## 1. Introduction

A better understanding of the Lagrangian transport of tracers, energy, and enstrophy in the ocean is relevant to a wide range of problems. It helps describe, for example, how pollutants (e.g., Rypina et al. 2013; Poje et al. 2014), freshwater (e.g., Mahadevan et al. 2016), or biological organisms (e.g., Rypina et al. 2014) disperse and how energy and enstrophy cascade across scales. The two-dimensional Lagrangian statistics of the near-surface oceanic flow and its associated transport can be explored using pairs of drifters, which float at the surface while being advected by the flow a few meters below.

In a turbulent ocean, the separation of pairs of drifters can theoretically be predicted when the kinetic energy spectrum is known (LaCasce 2008, 2016) and, furthermore, depends on the separation distance *r*. On scales above about 100 m, the vertical velocity makes a negligible contribution to the kinetic energy, which is dominated by the horizontal velocity. The horizontal pair dispersion, defined as the mean square separation of pairs of drifters, describes how a group of drifters disperses around its center of mass. If energy spectra are steep, like the *nonlocal* (Bennett 1984). If energy spectra are flatter, like the *local* (Bennett 1984; Richardson 1926).

In the presence of two inertial spectral ranges, pairs of drifters transition from one dispersion regime to another as their separation distances grow (LaCasce 2008). As an illustration, consider a two-dimensional, quasigeostrophic model often used to represent mesoscale dynamics in the ocean. At small separations, initially, in the enstrophy-cascading range, drifters disperse nonlocally until they reach the scale at which energy is injected (often through baroclinic instability at the first baroclinic deformation radius *L*_{D}, which we estimate as 60 km for our observations; Chelton et al. 1998). Subsequently, in the energy-cascading range, they disperse locally. Unlike in the atmosphere (Nastrom et al. 1986; Er-El and Peskin 1981), the kinetic energy spectrum in the ocean and its inertial ranges are not well documented. Subdeformation scale dynamics are not thoroughly understood, are often dominated by ageostrophic, divergent motions (Bühler et al. 2014; Callies and Ferrari 2013; D’Asaro et al. 2018), and vary significantly in time and space (Callies et al. 2015).

At submesoscales, considered as length scales of 0.1–20 km in our observations, a transition from nonlocal to local dispersion is likely to occur: The two-dimensional, quasigeostrophic flow, with energy spectra of

The Bay of Bengal, the location of this study (Fig. 1), hosts energetic submesoscale dynamics, which are evident in observations (e.g., Ramachandran et al. 2018) and in numerical simulations (e.g., Sarkar et al. 2016). These likely occur at freshwater-dominated density fronts, which are generated by massive seasonal freshwater fluxes, mainly from major rivers in the north, and intense precipitation during the southwest monsoon. The shallow freshwater cap affects the evolution of the sea surface temperature (SST; Jaeger and Mahadevan 2018) and the upper-ocean’s heat content (Shroyer et al. 2016; Mahadevan et al. 2016), both of which can alter the air–sea fluxes and, hence, affect the monsoon dynamics. The Air–Sea Interaction Regional Initiative (ASIRI; Lucas et al. 2014; Wijesekera et al. 2016) aims to understand the upper-ocean dynamics in the Bay of Bengal by extensive high-resolution measurements and modeling to eventually improve the monsoon forecasts.

The primary objectives of this study are to characterize the near-surface dispersion in the Bay of Bengal (specifically the dispersion of pairs of drifters) and to identify the dominant dispersion regime at submesoscales.

We use high-resolution data from a large cluster of drifters deployed as part of ASIRI in the Bay of Bengal (Hormann et al. 2016) to characterize the near-surface dispersion. Close pairs of drifters that report their position with high temporal resolution allow studying the submesoscale range of motions, which is challenging to observe synoptically with shipboard measurements because of the fast evolution of kilometer-scale features within time scales of hours to days. We compare the statistics of separations and velocities of drifter pairs to the theoretical predictions of local and nonlocal dispersion. Trajectories are low-pass filtered to determine the impact of inertial oscillations and small-scale processes on the dispersion statistics. For comparison, we simulate drifter trajectories using the satellite-derived, geostrophic flow field in the same region and contrast statistics from simulated, “AVISO-advected” drifters to the observed drifters. To identify the dispersion by spatially uncorrelated motions, we add a stochastic closure to the AVISO-advected drifters at small scales which is correlated in time.

In what follows, section 2 introduces the drifter dataset collected in the Bay of Bengal, the simulated drifter experiment based on satellite-derived flow fields, and the metrics used to characterize the near-surface dispersion. Section 3 presents the results for complementary dispersion statistics and compares drifter statistics to theoretical expectations. Section 4 offers a discussion of the results followed by a conclusion in section 5.

## 2. Data and approach

### a. Drifter deployments

We launched 46 surface drifters (one of which failed after deployment) during an extensive measurement campaign in the Bay of Bengal in September 2015 as part of ASIRI (Wijesekera et al. 2016; Hormann et al. 2016). The drifters were Surface Velocity Program (SVP) drifters (Niiler 2001; Maximenko et al. 2013) that consist of a buoy and a holey-sock drogue at 15-m depth. SVP drifters are part of the Global Drifter Program (Niiler 2001; Maximenko et al. 2013; Centurioni 2018) that aims to map the near-surface circulation of the global ocean and to provide SST and sea level pressure data. These data are important for calibration and validation of satellite-derived SST datasets (e.g., Zhang et al. 2009) and for numerical weather prediction (Centurioni et al. 2017; Horányi et al. 2017).

Drifters were released at the edge of a mesoscale cyclonic eddy and across a strong salinity and density front (Figs. 1a,b). With the goal of resolving motions over a wide range of length scales, we deployed drifters such that pair separations ranged from 0.5 to 30 km. We achieved this by deploying 10 clusters of four drifters (Hormann et al. 2016) as shown in the inset in Fig. 1. Each cluster of four drifters was deployed at almost the same time and provides six drifter pairs with a minimum separation of less than 1 km. The entire array of 10 clusters, with an intercluster separation of 5 km, was deployed over a period of 48 h.

During the first month after deployment, drifters reported their positions every 5 min (Hormann et al. 2016) giving a particularly high temporal resolution. After that, drifters reported every 30 min.

A conservative estimate of the position error is 50 m, which is likely a function of the region, sea state, and GPS coverage. The position data were quality controlled to remove erroneous GPS fixes and median filtered with a 1-h window to remove spurious events of acceleration. The velocity data of each drifter were then calculated by centered differencing. We bin the data to a 30-min grid for the first month and a 1-h grid after that by taking the median value of all points in this period. For the 90 days considered in this study, all drogues stayed attached, thus ensuring that they followed 15-m depth currents with an accuracy of ~0.1 m s^{−1} in winds up to 10 m s^{−1} (Niiler et al. 1995).

Since the drifters have a high temporal resolution, they resolve processes such as near-inertial oscillations and tides, and Langmuir turbulence. We assume that the effect of wind and surface waves on the drifter positions is small since their drogues are at 15-m depth and the Stokes’ drift is a second-order effect on the drifter displacement (Niiler et al. 1995). The Lagrangian frequency spectra of the observed drifters show high energy density at the inertial frequency and the M_{2} tidal frequency (Fig. 2; cf. Hormann et al. 2016), however, substantially less than at subinertial frequencies. For part of the subsequent analysis, we low-pass filter the position data using a fifth-order Butterworth filter with a cutoff of 1.5 times the inertial period

(a) Example trajectory for one drifter experiencing inertial oscillations. Unfiltered and low-pass-filtered trajectories are indicated in blue and orange, respectively. (b) Ensemble mean velocity power spectral density, before (blue) and after (orange) low-pass filtering with a cutoff of 1.5 times the inertial period. Dotted lines indicate the inertial and M_{2} tidal frequencies. The inertial and tidal spectral peaks seen in the original data (blue) are removed from the processed data by filtering (orange).

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

(a) Example trajectory for one drifter experiencing inertial oscillations. Unfiltered and low-pass-filtered trajectories are indicated in blue and orange, respectively. (b) Ensemble mean velocity power spectral density, before (blue) and after (orange) low-pass filtering with a cutoff of 1.5 times the inertial period. Dotted lines indicate the inertial and M_{2} tidal frequencies. The inertial and tidal spectral peaks seen in the original data (blue) are removed from the processed data by filtering (orange).

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

(a) Example trajectory for one drifter experiencing inertial oscillations. Unfiltered and low-pass-filtered trajectories are indicated in blue and orange, respectively. (b) Ensemble mean velocity power spectral density, before (blue) and after (orange) low-pass filtering with a cutoff of 1.5 times the inertial period. Dotted lines indicate the inertial and M_{2} tidal frequencies. The inertial and tidal spectral peaks seen in the original data (blue) are removed from the processed data by filtering (orange).

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

### b. AVISO-advected drifters

As a reference dataset for the large-scale, geostrophic circulation, we simulate drifters that are advected by satellite-derived, geostrophic currents *L*_{D}.

We trace the trajectories of the AVISO-advected drifters by time-integrating the currents using a fourth-order Runge–Kutta scheme and an hourly time step (Fig. 1c). This time step is well below the temporal resolution of AVISO; the geostrophic velocities

To compare with the observed trajectories, we release the AVISO-advected drifters in the same region in the Bay of Bengal and at the same time as the real drifters. Since we are interested in pairs of drifters, we initialize pairs of drifters with fixed initial separation creating a grid of *N* drifters and adding a second grid offset by a distance

### c. Stochastic drifters

Lagrangian trajectories of fluid parcels are significantly altered by small-scale processes that influence the dispersion properties, for example, how fast they spread from their source region or for how long they are trapped in a flow feature.

*W*(

*t*) [second term on the right-hand side in Eq. (1)]. The new drifter position

*x*component (and analogously for the

*y*component) of the position:

*e*-folding time of the memory loss.

We take ^{−1} for the turbulent velocity fluctuation, which is similar to the value found by Haza et al. (2012).

We use the first-order stochastic model to account for the subgrid processes and superpose it onto the AVISO-derived background currents

In the following sections, we will refer to three different datasets as 1) the observed drifters, 2) the AVISO-advected drifters, and 3) the stochastic drifters, which are the AVISO-advected drifters with stochastic noise.

### d. Dispersion metrics and structure functions

#### 1) Relative dispersion

A common metric for the dispersion of two particles is the mean square separation, known as relative dispersion *δ* at some time during the drift.

*N*pairs:

*r*.

Relative dispersion

At intermediate scales, however, the relative dispersion can be predicted from turbulence scaling laws given the energy spectrum,

We estimate the 95% confidence interval of the relative dispersion and diffusivity by bootstrapping. For each time, the population of available pairs is sampled 1000 times with replacement and the statistics are computed on each subsample. The confidence interval is then determined from the distribution of values that the subsamples generate.

#### 2) Finite-size Lyapunov exponent

Since pair dispersion is not only a function of the velocity field, but also of the pair separation itself, it is necessary to adopt metrics that invoke distance as their independent variable. Distance-averaged statistics treat the positions and velocities as Eulerian point measurements on an unstructured grid. Although Eulerian and Lagrangian statistics should be equivalent, sparse drifter trajectories often yield different results between time-averaged and distance-averaged metrics.

In theory, Lyapunov exponents are used to study exponential growth of pair separations. If the FSLE is constant over a range of distance classes, the *e*-folding time scale is constant, which is equivalent to an exponential growth of pair separations. However, FSLEs are also useful to study growth of separations that is not exponential; we can use scaling arguments to relate the FSLE power law exponent *β* to the mean square separation

Previous studies have shown that FSLEs are sensitive to the temporal evolution of the data (LaCasce and Ohlmann 2003; Poje et al. 2010), as well as the implementation of the method (Lumpkin and Elipot 2010; Haza et al. 2008). If the temporal resolution is not high enough, FSLEs underestimate the maximum Lyapunov exponent and miss transitions between regimes. Similarly, the quality of drifter data affects the estimate, as small-scale noise in the position data contaminates the FSLE at scales that are up to 6 times larger than the noise scale (Haza et al. 2014).

Here, we use the method of fastest crossing (e.g., Lumpkin and Elipot 2010; Haza et al. 2008) to determine

We estimate the 95% confidence interval of the FSLE by bootstrapping. For each separation bin, we generate 1000 subsets of the available crossing times by randomly resampling the data with replacement. For each subset, the FSLE is computed with the method described above. The confidence interval is then determined from the distribution of values that the subset generate.

#### 3) Pair separation PDF

Richardson (1926) was the first to distinguish dispersion regimes by studying pair separation probability density functions (PDFs) *r* at time *t*. PDFs and their statistics are powerful because they illuminate the pair dispersion process (Sullivan 1971) and encapsulate conventional pair dispersion statistics; the second moment of the pair separation PDF, for instance, is the relative dispersion. By comparing PDFs to the theoretically expected solutions for a given flow field, we can distinguish between different turbulent dispersion regimes (LaCasce 2010; Beron-Vera and LaCasce 2016; Bennett 1984; Graff et al. 2015).

To compare our data to the theoretical PDFs of the Lundgren, Richardson, and Rayleigh regimes, we first estimate *β* is related to the third root of the energy dissipation rate (Graff et al. 2015; LaCasce 2010; Beron-Vera and LaCasce 2016). We find *β* by fitting the theoretical prediction for the relative dispersion to the data. These fits are only computed over the initial period until separations are 10 times as large as the initial separation *T* is related to the third root of the enstrophy dissipation rate (Graff et al. 2015; Beron-Vera and LaCasce 2016; LaCasce 2010). Parameter *T* is found similarly, by fitting the theoretical prediction of relative dispersion to our data for scales *t* > 20 days.

Using

#### 4) Structure functions

**u**into its longitudinal,

*r*is the magnitude of the separation vector. Here, we take the average of the squared velocity difference for the computation of the structure function.

Due to the difficulty of observing the energy spectrum in the ocean, structure functions have been used to understand the distribution of energy across scales. As LaCasce (2016) points out, however, the calculation of energy spectra from drifter-derived structure functions is not practicable. Particularly at large scales, where a limited number of pairs is available and only a few realizations of the flow are sampled, the transformation to energy density produces large uncertainties. Nonetheless, structure functions give valuable information about the distribution of energy as a function of scale (Balwada et al. 2016). In particular, they are able to reproduce the scale-dependent transitions from one inertial subrange to another in the energy spectrum.

*r*is the pair separation distance,

*k*is the horizontal wavenumber. In the limit of

*r*.

We can predict the power law behavior of structure functions from Eq. (10). Given a self-similar energy spectrum

The longitudinal and transverse components of the structure function are not independent. Assuming that the structure function has a power law dependence

#### 5) Helmholtz decomposition

In a two-dimensional, quasigeostrophic regime, we expect the flow to be nearly nondivergent. In this case the rotational component of the structure function will be

## 3. Results

In this section, we characterize the dispersion in the upper ocean using the pair dispersion statistics and structure functions described above. We focus on the early-time and small-scale behavior that falls into the spatiotemporal regime of submesoscale motions.

### a. Relative dispersion

The observed relative dispersion *e*-folding time scale of about 1.25 days. As separations reach the deformation radius

Relative dispersion

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Relative dispersion

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Relative dispersion

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

At long times, *t* > 100 days, relative dispersion does not flatten to linear growth. Linear growth is expected for constant relative dispersion if pair separations grow larger than the dominant eddy size (≈100 km) and become uncorrelated (Fig. 4). A possible explanation is that the Indian coast prevents drifters from spreading isotropically and drifters become entrained into the boundary current (Fig. 1). Furthermore, dispersion calculated for large separations above 100 km has uncertainty, as only a small number of drifter pairs is available at this scale.

Pair velocity correlation normalized by the single particle velocity variance for three different initial separations: *e* from its maximum.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Pair velocity correlation normalized by the single particle velocity variance for three different initial separations: *e* from its maximum.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Pair velocity correlation normalized by the single particle velocity variance for three different initial separations: *e* from its maximum.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

The simulated, AVISO-advected drifters are explicitly advected with geostrophic currents that lack variability at scales smaller than 1/4°. AVISO-advected drifters are able to reflect the overall drift pattern of the observed drifters. Their trajectories, however, are significantly smoother and their rates of dispersion are slower than for the observed drifters. After the same period of drift, AVISO-advected drifters stay much more coherent and spread over a much smaller area than the observed drifter trajectories (Fig. 1).

Estimates of relative dispersion further illustrate the difference between the AVISO drifters and the observed drifters (Fig. 3). Since the AVISO energy spectrum is steep, likely

The stochastic drifters that include superposed noise in addition to the AVISO currents improve both the qualitative trajectories (Fig. 1), as well as the relative dispersion estimates. Generally, the stochastic noise increases the magnitude of relative dispersion compared to the AVISO drifters, such that the behavior is closer to the observed drifters. During the first 4 days, however, stochastic drifters show a different behavior than the observed drifters. Their relative dispersion grows more rapidly, possibly like

The relative diffusivity

In Fig. 5, the relative diffusivity is shown as a function of pair separation. For the observed drifters, a

Relative diffusivity, the time derivative of the relative dispersion (Fig. 3), as a function of separation *r*. The blue, green, and red curves denote the observed, AVISO-advected, and stochastic (AVISO-advected with stochastic closure) drifter trajectories, respectively. The shaded area is the bootstrapped 95% confidence interval of the relative diffusivity. The dotted lines indicate

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Relative diffusivity, the time derivative of the relative dispersion (Fig. 3), as a function of separation *r*. The blue, green, and red curves denote the observed, AVISO-advected, and stochastic (AVISO-advected with stochastic closure) drifter trajectories, respectively. The shaded area is the bootstrapped 95% confidence interval of the relative diffusivity. The dotted lines indicate

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Relative diffusivity, the time derivative of the relative dispersion (Fig. 3), as a function of separation *r*. The blue, green, and red curves denote the observed, AVISO-advected, and stochastic (AVISO-advected with stochastic closure) drifter trajectories, respectively. The shaded area is the bootstrapped 95% confidence interval of the relative diffusivity. The dotted lines indicate

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

The AVISO drifters and stochastic drifters show a similar behavior but with lower overall diffusivity. The stochastic noise increases the diffusivity compared to the AVISO drifters which is particularly pronounced at small scales below 5 km. Curve fitting results suggests that the AVISO drifters follow

### b. Finite-size Lyapunov exponent

The observed FSLEs (Fig. 6) are largest for the smallest separations. The associated *e*-folding time is about one day which is close to the *e*-folding time estimated from the initial phase of the relative dispersion. Due to the limited number of pairs available, however, uncertainties are large at the smallest scales. With the number of data points available, it is difficult to validate, whether the FSLE is constant over a range of scales as predicted for exponential pair dispersion. At intermediate scales, below

FSLE as a function of separation *r*. The blue, green, and red curves denote the observed, AVISO-advected, and, stochastic (AVISO-advected with stochastic closure) drifters, respectively. The shaded area is the bootstrapped 95% confidence interval of the FSLE. Dotted lines denote power law dependencies.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

FSLE as a function of separation *r*. The blue, green, and red curves denote the observed, AVISO-advected, and, stochastic (AVISO-advected with stochastic closure) drifters, respectively. The shaded area is the bootstrapped 95% confidence interval of the FSLE. Dotted lines denote power law dependencies.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

FSLE as a function of separation *r*. The blue, green, and red curves denote the observed, AVISO-advected, and, stochastic (AVISO-advected with stochastic closure) drifters, respectively. The shaded area is the bootstrapped 95% confidence interval of the FSLE. Dotted lines denote power law dependencies.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

The AVISO drifters show constant FSLEs from about 1 km to the deformation radius. If the FSLE is constant across a range of scales, the *e*-folding time is constant, and pairs separate exponentially fast. The smallest scales below 1 km, however, suffer from large uncertainty. Above *r*, such that the stochastic drifters behave like the AVISO drifters above

Since time resolution is important in the computation of FSLEs and the sea level anomaly data is updated daily, it is expected that FSLE estimates from AVISO-advected drifters cannot resolve the *e*-folding time scale of 1–2 days below 5 km. Similarly, the 30-min resolution of the observed drifters might not be able to resolve the true maximum FSLE.

### c. Pair separation PDFs

The pair separation PDFs (at

Pair separation PDFs at

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Pair separation PDFs at

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Pair separation PDFs at

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Estimated parameters for the theoretical pair separation PDFs.

The PDFs for the AVISO separations (Fig. 7b) occupy a much smaller range of scales. It is evident that the Lundgren distribution is the best fit for all three initial separation classes. The stochastic drifters, however, behave differently from the two other datasets. The stochastic noise causes a fast widening of the PDFs, such that both Lundgren and Richardson solutions fail to predict the distribution. The widening of the PDF is caused by uncorrelated motions and is comparable to the Rayleigh dispersion. Since we estimate the diffusion coefficient from the late-time behavior (*t* > 10 days) of the relative dispersion, this diffusivity is not captured. The fast dispersion of the stochastic drifters at small scales (*r* < 10 km) has also been observed in the relative diffusivity (Fig. 5) and the FSLE (Fig. 6). The character of this dispersion is different from Lundgren and Richardson as well as the late-time Rayleigh regime.

The theoretical solution for the Rayleigh regime, that is, dispersion by uncorrelated pair velocities, does not describe any of the datasets. This is expected for these small initial separations and after 3 days, in particular, because the diffusivity for the Rayleigh regimes is estimated from the late-time relative dispersion. As we see in the velocity cross correlation (Fig. 4), the velocities are correlated below the deformation radius

The kurtosis, the fourth moment of the PDFs, is a metric to quantify the peakedness of a distribution. We group the data into the three classes of initial separation

Kurtosis of the pair separation PDFs as a function of time for the (a) observed, (b) AVISO-advected, and (c) stochastic drifters (AVISO-advected with stochastic closure). Colors indicate the three classes of initial pair separation

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Kurtosis of the pair separation PDFs as a function of time for the (a) observed, (b) AVISO-advected, and (c) stochastic drifters (AVISO-advected with stochastic closure). Colors indicate the three classes of initial pair separation

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Kurtosis of the pair separation PDFs as a function of time for the (a) observed, (b) AVISO-advected, and (c) stochastic drifters (AVISO-advected with stochastic closure). Colors indicate the three classes of initial pair separation

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

In case of the observed drifters (Fig. 8a), pairs that are initially close (

The AVISO drifters (Fig. 8b) produce less of an exponential growth (at least at these initial separation scales) with maximum values that are larger than 5.6. The late-time asymptotic behavior of the AVISO drifters could be identified as Richardson-like, as values oscillate around 5.6. Interestingly, and opposed to the observed drifters, values are larger than 2, suggesting that a Rayleigh regime is unlikely.

Contrary to the AVISO drifters, the stochastic drifters generate an exponentially growing kurtosis (Fig. 8c). Particularly the smallest separation class grows to a maximum value of 25, which can clearly be attributed to nonlocal behavior. The difference between AVISO drifters and stochastic drifters suggests that the stochastic noise facilitates faster separation. The exponential growth of the kurtosis also occurs at a later time than for the observed drifters which is an artifact of the resolution of the AVISO currents.

### d. Structure functions

The structure functions *r*. At intermediate scales, 10–100 km, *r* < 10 km) and at large scales (*r* > 100 km),

(a) Second-order velocity structure functions as a function of separation *r* for the observed, AVISO-advected, stochastic (AVISO-advected with stochastic closure), and low-pass-filtered trajectories. (b) Velocity structure functions after decomposition into longitudinal and transverse, and rotational and divergent components. The total observed structure function is the same in (a) and (b). The solid vertical line indicates the length scale at which the rotational component becomes dominant. The dotted lines indicate the theoretical slopes of nonlocal (

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

(a) Second-order velocity structure functions as a function of separation *r* for the observed, AVISO-advected, stochastic (AVISO-advected with stochastic closure), and low-pass-filtered trajectories. (b) Velocity structure functions after decomposition into longitudinal and transverse, and rotational and divergent components. The total observed structure function is the same in (a) and (b). The solid vertical line indicates the length scale at which the rotational component becomes dominant. The dotted lines indicate the theoretical slopes of nonlocal (

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

(a) Second-order velocity structure functions as a function of separation *r* for the observed, AVISO-advected, stochastic (AVISO-advected with stochastic closure), and low-pass-filtered trajectories. (b) Velocity structure functions after decomposition into longitudinal and transverse, and rotational and divergent components. The total observed structure function is the same in (a) and (b). The solid vertical line indicates the length scale at which the rotational component becomes dominant. The dotted lines indicate the theoretical slopes of nonlocal (

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

Low-pass filtering the trajectories affects

It is evident in Fig. 9b that the rotational component and the divergent component cross at approximately 75 km, which is close to the local deformation radius

The ratio of transverse to longitudinal components *r* < 50 km and increases to about 1.5 for *r* > 75 km. The ratio

(a) Ratio of the transverse and longitudinal component of the velocity structure function (*r*. (b) Ratio of the rotational and divergent component of the velocity structure function after Helmholtz decomposition (*r*. Colors denote the observed, AVISO-advected, stochastic (AVISO-advected with stochastic closure), and low-pass-filtered datasets. Horizontal dotted lines are for reference.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

(a) Ratio of the transverse and longitudinal component of the velocity structure function (*r*. (b) Ratio of the rotational and divergent component of the velocity structure function after Helmholtz decomposition (*r*. Colors denote the observed, AVISO-advected, stochastic (AVISO-advected with stochastic closure), and low-pass-filtered datasets. Horizontal dotted lines are for reference.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

(a) Ratio of the transverse and longitudinal component of the velocity structure function (*r*. (b) Ratio of the rotational and divergent component of the velocity structure function after Helmholtz decomposition (*r*. Colors denote the observed, AVISO-advected, stochastic (AVISO-advected with stochastic closure), and low-pass-filtered datasets. Horizontal dotted lines are for reference.

Citation: Journal of Physical Oceanography 49, 9; 10.1175/JPO-D-18-0181.1

The ratio of rotational to divergent component *r* > 75 km. Filtering removes energy predominantly from the divergent component and shifts

In the limit of divergence-dominated flow (*r* < 10 km), the ratios agree with the theoretical expectations. The total structure function of the observed drifters has a slope of

The finding that motions are divergent at small scales suggests that two-dimensional, geostrophic dynamics are no longer dominant. Candidates for divergent motions could be the internal wave field, inertial oscillations, and Langmuir turbulence as well as the horizontally divergent submesoscale flow.

## 4. Discussion

### a. Dispersion regimes

The results of the dispersion statistics (relative dispersion and diffusivity, FSLE, pair separation PDFs, and kurtosis) describe the dispersion characteristics of the flow in the Bay of Bengal. The combination of distance-averaged and time-averaged metrics allows for a more complete description of the flow. Additionally, the structure functions contribute to the understanding of the distribution of energy across scales. In the absence of knowledge about the energy spectra, this can be useful despite the fact that structure function can be affected by nondispersive modes, which are part of the true, observed flow.

The drifter-derived relative dispersion and FSLE (for pairs with *e*-folding time scale predicted by the FSLE is half as large as for the relative dispersion and the regime shift from exponential to power-law growth is at 20 km (Fig. 6) as opposed to 10 km shown in the relative dispersion (Fig. 3). The relative diffusivity is consistent with this result and grows like

Dispersion regimes in Lagrangian statistics at submesoscales,

The pair separation PDFs and their kurtoses suggest that a Lundgren distribution is the best fit for initially close pairs (

On the contrary, structure functions show a

The decomposition of structure functions indicates that motions are divergent at scales below 75 km and the ratio between rotational and divergent motions

Energy at small scales that causes the structure function to indicate a local regime can likely not be explained by inertial oscillations, which begs the question as to which processes are responsible for the variability at those scales. We find that motions become increasingly divergent below the deformation radius and clearly deviate from two-dimensional, quasigeostrophic dynamics below 20 km. The kurtosis supports this at scales above 5 km, which quickly falls off to a value of 2 (as the Rayleigh asymptotic limit). The ratios between longitudinal and transverse structure functions, and between rotational and divergent structure functions, further suggest that those motions are weakly rotational, a property that applies to the internal wave continuum as well as to balanced dynamics that have a considerable ageostrophic component.

### b. Consequences of stochastic noise

While at global or basin scale, AVISO-advected drifters can reproduce the observed relative dispersion, they are not sufficient to model dispersion on the regional scale,

The first-order model implemented on the AVISO-advected trajectories has two primary effects, both of which are illuminating when interpreting the dispersion of observed drifters. First, diffusive growth tends to be faster than exponential spreading at early times. The dominant drivers for pair separations are, thus, the uncorrelated motions due to the stochasticity in the velocity. This can most clearly be seen in the pair separation PDFs that resemble dispersion due to uncorrelated velocities (Fig. 7), the diffusive growth of the relative dispersion (Fig. 3) as well as the flattening of structure functions at the smallest scales. Second, random motion in a constant shear flow leads to a relative dispersion that grows like

The subgrid-scale noise affects scales much larger than the noise scale. In fact, FSLE and structure functions suggest that the stochastic noise affects drifter motions at scales up to 10 km. This scale can be identified by comparing the stochastic drifters with the AVISO drifters. Haza et al. (2014) found that uncertainty in the drifter position can affect the dispersion statistics at scales up to 6 times the magnitude of the position error.

### c. Limitations of the dataset

The results have to be interpreted taking into account the shortcomings of each metric and the limitations of the dataset itself. Since drifters were drogued at 15-m depth, mixed layers were possibly shallower than that during the first month (Hormann et al. 2016), and submesoscale turbulence decays away from the surface, we expect that the drifters measure less small-scale variability and steeper spectra than they would at the very surface. The fact that the drogue is deeper than the mixed layer, furthermore, reduces the bias due to surface wave motions, and most likely reduces the effect of convergent flow such as Langmuir circulation on the distribution of drifters. These convergent flows otherwise tend to align drifters and bias the regions they sample.

Additionally, a sampling bias occurs because of the choice of deployment site. Certain features in the flow are sampled more extensively, rather than a representative sampling of the velocity field. The release location causes drifters to be entrained in a cyclonic eddy for the first 10 days. During this period, drifters traveled as a coherent cluster, reducing the degrees of freedom of our statistics due to dependent pairs.

The energy at the mesoscale might also overshadow any coherent small-scale motion and result in nonlocal dispersion. This has severe consequences when inferring the energy spectrum from dispersion statistics. Additionally, as our results suggest, there is a large uncertainty associated with the largest scales. At large scales, a limited number of drifters samples the mesoscale eddies, providing only a limited number of realizations.

## 5. Conclusions

The dispersion study presented here identifies a contradiction between pair dispersion statistics (relative dispersion, FSLE, pair separation PDFs) and an analysis of structure functions. Pair dispersion statistics consistently identify a nonlocal dispersion regime at scales below 20 km that is associated with an exponential rate of pair separation, and predicted by energy spectra that follow a

To answer the question we posed in the title, submesoscale flows are possible to observe from pair dispersion statistics, however, the interpretation of the data can be difficult due to the richness of processes occupying the same spatiotemporal band. In particular, we find that there are motions such as near-inertial oscillations and tides that affect the energy spectrum, but are inefficient at dispersion. Theoretical predictions for the submesoscale range, therefore, do not hold for the observed data. The limitations of our dataset further constrains our ability to resolve the submesoscale range.

Our findings are relevant when studying drifter dispersion at submesoscales, especially in the presence of an energetic mesoscale circulation, as they question the ability of pair dispersion statistics to capture submesoscale flows. Alternatively, in a region of thin mixed layers, submesoscale flows could be inefficient at dispersing drifters as there is less available potential energy to be extracted by mixed layer instabilities.

Since dispersion statistics are often not conclusive in inferring an energy spectrum from pair statistics, particularly at small scales, more information about the flow field is needed. Multiple drifter statistics and clusters can help to map velocity gradients to further characterize the kinematics of a flow field. Velocity gradients are especially important in submesoscale flows, where the local Rossby number becomes

## Acknowledgments

This research was supported by the Air Sea Interaction Regional Initiative (ASIRI) under ONR Grant N00014-13-1-0451 (SE and AM) and ONR Grant N00014-13-1-0477 (VH and LC). Additionally, AM and SE thank NSF (Grant OCE-I434788) and ONR (Grant N00014-16-1-2470) for support; VH and LC were further supported by ONR Grant N00014-15-1-2286 and NOAA GDP Grant NA10OAR4320156. We thank Joe LaCasce, Dhruv Balwada, and one anonymous reviewer for helpful comments and discussions that significantly improved this manuscript. The authors thank the captain and crew of the R/V *Roger Revelle*. The SVP-type drifters are part of the Global Drifter Program and supported by ONR Grant N00014-15-1-2286 and NOAA GDP Grant NA10OAR4320156 and are available under http://www.aoml.noaa.gov/phod/dac/. The Ssalto/Duacs altimeter products were produced and distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS, http://www.marine.copernicus.eu).

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