## 1. Introduction

Surface drifters and subsurface floats have been widely used in the last few decades to study transport properties in the mesoscale range (e.g., Freeland et al. 1975; Riser and Rossby 1983; Rossby et al. 1983; Colin de Verdière 1983; Krauss and Böning 1987; Figueroa and Olson 1989; Zhang et al. 2001; Bauer et al. 2002; Zhurbas and Oh 2003). The main objective of these works was the characterization of the correlation properties of the particle velocities, the study of their dependence on the eddy kinetic energy and the estimate of the absolute dispersion in the mesoscale range. Many of these studies were based on the analysis of single experiments. However, even when considering data collected at the same time in a given geographical area, the analysis of the correlation properties of the Lagrangian velocity field is complex. In fact, the theoretical framework used in these analyses is Taylor’s (1921) theory of stationary and homogeneous turbulence but, even inside relatively small subdomains, different regimes of dispersion may exist, because of the huge variability of Lagrangian behaviors (e.g., the coexistence of looping and nonlooping trajectories; Veneziani et al. 2004). Consequently, the computation of statistical parameters by means of simple and unconditioned means among Lagrangian trajectories experiencing different flow regimes may give rise to misleading results. The complicated nature of Lagrangian dispersion has been evidenced also using simplified conceptual models. For instance, Elhmaidi et al. (1993) and Pasquero et al. (2001), in the context of quasigeostrophic 2D turbulence, have shown that to correctly model dispersal properties, one should consider at least two different regimes. Namely, distinguishing particles trapped in energetic coherent structures from those floating in a more quiescent turbulent background. More recently, Berloff and McWilliams (2003), to represent the huge Lagrangian variability, have used stochastic models in which the decorrelation parameters vary with statistical properties given by a probability distribution obtained from data. In the past there was some effort to avoid a mixture of different regimes computing statistical quantities averaging between Lagrangian trajectories that were previously subdivided in kinematically quasi-homogeneous subdatasets considering as screening indices either the eddy kinetic energy and the Lagrangian correlation time (Rupolo et al. 1996) or the mean vorticity of the trajectory (Veneziani et al. 2004).

Nowadays, public archives contain a massive quantity of Lagrangian data collected since the late 1970s at different depths and in all World Ocean basins. An integrated analysis of this dataset, even if difficult for the aforementioned problems, would be of fundamental importance for the understanding of the ocean transport properties. Hence, considering important to define an experimental procedure to compute Lagrangian statistical properties through averages on quasi-homogeneous subdataset, we propose here a data processing technique to separate the whole dataset in classes of trajectories morphologically homogeneous and characterized by different regimes of dispersion and by different qualitative behavior of the velocity correlation function. The statistical properties computed averaging on such subdatasets are then used, first, to investigate kinematics relations in the different regimes and, second, to map in the geographical space the quantitative relevance of these different regimes.

The screening of the trajectories is done using the ratio between the acceleration and velocity time scales *y* = *T _{a}*/

*T*. The motivation for the choice of the index

_{υ}*y*=

*T*/

_{a}*T*is simple and is based upon the following considerations. Any real particle, because of its inertia, cannot be subjected to infinite acceleration and consequently the correlation time scale of the Lagrangian acceleration has to be different from zero, perhaps in a time range smaller than the observational time sampling. On the other hand, many previous studies (e.g., Zambianchi and Griffa 1994; Griffa 1996) have shown that, especially the motion of surface drifters but in some case also of subsurface floats, may be well represented by first-order stochastic model, in which

_{υ}*T*≠ 0 but

_{υ}*T*= 0. In this case, the motions that in the time sampling scale appear to be dynamically uncorrelated are highly energetic; the corresponding Lagrangian trajectory is often characterized by a well-defined slow frequency motion perturbed by jumps in the position and discontinuities in the velocity field. In contrast, if also

_{a}*T*≠ 0 a relevant part of the energy of the motion of the Lagrangian device is absorbed by dynamically correlated movements, the trajectory is smoother and the corresponding velocity field does not show discontinuities.

_{a}Hence, considering all the available world’s ocean surface drifters and the Atlantic Ocean subsurface floats data of the World Ocean Experiment (WOCE) archive, we will compute for each trajectory the ratio *y* = *T _{a}*/

*T*and we will separate the trajectories in different classes characterized by different values of

_{υ}*y*, showing the greater homogeneity of trajectories belonging to the same class in terms of shape, velocity correlation properties, and of relationships between eddy kinetic energy and correlation length scale. Then, we will show that the simple computation of the relative importance of the different classes of trajectories in the different geographical areas allows for a characterization of the dispersal properties of the main oceanic currents. This analysis is useful to detect how and where the presence of coherent structures influences the Lagrangian dispersion, something that may advance the representation of the eddy diffusivity in the ocean general circulation models (OGCMs).

The estimate of the velocity and acceleration correlation time scales will be done using, as a paradigm of the observed ocean variability, a second-order Lagrangian Stochastic Model (LSM), since this simple model directly allows for a simple estimation of the characteristic time scales from individual trajectories. All of the analysis of the present work will be done assuming that also the subsurface float moves on a two-dimensional surface.

The work is organized as follows. In section 2 we define the statistical functions to be used and derive some simple algebraic relations from the second-order LSM. In section 3 we select the different classes of trajectories by means of the dimensionless index *y* = *T _{a}*/

*T*, showing that they can be considered as representative of different regimes of dispersion. In section 4, studying the geographical distribution of the relative weight of each class, we will characterize the main oceanic currents identifying where the role of the coherent structures is important. In section 5 we give a summary and further discuss the results in the context of eddy diffusivity parameterization in OGCMs.

_{υ}## 2. Functions definition and second-order LSM

In this section, we shall introduce the statistical functions that are commonly used when analyzing Lagrangian data and we shall derive some simple algebraic relations from the second-order LSM, considering, for simplicity, the 1D space. The generalization to the 2D case is straightforward assuming the homogeneity of the velocity field.

*u*(

*t*), the velocity correlation function (1), its power spectrum (2), and the single-particle dispersion (3) are descriptions that are mathematically equivalent but complementary:Here

*x*(

*t*) is the position,

*u*′(

*t*) =

*u*(

*t*) −

*U*, where

*U*= (

*N*− 1)

^{−1}Σ

^{N−1}

_{j=1}

*u*(

*j*) is the time-averaged velocity,

*x*′(

*t*) =

*x*(

*t*) −

*Ut*,

*σ*

^{2}

_{U}is the eddy kinetic energy, and the angle brackets indicate an average over the time lag

*t*.

*T*is a measure of the time correlation of the velocity time series and is defined asThe single-particle dispersion (3) is related to the velocity correlation function

_{L}*R*(

*t*) and to the power spectrum

*P*(

*υ*) by the relationsTaylor’s (1921) result in form (5) and (6) is due to Kampé de Fériet (1939).

^{2}(

*πνt*) ∼ (

*πνt*)

^{2}and (6) becomes (ballistic regime)and, for large times, it can be shown (e.g., Krauss and Böning 1987) that the other asymptotic behavior holds:For small times (7), all frequencies equally contribute to particle displacements, whereas, for large times (8), the lower frequencies dominate and, if the velocity spectrum saturates at small frequencies, the Lagrangian diffusivity

*K*(

*t*) = (1/2)(

*d*/

*dt*)

*A*(

*t*) asymptotically approaches the constant value (random walk or Brownian regime):In a real trajectory, the variances of the velocity and acceleration time series are finite and consequently the following constraints on the spectral shape hold:This means that the high-frequency spectral decay is bounded by a logarithmic spectral slope

*α*that satisfiesThis high-frequencies asymptotic behavior, which may occur in a frequency range not resolved by the time sampling interval, is due to the inertia of the Lagrangian particle that cannot be subjected to infinite accelerations. Condition (11) is equivalent to a constraint for the velocity correlation function that in this case is characterized by a smooth quadratic behavior for small times.

Experimental spectra from Lagrangian floats (Colin de Verdière 1983; Krauss and Böning 1987; Rupolo et al. 1996; Lumpkin and Flament 2000) and from long time series of altimetric Ocean Topography Experiment (TOPEX)/Poseidon data in the extratropical ocean (Stammer 1997) are characterized by a low-frequency (wavenumber) plateau and by a decay in the mesoscale range with bilogarithmic spectral slope 2 < *α* < 3. In regions characterized by strong mesoscale activity the asymptotic behavior (11) are often observed and the velocity correlation function may also show oscillations, due to the spiraling of fluid particles trapped in coherent structures.

### a. Second-order LSM

*a*and the incremental equations arein which, using the previous definitions, the stochastic impulse is represented by a random increment

*dw*(

*t*) from a normal distribution with zero mean and variance

*dt*. Equations (12)–(14) are equivalent to the second-order stochastic processwith

*α*

_{1}= (1/

*T*) + (1/

_{υ}*T*) and

_{a}*α*

_{2}= 1/(

*T*).

_{a}T_{υ}This model has been introduced by Sawford (1991) to correct the small-scale behavior of the first-order LSM in the description of flows at high, but finite, Reynolds number. The two characteristics time scales *T _{a}* and

*T*respectively represent the correlation of the forcing and the memory of the velocity induced by, high-frequency, uncorrelated impulses (Griffa 1996).

_{υ}*z*

^{2}+

*α*

_{1}

*z + α*

_{2}= 0 and are equal to

*β*

_{1}= −(1/

*T*) and

_{υ}*β*

_{2}= −(1/

*T*) (Sawford 1991). It is easy to show (Berloff and McWilliams 2002) that the following relation holds:where

_{a}*σ*

^{2}

_{U}and

*σ*

^{2}

_{A}are the velocity and acceleration variances. The Lagrangian correlation time

*T*, obtained integrating (16), is equal towhich means that when

_{L}*T*≠ 0 the finite acceleration contributes to increase the correlation of the velocity field.

_{a}The appearance of a complex conjugates pair of time scales is symptomatic of a missing but dynamically important spatial dimension. Reynolds (2002) has examined the correspondence between multidimensional (*n* − 1)th-order LSM and one-dimensional *n*th-order LSM. In the parameter range in which *T _{υ}*

_{,}

*T*∈

_{a}*C*, the 1D second-order LSM is then equivalent to a 2D first LSM with spin, where the nonisotropy of the velocity field is expressed by a preferred sense of rotation about the vertical axis to simulate spiraling particles motion (Borgas et al. 1997).

For *t* ∼ 0, *R*(*t*) defined by (16) is characterized by a smooth behavior, as is expected in flow with finite acceleration, while for *t* ≫ *T _{a}* the correlation function asymptotes to the exponential form characteristics of the first-order LSM. In this model, when

*T*> 0, the variance of both velocity and acceleration is finite, and for

_{a}*T*→ 0 relation (15) becomes a first-order equation and the model collapses to a first-order LSM.

_{a}## 3. Dataset and trajectories taxonomy

### a. Subsurface floats dataset

Since we are interested in computing eddy kinetic energy and velocity decorrelation time scales in this work we have to consider only continuously (acoustically) tracked SOFAR and RAFOS floats. In particular we analyze subsurface floats downloaded from the WOCE (2002) archive moving at a nominal depth between 400 and 1000 m in the Atlantic Ocean, since only there does the data density allows for a reliable statistics. In the WOCE dataset, the data are divided per experiment and in Table 1 we list the experiments considered in this work. The floats are neutrally buoyant, ballasted for a specific pressure or density surface. Most of the floats considered here were isobaric, but those in the North Atlantic Current, Western Boundary Current, Atlantic Climate Change Experiment (ACCE), Canary Islands Azores Gibraltar Observations (CANIGO), and Cape of Good Hope Experiment (KAPEX) were quasi-isopycnal and thereby occupying a range of depth. Recent analyses including both isobaric and isopycnal floats (Lacasce 2000; Lacasce and Bower 2000) showed no systematic differences between the two datasets, even considering relative dispersion, that may be amplified by the vertical shear. So, in the following, because of the lack of continuously tracked subsurface trajectories, we will consider data from both isopycnal and isobaric instruments.

The depth interval (400–1000 m) was chosen since we are interested in studying the geostrophic turbulence that at this depth is intensified by the energetic transfer from the high to the first baroclinic mode (Shafer Smith and Vallis 2001). At greater depth, below the permanent thermocline, the turbulent activity is smaller and the float motion is dominated by low-frequency wave and by topographic constraint.

Subsurface floats data positions were mostly recorded every 24 h and when a different time step was used, positions and velocity were interpolated to have a daily time sampling. Gaps in the original time series smaller than four consecutive points were linearly interpolated. When larger gaps were found, the original trajectory was divided in more pieces.

To fully resolve the mesoscale frequency range and to minimize the effects of the nonhomogeneity of the velocity field, we perform correlation analysis on pieces of trajectories of uniform length of 64 days. For the Atlantic Floats (AF) dataset, our analysis is the based on 2074 pieces of 64-day-long trajectories, for a total of 132 736 daily observations (see Table 3).

### b. Surface drifters dataset

In this work we analyze surface drifters data collected from 1979 to October 2005 in the context of the Global Drifter Program (GDP; online at http://www.aoml.noaa.gov/phod/dac/gdp.html), formerly WOCE Surface Velocity Program (WOCE/SVP). The Marine Environmental Data Services (MEDS; http://www.meds-sdmm.dfo-mpo.gc.ca) archives and posts all data collected and processed by the Atlantic Oceanographic and Meteorological Laboratory (AOML; http://www.aoml.noaa.gov) under the GDP. We downloaded data from 1979 to 2003 from the WOCE (2002) and MEDS archives in which the Lagrangian data are grouped for year and divided geographically in four subdatasets for the Atlantic, Arctic, Pacific, and Indian Oceans (see Fig. 1). Data from 2004 to 2005 were downloaded directly from the AOML archive.

Drifters are tracked by the Argos satellites with an estimate of accuracy of position of about 150–1000 m. The AOML’s Drifter Data Assembly Center (DAC) perform a first quality control of the data (see, e.g., Lumpkin and Pazos 2007) and then interpolates the raw fixes to uniform 6-h interval using an optimal procedure known as kriging (Hansen and Poulain 1996). Besides the interpolated positions the DAC provides also an estimate of the error on the position, which becomes very useful in identifying accidental or programmed gaps on the data transmission. Measuring surface currents with Lagrangian instruments was an important challenge for centuries and instruments and techniques continuously evolved with time until the modern satellite-tracked surface drifters. History of the Lagrangian surface devices and the description of the evolution of different techniques can be found in Lumpkin and Pazos (2007). Here it is interesting to note that from the eighties, and under the request of the World Climate Research Program, efforts were made to deploy standardized surface drifters. In particular, the modern set of SVP data is made with drifters drogued at a depth of 15 m. The data downloaded from MEDS (2001–03) and AOML (2004 and 2005) are all collected using drifter drogued at this depth. Nevertheless, in the WOCE (2002) archive from 1979 to 2000 are included also few drifters drogued at different depth. The first screening of the downloaded dataset then consists in retaining only drifters with the drogue between 0 and 50 m. Doing that we reject 403 drifters (over a total of 8091 from 1979 to 2000) for which the drogue depth is not specified or it is greater than 50 m. The 92% of the retained drifters have a drogue depth between 15 and 15.4 m and only about the 2.8% (1.4%) are drogued at a depth smaller than 10 m (greater than 20 m).

Coherently with the AF dataset, also for the surface drifters we consider pieces of trajectories of uniform length of 64 days. This relatively large dataset of pieces of 64 days long of trajectories is subsequently screened in order to eliminate segments affected by accidental or programmed gaps in the transmission of data. In fact, to save battery power, older drifters often used a duty cycle of ⅓, typically with the transmitter spent one day on then two days off. Many drifters were also programmed to transmit continuously for a first period to subsequently switch to an alternating mode of transmission. Moreover, the DAC interpolates positions, when occasional gaps in the transmission days are present. Since we are interested in computing decorrelation time scales, the presence of such drifters could lead to significant biasing in our results. Unfortunately, the metadataset does not contain exhaustive information about transmission gaps for all the drifters. Consequently, we adopt the technique of individuate drifters (or pieces of trajectory of drifters) transmitting not continuously looking at the power spectrum of the errors on the position. As an example, drifters working with a 1-day on 2-days off ⅓ duty cycle, display a well-defined peak at the frequency ⅓ days^{−1}. In particular, inspecting trajectories for which information on the duty cycle is given (using a routine kindly furnished by R. Lumpkin) we select threshold values useful to automatically detect on the spectrum of the error the peaks induced by the transmission gaps. Accidental and isolated gaps in the transmission are identified imposing threshold values both on the magnitude of the error and in the low frequency content of the error spectra. The entire dataset (42 813 pieces of 64-day-long trajectories, for the four basin) is then screened in order to eliminate peaks and to obtain an approximately white spectrum of the position error, for frequency range smaller than 1 day^{−1}. With the chosen threshold values about the 40% of 64-day-long trajectories are rejected.

*ν*> day

^{−1}, a further smoothing of the positions time series

*y*(

_{i}*j*) is performed through a boxcar average of width 2,before computing the velocities from the smoothed time series

*x*(

_{i}*j*) bywhere Δ

*t*= 6 h and the index

*i*= 1, 2 refers to the zonal and meridional directions. This further smoothing was performed in order to avoid the influence of inertial and subinertial motions on the experimental estimate of the correlation time scale and its effect on the results of this work will be discussed in the following.

Last, to reject trajectories affected by anomalous noise in the high-frequency range, we reject all the (surface and subsurface) trajectories (<1%) with *T _{L}* < 0.75 days (see below for the experimental definition of the Lagrangian correlation time

*T*). We end up with 2074, 7333, 561, 12 154, and 2969 pieces of 64-day-long trajectories for the Atlantic Floats (AF), Atlantic Drifters (AD), Arctic Drifters (ARD), Pacific Drifters (PD), and Indian Drifters (ID) datasets, for a total of 132 736 and 1 473 088 daily observations for the subsurface floats and surface drifters datasets, respectively. Details of the datasets are given in Tables 1 –3, and in Figs. 1 and 2 we show all Lagrangian trajectories (black lines for the subsurface ones) considered in this work and a map of daily observations per 1° square bin.

_{L}In Fig. 3 we show the mean zonal and meridional spectra and in Fig. 4 we plot the mean total velocity correlation functions (semisum of the correlation functions of the zonal and meridional velocities) for the five datasets. Although these quantities are obtained by averaging on a mixture of dynamics distributed geographically, some considerations can be made:

- The subsurface floats are less energetic than the surface drifters (see also Table 4) and the latter ones, especially in the Pacific, display a well-defined anisotropy in the low-frequency energy content (Fig. 3), showing that the zonal variability is inherently more energetic at lower frequencies (Krauss and Böning 1987).
- The Lagrangian meridional velocity spectra are similar to the generic shape evidenced by Stammer (1997) analyzing TOPEX/Poseidon altimetric data and are characterized by a low-frequency plateau followed by a spectral decays. Since the Lagrangian time scales are typically smaller than the Eulerian ones (e.g., Middleton 1985), in the Lagrangian spectra the break between the plateau and the spectral decays is observed at frequencies [∼(20 days)
^{−1}] higher than in the Eulerian ones [∼(250 days)^{−1}]; - even if subsurface floats data were mostly recorded in areas where the motion is characterized by a massive presence of coherent structures, like the Gulf Stream and Agulhas currents or the Mediterranean outflow area, no striking qualitative differences (except for the anisotropy) are observed between the spectra and velocity correlation functions of the float and drifters datasets, nor in the spectral slope decay or in the shape of the velocity correlation function (oscillations and small times behaviors). In particular, all the mean velocity correlation functions have an exponential like behavior for
*t*→ 0.

### c. Trajectories taxonomy

Here, we will not discuss the anisotropy of the spectral and transport properties but we will use the index *y* = *Ta/Tυ*, computed for each trajectory, to individuate subdatasets of trajectories morphologically similar and characterized by having more homogeneous correlation and dispersion properties.

The estimate of the Lagrangian correlation time *T _{L}* is obtained from Eq. (4) integrating in the first positive lobe the zonal and meridional components of the experimental velocity correlation (1) and then considering

*T*as the semi sum of the two values. This first zero crossing technique, even if ill founded, is commonly used [for a discussion of the different techniques used in the literature see Lumpkin et al. (2002)]. In particular, the use of the first zero crossing technique may lead to overestimate time scales, especially in presence of spiraling trajectories characterized by having velocity autocorrelation function with deep negative lobes. On the other hand, dropping the mean velocity of the time series will tend to underestimate the time scale, since low-frequency variability is eliminated. Last, when computing Lagrangian time scales with trajectory data one has to consider that, whatever the technique, the final estimate is also depending on the size of the time sampling. Obviously a better estimate is obtained with a smaller size of the time sampling. It can be shown subsampling original trajectories with small sampling time (e.g., Storch and Zwiers 1999), that a reliable estimate is obtained with a ratio Δ

_{L}*t*/

*T*≤ 0.2–0.25. When this ratio is greater, the computation of

_{L}*T*will give raise to an overestimate. Looking at the values of

_{L}*T*in Table 4 and considering that the time sampling is 24 and 6 h for the floats and drifters data we can see that in some cases we are near the threshold limit. In general, when calculating Lagrangian time scales some approximations are inevitable and the specific value of the decorrelation time may be not very informative. However, comparison between time series with the same time sampling and uniformly computed indicates which processes have large memory.

_{L}*T*and

_{υ}*T*using relations (18) and (19), which givewhere, for each trajectory, the Lagrangian eddy kinetic energy

_{a}*σ*

^{2}

_{U}= (1/2)(

*σ*

^{2}

_{u1}+

*σ*

^{2}

_{u2}) and the acceleration variance

*σ*

^{2}

_{A}= (1/2)(

*σ*

^{2}

_{a1}+

*σ*

^{2}

_{a2}) are computed from the velocity and acceleration time series

*u*(

_{i}*j*) and

*a*(

_{i}*j*) as follows:where

*N*− 1)

^{−1}Σ

^{N−1}

_{j=1}

*u*(

_{i}*j*),

*N*− 2)

^{−1}Σ

^{N−2}

_{j=1}

*a*(

_{i}*j*), and

*N*is the number of points of the trajectory.

*y*=

*T*/

_{a}*T*. When

_{υ}*T*

^{2}

_{L}< 4(

*σ*

^{2}

_{U}/

*σ*

^{2}

_{A}) (which is equivalent to

*α*

^{2}

_{1}< 4

*α*

_{2}, in the notation of section 2), we have that

*T*,

_{υ}*T*=

_{a}*a*±

*ib*∈

*C*, |

*T*| = |

_{υ}*T*|, and |

_{a}*y*| = 1. In this case, the trajectory is characterized by an oscillatory behavior and it is interesting to compute also the dimensionless parameterwhich represents the ratio between the oscillatory and memory time scales. It may be analytically shown that when 2

*α*

_{2}<

*α*

^{2}

_{1}<4

*α*

_{2}the velocity correlation function (16) oscillates but the corresponding velocity spectrum has no peak. An illustrative example, kindly suggested by an anonymous reviewer, directly shows how the experimental computation of the ratios

*y*and

*x*identifies a looping trajectory. In this case, the Lagrangian velocity components (

*υ*

_{1},

*υ*

_{2}) may be approximated aswhere

*V*is the constant scalar velocity,

*ω*=

*V*/

*r*is the angular velocity, and

*r*is the radius of the loop of the trajectory. It can be easily shown that(in view of the first zero crossing technique),Therefore we have that |

*y*| = 1 and

*x*= (3)

^{1/2}> 1.

From a technical point of view, the computation of *T _{a}* and

*T*strongly depends on the interpolation and smoothing procedures applied to the original drifter data. We performed the same analysis presented here without smoothing the data by the boxcar average, and, even if the relative population of the different classes of trajectories is different (see Fig. 7), the overall description and the geographical distribution (see section 4) of the different turbulence regimes are qualitatively similar. Last, we tested the sensitivity of the method on the length of the time series, performing the same analysis using 128-day-long trajectories, and even in this case we found slightly different relative population in each class with unchanged general results.

_{υ}In Figs. 5, 6 and 7 we show, for the five datasets, the probability density functions (PDF) of the computed values of *T _{L}*,

*T*, and

_{a}*y*; as expected (see also Table 4), the AF dataset shows for

*T*,

_{L}*T*, and

_{a}*y*a distribution shifted toward higher values. In Fig. 8 we show, for the drifters and subsurface floats datasets, the scatterplot of the decorrelation length

*L*=

_{L}*σ*versus the square root of the eddy kinetic energy

_{U}T_{L}*σ*. The black and yellow crosses indicate values of drifters and floats characterized by having

_{U}*y*< 0.2 and 0.4 <

*y*< 0.8, and the blue and red crosses are relative to the floats and drifters whose time scales lie in the complex conjugate space with, respectively, |

*y*| = 1,

*x*< 1, and |

*y*| = 1,

*x*> 1. For small

*y*, the decorrelation length is approximately linearly dependent on

*σ*, whereas when the time scales become complex the dependence on the eddy kinetic energy is definitely weaker and, for large

_{U}*σ*,

_{U}*L*is nearly constant.

_{L}This simple observation may have important consequences; in fact a qualitative view of the source of the Lagrangian variability may be obtained considering the two limiting cases in which the Eulerian field evolves slowly (*σ _{U}* ≫

*L*/

_{E}*T*, strong spatial structure or frozen turbulence regime) or rapidly (

_{E}*σ*≪

_{U}*L*/

_{E}*T*, weak spatial structure or fixed-float regime), where c

_{E}_{*}=

*L*/

_{E}*T*is the evolution speed of the eddy field (Lumpkin et al. 2002) and

_{E}*T*and

_{E}*L*are the Eulerian counterparts of

_{E}*T*and

_{L}*L*.

_{L}*L*∝

_{L}*σ*, and the Eulerian and Lagrangian time statistical properties are similar, while in the frozen turbulence

_{U}*L*∼

_{L}*L*and the frequency Lagrangian energy spectrum tends to the wavenumber Eulerian energy spectrum (Middleton 1985).

_{E}We know that the Eulerian length scales vary as a function of latitude from about 100 km at 50°N/S to greater than 160 km within 8° of the equator (e.g., Stammer 1997). The behavior stated by (24) and (23) essentially is that, at a given latitude, variations of *L _{L}* are primarily due to changes in turbulence regimes and not to variations of

*L*.

_{E}Middleton (1985), studying different experimental datasets, has shown that the case *T _{L}*/

*T*< 1 [(22)] is the most generic one; more recently Lumpkin et al. (2002), analyzing experimental and numerical data in the North Atlantic, have shown that neither a fixed-float or frozen field regime characterizes the full set of observations. Nevertheless, results shown in the same paper as well as several other regional studies (Rossby et al. 1983; Krauss and Böning 1987; Poulain and Niiler 1989; Figueroa and Olson 1989; Brink et al. 1991) indicate that for energetic mesoscale flows, the regime (22) with

_{E}*L*∼ const and

_{L}*K*∝

*σ*seems to be the most appropriate. Hua et al. (1998), by means of numerical experiments, found that also the geostrophic turbulence seems to be more consistent with the frozen turbulence regime.

_{U}Based upon these considerations, we think that the plot of Fig. 8 shows that the ratio *y* = *T _{a}*/

*T*can be considered as a useful index to discriminate between the frozen turbulence and fixed-float regime. Moreover, plotting the mean position of the trajectories belonging to the different classes (plot not shown here) it is possible to see that the trajectories that have complex conjugates time scales are mainly located in the regions characterized by strong currents.

_{υ}To get a direct feeling about the qualitative difference between trajectories characterized by different values of the ratio *y* = *T _{a}*/

*T*, we plot in Figs. 9 –11 (only for AF, AD, and PD) typical trajectories of the four classes defined by

_{υ}*y*< 0.2 (class I), 0.4 <

*y*< 0.8 (class II), |

*y*| = 1

*x*< 1 (class III), and |

*y*| = 1

*x*> 1 (class IV). Trajectories belonging to the two extreme classes I and IV are considerably different. In class I, there are grouped floats and drifters characterized by large-scale meandering influenced by high-frequency variability. In class IV, trajectories are clearly influenced by the presence of coherent structures and very often they rapidly whirl, trapped inside an eddy with a well-defined length scale (looping behavior). This is partly true also for the trajectories characterized by |

*y*| = 1 and

*x*< 1 (class III), even if in this case the length scale of the loops is not well defined and often the floats/drifters seem to jump between eddies of different size. Last, floats/drifters of the class II (0.4 <

*y*< 0.8) have trajectories showing intermediate characteristics and, meandering around large-scale structures, they did not show a clear influence of high-frequency motions.

In the presence of coherent structures only particles trapped in the eddies quickly rotate, while the rest of the particles meander in the less energetic turbulent background experiencing only sporadic looping behavior. Richardson (1993) has introduced the simple criterion of defining a looper as a trajectory that undergoes at least two consecutive loops in the same direction. Veneziani et al. (2004), analyzing subsurface floats in the northwest Atlantic, found that in this area the percentage of looping floats varies from 15% to 40%.

Even if an exhaustive analysis is difficult, because of the huge number (25 091) of trajectories analyzed in this work, we believe that the method proposed here, and based on the computation of the ratio *y* = *T _{a}*/

*T*, is capable of identifying looping trajectories. An inventory (postscript file) of all the trajectories analyzed here divided per classes was available at http://clima.casaccia.enea.it/staff/rupolo/trajectorybook/index.html.

_{υ}In the following we will designate as turbulent the trajectories belonging to classes III and IV that, coherent with previous observations (e.g., Krauss and Böning 1987) and results from 2D turbulence studies (e.g., Babiano et al. 1987; Pasquero et al. 2001), are characterized by smaller correlation times and greater energetic contents (see Tables 5 and 6).

It is crucial to note the different relative importance (Table 7) of these classes in the five datasets. While in AF more than one-half of the trajectories (60%) belong to the turbulent classes III and IV, in the main world’s ocean basins (AD, PD, and ID) the surface trajectories are mainly influenced by high-frequency motions (class I) and the percentage of turbulent trajectories (classes III and IV) vary from a minimum of 17% (PD) to a maximum of 44% (ARD), the most turbulent surface dataset. The same relative importance, with different values of percentage, it is observed when analyzing drifter data without the boxcar smoothing procedure, as it is possible to see in Fig. 7 where are reported the percentages of drifters belonging to classes III and IV.

The capability of the proposed method of separating the whole dataset in more homogeneous classes is also evident from Fig. 12, where we plot for the five datasets the correlation functions of each class with the time axis normalized by *T _{L}*. Correlation functions of the same class, but that belong to different ocean basins, display very similar shapes, while correlation functions of the different classes are characterized by qualitatively dissimilar time behavior. In particular, it is worth noting the clear oscillations in the class IV (red curves in Fig. 12) and the smooth behavior for

*t*∼ 0 for the classes II, III, and IV, as expected for trajectories characterized by a finite variance of the acceleration. Looking in more detail, we may also notice that there are minor differences between the different datasets (as the great negative lobe for the velocity correlation functions of class I of AF) and discrepancies between the expected theoretical behavior and the computed normalized velocity correlation functions, as, for example, the weak oscillatory behavior for functions belonging to class III. This is not surprising since the analysis is based upon the estimate of the Lagrangian time scale directly from the single trajectory, through the first zero crossing technique. Nevertheless, the qualitative behavior of the dimensionless single-particle dispersion, that is obtained integrating the Taylor relation (5) without the dimensional value

*σ*

^{2}

_{U}and rescaling time by

*T*, is similar in the different datasets. In Fig. 12f we plot this function only for the AD dataset; this figure shows that the influence of the coherent structures (classes IV and III, red and blue lines) appears to be more important only for times smaller than ∼4–5

_{L}*T*. It is also interesting to note that for larger times only the single-particle dispersion of class IV clearly exhibits the transition from ballistic motion to the random walk regime, which is also found in 2D turbulence simulations (Babiano et al. 1987; Pasquero et al. 2001).

_{L}Quantitative information on the real dispersal properties is given by the dimensional single particle that obviously depends both on the kinetic energy and on the relative population of each class that, as it will be discussed in the next section, strongly varies in the geographical space and in the different current systems.

In a recent paper Veneziani et al. (2004) propose a quantitative criterion to define looping trajectories, based on the computation, for each trajectory, of the mean angular velocity Ω. Trajectories are classified as loopers when Ω exceeds a given threshold value (|Ω| > 0.1 days^{−1}). This criterion, applied to the subsurface floats in the northwestern Atlantic, give similar results in terms of number of looper floats. They then interpret the results in terms of a 2D first-order model with spin [formally equivalent to a 1D second-order model (Reynolds 2002)] in which the two characteristic time scales (*T _{L}* and |Ω|

^{−1}) are formally independent and the Lagrangian motion is due to a superimposition of a correlated noise and a spiraling component. As a contrast, the 1D second-order model presented here allows for a continuous transition from a situation characterized by correlated noise to looping trajectories. Another difference consists in the fact that directly computing Ω it is possible to distinguish cyclonic and anticyclonic loopers. Since both criteria need to estimate the derivative of the velocity, they are probably characterized by similar uncertainties, because of technical difficulties in determining the right smoothing procedure of the original trajectory. We think that a comparison of these two screening criteria could be of interest, and we believe that besides the limitation given by the difficult estimate of the time scales, the coherence of the results presented here confirms the validity also of the approach based upon the screening of the trajectories by means of the dimensionless index

*y*=

*T*/

_{a}*T*. In the next section, we will exploit these results to characterize the main oceanic current systems in terms of turbulence regimes.

_{υ}## 4. Dispersal properties and turbulence regimes in the main oceanic currents

Until now, all the analyses were performed carrying out means on quasi-homogeneous trajectories covering huge geographical areas. In this section we will study the main oceanic current systems mapping on the geographical space parameters computed from the trajectories. All these maps are obtained by averaging, on a grid of 1° × 1°, between all the points falling in a given bin, and assigning to each point, the trajectory value. This technique, which was already used by Lumpkin et al. (2002) and partly tested by Berloff et al. (2000), allows us to maximize the amount of data and to map quantities computed along the entire trajectory (as correlation parameters). On the other hand, the nonlocality of the considered values can affect the results in case of trajectories spanning a very large geographical area.^{1}

The geographical distribution of the Lagrangian eddy kinetic energy (EKE) at surface (Fig. 13a) highlights, similar to previously published results (e.g., Stammer 1997; Lumpkin et al. 2002), many of the main oceanic currents: the equatorial bands, the Gulf Stream and North Atlantic Current systems (GS/NAC), the Loop Current in the Gulf of Mexico, the Agulhas retroflection area and the Brazil–Malvinas confluence in the Atlantic Ocean; the Mozambique-Agulhas and the Leuuwin currents in the Indian Ocean; the Kuroshio and the East Australia Current (EAC) together with the area west of Central America in the Pacific. However, other important current systems do not show high values of EKE as, for example, the Antarctic Circumpolar Current (ACC), which is only partly marked by high values of EKE, or the North Equatorial Current at around 10°N in the western Pacific, which is even characterized by a minimum of EKE (Zhurbas and Oh 2004), because of their strong component of the mean zonal flow. At subsurface in the Atlantic (Fig. 13 b) the Gulf Stream, the Agulhas retroflection and the Brazil–Malvinas confluence areas are clearly evident.

The geographical distribution of *T _{L}* (Fig. 14) shows a general tendency, both at surface and subsurface, to have larger values in the less energetic interior of the oceans in opposition to the energetic currents that are characterized by shorter decorrelation velocity time scales (the opposite is true for the acceleration time scale, map not shown here). Some energetic areas, such as the GS, the EAC, the Loop Current, the Agulhas system and the Brazil–Malvinas confluence, are clearly marked by relative minima of

*T*. However, it is interesting to note that this general tendency it is not observed in the equatorial bands and is less evident in the Kuroshio Current [the same result was obtained by Stammer (1997)]. At surface the Lagrangian correlation length (map not shown) is less than 50–60 km almost everywhere, except in the Kuroshio region and in the tropical bands, where it reaches values greater than 100 km; in the energetic regions marked by a relative minimum of

_{L}*T*the Lagrangian correlation is only slightly greater than the background value.

_{L}The geographical distribution of the ratio *y* = *T _{a}*/

*T*shows that the subsurface AF dataset (Fig. 15b) is the most homogeneous and is characterized by a mean value of

_{υ}*y*that is relatively large everywhere. The highest values of

*y*=

*T*/

_{a}*T*(>0.8) are observed in the GS/NAC and Agulhas retroflection area, in the Brazil–Malvinas Current and, not uniformly, in the Mediterranean outflow area. With the exception of this latter area, the same regions are evidenced in the surface AD dataset (Fig. 15a), where high values of

_{υ}*y*are observed also between the NAC and Azores Current, in the Norwegian Current, in the Gulf of Mexico, in the central equatorial band and near the northern coasts of South America.

At the surface of the Pacific Ocean, one can distinguish, in a more quiescent background, the turbulent areas east of New Guinea, where the westward flow of the South Equatorial Current (SEC) turns to the north and east to feed the North Equatorial Counter Current (NECC), east of Australia (EAC), and west of Mexico. Moreover, it is possible to observe a large tongue with intermediate values of *y* within 20°–30°N southeast of the Kuroshio area, linked to the baroclinic instabilities in the seasonal North Subtropical Counter Current (NSTCC); a less-prominent counterpart it is observed also in the South Pacific where a strong seasonal signal is also observed (Zhurbas and Oh 2004).

In the Indian Ocean, relatively high values of *y* are observed along the eastern coasts of the Peninsula Arabica and of Africa till the Mozambique–Agulhas system, in the equatorial band and in the Leuuwin current west of Australia. Last, the few drifters in the ACC are characterized by relatively high values of *y*.

Even if the finite time and space scales of ocean variability do not support the eddy diffusivity model (e.g., Davis 1987) and the Lagrangian diffusivity is not defined if the velocity spectrum does not saturate at low frequencies, the use of an eddy diffusivity model is common in OGCMs. In recent years, efforts have been devoted to calculate lateral diffusivity from Lagrangian data, or directly from the Taylor relation (5) or following other approaches (e.g., Griffa et al. 1995; Davis 1987, 1991; Oh et al. 2000; for a review see Zhurbas and Oh 2003). The interest of this kind of study relies mainly in the investigation of possible relationship between diffusivity and kinematic or dynamical quantities, since in the OGCMs the actual value of the eddy diffusivity depends on many technical details and above all on the grid space resolution. Consequently, to characterize the different oceanic currents in terms of diffusivity, we simply map in Fig. 16 the quantity Γ = *σ*^{2}* _{U}T_{L}* =

*σ*that has to be considered as a, dimensionally homogeneous, proxy (Stammer 1998; Lumpkin et al. 2002) of the Lagrangian eddy diffusivity

_{U}L_{L}*K*defined in section 2.

The surface Γ field shown in Fig. 16 is very similar to the diffusivity field estimated by Zhurbas and Oh (2004) using a different approach (Zhurbas and Oh 2003). The maximum values of Γ presented here slightly exceed the maximum values of their estimate but the overall paths coincide. The only worth noting difference is that in their analysis is less evident the suppressed rate of diffusivity (discussed below) in the areas characterized by the presence of coherent structures.

At surface, Γ is higher in the main energetic current paths but, consistently with the previous results, the maximum values are observed in the area characterized by high energy and intermediate and low values of *y*, such as the Kuroshio region and the equatorial area. On the contrary, where turbulent trajectories are predominant, as for instance east of Australia or in the Malvinas–Brazil confluence, the same energetic content gives rise to a lower value of Γ.

In the Pacific, the NEC is characterized by a minimum of Γ that shows a general tendency to increase westward (eastward) in the midlatitudes (equatorial regions) and toward the equator in both hemispheres. The highest values are observed in the eastern equatorial Pacific and the lowest ones in the northeast Pacific, northward to the subpolar front and in the eastern subtropics in the South Pacific. In the Atlantic, the main energetic systems are characterized by relative maxima of Γ both at surface and subsurface (Fig. 16b), while in the Indian Ocean maximum values are found in the eastern equatorial regions and in the western boundary currents along the African coasts.

## 5. Summary and discussion

In this work, we have shown that in many oceanic current systems, both at subsurface and at surface, the Lagrangian motion is characterized by having a nonnegligible (with regard to the observational time sampling) acceleration time scale *T _{a}* that may also assume values similar to the velocity time scale

*T*. Moreover, we proposed the use of the ratio

_{υ}*y*=

*T*/

_{a}*T*to rationalize the analysis of huge Lagrangian dataset. In particular, when analyzing the GDP surface dataset and WOCE subsurface floats in the Atlantic, it was shown that trajectories having different values of

_{υ}*y*are characterized by different shapes, correlation and dispersal properties. This simple observation has lead to a first characterization of the dispersal properties in the main oceanic current systems.

The subsurface AF dataset is characterized by the highest values of the ratio *y* = *T _{a}*/

*T*and by having minimal values of

_{υ}*T*in the most turbulent and energetic areas (Gulf Stream, the Brazil–Malvinas confluence, and in the Agulhas retroflection). At surface, the less-energetic inner oceans are generally characterized by low values of

_{L}*y*=

*T*/

_{a}*T*and by mostly constant value of

_{υ}*T*.

_{L}Otherwise, the main energetic surface systems in the World Ocean basins can be divided in two categories. The first is characterized by a Lagrangian correlation time decreasing with the eddy kinetic energy and is massively populated by turbulent trajectories. To this category belong the Gulf Stream, Agulhas–Mozambique and the currents east and west of Australia, the Gulf of Mexico and the region west of Mexico. The second is characterized by a Lagrangian correlation length increasing with the eddy kinetic energy and is populated by trajectories having a relatively low value of the ratio *y* = *T _{a}*/

*T*. The energetic equatorial bands and partly the Kuroshio current belong to this category.

_{υ}The analysis performed in the previous sections indicates that in the first category the frozen-turbulence regime prevails, and in the second one the fixed-float regime prevails. This kind of a geographical characterization of the main oceanic currents can be useful to advance the representation of the eddy diffusivity in the OGCMs because, in these two regimes, the Lagrangian diffusivity varies differently according to the Lagrangian eddy kinetic energy.

In recent years, a large effort has been devoted by the scientific community to improve the parameterization of eddy diffusivity. This effort was mainly focused on the attempt of mimicking, in coarse resolution models, the effect of baroclinic instability by means of a diffusion coefficient (or an eddy-induced velocity) proportional to some index representative of the isopycnal slope. The justification for this approach is that baroclinic instability is the main mechanism of energy extraction and is the primary source of eddies (e.g., Treguier et al. 1997; Visbeck et al. 1997; Bryan et al. 1999). Nevertheless, in the oceans exist also regions characterized by large-eddy activity in the absence of strong density gradients. Stammer (1997) has shown (with hydrological and altimeter data) that in the vicinity of the East Australia Current, in the Brazil–Malvinas confluence and in the Agulhas Retroflection, a strong eddy field is present even in absence of important horizontal density gradients. This suggests that the scenario in which the source of eddies is the kinetic, rather than the potential, energy cannot be totally neglected, and that a complete parameterization of eddies should take into account complementary approaches. For instance, McWilliams and Gent (1994) using a double-gyre GCM proposed diffusivity to be proportional to the local kinetic energy times a tapering time dimensional factor, with prescribed spatial form, to avoid an overestimate of diffusivity in the energetic border currents. In the present work, we found strong indications that in the regions evidenced in the analysis of Stammer (1997) and in most of the western boundary currents the Lagrangian diffusivity is proportional to the square root of the eddy kinetic energy. This shows how Lagrangian data, that directly record both kinetic energy and the existence of eddies, may help to indicate in which geographical area a particular parameterization is appropriate.

As an example of this method, we mention the parameterization of the eddy diffusivity proposed by Babiano et al. (1987) that does not account for the contribution of baroclinic instability to the diffusivity fields but describes nonlocal effects of geostrophic turbulence. Such parameterization has been implemented on a regional OGCM by Rupolo et al. (2003) and results into a scaling of the eddy diffusivity coefficient with the square root of the eddy kinetic energy (Fig. 17). We think that the results discussed in this work suggest where it could be useful to improve OGCM results.

## Acknowledgments

Drifters and floats data were obtained from the GDP/SVP and WOCE Data Products Committee. I thank Cara Schock (MEDS) and M. Pazos and R. Lumpkin (NOAA/AOML) for their help in the preprocessing phase of the data. I am also grateful to B. L. Hua, A. Griffa, V. Artale, A. Babiano, and R. Iacono for stimulating discussions. I also acknowledge two anonymous reviewers, since their constructive comments and criticisms helped to improve this work, which was partly supported by the Italian Research Program in Antarctica (CANOPO-PNRA) and by the UE-funded project MFSTEP.

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List of SOFAR and RAFOS experiments. For each experiment are given the principal investigator, institution, geographical area, and number of 64-day-long trajectories considered here; S is SOFAR and R is RAFOS.

Number of original trajectories of drifters drogued at depth *z:* 0 < *z* < 50 m for each year.

Number of 64-day-long trajectories considered in this work and daily observations for the five datasets and for all the surface drifters data.

For each dataset: mean values and std dev of *T _{L}*,

*T*,

_{υ}*T*(expressed in days),

_{a}*y*, and eddy kinetic energy (cm

^{2}s

^{−2}). The number in parentheses is the median of the relative PDF.

Mean value and std dev of the correlation time *T _{L}* (days) of the different classes of trajectories of the five datasets.

Mean value and std dev of the eddy kinetic energy *σ*^{2}_{U} (cm^{2} s^{−2}) of the different classes of trajectories of the five datasets.

Mean values and std dev of *y* (classes I and II) and *x* (classes III and IV), number of trajectories, and relative percentage for each class of the five datasets.

^{1}

An alternative approach consists in binning Lagrangian velocities in the physical space and then to compute statistics in the obtained Eulerian map. In this case, when dealing with time correlation properties, one has first to compute a mean correlation function and then to estimate a single correlation time for the entire area.