersand (c) the anticyclonic loopers. The 95% CL is drawn again as dottedlines. The y-axis units are centimeters squared per second squared.

tribution in a double-gyre model. They analyzed theautocorrelation and dispersion functions in different re-gions of the western boundary current extension andsubtropical gyre, providing a complete description ofthe eddy variability for an idealized ocean. In particular,the velocity autocorrelation functions in their centralsubtropical gyre show superdiffusive behaviors that arequalitative very comparable with our results (see Fig.10 in Berloff et al. 2002). In a companion paper, Berloff

and McWilliams (2002) applied higher-order Lagrang-ian stochastic models to describe these anomalous fea-tures of the eddy field.

In the present work, we suggest that the ‘‘superdif-fusivity’’ arises from the superposition of the two dif-ferent eddy regimes described by the nonlooping andlooping floats. In terms of autocovariance functions, thisis clear in Fig. 4, where the nonloopers give rise to anexponentially decaying velocity autocovariance (Fig.


4b) that, when superposed onto the oscillatory auto-covariance function obtained from the anticyclonicfloats (Fig. 4c), yields the overall results shown in Fig.4a. The nonlooping trajectories are associated with aneddy kinetic energy of ø100 cm2 s22 (Vrms ø 14 cms21) and are characterized by a Lagrangian decorrelationtime scale TL ø 7 days. The anticyclonic loopers arerepresentative of a more energetic eddy field than thatdescribed by the nonloopers, exhibiting EKE levels onthe order of 400 cm2 s22 (Vrms ø 29 cm s21). They alsofeature a longer decorrelation scale ø12 days and anoscillation time scale Tw ø 10 days.

The coexistence of two separate eddy regimes is fur-ther shown by the results in terms of velocity cross-covariance functions. In this case, the pattern of theoverall statistics depicted in Fig. 5a indicates a strongcontribution of the oscillating cross covariances asso-ciated with the anticyclonic loopers (Fig. 5c), while thecross-covariance functions obtained from the nonloop-ing floats (Fig. 5b) are not significantly different fromzero.

The scenario can be summarized as follows. The non-loopers are representative of a background flow fieldwhose instantaneous velocities decorrelate exponential-ly with time, each component independently from eachother. The looping trajectories give rise to oscillationsin both the autocovariance and cross-covariance func-tions, indicating that they are affected by a rotating eddyvelocity field that is more energetic than the backgroundeddy regime.

The oscillatory pattern in the looper statistics resem-bles the behavior dictated by Eq. (4), suggesting thatthe eddy features can be reproduced by a two-dimen-sional, first-order Lagrangian stochastic model of thekind described by Eq. (2) with a finite value of spin,V. Similar oscillatory features in the autocovariancefunction are also predicted by a one-dimensional, sec-ond-order model that was applied by Berloff andMcWilliams (2002). This model reproduces the oscil-latory subdiffusive behavior in the autocovariances byintroducing appropriate time scales for the accelerationfield, but it does not exhibit oscillatory cross-covariancefunctions typical of rotating coherent structures. One-dimensional LS models are therefore not suitable to de-scribe our statistical results for the loopers in RECWand in the other regions of the northwestern Atlantic.

The two-dimensional model of Eq. (2) can be usedto interpret the nonlooper statistics as well, by adoptingV 5 0 in the model equations. Therefore, we hypoth-esize that the overall statistical behavior can be repre-sented by a two-dimensional, first-order LS model witha bimodal distribution of spin, that is, with distinct val-ues of spin associated with the nonlooping and loopingtrajectories. More generally, we should consider tri-modal distributions of spin, since we associate positive(negative) values of V to cyclonic (anticyclonic) loop-ers. However, each region is characterized by a domi-nant sign of angular velocity, and for this reason we

will mostly refer to the bimodality hypothesis. The prob-ability associated with each family of trajectories is dic-tated by the actual data distribution (percentage of loop-ers in Table 2).

To test the applicability of the model represented byEq. (2), the parameters s 2, TL, and V are estimatedfrom the looper and nonlooper datasubsets, and simu-lations of numerical trajectories are carried out by in-tegrating Eq. (2) with the estimated parameters. Thestatistics obtained from the simulated trajectories arecompared with the autocovariances and cross covari-ances computed from the real floats.

The method used to estimate the model parametersis described in appendix B. Here we concentrate ondiscussing the results that are reported in Table 3. Wenotice that RECW is characterized by turbulent dynam-ics that are nearly isotropic, with comparable zonal andmeridional parameters within the limits of the error bar.The angular velocity for the nonloopers is approxi-mately 0, while the anticyclonic loopers have V ø20.54 days21, corresponding to an oscillation time scaleTw on the order of 11 days (for the cyclonic loopers Vø 0.31 days21 and Tw ø 20 days). The average vortexradius r computed for the looping floats is approxi-mately equal to 47 (66) km for the anticyclones (cy-clones).

The parameters listed in Table 3 are used to numer-ically integrate Eq. (2) and simulate three groups oftrajectories representative of the nonlooping, the cy-clonic, and the anticyclonic looping real floats. Thenumber of trajectories and their duration are set in sucha way to reproduce the total number of real float daysavailable in each datasubset. From the synthetic trajec-tories, we carried out similar statistical analyses as per-formed for the real Lagrangian data. The results in termsof autocovariance functions are shown in Fig. 6 for theoverall simulated floats and for the nonlooping and an-ticyclonic numerical trajectories. The cross-covariancefunctions are presented in Fig. 7.

The most important outcome is that the autocovari-ance and cross-covariance functions computed from thecombined three groups of simulated floats (Figs. 6a and7a) are very comparable to the corresponding statisticsobtained from all the real data available in region RECW(Figs. 4a and 5a). This means that the Lagrangian sto-chastic model with the bimodal (trimodal) spin distri-bution did succeed in reproducing the statistical resultsthat were previously investigated. The major eddy fea-tures, such as the superdiffusive characteristic of theoverall autocovariance function and the oscillating pat-terns in both the autocovariances and cross covariances,are predicted by the synthetic trajectories. Furthermore,the statistics from the numerical nonlooping and anti-cyclonic floats depicted in Figs. 6b,c and 7b,c are alsovery similar to their counterparts obtained from the realfloat subsets (Figs. 4b,c and 5b,c, respectively).


TABLE 3. Estimates of the first-order two-demensional Lagrangian stochastic model parameters, in the five subregions of the northwesternAtlantic, for the nonlooping (no loop), the cyclonic (cycl), and the anticyclonic (acycl) trajectory subsets. The computation of the parametersand their errors has been explained in appendix B. The symbols stand for the velocity variance and the decorrelation time scale2 u,ys Tu,y L

(subscript/superscript u, y for either zonal or meridional estimates), the rms velocity Vrms, the angular velocity V, the oscillation time scaleTw, and the average radius r, respectively.

s (cm2 s22)2u,y Vrms (cm s21) T (days)u,y

L V (days21) Tw (days) r (km)

RECWNo loop

Cycl

Acycl

103.7 6 9.8100.3 6 13.4278.0 6 45.0278.0 6 54.2417.0 6 79.0447.0 6 86.4

14.3 6 0.6

23.6 6 2.0

29.4 6 2.6

6.8 6 1.07.9 6 1.58.0 6 3.48.2 6 3.7

12.1 6 2.512.3 6 3.4

0.02 6 0.01

0.31 6 0.05

20.54 6 0.06

O(300)

20.3 6 3.8

11.6 6 1.5

O(600)

65.7 6 11.1

47.0 6 7.7

RECENo loop

Cycl

Acycl

163.3 6 15.5135.5 6 5.4234.2 6 29.0231.8 6 39.5143.1 6 37.1273.8 6 111.4

17.3 6 0.5

21.6 6 1.5

20.4 6 3.6

8.3 6 0.97.0 6 0.46.5 6 1.17.7 6 2.06.7 6 2.8

14.4 6 6.8

0.04 6 0.01

0.34 6 0.04

20.16 6 0.06

O(150)

18.5 6 2.8

39.3 6 26.5

O(400)

54.8 6 10.0

110.3 6 54.4

GSWNo loop

Cycl

Acycl

741.0 6 51.8550.0 6 50.2294.0 6 52.3312.0 6 49.8200.0 6 85.2181.0 6 62.8

35.9 6 1.1

24.6 6 1.9

19.5 6 3.5

7.8 6 1.17.3 6 1.13.7 6 0.96.7 6 1.7

12.8 6 8.115.3 6 29.0

0.02 6 0.01

0.21 6 0.04

20.16 6 0.05

O(300)

30.0 6 8.1

39.3 6 21.3

O(1300)

101.3 6 28.5

105.4 6 59.6

GSENo loop

Cycl

Acycl

298.0 6 18.0294.0 6 26.6412.0 6 93.8276.0 6 89.4440.0 6 263.1233.0 6 164.2

24.3 6 0.7

26.2 6 1.9

25.9 6 8.4

6.9 6 0.66.1 6 0.69.8 6 2.06.6 6 3.4

20.3 6 17.517.2 6 16.3

0.06 6 0.02

0.39 6 0.07

20.47 6 0.11

O(100)

16.1 6 3.5

13.4 6 4.8

O(350)

58.1 6 12.3

47.7 6 10.6

AZNo loop

Cycl

Acycl

64.6 6 5.370.8 6 3.567.6 6 6.881.0 6 9.155.9 6 14.056.5 6 7.0

11.6 6 0.3

12.2 6 0.6

10.6 6 0.8

14.7 6 1.715.3 6 1.711.1 6 1.3

7.9 6 1.015.0 6 3.615.7 6 3.1

20.01 6 0.01

0.19 6 0.02

20.12 6 0.02

O(800)

33.1 6 4.8

52.3 6 16.7

O(1200)

55.4 6 7.0

76.3 6 25.3

b. The other northwest Atlantic regions

1) REGION RECE

This area (Fig. 2b) is located south of the Gulf Streamaxis between 628 and 438W, therefore encompassing thewhole recirculation branch of the northern subtropicalgyre. It comprises the southern half of the Worthingtongyre, while its western part is dominated by a clockwisemean circulation suggesting the presence of another re-circulation gyre to the west. The region zonal mean flowis 22.6 6 0.7 cm s21, whereas the meridional com-ponent is 21.4 6 0.7 cm s21, thus forming a west–southwest mean circulation. The eddy kinetic energy is156 6 8.8 cm2 s22, with a corresponding rms velocityof ø17.7 cm s21.

Most of the Lagrangian data in this region are 700-m SOFAR floats released during the Gulf Stream Re-circulation Experiment (Owens 1984) aimed at givingthe first long-term thermocline description of the Gulf

Stream and its recirculation dynamics, and floats fromTOPOGULF, which investigated the influence of theMid-Atlantic Ridge on large-scale and mesoscale mo-tions (Ollitrault and Colin de Verdiere 2002a,b). Similarto RECW, 75% of the floats leave the area in less than40 days, while about 14% of them remain for periodsbetween 2 and 8 months. The total number of indepen-dent measurements, n*, is equal to 316. The number oflooping trajectories in RECE constitutes 28% of the totalfloat days (Table 2), and they are mainly cyclonic par-ticles trapped inside Gulf Stream cold rings or in ring–stream interactions. Some loopers are in eddies spawnedthrough interactions with the Corner Rise seamounts,which consist of a cluster of six individual seamountscentered at 368N, 528W rising from the ocean floor upto 650-m depth.

The statistical results in terms of autocovariance andcross-covariance functions computed from the overallLagrangian data and separately from the nonlooping and


FIG. 6. Velocity autocovariance functions (the different line typesare the same as for the previous figures) obtained from the simulatedLagrangian datasets in RECW. Results are from (a) the total synthetictrajectories and from (b) the simulated nonloopers and (c) the anti-cyclonic loopers, respectively. These modeled statistics must be com-pared with the results computed from the real data in Fig. 4.

FIG. 7. Same as in Fig. 6, but for the simulated velocity cross-covariance functions. The modeled statistics can be compared withthe results obtained from the real data in Fig. 5.

the cyclonic rotating trajectories in RECE are shown inthe upper three panels of Figs. 8 and 9 (each panel isclearly identified by text). Hereinafter, only the statisticsfrom the dominating loopers (either cyclonic or anti-cyclonic) will be displayed for each region.

Similarly to what was found in RECW, this regionexhibits oscillating autocovariances with pronouncedpositive lobes and weaker negative lobes. Despite thefact that some features remain within the 95% confi-

dence limits, a small curve protuberance around 6–7days does exist, especially in the zonal autocovariancefunction, which is indicative of a superdiffusive eddyfield. Furthermore, the oscillations in the cross-covari-ance functions are statistically significant, suggestingthat the eddy flow characteristics are not too differentfrom those identified in region RECW. Some distinc-tions arise, however, in the statistics obtained from thecyclonic looping trajectories (right upper panel in bothFigs. 8 and 9). Unlike the straight oscillatory autoco-variance and cross-covariance functions calculated for


FIG. 8. Velocity autocovariance functions computed in the recirculation region RECE. (top) Results from (left) the real overall Lagrangiandata, (middle) the nonlooping floats, and (right) the cyclonic looping floats. (bottom) Results from (left) the simulated total trajectories,(middle) the synthetic nonloopers, and (right) the cyclonic loopers. Direct comparisons can be made between the real and the modeledstatistics by comparing the top panels with the bottom panels.

FIG. 9. Same as in Fig. 8, but for the velocity cross-covariance functions.

the looping floats of RECW, the RECE statistics showa more complex behavior suggesting the presence ofvarious coherent structures with different spatial androtational time scales. This point will be further devel-oped and discussed in section 5.

By continuing to assume a simple bimodality hy-pothesis in this region, we test the applicability of afirst-order Lagrangian stochastic model with different

values of spin to be attributed to the nonlooping andlooping floats. The model parameters are computed fromthe real Lagrangian data as for RECW (Table 3). Wenotice that the oscillation time scale Tw is slightly longerthan the one resulting from the anticyclonic loopers ofregion RECW (18 vs 11 days), while the nonrotatingfloats continue to exhibit approximately 0 angular ve-locity. Similarly to what was performed for RECW, we


FIG. 10. Velocity autocovariance functions determined in the Gulf Stream extension region GSW, as in Fig. 8.

used these parameters to numerically integrate Eq. (2)and simulate three groups of trajectories representativeof the real nonloopers, cyclonic loopers, and anticy-clonic loopers. The autocovariance and cross-covariancefunctions obtained from the synthetic trajectories aredisplayed in the lower panels of Figs. 8 and 9. The mainoscillation patterns of these statistics for the overall realfloats (left upper panel in Figs. 8 and 9) are clearlyreproduced by the synthetic trajectories, although thedetailed structure appears different and less smooth thanthat predicted by the simulations. This is presumablydue to coherent structures characterized by variousscales not entirely represented by the single V and TL

of the simulated cyclones.

2) REGIONS GSW AND GSE

The two regions (Fig. 2b) are the most energetic areasof the northwestern Atlantic because they are locatedalong the Gulf Stream axis. GSE is the eastward con-tinuation of GSW and extends slightly east of the jetbifurcation where the Gulf Stream starts flowing north-ward as North Atlantic Current. Both regions are char-acterized by a mainly eastward mean flow, equal to 126 2.4 cm s21 in GSW and to 6.3 6 1.8 cm s21 in GSE(with nonsignificant mean meridional component). Theeddy kinetic energy reaches 547.7 6 52.6 cm2 s22 inGSW, corresponding to a rms velocity of ø33 cm s21,while the eddy energy level is lower in GSE and equalto 304.3 6 31.3 cm2 s22 (Vrms ø 25 cm s21). As ex-pected, the two regions are characterized by strongermean circulation and higher eddy kinetic energy thanthose found in the recirculation areas RECW and RECEanalyzed previously. However, the fundamental differ-ence between the Gulf Stream and the recirculation re-

gions lies in the dynamics, since GSW and GSE aredominated by two kinds of coherent structures, the jetmeanders and the Gulf Stream rings that are shed inresponse to the current instabilities. The Lagrangian dataare affected by both types of coherent features, andconsequently their eddy statistics contain the effects ofthe unstable jet and the oscillating characteristics dueto the rotating vortices.

Most of the floats in GSW and GSE are from the GulfStream Recirculation and the Site L experiments. About85% of them leave the regions in less than 30 days, themajority being advected quickly away by the strongGulf Stream current. The looping trajectories tend torotate mainly in the counterclockwise direction, al-though they may not correspond always to cyclonic ed-dies. In region GSW, for example, some loopers arewaving and moving with the jet, and it is difficult todiscern whether the rotation is due to the trapping actionof semidetached rings or to the fluctuation of the me-andering jet. In region GSE, rotating vortices are moreeasily recognizable and are possibly due to Gulf Streamcold rings. The total number of independent measure-ments is 111 and 96 for GSW and GSE, respectively.

The autocovariance and cross-covariance functionscomputed in region GSW are displayed in the upperthree panels of Figs. 10 and 11. The same statistics forGSE are shown in the upper panels of Figs. 12 and 13.In region GSW, the autocovariance functions for theoverall float data seem to be dominated by the non-looping trajectories (left and middle upper panels in Fig.10). Furthermore, the turbulent field associated with thecyclonic floats is less energetic than that represented bythe remaining trajectories. The results suggest that theanomalous behavior of the overall statistics is not only



FIG. 12. Velocity autocovariance functions obtained in the Gulf Stream extension region GSE.

due to the looper’s influence, but also to the effect ofadditional eddy mechanisms such as jet meanderings orfluctuations of the mean field. These assertions, how-ever, are merely speculative, since the oscillations ofboth the autocovariance and cross-covariance functionsremain within the 95% CL. The situation is simpler inregion GSE where the superdiffusive autocovariance ob-tained from the complete float dataset (left-upper panelin Fig. 12) seems once again the result of the super-position of the statistics computed from the nonloopersand the cyclonic looping trajectories (middle and right

upper panels in the same figure). A common feature ofGSW and GSE is that the loopers and the nonloopersexhibit comparable eddy kinetic energy levels, indicat-ing the presence of a background flow field as energeticas the rotating coherent structures.

The estimate of the parameters derived by assumingthe applicability of a first-order, two-dimensional La-grangian stochastic model in GSW and GSE is againreported in Table 3. The rms velocities of the loopingfloats are slightly higher than those characterizing theloopers of the recirculation regions RECE and RECW.



The Vrms values associated with the nonlooping trajec-tories are higher than or on the same order of magnitudeas the looper velocities, as anticipated by the autoco-variance function results. The oscillation time scale Tw

is rather noisy in GSW, while it ranges between 13 and16 days in GSE.

We simulated only two groups of floats in these re-gions, the nonloopers and the cyclonic trajectories, be-cause the anticyclonic data were too few to modify sig-nificantly the overall results. The autocovariance andcross-covariance functions computed from the synthetictrajectories are displayed in the three lower panels ofFigs. 10 and 11 for region GSW, and in the lower panelsof Figs. 12 and 13 for GSE. When comparing the out-come with the corresponding statistics from the real datain region GSW, we notice that the simulations are notvery successful at reproducing the overall autocovari-ances and cross covariances, although the errors arelarge for any definitive conclusion to be drawn. Thissuggests that in areas dominated by highly coherent jetsrather than vortices and eddies, different parameteri-zation schemes should be employed. The applicabilityof a nonlinear two-dimensional LS model to wave struc-ture dominated dynamics is investigated for example byReynolds (2002b).

The simulations performed in region GSE, however,provide statistics very comparable to those computedfrom the real Lagrangian data, suggesting that the bi-modal spin distributed model applies properly in thisarea.

3) REGION AZ

This area (Fig. 2b) is located north of the AzoresCurrent (Ollitrault and Colin de Verdiere 2002b) and is

characterized by a weak eastward mean flow to the northand by an eastward circulation (Azores Current signal)to the south. The regional average flow has a zonalcomponent of 21.1 6 0.5 cm s21 and an even weakermeridional mean velocity equal to 0.8 6 0.5 cm s21.The area is the most quiescent of all the northwesternAtlantic regions considered in this paper, with an over-all eddy kinetic energy approximately equal to 68 64 cm2 s22 and an rms velocity ø 11.7 cm s21. Therefore,the eddy field is stronger than the mean flow also forthis case.

Most of the Lagrangian data in AZ are 700-m floatsreleased during the TOPOGULF and the NewfoundlandBasin Experiment (Schmitz 1985). Since many loopingfloats were launched spatially close to each other andwithin short time scales, we performed a subsamplingprocedure by considering only those trajectories thatwere initially farther than 100 km apart. About 74% ofthe trajectories remain in the region for less than 40days, while 14% of them stay for periods between 3and 9 months mainly because they are trapped insidenearly steady coherent vortices. The region has the high-est percentage of loopers (40%, see Table 2), mostlycyclonic and possibly due to rotating structures origi-nating through instabilities of the subtropical front (Pin-gree and Sinha 2001). The number of independent data,n*, is equal to 296.

The autocovariance and cross-covariance functionscomputed from the whole Lagrangian dataset in thisregion and from the separated subsets of the nonloopersand cyclonic loopers are displayed in the upper threepanels of Figs. 14 and 15, respectively. The overall ve-locity autocovariance shows a weak subdiffusive be-havior, while the corresponding cross-covariance func-


FIG. 14. Velocity autocovariance functions computed in the Azores Current region AZ.

tion is approximately sinusoidal with oscillation timescale ø20 days. This pattern appears to be induced bythe dominant cyclonic trajectories (whose statistics areshown in the upper-right panels of Figs. 14 and 15),since the nonlooping floats give rise to exponentiallydecaying autocovariances and statistically nonsignifi-cant cross covariances (upper-middle panels in Figs. 14and 15). An important difference between AZ and thepreviously analyzed regions is that the loopers are asenergetic as the nonloopers, suggesting that their for-mation mechanism is associated with more quiescentdynamical regimes.

We applied the first-order model with the bimodalspin distribution also in this region. The results in termsof model parameters (Table 3) are characterized by lon-ger decorrelation and oscillation time scales than thoseobtained for the other North Atlantic areas. The cyclonicloopers, for example, have TL ranging between 8 and11 days, and Tw ø 30 days. The nonlooping floats ex-hibit even longer Lagrangian decorrelation scales on theorder of 14 days.

The results from the numerical integration of Eq. (2)give rise to autocovariance and cross-covariance func-tions that are shown in the lower three panels of Figs.14 and 15. We notice that the simulated trajectories areable to reproduce the statistics obtained from the realdata, proving that, also for region AZ, the bimodalityhypothesis is sufficient to describe the main features ofthe eddy field.

5. Parameter distribution and bimodality

The statistical results of section 4 have shown thatthe approach of applying a first-order Lagrangian sto-chastic model with a bimodal distribution of the spin

parameter is successful in describing the main featuresof the mesoscale eddy field. Some quantitative differ-ences persist, however, between the modeled statisticsand the results obtained from the real data. This raisesthe question of how appropriate the bimodal distributionis in quantitative terms, and whether other more com-plex distributions could improve the results. In an at-tempt to address this question, we have estimated theparameter V and the eddy kinetic energy for each tra-jectory (longer than 20 days) using the method de-scribed in appendix B.

We have then considered their resulting distribution,by plotting V versus EKE for each northwestern At-lantic region (Fig. 16). The various colors in the plotdenote different ranges of trajectory length: black forfloats shorter than 50 days, blue for floats ranging be-tween 50 and 100 days, and red for trajectories longerthan 100 days. Only the V error bars are displayed forthe sake of clarity, but the EKE errors are generallysmaller. The error over the angular velocity does notdepend only on the number of float days, but also onthe sampling interval Dt (see appendix B). We identifythe loopers as the trajectories with V significantly dif-ferent from 0 ( | V | $ 0.1 days21, corresponding to anoscillation time scale Tw # 60 days). Notice that thisdefinition is not exactly equivalent to the Richardson(1993) criterion. There are in fact a few trajectories(18% of the total loopers) that have significant V with-out looping twice in the same direction, especially incorrespondence of the meanders in the Gulf Stream re-gions. However, as mentioned in section 4, the statisticalresults are robust and independent from the specificlooper definition used.

The results in Fig. 16 show an interesting scenariothat can be outlined as follows.



1) In the recirculation regions RECW and RECE, andin region AZ (top two panels and lower panel in Fig.16, respectively), there are clearly a number of floatscharacterized by nonsignificant V and weak EKE (inthe range of 80–100 cm2 s22, with typical Vrms # 10cm s21). They represent what we have called thefamily of nonloopers.

In RECW and RECE distinctive loopers are alsoidentifiable, with high values of both V (ø0.5 days21

in amplitude) and EKE (higher than 600 cm2 s22,Vrms ø 35 cm s21). They are representative of cold-core Gulf Stream rings (cyclonic floats) and ener-getic lenses (anticyclonic floats). Although theselooping features are not numerous, they dominatethe statistics of the loopers because they are energeticand able to trap the floats for long periods of time.

Other trajectories in the recirculation regions arecharacterized by intermediate values of both V(ø0.2–0.4 days21 in amplitude) and EKE (rangingbetween 100 and 300 cm2 s22, with typical Vrms ø20 cm s21). These loopers are less distinctive andshorter in time, although they are most probably re-sponsible for the complex behavior of the loopingfloat statistics in region RECE (section 4b).

2) In the Azores Current region AZ, there are also anumber of looping trajectories, but they exhibit lowvalues of V (ø0.2 days21) and EKE (60–80 cm2

s22).3) In the Gulf Stream extension regions (two middle

panels in Fig. 16), the scenario is more complex,probably because of the influence of fluctuating me-anders and wave structures. In GSE, it is easier toidentify a family of nonloopers (characterized byhigher EKE levels than those typical of the non-

looping floats in the other regions) and a group ofloopers exhibiting scattered values of V (ø0.2–0.6days21).

The scattering of V and EKE for the loopers, observedin the various regions, could be due to the coexistenceof vortices with differing dynamics and/or to the dif-ferent type of sampling within vortices of the same kind.Hydrographic surveys of Gulf Stream rings, in fact, haverevealed that mainly cold-core, but also warm-core ringsare characterized by an inner core rotating approxi-mately as a solid body (Olson 1980; Joyce 1984). Thismeans that the azimuthal velocity increases linearly withthe distance from the ring center, and the angular ve-locity remains constant within the vortex core. Outsidethe core, however, the azimuthal velocity falls expo-nentially away from the ring center, and a similar be-havior is expected for V. Therefore, if the float is mov-ing on the edge of a vortex, its angular velocity andEKE are lower than what they would be in case thefloat were sampling the eddy core.

To distinguish between sampling effects and the ac-tual influence of different vortices is not an easy taskwith the available data. We have compared Vrms, r, andV estimated from the loopers characterized by inter-mediate EKE levels with the same parameters obtainedfrom the highly energetic looping floats, and we havealso looked closely at their looping trajectories. Theconclusion is that both possibilities are likely to occur.Some of the loopers with intermediate values of EKE/V provide significantly higher average radii than thoseestimated by the high-EKE loopers, suggesting that theyare moving on the edge of vortices rather than insidethe eddy cores. In contrast, other looping trajectories do


FIG. 16. Plot of V vs the eddy kinetic energy computed from single trajectories in the five subregions of thenorthwestern Atlantic. The angular velocity error bars are also drawn. Various colors have been used to denote theactual length of the trajectories: black for floats shorter than 50 days, blue for floats ranging between 50 and 100 days,and red for trajectories longer than 100 days.

seem to represent different types of eddy structures,since they exhibit non–significantly different averageradii and a proportionality relationship between Vrms

and V.In summary, these results indicate that, although the

simplifying assumption of bimodality is able to capturethe main statistical characteristics of the data, the actualparameter distribution is probably more complex. Fromthe theoretical and numerical point of view, there is noproblem in using different types of distributions (Berloffand McWilliams 2003; Reynolds 2002b, 2003b). Forexample, the results shown in Fig. 16 could be useddirectly to simulate trajectories and reproduce the detailsof the statistics in section 4. We believe, however, that

this would not really contribute to our understanding ofthe phenomenon because the distribution from the pre-sent data is too incomplete and too strongly influencedby the specific sampling to be considered general.

Another point that was not addressed in the presentwork because of lack of data is the application of anLS model that takes into account the possible transitionof floats between looping and nonlooping regimes. Sucha model would be more representative of the real oce-anic processes over time scales comparable to, or greaterthan, the transition time scale. However, we feel thatthe present data coverage is not sufficient to providereliable statistics of number of particles that enter/exitthe looping structures. Further investigations will be


necessary to identify the actual parameter distributionin some detail, and the float transition between loopingand nonlooping events. A useful tool will be the use ofsynthetic Lagrangian data simulated by high-resolutionocean models, so that sampling limitation will not be aproblem, and Eulerian velocity fields will also be avail-able (Garraffo et al. 2001a,b).

6. Summary and concluding remarks

In this paper, an analysis of mesoscale Lagrangianturbulence in the subsurface North Atlantic is presented.The basic strategy, commonly used in oceanography,consists in first identifying geographical regions that areapproximately homogeneous in terms of both dynamicaland statistical properties (EKE distribution). The meanflow U is then computed in these regions and subtractedfrom the total velocity. The residual velocity u is con-sidered as representative of mesoscale eddy field andanalyzed as homogeneous turbulence. The outstandingquestion behind this approach is of course whether thehypothesis of scale separation between U and u and thehypothesis of u homogeneity actually hold true in theconsidered regions (Landau and Lifshitz 1981). Of thefive subregions identified in this paper, the most westernjet extension area (GSW) is probably the most criticalin terms of scale separation, since the variability is sig-nificantly influenced by the meandering jet. In fact, theresults of the analysis seem to be less clear and reliablein this specific region. In all of the other areas, on theother hand, the decomposition appears solid, with aweak mean flow and a mesoscale turbulent field withvery well defined properties.

Our main result is that mesoscale turbulence can beviewed in each region as the superposition of two dif-ferent regimes, corresponding to looping and nonloop-ing trajectories, and that both can be parameterized us-ing a simple first-order LS model with spin V. The spinis considered as a random parameter (Reynolds 2002a;Berloff and McWilliams 2003) and approximately as abimodal process, reflecting the distribution of loopers(finite V) and nonloopers (zero V). The loopers can befurther separated in cyclonic and anticyclonic trajec-tories, but only one family is found to be dominant ineach region. The first-order Lagrangian stochastic modelis found to be very effective in reproducing the mainstatistical properties observed from the data.

Physically, the nonlooping trajectories characterizethe background flow field, and they tend to dispersediffusively as in random turbulence. They correspondto the majority of the floats (ø85%–60% of the data,depending on the region), but they are generally lessenergetic and shorter. The loopers characterize the co-herent vortices and, although less numerous, they tendto dominate the statistics because they are generallymore energetic and tend to trap particles for a long time,exhibiting a subdiffusive behavior. The superposition of

loopers and nonloopers can produce superdiffusive sta-tistics for the overall dataset.

At the conceptual level, the resulting scenario isstrongly reminiscent of two-dimensional turbulence(Elhmaidi et al. 1993; Provenzale et al. 1995; Proven-zale 1999). The statistics appear highly influenced bycoherent structures, which are likely to be also respon-sible for the observed non-Gaussianity in the velocityfield (Bracco et al. 2000). At a more detailed level,however, there might be significant departures from two-dimensional turbulence dynamics. In particular, the na-ture of coherent vortices may be different. Vortex char-acteristics as shown by the data are highly local andchange significantly in the various regions, probablybecause of the action of different forcings. Energeticlenses (anticyclonic) and Gulf Stream rings (cyclonic)are found in the recirculation regions, while much lessenergetic vortices characterize the Azores Current area.Depending on the vortex characteristics, the propertiesof particle trapping and mass exchanges might also besignificantly different from those found in two-dimen-sional turbulence (Elhmaidi et al. 1993).

The present results open a new avenue for the inter-pretation and parameterization of Lagrangian mesoscaleturbulence in the ocean in presence of coherent struc-tures. The identification of a suitable LS model in re-gions characterized by complex eddy dynamics is, infact, an important step forward in local parameteriza-tions of turbulent transport. Future investigations canbe foreseen, aimed at refining the appropriate modelsand improving their quantitative performance. The bi-modal assumption for the LS model parameter distri-bution, for instance, is certainly oversimplified, as sug-gested by the plot of V against EKE computed fromthe single trajectories. The present data coverage, how-ever, is not sufficient to provide more detailed infor-mation on the actual distribution structure and on theprobability of a float to transit between looping andnonlooping flow regimes. Further investigations usingsynthetic Lagrangian data in high-resolution modelsmight help in addressing these points.

Also, an important point to be assessed in the futureis whether the relevance of coherent structures is specificto highly energetic, strongly sheared regions such as theones considered here, or whether it is a more pervasivefeature of subsurface flows. Below the ocean surface,in fact, coherent structures might survive longer than atthe surface, because of the absence of direct random-izing forcing such as synoptic winds. Further subsurfacedata and turbulent statistical analyses are needed to an-swer this question.

The present analysis provides indications for transportparameterizations in coarse-grain GCMs. Lagrangianstochastic models are often used in conjunction withOGCM velocity outputs to simulate transport of pol-lutants or of biological quantities that can be consideredapproximately as passive tracers (Cowen et al. 2000,2003). Very simple first-order, one-dimensional LSMs


with constant parameters are often used (e.g., Dutkie-wicz et al. 1993; Falco et al. 2000). The present resultsindicate that the use of two-dimensional models withdistributed spin is more appropriate in presence of co-herent structures and it provides information on param-eter distribution in the subsurface North Atlantic. Whenconsidering global (i.e., basinwide) applications, the ef-fect of inhomogeneities will have to be introduced bymeans of modified LSMs, whose drift terms take intoaccount the spatial distribution of the parameters (e.g.,Berloff and McWilliams 2002). The approach is ex-pected to provide useful results in regions where thescale separation holds true and where the local homo-geneity allows one to correctly identify the LSM pa-rameters, as in most of the subregions considered here.Applicability to highly inhomogeneous and nonlinearareas, such as the jet and its extension, are potentiallymore problematic.

Acknowledgments. The authors thank sincerely Leo-nid Piterbarg for important suggestions on statistics anderror estimations, and Bill Johns, Don Olson, EnricoZambianchi, Zulema Garraffo, Eric Chassignet, andClaudia Pasquero for many stimulating discussions. Au-thors M. Veneziani and A. Griffa were supported by theNational Science Foundation Grant OCE-9811358, theOffice of Naval Research Grant N00014-97-1-0620, andthe Italian National Research Council (CNR) throughthe SINAPSI Project. Author A. M. Reynolds was par-tially supported by the BBSRC through a core strategicgrant. Author A. J. Mariano was supported throughONR Grant N00014-99-1-0049.

APPENDIX A

Estimate of the Mean Flow by BicubicSpline Interpolation

The results of the bicubic spline interpolation dependon four parameters, the knot spacing, which is relatedto the number of finite elements of the interpolationfunction, and three weights, which are associated withuncertainties in the data and in the first and secondderivatives of the interpolated field (Inoue 1986). Thedata weight depends on our confidence of the Lagrang-ian velocity data and can be chosen proportional to theinverse of the estimated variance of the eddy field. Thetension is associated with the first derivative of the sp-lined flow, and its value has to minimize fluctuations atthe boundaries. It is typically fixed at 0.99 (Mariano andBrown 1992). The roughness is related to the secondderivative—that is, to the ‘‘curvature’’ of the interpo-lated field—and controls the wavenumber content of theresults. Bauer et al. (1998) showed that, at fixed knotspacing, data weight, and tension, the roughness r canbe optimized in such a way to minimize the energy inthe fluctuation field at low frequency.

We have performed a sensitivity analysis by varying

the spline parameters and testing the robustness of theeddy statistics. The knot spacing has been varied be-tween 18 and 38, while two values for the data weighthave been chosen, corresponding to different binnededdy kinetic energies. The roughness has been consid-ered in the range 1022–1000, changing its value by oneorder of magnitude at a time.

It has been found that, for all the knot spacing anddata weight choices, the eddy statistics tend to behaveasymptotically at high values of r, with the autocovari-ance and cross-covariance structures becoming indepen-dent from the specific value of roughness. This suggeststhat the eddy statistics are robust, independent from theparticular estimate of the interpolated mean flow.

APPENDIX B

Estimate of the Model Parameters

The parameters of the two-dimensional, first-orderLagrangian stochastic model described by Eq. (2) arethe velocity variance s 2, the decorrelation time scaleTL, and the angular velocity V. They can be estimatedas follows.

The velocity variance is given by the autocovariancefunction at 0 time lag, while TL is evaluated by makingthe a priori assumption that a two-dimensional first-order model applies. The decorrelation scale is then es-timated by using the theoretical autocorrelation functiongiven in Eq. (4). We obtain the formula

212R(0)

T 5 2t ln , (B1)L 25 6[ ]2R(t) 2 R(0)R(2t)

where R is either the zonal or the meridional autocor-relation function according to what component of thedecorrelation scale we are computing. Because of largeerrors of the autocorrelation estimate, we calculated dif-ferent values of TL by choosing t 5 Dt, 2Dt, 3Dt, . . . ,until we obtained a saturation level for the decorrelationscale.

The angular velocity V is evaluated through Eq. (3),where the mean spin ^ds& is estimated as

^ds& [ ûdy 2 ydu& ; û(t)[y(t 1 Dt) 2 y(t)]

2 y(t)[u(t 1 Dt) 2 u(t)]&

5 û(t)y(t 1 Dt) 2 y(t)u(t 1 Dt)&

5 R (Dt) 2 R (Dt). (B2)uy yu

The accuracy of this estimate depends not only on thetotal number of observations, but also on the samplingperiod Dt. In fact, Dt must be much shorter than V21

in order for the oscillations in the data statistics to beresolved.

The oscillation time scale Tw is simply given by2pV21, while the average vortex radius r is estimatedas VrmsV21.

Error estimation of these parameters is not straight-


forward. We adopted the ‘‘bootstrap’’ technique,through which a series of numerical experiments is car-ried out to simulate the independent realizations nec-essary for the error evaluation. Specifically in our case,we simulated a certain number of float trajectories soas to reproduce the number of observations character-istic of either the looper or the nonlooper dataset. Thesimulation is repeated as many times as the number ofindependent realizations we wish to reproduce (typically10). A lower bound for the parameter error is computedas standard deviation of the estimates carried out in eachrealization.

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