## 1. Introduction

The turbulent dispersion of tracers is one of the fundamental problems of physical oceanography and is relevant at a wide range of scales. Tracer dispersion can be quantified from the motion of particles advected by the flow. From observations, estimates of oceanic dispersion have been calculated on the mesoscale (≳10 km; e.g., Zhurbas and Oh 2004; Rypina et al. 2012), on the submesoscale (0.1–10 km; Poje et al. 2014), and within the surfzone (10–100 m; Spydell et al. 2007; Brown et al. 2009) using surface drifters and subsurface floats (e.g., Rupolo 2007). LaCasce (2008) offers a thorough review.

*k*

^{−5/3}turbulent wavenumber spectra (Kolmogorov 1941). The essence of turbulent relative dispersion is that as particles separate to larger length scales, larger more energetic eddies are more effective at dispersing the particles. For two particles separated by the distance

*r*in an inertial subrange, this results in a mean squared two-particle separation, or dispersion

*t*since release as

*K*

_{r}:

*r*

_{0}(e.g., Salazar and Collins 2009), hence the coefficients of proportionality in (1) and (2) will not depend on

*r*

_{0}but rather depend only on the energy dissipation rate

*t*after release large enough so that the two-particle separation vector

**r**(

*t*) no longer depends on the initial separation vector

**r**

_{0}(e.g., Salazar and Collins 2009).

*t*just after release such that the separation

**r**(

*t*) depends on the initial separation

**r**

_{0}. From dimensional considerations and a Taylor expansion of the velocity field about one of the particles (e.g., Ouellette et al. 2006), the perturbation dispersion

*r*′ = ||

**r**(

*t*) −

**r**

_{0}||, grows ballistically in time

*t*≫

*t*

_{B}, memory of

*r*

_{0}is lost. A different Batchelor time scale

*t*

_{B}(4) are equivalent (

*t*

^{3}growth (e.g., LaCasce 2008).

**r**

_{1}and

**r**

_{2}) separated by

*r*= ||

**r**

_{2}−

**r**

_{1}||. For isotropic, homogenous, and stationary flow, Eulerian statistics of (random) initial separations

*r*

_{0}are equivalent to Eulerian statistics of later (random) separations

*r*such that

*S*(

*r*

_{0}) =

*S*(

*r*). The structure function characterizes the spatial structure of the velocity field and is an alternative equivalent to the velocity wavenumber spectra

*E*(

*k*). In a

*k*

^{−5/3}inertial subrange,

*S*(

*r*

_{0}) =

*S*(

*r*) due to flow stationarity,

*S*(

*r*

_{0}) ≠

*S*(

*r*) from Lagrangian observations as drifters tend to preferentially sample regions of convergence (Pearson et al. 2019). Furthermore, although it is possible to infer Eulerian wavenumber spectra from structure functions based on Lagrangian data for nondivergent flows, the spectra can be inaccurate due to the filtering inherent in the transform (LaCasce 2016).

Some previous research has suggested ballistic (~*t*^{2}) growth for the full dispersion

Early oceanographic observations of scale-dependent diffusivity were suggestive of RO scaling over a very wide range of length scales 10^{2} < *r* < 10^{5} m (Okubo 1971) implying a *k*^{−5/3} wavenumber spectra. In the Gulf of Mexico, two-particle diffusivities *K*_{r}, derived from many drifters show clear evidence of ^{5}-m length scales; Poje et al. 2014). Scale-dependent diffusivities were also found for 10–100-m surf zone observations (Spydell et al. 2007). The dispersion of drifter separations is also consistent with *D*_{r} ~ *t*^{3} in the Gulf of Mexico for *t* ≥ 10 days (Ollitrault et al. 2005), in the North Atlantic for *t* > 1 day (Lumpkin and Elipot 2010), and in the Nordic Sea for 2 < *t* < 10 days (Koszalka et al. 2009).

At the same space (≲10 km) and time (≲10 days) scales for which some observations have found RO scaling, other observations suggest *D*_{r} ~ exp(*t*) and *K*_{r} ~ *D*^{2}. As dispersive scalings are linked to the underlying wavenumber spectra (e.g., Foussard et al. 2017), these observations therefore suggest a steeper *k*^{−3} wavenumber spectra at these scales implying a 2D turbulence enstrophy cascade (Lin 1972). For example, LaCasce and Ohlmann (2003) reported *t* ≤ 10 days and *r*_{0} ≤ 10 km using drifters deployed in the Gulf of Mexico. Drifters deployed in the Nordic Sea also suggest *D*^{2} ~ exp(*t*) for 0.5 ≤ *t* ≤ 2.5 days and *D* < 10 km before dispersion transitioning to RO scaling (Koszalka et al. 2009). Relative dispersion in the Benguela Upwelling Region shows similar enstrophy scaling at smaller space and time scales which then transitions to RO scaling at larger space and time scales (Dräger-Dietel et al. 2018). In these studies, the dispersion *r*_{0} (e.g., Ohlmann et al. 2012; Romero et al. 2013; Dauhajre et al. 2019), precluding an examination of Batchelor scaling which depends explicitly on *r*_{0}. As the dispersive scaling is directly linked to the turbulence wavenumber spectra (e.g., Foussard et al. 2017), determining the correct dispersive scaling is critical to properly inferring the turbulence responsible for the dispersion.

Although dispersion is generally investigated from the perspective of turbulence, nonturbulent motions can lead to dispersion. For example, the effect of internal wave processes on dispersion has been examined (e.g., Young et al. 1982; Suanda et al. 2018). However, in these studies vertical tracer or particle motion is also required as the enhanced horizontal dispersion results from vertically sheared currents. For surface trapped tracer, convergent motions at the surface lead to spatially localized tracer concentrations (e.g., Okubo 1980; Maximenko et al. 2012; D’Asaro et al. 2018). For various flow situations, various clustering rates due to convergent motions have been estimated using a variety of methods (e.g., Huntley et al. 2015; Gutiérrez and Aumaître 2016; Koshel et al. 2019). How convergence and divergence affects surface drifter dispersion is not well understood, however, the presence of convergence/divergence may affect the dispersion relative to RO scaling (Cressman et al. 2004).

In this paper, surface drifter dispersion is examined for drifters deployed in shallow continental shelf waters (*h* < 40 m) off of Pt. Sal, CA. The dispersion is examined at short enough time and length scales to resolve Batchelor scaling. This paper is organized as follows. First, two-particle dispersion statistics and structure functions are defined (section 2). For example, proper definitions of the dispersion

## 2. Background

### a. Two-particle dispersion statistics

*t*after release of drifter

*m*be given by

*X*

_{m}(

*Y*

_{m}) are the easting (northing) of the drifter. The initial position of the drifter is

**X**

_{m,0}≡

**X**

_{m}(

*t*= 0). The vector between two drifters

*m*and

*n*is

**r**

_{mn}(

*t*) =

**X**

_{m}(

*t*) −

**X**

_{n}(

*t*) with the initial separation vector given by

**r**

_{mn,0}=

**X**

_{m,0}−

**X**

_{n,0}. The squared separation for an individual drifter pair is given by

*m*, the position relative to the initial position, the drifter displacement, is indicated by a superscripted prime

**X**

_{m}(

*t*) =

**X**

_{m}(

*t*) −

**X**

_{m,0}. The difference in displacements between drifter

*m*and

*n*is considered the perturbation separation

*θ*

_{mn}is the angle between

**r**

_{mn,0}and

*r*

_{mn}and

*r*

_{0}are now defined. The dispersion

*r*

_{0}. The number of pairs with

*r*

_{0}initial separation is time dependent and denoted

*N*

_{p}=

*N*

_{p}(

*t*,

*r*

_{0}). Similarly, the perturbation dispersion is

*K*

_{r}measures the average squared separation rate and is defined as

*K*

_{r}equals Taylor’s single particle diffusivity (Taylor 1922). This occurs, for example, for drifters separated by a distance larger than the largest eddy length scale. The perturbation diffusivity is

*r*

_{0}, the diffusivity and perturbation diffusivity are related by

*d*Φ(

*t*,

*r*

_{0})/

*dt*is typically assumed to be zero, it will be shown that this term is important to the dispersion investigated here similarly to laboratory dispersion (Ouellette et al. 2006).

### b. The Batchelor regime

*C*

_{R}is a nondimensional constant (e.g., Salazar and Collins 2009). Thus, RO scaling is independent of

*r*

_{0}and only formally valid if

*t*≫

*t*

_{B}) where perturbation dispersion is much larger than the initial separation

*k*

^{−5/3}turbulence, Batchelor (1950) derived scaling for the perturbation dispersion

*r*

_{0}. The time scale separating RO (1) and Batchelor scaling (3) is the Batchelor time

*t*

_{B}(e.g., Salazar and Collins 2009), here defined as

*γ*= 2 to

*γ*= 3. However, laboratory experiments indicated that the dispersion is weaker than Batchelor (

*γ*< 2) as the dispersion transitions out of the Batchelor regime (Ouellette et al. 2006). Although, the Batchelor regime was originally derived for

*k*

^{−5/3}inertial subrange turbulence, for

*t*≪

*t*

_{B}Batchelor scaling

*E*~

*k*

^{−β}as long as

*β*< 3.

### c. Structure function definitions

Here the separation *r* dependent velocity structure function *S*(*r*) is used to examine the spatial structure of the flow. Structure functions are fundamentally Eulerian statistics but can be estimated from drifters (e.g., Poje et al. 2017), although the results may be biased relative to *S*(*r*) calculated from Eulerian data as divergent motions may preferentially place drifters in convergence zones (Pearson et al. 2019). We calculate *S*(*r*) from drifter pair trajectories using (5). For *S*(*r*), the averaging ⟨·⟩ in (5) is over all drifter pair velocity increments *δ***u**_{mn} = **u**_{m}(**r**_{m}) − **u**_{n}(**r**_{n}) separated by *r* = ||**r**_{mn}||. Thus, the averaging for structure functions differs from the averaging for dispersion statistics. Unlike *D*^{2}(*t*, *r*_{0}) and *K*(*t*, *r*_{0}), where averaging is done over pairs of drifters at time *t* separated initially by *r*_{0}, structure function *S*(*r*) averaging is over drifter pairs and times for which the drifters are separated by *r*.

*S*(

*r*) is related to the wavenumber spectra

*E*(

*k*) by Fourier transform as the space-lagged correlation function and

*E*(

*k*) are Fourier transform pairs (Babiano et al. 1985). Thus, energy spectra, structure functions, and dispersion are linked via

*β*< 3 wavenumber spectra,

*t*≫

*t*

_{B}corresponding to

*t*≪

*t*

_{B}, the perturbation dispersion follows a Batchelor scaling

*β*< 3 with structure function

*β*= 3 in the 2D turbulent enstrophy cascade,

*S*~

*r*

^{2}and the dispersion for all

*t*is given by

*k*

^{−3}spectra.

## 3. Methods

### a. Lagrangian data

Drifter releases were performed during September and October of 2107 as part of the Office of Naval Research–funded Inner Shelf experiment conducted near Pt. Sal, CA (Kumar et al. 2021; Spydell et al. 2019). Unlike many previous studies (e.g., Ollitrault et al. 2005; Koszalka et al. 2009; Lumpkin and Elipot 2010; Beron-Vera and LaCasce 2016), but similar to Ohlmann et al. (2012) where there were ≈12 drifter release realizations, drifters were repeatedly released in the same geographic area increasing the number of independent drifter release realizations. Coastal Ocean Dynamics Experiment (CODE) drifter bodies (Davis 1985) were equipped with SPOT Trace GPS receivers (Subbaraya et al. 2016; Novelli et al. 2017) nominally sampling every 2.5 min. SPOTs have been used in other oceanographic drifter studies (Beron-Vera and LaCasce 2016; Pearson et al. 2019) and methods to reduce their errors developed (Yaremchuk and Coelho 2015). Consistent with previously reported SPOT position errors between 2 and 10 m (Yaremchuk and Coelho 2015; Novelli et al. 2017), we estimate SPOT errors to be ≈4 m based on comparing SPOTs that were codeployed with higher-accuracy GPSs on some drifters. Drifters followed the mean surface horizontal flow between approximately 0.3 to 1.2 m below the surface. The water following properties of CODE drifters is well established (Poulain 1999; Novelli et al. 2017). Wind-induced drifter slips (<0.01 m s^{−1}, 0.1% of wind speed; Poulain 1999; Poulain and Gerin 2019) were small compared to the currents (≈0.15 m s^{−1}) as wind speeds during drifter releases were much less than the maximum midafternoon wind speed 10 m s^{−1} recorded on two of the days.

Drifters were released in 10–40-m water depths (Fig. 1a). Here, 12 drifter releases are analyzed: 8 releases were off of the rocky Pt Sal headland (see Fig. 1b for an example release) and 4 releases were off the long sandy beach area called Oceano (dots in Fig. 1a are initial drifter positions for each release colored by latitude, blues to the south, reds to the north). Trajectories varied from 3 to 23 h long depending on the release leading to the median length of drifter pair trajectories ranging from 3.2 to 22.9 h (Table 1). The relatively long pair trajectories from release 4 and 9 are due to some drifters being left out overnight. The drifter deployment pattern, multiple groups of 9 drifters arranged in a plus pattern (blue dots Fig. 1b), consists of various initial separations *r*_{0} from 100 to 2000 m. For each release, all drifters were deployed within approximately 35 min—the mean over the 12 releases of the time it took to deploy the drifters for each release. As the deployments were quickly performed, time *t* for drifter statistics, such as

Pt. Sal drifter release information. The starting time (column 2) of each release, the number of drifters (column 3), the total number of drifter pairs (column 4), initial separation information (columns 5–7), and pair trajectory length *T*_{p} (columns 8–10) information.

SPOT GPS data contain gaps (e.g., Yaremchuk and Coelho 2015) with the time between fixes *δτ* ≥ 2.5 min. For the trajectories used here, 91% of all *δτ* is ≤5 min with the mean time between fixes *δτ*_{mx} = 45 min are not included in the analysis. For each release, the number of trajectories *N* meeting this requirement (all *δτ* < *δτ*_{mx}), and the number of drifter pairs *N*_{p}, are shown in Table 1. The number of pairs *N*_{p} sometimes differs from *N*(*N* − 1)/2, the number of pairs given *N* drifters, because, infrequently, two trajectories from the same release do not overlap in time. The results reported here do not depend on the gap criterion *δτ*_{mx} and nearly identical results are obtained if the requirement is loosened (e.g., *δτ*_{mx} = 1 h) or tightened (e.g., *δτ*_{mx} = 15 min).

Drifter trajectories are processed as follows. 1) GPS latitude–longitude fixes, sampled at 2.5 min, are projected onto the local UTM plane. 2) Easting and northing drifter positions with gaps are differenced to obtain velocities. 3) Velocities are then linearly interpolated to times separated by 2.5 min filling any gaps. 4) Velocities are integrated to obtain positions with the constant determined such that the mean square difference between the original trajectory and the interpolated trajectory is minimized. 5) Position spikes are removed by linearly interpolating positions for which acceleration (velocity differences) magnitudes are >0.0387 m s^{−2}. This acceleration magnitude removes all obvious outlier positions. Only 0.13% of all positions required despiking. Velocities are then recomputed from despiked positions. 6) A 5 min (3 point) moving boxcar average is then applied to positions and velocities resulting in the drifter positions **X**(*t*) and velocities **U**(*t*) analyzed here. Assuming independent (every 2.5 min) position errors of 4 m, 5-min averaged positions **X**(*t*) have approximately 2.3 m errors and 5-min averaged velocity **U**(*t*) errors are approximately 0.015 m s^{−1}.

Results are presented for two different averages. First, experiment averaged (EA) drifter statistics are presented for which the averaging ⟨·⟩ in (8), (9), and (15) is over all possible drifter pairs from the entire experiment, i.e., averaging over all 12 drifter releases. Second, single release (SR) averaged statistics are presented for which the averaging is only over drifter pairs for a particular release. Thus, SR statistics are based on averages over fewer drifter pairs than EA statistics. For EA statistics, there are a total of 2187 drifter pairs (*N*_{p} in last row of Table 1). For the entire experiment, the majority of initial separations *r*_{0} are ≤1500 m (Fig. 2). Initial separations *r*_{0} are binned every 250 m from 250 to 3000 m (bin centers) with >200 drifter pairs for *r*_{0} ≤ 1500 m, whereas there are fewer drifter pairs (*N*_{p} ≤ 100) for *r*_{0} ≥ 1750 m. The mean *r*_{0} within each bin, for 250 ≤ *r*_{0} ≤ 1500 m, is very close to the bin center (see red plus signs in Fig. 2). For this reason, and because there are few pairs for *r*_{0} ≥ 1750 m, only results for 250 ≤ *r*_{0} ≤ 1500 m are presented for which there are a total of 1998 drifter pairs. The number of drifter pairs used for experiment averaged statistics at each *r*_{0} is larger than some previous studies (e.g., Koszalka et al. 2009; Ohlmann et al. 2012) with similar *r*_{0}, thus, the experiment averaged statistics reported here are robust relative to previous estimates for *r*_{0} ≤ 1500 m. Statistics depend on time *t* where *t* = 0 is the first time for each drifter pair trajectory, i.e., the time of the first GPS fix for which both drifters are in the water. Relative dispersion statistics are not affected by the time gap between drifter deployments as statistics are only a function of time *t* since both drifters are deployed. For experiment averaged statistics time *t* does not correspond to a UTC time whereas for single release statistics, assuming drifters were rapidly deployed, *t* is the time in UTC since deployment.

### b. Eulerian data

For the two months of the Pt. Sal experiment, 46 collocated upward-looking ADCPs and temperature moorings were deployed. Velocity and temperatures averaged to 10 min resolution from one mooring deployed in 30 m water depth close to the drifter releases [pink asterisk at ≈(−2, 0) km in Figs. 1a,b] are used in the analysis. McSweeney et al. (2020) provides a thorough description of the moorings.

## 4. Results

Here we examine two-particle dispersion statistics. In particular, we examine the effects of drifter initial separation, the difference between perturbation and total separation statistics, and the relationship between dispersion and structure functions. Results are presented first for experiment averaged (EA) dispersion and then for single release (SR) dispersion.

### a. Experiment averaged two-particle statistics

Experiment averaged (EA) dispersion statistics are calculated as long as there are a sufficient number of drifter pairs. The number of drifter pairs *N*_{p} depends on the initial separation *r*_{0} and time *t* (Fig. 3a). The number of drifter pairs is constant in time for *t* < 10^{4} s and equal to the initial number of pairs (Fig. 2) before rapidly dropping as drifters were picked up. For each *r*_{0}, the EA dispersion statistics are displayed only for *N*_{p}(*t*) > 200 (gray line, Fig. 3a) which is effectively *t* < 10^{4} s. Including fewer drifter pairs yields noisy statistics for these times.

The experiment averaged (EA) perturbation dispersion scales like *r*_{0} ≥ 500 m (colored curves in Fig. 3b) for approximately *t* < 5000 s. For the smallest *r*_{0} = 250 m, the growth is slightly slower than *t*^{2} (≈*t*^{1.85}). The slightly slower than ~*t*^{2} perturbation dispersion growth for *r*_{0} = 250 m (and to a lesser extent *r*_{0} = 500 m) could be due to GPS position errors that are inversely correlated with drifter separation. Such GPS correlated GPS position errors have been observed for another type of GPS receiver (Spydell et al. 2019). Such correlated GPS position error may cause the estimated *t*^{2}. The ballistic growth *r*_{0} (stacking of colored curves in Fig. 3b). Thus, the initial EA perturbation dispersion is consistent with a Batchelor regime as *r*_{0} (Salazar and Collins 2009). Although some oceanographic studies have found RO scaling over multiple decades (e.g., Poje et al. 2017), the durations of these drifter releases were too short to observe classic RO scaling *r*_{0} curves in Fig. 3b would collapse to a single curve. Thus, the focus here is on Batchelor scaling rather than RO scaling. However, the steepening of the *t* > 10^{4} s Fig. 3b) suggests that the dispersion maybe transitioning from *t*^{2} to *t*^{3} growth.

*t*

^{2}clearly shows a Batchelor scaling for

*r*

_{0}≥ 500 m (Fig. 3c). Assuming Batchelor scaling for the EA perturbation dispersion, the Batchelor velocity

*r*

_{0}(circles placed at

*t*= 10

^{2}s in Fig. 3c) from 0.0026 m

^{2}s

^{−2}at

*r*

_{0}= 250 m to 0.0149 m

^{2}s

^{−2}at

*r*

_{0}= 1500 m. Also consistent with theoretical expectations, comparing colored curves in Fig. 3c shows that the duration of Batchelor scaling increases with

*r*

_{0}as cooler colored curves depart from a constant sooner than the warmer colored curves (Ouellette et al. 2006). Thus, the Batchelor time

*r*

_{0}(thin vertical lines in Fig. 3c). Except for the smallest initial separation

*r*

_{0}= 250 m, the Batchelor time

*t*

_{B}is consistent with

*r*

_{0}. The decreasing in time

*t*compensated EA perturbation dispersion

*t*= 200 s for

*r*

_{0}= 250 m and

*t*= 4000 s for

*r*

_{0}= 1500 m (Fig. 3c), indicates that the dispersion is weaker than Batchelor (

*t*

^{γ}with

*γ*< 2) when transitioning out of Batchelor scaling.

Scaling EA *r*_{0} by *t* by the Batchelor time *t*_{B}(*r*_{0}), collapses the perturbation dispersion fairly well overall (Fig. 3d). The collapse is very good for 750 ≤ *r*_{0} ≤ 1500 m (cyan to orange curves) for which the scaled EA perturbation dispersions are all generally similar to each whereas for *r*_{0} = 250 m and 500 m the collapse is not as good. For *r*_{0} ≥ 750 m, *t*_{B} well predicts the time when *t*/*t*_{B} ≈ 2. Thus, for the *r*_{0} ≥ 750 m that show the best Batchelor scaling (*t*_{B} indicates well the duration of *r*_{0} ≤ 500 m that show weaker than ~*t*^{2} growth, *t*_{B} does not correspond to the duration of the initial growth. The scaled dispersions for *r*_{0} = 250 m and 500 m (blue curves) drop less rapidly than for *r*_{0} ≥ 750 m. For all *r*_{0}, the departure from Batchelor scaling, and subsequent *t*^{2}, results in perturbation dispersions for long times that are approximately 50% of the dispersion that would result if Batchelor scaling *t*, i.e., for all curves, eventually *t*/*t*_{B} > 1 (Fig. 3d). This also indicates that the dispersion weakens when transitioning out of the Batchelor regime similar to laboratory experiments (Ouellette et al. 2006).

In contrast to EA *t* < 10^{4} s (Fig. 4a). For each *r*_{0}, EA *r*_{0} = 250 m and smallest (*r*_{0} = 1500 m. For each *r*_{0}, the total dispersion *t*, *r*_{0}), see (10) and (11), hence Φ is estimated as **r**_{0} ⋅ **r**′(*t*)⟩ directly calculated using (11) (thin black curves in Fig. 4b). In accordance with the theory, directly calculating Φ using (11) is identical to calculating Φ from dispersion residuals (thin black and colored curves are indistinguishable in Fig. 4b). For all *r*_{0}, initially Φ is larger than *t*. The time *t*_{Φ}(*r*_{0}) when *r*_{0} (circles in Fig. 4b). For *t* < *t*_{Φ}, Φ initially contributes more to *t* < *t*_{Φ} the quantity *t* due to the Φ contribution. The total dispersion *d*Φ/*dt* > 0 represents particles on average moving away from each other. For Φ ~ *t*, particles are on average moving away from each other at a constant velocity.

*t*< 4000 s,

*K*

_{r}(

*t*,

*r*

_{0}) is considerably different (Fig. 5b) than

*r*

_{0}= 1250, 1500 m,

*K*

_{r}(

*t*) is nearly constant for all

*t*whereas for

*r*

_{0}= 250, 500 m,

*K*

_{r}(

*t*) increases more quickly in time. Specifically,

*K*

_{r}(

*t*) changes by a factor of ≈1.7 times for

*r*

_{0}= 1500 m and by ≈12.5 times for

*r*

_{0}= 250 m. The difference between

*K*

_{r}is due to

*d*Φ/

*dt*where

*d*Φ/

*dt*) is constant or slowly decreasing in time for all

*r*

_{0}(Fig. 5c). Owing to the different growth rates for

*d*Φ/

*dt*), for

*t*<

*t*

_{0}, where

*t*

_{0}is the time when

*d*Φ/

*dt*) contributes more to

*K*

_{r}(

*t*) than the perturbation diffusivity

*t*>

*t*

_{0},

*d*Φ/

*dt*contributes less to

*K*

_{r}(

*t*) than the perturbation diffusivity

*t*

_{0}generally increases with

*r*

_{0}(circles in Fig. 5c). The mean separation velocity

*υ*

_{r}is found from Φ

*t*≈ 100 s) increases with

*r*

_{0}from about 0.004 m s

^{−1}to ≈0.03 m s

^{−1}(Fig. 5d). The initially constant

*r*

_{0}results from Φ ~

*t*and indicates that particles on average are moving away from each other at a constant velocity that increases with initial particle separation. For all

*r*

_{0},

*t*≈ 4000 s before generally increasing in time from 4000 to 10 000 s.

The dependence of the EA diffusivity *r*_{0} consistent with the requirement that *r*_{0} = 250 m than for *r*_{0} = 1500 m (cf. blue and orange curves in Fig. 6a). The *r*_{0}-dependent *r*_{0} (thick solid gray line in Fig. 6a). However, the perturbation diffusivity *r*_{0}-independent scaling when the total separation *s*^{4/3} scaling only occurs after the Batchelor time *t*_{B} for each *r*_{0} (circles in Fig. 6b indicate *t*_{B}). In Fig. 6b, before *t*_{B} each *r*_{0} curve has vertical tails due to *s* ≈ *r*_{0} in the Batchelor regime (*s*^{4/3} scaling found in previous studies (e.g., Poje et al. 2014). Specifically, the thick gray curve in Fig. 6b is very close to the drifter data curve in Fig. 6 of Poje et al. (2014). Using the dispersion *D*_{r}, rather than *s*, results in the same scale dependence because *K*_{r} on scale *s* is approximately the same as *s* except *K*_{r} versus *s* lacks the vertical Batchelor regime tails for each *r*_{0} as *K*_{r} lacks an initial Batchelor regime. Although each *r*_{0} curve appears to approach the same RO *s*^{4/3} scaling (thin gray line in Fig. 6b), drifter trajectories were too short to definitively observe RO scaling which would have the different *r*_{0} curves collapse in Fig. 6b. Thus, for this dataset, the scale dependence of the diffusivity shows clear evidence of a Batchelor regime and suggests that the dispersion maybe approaching RO scaling.

As indicated by the circles in Fig. 3c, squared Batchelor velocities *r*_{0} (colored circles in Fig. 7). Overall, there is no consistent *r*_{0} and neither enstrophy cascade (*S*(*r*) (black curve in Fig. 7) calculated using (5) for all drifter pair data is ~*r*^{2/3}. For *r* ≥ 1000 m, the structure function and squared Batchelor velocities are similar, i.e., *r* ≤ 750 m, *r*_{0} ≤ 750 m, *r*_{0} scaling for *r* scaling for *S*. The difference between *S* is due to *S* including all (≈10^{4} s) data for each drifter pair trajectory whereas *t* ≤ 600 s) of each drifter pair trajectory. Because *r*_{0} and *r*. Indeed, the modified structure function *r* ≥ 1250 m.

Differences between *S*(*r*) and *r* ≤ 500 m are not due to sampling error, rather they are due to drifters preferentially sampling regions of convergence biasing *S*(*r*) at small *r* to larger values relative to the Eulerian estimate (Pearson et al. 2019). The differences are not due to sampling errors as the distributions of ||*δ***u**_{mn}|| used to estimate *S*(*r*) and *r* ≤ 500 m are different at the 99% confidence level according to a Kolmogorov–Smirnov test. Thus, as drifter positions evolve in time, drifters tend to sample regions of surface convergence with larger velocity variance and are thus no longer unbiasedly sampling the flow (as in Pearson et al. 2019).

### b. Single release statistics

We now present the results for drifter statistics averaged over each release (Table 1). Statistics averaged over each release are calculated for the following reasons: 1) to examine the variation in the dispersion between releases, 2) to illustrate the degree to which single release (SR) statistics can differ from EA statistics, and 3) to examine the effect of particular flow events, in particular nonlinear internal waves, on dispersion statistics. In addition to finite sampling effects, the statistics for each release will differ from each other because the flow was not stationary over the 12 releases, i.e., dispersion differences between releases are not just noise but can be due to flow differences. With 12 releases, 12 SR perturbation dispersions *r*_{0} bin. For the SR statistics, *t* is roughly the time since release (column 2 in Table 1) because drifters for each release were rapidly deployed (within ≈35 min). Because SR statistics for each *r*_{0} are based on fewer drifter pairs (~20) than EA statistics (≥200), SR statistics are inherently noisier than EA statistics. However, the number of drifter pairs used in the analysis of SR statistics here is similar to some previously reported oceanographic dispersion statistics (e.g., Ollitrault et al. 2005).

#### 1) The 13 September 2017 release

To illustrate individual pair (IP) and the single release (SR) dispersion, the dispersion for the 13 September 2017 release (Fig. 1b) is presented. The initial drifter pair number *N*_{p} for *r*_{0} = 500 m is *N*_{p} = 27 before dropping dramatically for *t* ≥ 1.5 × 10^{4} s (red curve, Fig. 8a). Averaging over each release separately greatly reduces *N*_{p} for each *r*_{0} compared to averaging over the entire experiment (i.e., compare this *N*_{p} to that in Fig. 3a). For this *r*_{0} with 27 initial pairs, IP perturbation dispersions *r*′^{2} are variable in time and between drifter pairs (thin blue lines Fig. 8a). For instance, all *r*′^{2} initially grows in time reaching a local maximum at 5000–8000 s (depending on which *r*′^{2}), then dropping for 1000–2000 s before increasing again. The variability between *r*′^{2} is considerable with *r*′^{2} ranging from ≈0 to 5 × 10^{5} m^{2} for *t* ≈ 7000 s.

The time variability for this *r*_{0} = 500 m is also evident in the single release (SR) perturbation dispersion *r*′^{2}, thick black curve Fig. 8a) with *t* = 0 to 0.2 × 10^{6} m^{2}at *t* = 7000 s then dropping until *t* ≈ 9000 s before rising to ≈0.4 × 10^{6} m^{2}(at *t* = 1.5 × 10^{4} s). Overall, for *r*_{0} = 500 m the growth of *t*^{2} despite the local maximum at *t* ≈ 7000 s. The variability across individual drifter pairs at time *t* is quantified by the standard deviation of *r*_{0} scales directly with *t* ≤ 1.5 × 10^{4} s, i.e., *t* ≤ 1.5 × 10^{4} s).

The SR total dispersion *r*_{0} = 500 m shows similar time dependence to the SR perturbation dispersion *y* scales between Figs. 8a and 8b). Thus, for this release and *r*_{0}, Φ ≠ 0. For instance, at the local maximum (*t* ≈ 7000 s), Φ ≈ 0.2 × 10^{6} m^{2}, a value greater than *r*_{0} = 500 m scales with the total dispersion but offset by the initial total dispersion such that

For the 13 September 2017 release, the pair number *N*_{p} for *r*_{0} = 1000 m is initially *N*_{p} = 22 before dropping when *t* ≈ 1.5 × 10^{4} s (red curve Fig. 8c). The SR perturbation dispersion *r*_{0} = 1000 m shows similar time dependence to the *r*_{0} = 500 m SR *t*^{2}) but is approximately 8 times larger in magnitude (thick black curves in Figs. 8a and 8c). A point of difference is that *r*_{0} = 1000 m remains constant from 5000 to 10 000 s rather than decreasing like *r*_{0} = 500 m. Unlike the variability across pairs for *r*_{0} = 500 m in which *r*_{0} = 1000 m the variability is smaller than the dispersion with

The SR total dispersion *r*_{0} = 1000 m shows similar time dependence to the other 13 September 2017 dispersions (thick black curve Fig. 8d). Again, the magnitude of *r*_{0} = 1000 m (note different *y* scales between Figs. 8c and 8d) by a factor of about 2–3 indicating the Φ ≠ 0. For instance, at *t* ≈ 7500 s when ^{6} m^{2} a value more than 2 × *t*. Like *r*_{0}, the total dispersion variability across pairs *r*_{0} and the total dispersion is larger than the perturbation dispersion for a given *r*_{0}. However, for this release, the SR dispersion is much more variable in time than EA dispersion suggesting that dispersion for single releases may not show a clean scaling law dependence like EA dispersion. Although this is not surprising given that single release statistics are based on fewer drifter pairs, it is illustrative as to how different SR statistics can be from EA statistics.

#### 2) Dispersion for all releases

The perturbation dispersion *r*_{0} = 500 and 1000 m for the 13 September 2017 release (Figs. 8a,b), for all releases and for these *r*_{0}, the initial time dependence of *t*^{2} (Figs. 9a,b). Thus, the perturbation dispersion for each release is largely consistent with a Batchelor regime.

The time when *r*_{0} = 500 m (red and dark orange curves in Fig. 9a) both depart and grow more slowly than *t*^{2} at *t* ≈ 1000 s, whereas the departure from *t*^{2} growth is much later (*t* ≈ 10 000 s) for release 12 (light blue curve in Fig. 9a). Unlike the experiment averaged *r*_{0} = 500 m, and 2 for *r*_{0} = 1000 m) transition to faster than *t*^{2} growth rather than slower. For example, release 2 for *r*_{0} = 1000 m is faster than *t*^{2} at *t* = 2000 s before transitioning to slower than *t*^{2} for *t* > 4000 s (blue curve in Fig. 9b). Although the experiment averaged *t*_{B} is associated with departure from *t*^{2} growth (e.g., Figs. 3c,d), for the individual release *t*^{2} growth is not obvious. For the same *r*_{0}, the Batchelor time *r*_{0} = 500 m, release 4 (orange curve in Fig. 9a), with the largest *t*^{2} growth before the other releases and release 6 (dark red curve in Fig. 9a), with the smallest *t*^{2} growth after the other releases. No *t*_{B} pattern is apparent likely due to the individual release

The SR perturbation dispersion *t* = 150 s) the *r*_{0} = 500 m *r*_{0} = 500 m. For *r*_{0} = 1000 m, the largest initial ^{2}) is 36 times larger than the smallest ^{2}) (Fig. 9b). For both *r*_{0} = 500 and 1000 m, the difference between largest and smallest *t* (Figs. 9a,b). Although the initial growth of SR dispersion is generally similar to EA dispersion (~*t*^{2}), the different SR dispersion magnitudes indicate how much tracer dispersion can vary between releases and how much tracer dispersion can differ for a specific release from the ensemble mean. The SR dispersion differences generally correspond to the location of each release, as releases to the north (warm colored curves in Figs. 9a,b) have the smallest *r*_{0} = 500 m, release 4 (orange curve with largest

#### 3) Scaling the dispersion with ${U}^{2}$

The initial perturbation dispersion for each release generally follows Batchelor scaling *t* = 10^{4} s (i.e., *t* > *t*_{B}) depends on *r*_{0}. Squared Batchelor velocities *r*_{0} are determined using (18) as the perturbation dispersion for each release is generally consistent with a Batchelor regime. Thus, for *r*_{0} = 500 m, *r*_{0} = 500 m is generally associated with larger later dispersion and small *r*_{0} = 1000 m.

Directly comparing the perturbation dispersion *t* = 10^{4} s for each *r*_{0} and release to the squared Batchelor velocities ^{4} s is generally associated with *t* = 10^{4} s and southern releases (cool colored dots) have the largest *t* = 10^{4} s. Although the perturbation dispersion

The initial separation *r*_{0} affects the later perturbation dispersion. Generally, for a specific release, *r*_{0} (larger dots of the same color are above smaller dots in Fig. 10). Thus, the initial separation influences the perturbation dispersion after 10^{4} s indicating that *r*_{0}-independent RO scaling has not been established. However, the variation in *r*_{0} values for a given release is smaller than differences between releases for the same *r*_{0}, compare the spread of

## 5. Discussion

### a. Influence of internal bores on dispersion

The experiment averaged, and individual release perturbation dispersion *t* ≲ 5000 s (Figs. 3c and 9). However, some individual releases deviate from Batchelor scaling as seen in Figs. 8a and 8c where *t* < 10^{4} s for *r*_{0} = 1000 m, Fig. 8c) and even shrink (6000 < *t* < 9000 s for *r*_{0} = 500 m, Fig. 8a). Via the example of the 13 September 2017 release (Figs. 1b and 8), here, we examine how a particular flow event, namely an onshore propagating nonlinear internal wave (NLIW) affects the dispersion and how NLIWs contribute to deviations from EA Batchelor scaling. NLIWs are known to be significant in the Pt. Sal region (Colosi et al. 2018; McSweeney et al. 2020; Feddersen et al. 2020),

Temperature and velocity are examined from the mooring nearest to the 13 September 2017 drifter release in order to investigate the effect of NLIWs on dispersion. This mooring was located in 30-m water depth offshore of Pt. Sal [magenta asterisk in Figs. 1a and 1b at (*x*, *y*) ≈ (−2, 0.5) km]. As this mooring was inshore of the drifter cluster ≈1 km, and bores are known to propagate ≈0.25 m s^{−1}, we time adjust the mooring −1 h to better match the time when the bore passes the mooring and drifter center of mass and therefore to better highlight the effect of the internal bores on drifter dispersion. During this release, east–west velocities *u* and temperatures from the 30-m-depth Pt. Sal mooring indicate that a strong NLIW (classified as a warm bore; Colosi et al. 2018) arrived at 1700 UTC (Fig. 11a). Prior to the NLIW arrival (<1700 UTC), surface velocities are offshore (*u* ≈ −0.2 m s^{−1} blues Fig. 11a), deeper (*z* < −10 m) velocities are onshore (*u* ≈ 0.1 m s^{−1}), and the 16°C isotherm (thickest black contour) is near the surface (*z* ≈ −5 m). After the NLIW arrives, surface temperatures increase by ≈1.5°C and the 16°C isotherm drops to *z* ≈ −20 m. The NLIW arrival also switches the east–west velocities with surface velocities onshore (*u* ≈ 0.1 m s^{−1}) and deeper velocities offshore from 1700 to 1800 UTC. By 1830 UTC, the 16°C isotherm has relaxed back to *z* ≈ −8 m, near its prearrival position, and near surface velocities are again offshore (*u* ≈ −0.05 m s^{−1}). Beyond 1830 UTC the influence of the NLIW at the mooring is weak. Examination of the north–south velocities *υ* (not shown) indicates that the NLIW propagates to the east-southeast (18° from the east).

For the 13 September 2017 release, drifter cross-shore positions *X*(*t*) (colored curves, Fig. 11b) indicate that drifters generally move offshore accompanied by cross-shore spreading. The NLIW arrival, found from the maximum cross-shore drifter acceleration (indicated by circles for each drifter in Fig. 11b), interrupts and pauses the cross-shore spreading for ≈1 h. From a linear best fit to the timing and cross-shore location of the NLIW arrival, the onshore NLIW propagation speed of this bore is estimated as 0.29 m s^{−1} (slope of circles in Fig. 11b) consistent with regional NLIW phase speeds (Colosi et al. 2018; McSweeney et al. 2020). Thus, the ADCP time offset of −1 h used in Fig. 11 effectively places this mooring ≈1000 m farther offshore (at *x* ≈ −3 km) near the center of drifter cross-shore center of mass.

Prior to NLIW arrival, drifter velocities are consistent with cross-shore surface mooring velocities (gray curve Figs. 11b,c) with the most offshore drifters (blue curves, Fig. 11b) having the largest offshore velocities (most negatively sloped curves) and the most onshore drifters have the weakest (≈0 m s^{−1}) offshore velocities (red curves, Fig. 11b). This is reflected in the surface mooring velocity (gray curve, Fig. 11b) which becomes more negative as the bore approaches the mooring. Thus, before bore arrival at the most offshore drifter (≈1615 UTC), cross-shore drifter spreading is associated with diverging (*du*/*dx* > 0) cross-shore velocities and results in quickly growing *r*_{0} = 500 m individual separations *r*′^{2}(*t*) (blue curves in Fig. 8a reproduced as colored curves versus UTC time in Fig. 11c). The faster than linear initial growth for all *r*′^{2}(*t*) before 1615 UTC appears Batchelor-like, i.e., ~*t*^{2}. Although individual perturbation separations *r*′^{2} are due to both cross- and alongshore position differences (6), for this release, examining the cross-shore mooring velocities *u* and cross-shore drifter positions *X*(*t*) explains the general features of *r*′^{2} as the bore is propagating nearly due east.

For drifters separated in the cross-shore, the effect of the NLIW is sequential with the NLIW arrival halting and reversing the growth of individual perturbation dispersions *r*′^{2} first for the most offshore pairs (blue curves in Fig. 11c) and later for more onshore pairs (orange curves). When averaged this effect results in the local maxima of *t* ≈ 7000 s (thick black curve Fig. 8a). The details of the process begin with the NLIW passing the most offshore drifter at ≈1600 UTC which results in the drifter *X*(*t*) accelerating onshore (indicated by a circle in darkest blue curve in Fig. 11b) halting offshore movement. After the NLIW passes the most offshore drifter but before reaching the other drifters, the cross-shore separation between the most offshore and the other drifters decreases because the most offshore drifter is stationary in the cross-shore and the others are moving offshore toward it. Thus, for drifter pairs containing the most offshore drifter, perturbation separations *r*′^{2}(*t*) increase and decrease before and after, respectively, NLIW arrival (blue curves Fig. 11c) indicated by local maxima in *r*′^{2}(*t*) a little after 1600 UTC. The NLIW passes the most onshore drifters at ≈1745 UTC (red circles in Fig. 11b), 1.5 h after passing the most offshore drifter. Thus, the most onshore drifter pairs have the most time for *r*′^{2}(*t*) to grow before the bore arrives. This results in the largest *r*′^{2} maximum at ≈1700 UTC for pairs with cross-shore initial separations (orange curves in Fig. 11c). For onshore pairs with alongshore initial separations, *r*′^{2} is much smaller (red curves in Fig. 11c).

After this NLIW passes all the drifters (>1800 UTC), the cross-shore positions *X*(*t*) begin spreading (Fig. 11b) similarly to before NLIW arrival. The most offshore drifters (blue curves) move offshore (consistent with mooring velocities for 1830–2100 UTC) whereas the most onshore drifters move shoreward. Thus, the spreading is again due to *du*/*dx* (>0) and results in quickly growing *r*′^{2}(*t*) similar to, and an approximate continuation of, the initial growth (Fig. 11c). This is reflected in Fig. 10 where the perturbation dispersion *t* = 10^{4} s for all *r*_{0} for this release (darkest blue dots with

### b. Comparison to previous work

For the release-averaged and most individual releases, the perturbation dispersion is consistent with Batchelor scaling *t* < *t*_{B} with the Batchelor time *t*_{B} (16) increasing with initial drifter separation *r*_{0} in accordance with theory and previous studies (e.g., Ouellette et al. 2006). Batchelor scaling is well established for various nonoceanographic laboratory and numerical investigations of inertial subrange turbulent dispersion (e.g., Salazar and Collins 2009) as well as proposed for dispersion within an atmospheric simulation (Haszpra et al. 2012). For oceanographic dispersion, Batchelor scaling has not been examined, however, other dispersive scalings have been identified and examined. Because the dispersive scaling is linked to the background turbulence wavenumber spectra [(17); e.g., Foussard et al. 2017], identifying the correct dispersive scaling is critical to properly inferring the turbulence responsible for the dispersion. Batchelor scaling may not have been identified or examined in previous oceanographic studies due to limited data, examining incompatible time scales, and investigating a dispersion statistic (

Some oceanographic observations and modeling studies are suggestive of Batchelor scaling. For a few drifters repeatedly released in the Santa Barbara Channel, resulting in less than 75 total pairs, Ohlmann et al. (2012) found *t* < 84 h. In this study, a thorough examination of Batchelor scaling was not possible as only one small *r*_{0} ≈ 7.5 m was considered. In modeling studies of coastal Southern California, for a single *r*_{0} ≈ 500 m and *t* ≤ 1 day, the dispersion *t*^{2} in a 40-m resolution model (Dauhajre et al. 2019) and *r*_{0} ≈ 1 and 5 km and *t* > 10 days (Haza et al. 2008). In this study, time or space smoothing affects the results. On the inner shelf of the Gulf of Mexico, a modified perturbation dispersion ⟨*r*^{2}⟩ − ⟨*r*⟩^{2} shows ~*t*^{2} growth for *t* ≲ 3 × 10^{4} s (Roth et al. 2017). In these previous studies, Batchelor scaling was not examined. Some previous studies that have found ballistic *t*^{2} growth rate from this mechanism requires pure uniform horizontal shear (LaCasce 2008) which the drifter trajectories here do not exhibit (Figs. 1a,b). Moreover, the structure function in a uniform shear scales as *S*(*r*) ~ *r*^{2} for drifters released similarly to the drifters here. The requirement of pure shear is strong as any small-scale turbulence, in addition to the uniform shear, results in

A Batchelor regime may not have been identified in previous oceanographic observations because the drifter sampling rates were too slow or the time scales analyzed were too long to properly resolve a Batchelor regime. For example, in some studies (Ollitrault et al. 2005; Koszalka et al. 2009), the drifter sampling resolution (daily) was much greater than, the Batchelor time found here (*t*_{B} ≈ 1 h, green + in Fig. 3c) for the *r*_{0} ≈ 1 km initial separation considered. In Lumpkin and Elipot (2010), where *r*_{0} ≈ 1 km, the sampling rate was faster (1–2 h) but still insufficient to resolve *t* ≲ 1 h and hence a Batchelor regime for this *r*_{0}. In Beron-Vera and LaCasce (2016), where *r*_{0} ≈ 1 km, the drifter sampling rate (15 min) was fast enough to resolve a Batchelor regime, however, only times (≥2.4 h) beyond the Batchelor time were analyzed. In contrast to these studies, both the temporal resolution (5 min) and time scales (≲ hours) considered here were sufficient to resolve a Batchelor regime.

More importantly, even with sampling rates fast enough to resolve a Batchelor regime (e.g., Ohlmann et al. 2012; Beron-Vera and LaCasce 2016), in most studies (e.g., Lumpkin and Elipot 2010; Ohlmann et al. 2012; Romero et al. 2013; Beron-Vera and LaCasce 2016, etc.) identifying a Batchelor regime may have been hindered because the total dispersion *r*_{0} = 1 km (green curve in Fig. 3b and Fig. 4a). For *t* ≤ 10^{4} s, *e*-folding times *τ*. For instance, in the Santa Barbara Channel for *r*_{0} = 7.5 m, *t* < 5 h with an *e*-folding time of *τ* = 0.9 h (Ohlmann et al. 2012) as well as for *r*_{0} = 1 km in the Nordic Sea for *t* ≤ 2 days with a much larger *e*-folding time *τ* ≈ 12 h (Koszalka et al. 2009). Because *k*^{−3} wavenumber spectra (Lin 1972), *e*-folding times are independent of *r*_{0} in contrast to exponential fits to observed *τ* increases with *r*_{0}. Rather than exponential growth, for *r*_{0} = 1 km the offset perturbation dispersion *t* ≤ 5000 s (red and black curves in Fig. 12) before transitioning to RO scaling for *t* > 6000 s (red and dashed black curves in Fig. 12). Thus, for *t* > 6000 s, *k*^{−5/3} rather than *k*^{−3} turbulence as incorrectly suggested by the *t* ≈ 12 000 s. Here, *t* and *t* ≲ 3000 s. To properly estimate the dispersive scaling(s), and therefore correctly infer the background turbulent wavenumber spectra, the perturbation dispersion

Note that rather than transitioning directly from Batchelor to RO scaling, the offset perturbation dispersion (red curve Fig. 12) *slows* after Batchelor but before RO scaling similar to laboratory dispersion (Ouellette et al. 2006). Here, this slowing is due to the RO fit including an *r*_{0} offset since the fit for the perturbation dispersion is *C*_{1} ≠ 0 whereas for theoretical RO scaling *r*_{0} term and *C*_{1} = 0. However, in accordance with inertial subrange theory (Batchelor 1950), for large enough *t*, the *t*^{3} term is eventually much larger than

## 6. Summary

GPS-equipped surface drifters were repeatedly deployed on the Inner Shelf off of Pt. Sal, CA, in water depths ≤ 40 m. Relative dispersion statistics were calculated from 1998 drifter pairs from 12 releases of ≈18 drifters per release. Unlike most previous studies that focus on the dispersion *K*_{r} (12) and perturbation diffusivities *t* ≲ 4 h and for initial drifter separations 250 ≤ *r*_{0} ≤ 1500 m.

The EA perturbation dispersion follows Batchelor scaling *r*_{0} ≥ 750 m. Consistent with theory, both the duration of Batchelor scaling *t*_{B}(*r*_{0}) and squared Batchelor velocities *r*_{0}. EA squared Batchelor velocities *r*_{0} = 1500 m are 5 times greater than *r*_{0} = 250 m. For *r*_{0} ≤ 1000 m, EA *r*_{0}. After Batchelor scaling, i.e., *t* > *t*_{B}, scale *r*_{0} values considered here, the EA *t*_{B} is smaller than but correlated with the initial Batchelor scaling. Specifically, the dispersion at *t* = 10^{4} s is approximately 50% of that predicted by Batchelor scaling. This indicates that the dispersion does not transfer directly from Batchelor to RO scaling but rather slows when transitioning out of the Batchelor regime. For each release and all *r*_{0}, the SR *r*_{0} and time *t*, the SR *t* = 10^{4} s is correlated with but less than the Batchelor scaling prediction.

For an individual release, a NLIW modulated (enhancing and then reducing) the dispersion. Potential reasons why previous studies did not investigate a Batchelor regime include the following. 1) The focus was on time scales that were too long. 2) Only one initial separation *r*_{0} was considered. 3) Most importantly, investigating only the dispersion *t*) scale as ~*t* for short times. Thus, previous studies investigating

## Acknowledgments

The Office of Naval Research supported this research through Grants N00014-5-1-2631 (SIO) and N0001418WX00229 (NPS). This work was also supported by the National Science Foundation (OCE-1459389). For the Point Sal fieldwork, Bill Boyd, Greg Boyd, Tucker Freismuth, Casey Gon, Matt Gough, Rob Grenzeback, Derek Grimes, Ami Hansen, Paul Jessen, Michael Kovatch, Paul Lenz, Aaron Morrone, Andy O’Neill, Lucian Parry, Brett Pickering, Greg Sinnett, Kent Smith, Marla Stone, Ata Suanda, Jim Thomson, Brian Woodward, and Keith Wyckoff are acknowledged for their help in deployment and recovery. We thank John Colosi for generously providing mooring and ADCP data. The authors thank Derek Grimes, Thomas Zdyrski, and Xiaodong Wu for providing useful feedback. The authors thank three anonymous reviewers for numerous constructive comments that improved the manuscript.

## Data availability statement

Inner-shelf dynamics experiment drifter and mooring data are archived at UC San Diego Library Digital Collections (https://doi.org/10.6075/J0WD3Z3Q).

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