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    (a) The Pt. Sal drifter trajectories. There are 12 releases of 8–26 drifters. Dots are the initial positions colored according to the latitude of the release: blues to the south, reds to the north. (b) An example realization showing trajectories for drifters released on 13 Sep 2017 in coordinates centered on the tip of Pt. Sal. Magenta asterisks indicate the location of the ADCP and temperature mooring used in the analysis. Bathymetry is contoured in gray: thick contours are separated by 10 m, and thin contours are separated by 2.5 m (starting at 10 m).

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    Distribution of initial drifter separations r0 used for experiment averaged statistics. The actual bin centers are indicated by red plus signs and do not deviate much from multiples of 250 m, especially for r0 ≤ 1500 m.

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    (a)–(c) Experiment averaged (EA) two-particle statistics vs time t. (a) The number of drifter pairs (Np) vs time t for initial separations 250 ≥ r0 ≥ 1500 m (colors). Minimum number of pairs Np(min)=200 for analysis is indicated by the horizontal gray line. (b) The EA perturbation dispersion Dr2 (9) vs time t. (c) The compensated EA perturbation dispersion Dr2/t2 vs time t. (d) The scaled EA perturbation dispersion Dr2/(U2t2) vs scaled time t/tB. The dispersion curves are only shown for times with Np(t)>Np(min). In (b), the dashed gray line is t2 and the solid gray line is ~t3. In (c), for each r0, circles indicate squared Batchelor velocities U2(r0) (18) and tick marks indicate the Batchelor time tB (16).

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    (a) The experiment averaged dispersion Dr2 (8) and (b) Φ (11) vs time t. In (a), thin black curves are r02+Dr2(t). In (b), colored curves are Φ=Dr2Dr2r02, thin black curves are Φ = 2⟨r0r′⟩, and circles indicate the time tΦ when Φ(t)=Dr2(t). Colors correspond to initial separations 250 ≤ r0 ≤ 1500 m (see legend in Fig. 3). In (a) and (b), thin black curves are on top of colored curves for all r0.

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    (a) The experiment averaged (EA) perturbation diffusivity Kr (13), (b) the EA total diffusivity Kr (12), (c) the EA difference 1/4(dΦ/dt)=KrKr (14), and (d) the EA separation velocity υr (19) vs time t. In (a), the Batchelor scaling ~t (dashed gray) and the RO scaling ~t2 (solid gray) are shown. In (c), circles indicate the time when Kr=1/4(dΦ/dt). Colors correspond to initial separations 250 ≤ r0 ≤ 1500 m (see legend in Fig. 3).

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    (a) The EA perturbation diffusivity Kr vs (a) EA perturbation dispersion (Dr2)1/2 and (b) total separation s=(r02+Dr2)1/2. Colors correspond to initial separations 250 ≤ r0 ≤ 1500 m (see legend in Fig. 3). In (a), Batchelor scaling ~Dr (dashed gray) and RO scaling ~Dr4/3 (gray) are shown. In (b), the RO scaling ~s4/3 (thick solid gray) and the enstrophy scaling ~s2 (dash–dotted gray) are shown. Circles in (b) indicate the Batchelor time tB.

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    Drifter pair trajectory derived second-order experiment averaged (EA) structure functions S (thick black) (5) vs r. The EA structure function S˜ based on velocity increments from initial drifter pair trajectory times (see text) vs r are shown as “x” marks. Squared Batchelor velocities U2 (circles) (18) are derived from the Batchelor scaling of Dr2 and colored according to r0 (as in Fig. 3). Energy cascade ~r2/3, enstrophy cascade ~r2, and ~r scalings are shown as solid, dotted gray, dash–dotted lines, respectively.

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    Dispersion vs time for the 13 Sep 2017 release: (a),(c) perturbation dispersion Dr2(t), and (b),(d) total dispersion Dr2(t) for (left) r0 = 500 m and (right) r0 = 1000 m. Individual pair (IP) dispersion r2 (6) [in (a) and (c)] and IP r2 [in (b) and (d)] are colored curves. In each panel, the mean of all pairs (thick black), i.e., the single release statistic, and the mean ± one standard deviation (thin black) are shown. The number of drifter pairs Np(t) for r0 = 500 m in (a) and r0 = 1000 m in (b) is indicated by the red curve. Means and IP dispersions are shown as long as the number of drifter pairs is within five of the initial number of pairs, i.e., as long as Np(0) − Np(t) ≤ 5 (≈5 h for r0 = 500 m, ≈4 h for r0 = 1000 m). Thus, five IP dispersions are shorter in duration than the mean.

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    For each release (colors; consistent with Fig. 1), the single release perturbation dispersion Dr2 vs time t for (a) r0 = 500 m and (b) r0 = 1000 m initial separations. Red (blue) colors correspond to drifter releases in the northern (southern) portion of the experimental region off of Oceano (Pt. Sal). In (a) and (b), the gray line is ~t2. Statistics are shown only if the number of drifter pairs Np(t) has not dropped by more than 5 from the initial drifter pair number Np(0). Release 4 is not plotted in (b) because this release did not have any drifter pairs with initial separations r0 ≥ 1000 m.

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    The single release (SR) perturbation dispersion Dr2(t) at time t = 104 s vs the SR squared Batchelor velocity U2. Colors correspond to latitude of the release consistent with Fig. 1. Symbol size corresponds to increasing initial release separation r0 = 250, 500, 750, …, 1500 m (see legend). The dashed black line corresponds to exact Batchelor scaling (3) Dr2(t)=U2t2 at t = 104 s.

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    (a) East–west velocity u (colors) and temperature (black contours at 13°, 14°, 15°, and 16°C) vs time and depth z at the 30-m Pt. Sal mooring [magenta asterisk in Figs. 1a and 1b at (x, y) ≈ (−2, 0.5) km] during the fifth drifter release on 13 Sep 2017. The tide level is indicated by the black curve at z ≈ 0. (b),(c) Pt. Sal 30-m ADCP near surface (z ≈ −3 m) east–west velocity u vs time (thick gray curves; left y axis). Colored curves (right y axis) are the 15 individual drifter cross-shore position X(t) in (b) and individual drifter pair perturbation separations r2 for r0 = 500 m vs time in (c) (same as blue curves in Fig. 8a). The trajectories X(t) and individual perturbation dispersions r2(t) are shown for the first 5 h, the approximate time when the Np for this release and r0 has decreased from the initial Np(0) by 6 (see Fig. 8a). Colors correspond to the initial cross-shore drifter position in (b) and the initial drifter pair cross-shore midpoint in (c). Reds denote positions closer to shore, and blues denote farther offshore positions. Mooring times are offset by −1 h to better align mooring and drifter times as this mooring was onshore of the drifters.

  • View in gallery

    The experiment averaged (EA) total dispersion Dr2(t) (green) and offset EA perturbation dispersion r02+Dr2(t) (red) vs time t for the 13 Sep 2017 release and r0 = 1000 m. Solid and dashed black curves are fits to the offset perturbation dispersion: (solid) Batchelor r02+U2(r0)t2 for t < 5000 s, and (dashed) RO (1.3r0)2 + (2.5 × 10−7 m2 s−3)t3 for 6000 ≤ t ≤ 15 000 s. Dash–dotted curve is the exponential fit r02exp[t/(4.7h)] for t < 104 s to Dr2(t).

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Relative Dispersion on the Inner Shelf: Evidence of a Batchelor Regime

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  • 1 Scripps Institution of Oceanography, La Jolla, California
  • 2 Naval Post Graduate School, Monterey, California
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Abstract

Oceanographic relative dispersion Dr2 (based on drifter separations r) has been extensively studied, mostly finding either Richardson–Obukhov (Dr2~t3) or enstrophy cascade [Dr2~exp(t)] scaling. Relative perturbation dispersion Dr2 (based on perturbation separation rr0, where r0 is the initial separation) has a Batchelor scaling (Dr2~t2) for times less than the r0-dependent Batchelor time. Batchelor scaling has received little oceanographic attention. GPS-equipped surface drifters were repeatedly deployed on the Inner Shelf off of Pt. Sal, California, in water depths ≤ 40 m. From 12 releases of ≈18 drifters per release, perturbation and regular relative dispersion over ≈4 h are calculated for 250 ≤ r0 ≤ 1500 m for each release and the entire experiment. The perturbation dispersion Dr2 is consistent with Batchelor scaling for the first 1000–3000 s with larger r0 yielding stronger dispersion and larger Batchelor times. At longer times, Dr2 and scale-dependent diffusivities begin to suggest Richardson–Obukhov scaling. This applies to both experiment averaged and individual releases. For individual releases, nonlinear internal waves can modulate dispersion. Batchelor scaling is not evident in Dr2 as the correlations between initial and later separations are significant at short time scaling as ~t. Thus, previous studies investigating Dr2(t) are potentially aliased by initial separation effects not present in the perturbation dispersion Dr2(t). As the underlying turbulent velocity wavenumber spectra is inferred from the dispersion power law time dependence, analysis of both Dr2 and Dr2 is critical.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Matthew S Spydell, mspydell@ucsd.edu

Abstract

Oceanographic relative dispersion Dr2 (based on drifter separations r) has been extensively studied, mostly finding either Richardson–Obukhov (Dr2~t3) or enstrophy cascade [Dr2~exp(t)] scaling. Relative perturbation dispersion Dr2 (based on perturbation separation rr0, where r0 is the initial separation) has a Batchelor scaling (Dr2~t2) for times less than the r0-dependent Batchelor time. Batchelor scaling has received little oceanographic attention. GPS-equipped surface drifters were repeatedly deployed on the Inner Shelf off of Pt. Sal, California, in water depths ≤ 40 m. From 12 releases of ≈18 drifters per release, perturbation and regular relative dispersion over ≈4 h are calculated for 250 ≤ r0 ≤ 1500 m for each release and the entire experiment. The perturbation dispersion Dr2 is consistent with Batchelor scaling for the first 1000–3000 s with larger r0 yielding stronger dispersion and larger Batchelor times. At longer times, Dr2 and scale-dependent diffusivities begin to suggest Richardson–Obukhov scaling. This applies to both experiment averaged and individual releases. For individual releases, nonlinear internal waves can modulate dispersion. Batchelor scaling is not evident in Dr2 as the correlations between initial and later separations are significant at short time scaling as ~t. Thus, previous studies investigating Dr2(t) are potentially aliased by initial separation effects not present in the perturbation dispersion Dr2(t). As the underlying turbulent velocity wavenumber spectra is inferred from the dispersion power law time dependence, analysis of both Dr2 and Dr2 is critical.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Matthew S Spydell, mspydell@ucsd.edu

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.

Particle dispersion can be studied in the framework of absolute (single-particle; Taylor 1922) or relative (two-particle) dispersion. The pioneering work of Richardson (1926) investigating the relative motion of particles within isotropic turbulence established the foundations of relative dispersion. These results were later theoretically justified by Obukhov (1941) using inertial subrange theory, i.e., a 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 Dr2(t)=r2(t), that increases in time t since release as
Dr2(t)~t3.
Because the relative diffusivity is proportional to the time derivative of dispersion (Krd/dtDr2), (1) is equivalent to Richardson’s 4/3 law for the scale-dependent diffusivity Kr:
Kr~Dr4/3.
The classic Richardson–Obukhov (RO) scalings, (1) and (2), do not depend on the initial separation r0 (e.g., Salazar and Collins 2009), hence the coefficients of proportionality in (1) and (2) will not depend on r0 but rather depend only on the energy dissipation rate ϵ¯. Thus, RO scalings are only valid for times t after release large enough so that the two-particle separation vector r(t) no longer depends on the initial separation vector r0 (e.g., Salazar and Collins 2009).
Batchelor (1950) considered two-particle dispersion but for times t just after release such that the separation r(t) depends on the initial separation r0. 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 Dr2=r2, where the perturbation separation r′ = ||r(t) − r0||, grows ballistically in time
Dr2(t,r0)=U2(r0)t2.
The Batchelor regime (3) is only valid for times less than the Batchelor time
tBr0/U(r0)
(e.g., Salazar and Collins 2009), the time it takes the perturbation dispersion Dr2 to grow to the squared initial separation r02. For ttB, memory of r0 is lost. A different Batchelor time scale ~U2/(2ϵ¯) has been investigated (Bitane et al. 2012). However, in an inertial subrange where U2~(ϵ¯r0)2/3, this time scale and tB (4) are equivalent (~ϵ¯1/3r02/3) within a proportionality constant. The Batchelor scaling (3) is well established for laboratory and numerical experiments (e.g., Ouellette et al. 2006; Salazar and Collins 2009). Ballistic dispersion Dr2~t2 can also be found for particles released in pure uniform shear flow, however, the addition of small-scale turbulent diffusion causes t3 growth (e.g., LaCasce 2008).
Squared Batchelor velocities U2(r0) are equivalent to the Eulerian second-order velocity structure function U2(r0)=S(r0) (e.g., Salazar and Collins 2009) where the structure function is the mean squared velocity difference
S(r)=δu2
and δu=u(r2)u(r1) is the velocity difference, or increment, between two locations (r1 and r2) separated by r = ||r2r1||. For isotropic, homogenous, and stationary flow, Eulerian statistics of (random) initial separations r0 are equivalent to Eulerian statistics of later (random) separations r such that S(r0) = 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, U2(r0)~r02/3 (Kolmogorov 1941). Batchelor scaling Dr2~t2 does not by itself correspond to the background wavenumber spectra, rather that must be inferred from the structure function U2(r0). Although structure functions, like wavenumber spectra, are fundamentally Eulerian statistics, they can be estimated from Lagrangian drifters (Poje et al. 2017), but doing so leads to significant biases (Pearson et al. 2019). Thus, unlike Eulerian derived structure functions where S(r0) = S(r) due to flow stationarity, S(r0) ≠ 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 (~t2) growth for the full dispersion Dr2(t) (LaCasce and Bower 2000; Haza et al. 2008; Ohlmann et al. 2012; Dauhajre et al. 2019; Romero et al. 2013). However, a thorough examination of perturbation dispersion Dr2(t) and the Batchelor regime for oceanographic flows has not been performed. In contrast, Batchelor scaling has been thoroughly examined for other flows. For example, various laboratory (Ouellette et al. 2006) and numerical (Sawford et al. 2008; Bitane et al. 2012) investigations of turbulence show clear evidence of Dr2=U2(r0)t2. Additionally, dispersion for an atmospheric simulation suggesting Batchelor scaling (Haszpra et al. 2012).

Early oceanographic observations of scale-dependent diffusivity were suggestive of RO scaling over a very wide range of length scales 102 < r < 105 m (Okubo 1971) implying a k−5/3 wavenumber spectra. In the Gulf of Mexico, two-particle diffusivities Kr, derived from many drifters show clear evidence of Kr~Dr4/3 from sub- to mesoscales (200–105-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 Dr ~ t3 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 Dr ~ exp(t) and Kr ~ D2. 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 Dr2~exp(t) for 2 ≤ t ≤ 10 days and r0 ≤ 10 km using drifters deployed in the Gulf of Mexico. Drifters deployed in the Nordic Sea also suggest D2 ~ exp(t) for 0.5 ≤ t ≤ 2.5 days and KrDr2 for 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 Dr2 rather than the perturbation dispersion Dr2 was examined, potentially obscuring the proper dispersive scaling (Ouellette et al. 2006). In addition to only examining Dr2(t), many studies only considered one r0 (e.g., Ohlmann et al. 2012; Romero et al. 2013; Dauhajre et al. 2019), precluding an examination of Batchelor scaling which depends explicitly on r0. 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 Dr2 and perturbation dispersion Dr2 are provided. The in situ data are described in section 3, i.e., the multiple drifter release realizations that resolve well the short times (≤4 h) after release. Relative dispersion results, both Dr2 and Dr2 and their relationships, are presented in section 4. Results are first presented for experiment averaged (or realization averaged) statistics followed by results for each particular release (or realization). The results are then discussed in section 5. First, the role of nonlinear internal waves on the dispersion is investigated (section 5a). Second, how the results presented here fit within the existing literature is discussed (section 5b). The paper is summarized in section 6.

2. Background

a. Two-particle dispersion statistics

Here, we describe the two-particle (relative) dispersion statistics used in the analysis. Let the horizontal position at time t after release of drifter m be given by Xm(t)=Xm(t)i^+Ym(t)j^, where Xm (Ym) are the easting (northing) of the drifter. The initial position of the drifter is Xm,0Xm(t = 0). The vector between two drifters m and n is rmn(t) = Xm(t) − Xn(t) with the initial separation vector given by rmn,0 = Xm,0Xn,0. The squared separation for an individual drifter pair is given by
rmn2rmn2=||XmXn||2.
For drifter m, the position relative to the initial position, the drifter displacement, is indicated by a superscripted prime Xm(t) = Xm(t) − Xm,0. The difference in displacements between drifter m and n is considered the perturbation separation rmn(t)=rmn(t)rmn,0 with squared distance
rmn2||rmn||2=rmnrmn,02=XmXn2.
Separation magnitudes and separation perturbation magnitudes are related through the law of cosines
rmn2=rmn2+rmn,02+2rmn,0rmncosθmn,
where θmn is the angle between rmn,0 and rmn(t). Relative, or two-particle dispersion is described by the statistics of rmn2 and rmn2.
The statistics of rmn and rmn over drifter pairs for a given initial separation r0 are now defined. The dispersion Dr2(t) is mean squared separation
Dr2(t,r0)=rmn2(t|r0),
where the averaging ⟨·⟩ is over all drifter pair separations rmn2(t) with initial separation r0. The number of pairs with r0 initial separation is time dependent and denoted Np = Np(t, r0). Similarly, the perturbation dispersion is
Dr2(t,r0)=rmn2(t|r0).
Expressing the last term in (7) as a dot product, the dispersion and perturbation dispersion are related via
Dr2(t,r0)=Dr2(t,r0)+r02+2rmn,0rmn(t|r0),
where it is assumed that all drifter pairs have the same initial separation r02=rmn,02. The third term on the RHS of (10), here denoted
Φ(t,r0)2rmn,0rmn(t|r0),
is zero if the initial separation vector and the perturbation vector are uncorrelated. This is typically assumed (Batchelor 1950) in homogeneous isotropic turbulence and when the averaging is over drifter pairs from many independent realizations. However, in a laboratory experiment, this term was nonzero and could not be neglected (Ouellette et al. 2006).
The total diffusivity Kr measures the average squared separation rate and is defined as
Kr(t,r0)=14ddtDr2(t,r0).
The factor of 1/4 is used in (12) so that if each drifter in a pair is independent of the other, the two-particle diffusivity Kr 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
Kr(t,r0)=14ddtDr2(t,r0).
Using (10), and assuming all drifters are initially separated by r0, the diffusivity and perturbation diffusivity are related by
Kr(t,r0)=Kr(t,r0)+14ddtΦ(t,r0).
Although dΦ(t, r0)/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).
The variability of the dispersion and diffusivities is calculated as standard deviations. For the dispersion Dr2, it is denoted σDr2, and estimated as
σDr22(t,r0)=[rmn2(t|r0)Dr2(t,r0)]2.
The standard deviation of Dr2 is found and denoted similarly.

b. The Batchelor regime

The RO scaling for the dispersion (1) in inertial subrange turbulence only depends on the mean dissipation rate ϵ¯, i.e., Dr2=CRϵ¯t3 where CR is a nondimensional constant (e.g., Salazar and Collins 2009). Thus, RO scaling is independent of r0 and only formally valid if ΦDr2 and for times (ttB) where perturbation dispersion is much larger than the initial separation Dr2r02. When these conditions are met, Dr2(t)Dr2(t). For inertial subrange k−5/3 turbulence, Batchelor (1950) derived scaling for the perturbation dispersion Dr2(t,r0)=U2(r0)t2, Eq. (3), for short times in which the dispersion depends on r0. The time scale separating RO (1) and Batchelor scaling (3) is the Batchelor time tB (e.g., Salazar and Collins 2009), here defined as
tB(r0)=14r0U(r0),
where U(r0) is the Batchelor velocity. According to inertial subrange theory, the transition from Batchelor (3) to RO dispersion (1) should be direct, i.e., the dispersion Dr2~tγ ought to increase directly from 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 ttB Batchelor scaling Dr2~t2 (3) will be found for initial times for any velocity field with spectra E ~ kβ 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 δumn = um(rm) − un(rn) separated by r = ||rmn||. Thus, the averaging for structure functions differs from the averaging for dispersion statistics. Unlike D2(t, r0) and K(t, r0), where averaging is done over pairs of drifters at time t separated initially by r0, structure function S(r) averaging is over drifter pairs and times for which the drifters are separated by r.

The total structure function is related to the 1D spaced-lagged velocity correlation function ρ(r)=u(x+r)u(x)/σu2, where σu2 is the velocity variance. Assuming isotropy and homogeneity
S(r)=4σu2[1ρ(r)].
When properly normalized, the structure function 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
E(k)~kβ,
S(r)~rβ1,
Dr2~t4/(3β),
Kr~Dr(β+1)/2.
These relationships (17) are valid for β < 3 wavenumber spectra, ΦDr2, and for times ttB corresponding to Dr2r02 such that Dr2Dr2 (e.g., Foussard et al. 2017). At ttB, the perturbation dispersion follows a Batchelor scaling Dr2=S(r0)t2 for all β < 3 with structure function S(r0)~r0β1 (17b). For β = 3 in the 2D turbulent enstrophy cascade, S ~ r2 and the dispersion for all t is given by Dr2~exp(t) and Kr~Dr2 (Lin 1972). Thus, a Batchelor regime does not exist for a 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 r0 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 Dr2(t), can be considered the time since deployment (column 2 in Table 1) in UTC.

Fig. 1.
Fig. 1.

(a) The Pt. Sal drifter trajectories. There are 12 releases of 8–26 drifters. Dots are the initial positions colored according to the latitude of the release: blues to the south, reds to the north. (b) An example realization showing trajectories for drifters released on 13 Sep 2017 in coordinates centered on the tip of Pt. Sal. Magenta asterisks indicate the location of the ADCP and temperature mooring used in the analysis. Bathymetry is contoured in gray: thick contours are separated by 10 m, and thin contours are separated by 2.5 m (starting at 10 m).

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

Table 1.

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 Tp (columns 8–10) information.

Table 1.

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¯=192s (rather than 150 s as prescribed). Large gaps occurred more often during calm conditions because the SPOT Trace GPS units used here required accelerations (from surface gravity waves) to continuously transmit. Trajectories with gaps larger than δτmx = 45 min are not included in the analysis. For each release, the number of trajectories N meeting this requirement (all δτ < δτmx), and the number of drifter pairs Np, are shown in Table 1. The number of pairs Np 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 δτmx and nearly identical results are obtained if the requirement is loosened (e.g., δτmx = 1 h) or tightened (e.g., δτmx = 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 (Np in last row of Table 1). For the entire experiment, the majority of initial separations r0 are ≤1500 m (Fig. 2). Initial separations r0 are binned every 250 m from 250 to 3000 m (bin centers) with >200 drifter pairs for r0 ≤ 1500 m, whereas there are fewer drifter pairs (Np ≤ 100) for r0 ≥ 1750 m. The mean r0 within each bin, for 250 ≤ r0 ≤ 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 r0 ≥ 1750 m, only results for 250 ≤ r0 ≤ 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 r0 is larger than some previous studies (e.g., Koszalka et al. 2009; Ohlmann et al. 2012) with similar r0, thus, the experiment averaged statistics reported here are robust relative to previous estimates for r0 ≤ 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.

Fig. 2.
Fig. 2.

Distribution of initial drifter separations r0 used for experiment averaged statistics. The actual bin centers are indicated by red plus signs and do not deviate much from multiples of 250 m, especially for r0 ≤ 1500 m.

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

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 Np depends on the initial separation r0 and time t (Fig. 3a). The number of drifter pairs is constant in time for t < 104 s and equal to the initial number of pairs (Fig. 2) before rapidly dropping as drifters were picked up. For each r0, the EA dispersion statistics are displayed only for Np(t) > 200 (gray line, Fig. 3a) which is effectively t < 104 s. Including fewer drifter pairs yields noisy statistics for these times.

Fig. 3.
Fig. 3.

(a)–(c) Experiment averaged (EA) two-particle statistics vs time t. (a) The number of drifter pairs (Np) vs time t for initial separations 250 ≥ r0 ≥ 1500 m (colors). Minimum number of pairs Np(min)=200 for analysis is indicated by the horizontal gray line. (b) The EA perturbation dispersion Dr2 (9) vs time t. (c) The compensated EA perturbation dispersion Dr2/t2 vs time t. (d) The scaled EA perturbation dispersion Dr2/(U2t2) vs scaled time t/tB. The dispersion curves are only shown for times with Np(t)>Np(min). In (b), the dashed gray line is t2 and the solid gray line is ~t3. In (c), for each r0, circles indicate squared Batchelor velocities U2(r0) (18) and tick marks indicate the Batchelor time tB (16).

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

The experiment averaged (EA) perturbation dispersion scales like Dr2(t,r0)=U2(r0)t2 for r0 ≥ 500 m (colored curves in Fig. 3b) for approximately t < 5000 s. For the smallest r0 = 250 m, the growth is slightly slower than t2 (≈t1.85). The slightly slower than ~t2 perturbation dispersion growth for r0 = 250 m (and to a lesser extent r0 = 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 Dr2(t,r0) to grow slower than t2. The ballistic growth U2(r0) increases with r0 (stacking of colored curves in Fig. 3b). Thus, the initial EA perturbation dispersion is consistent with a Batchelor regime as Dr2=U2(r0)t2 (3) and U2(r0) increases with r0 (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 Dr2~t3 (solid gray line in Fig. 3b) for which the different r0 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 Dr2 curves for long times (t > 104 s Fig. 3b) suggests that the dispersion maybe transitioning from t2 to t3 growth.

The EA perturbation dispersion compensated by t2 clearly shows a Batchelor scaling for r0 ≥ 500 m (Fig. 3c). Assuming Batchelor scaling for the EA perturbation dispersion, the Batchelor velocity U(r0) is estimated here from drifter data using
U2(r0)=Dr2(t,r0)/t2¯,
where the overline represents a time mean from 150 to 600 s. Thus, U is estimated using only the first 600 s of drifter pair data. Consistent with theoretical expectations (Salazar and Collins 2009), squared Batchelor velocities increase with r0 (circles placed at t = 102 s in Fig. 3c) from 0.0026 m2 s−2 at r0 = 250 m to 0.0149 m2 s−2 at r0 = 1500 m. Also consistent with theoretical expectations, comparing colored curves in Fig. 3c shows that the duration of Batchelor scaling increases with r0 as cooler colored curves depart from a constant sooner than the warmer colored curves (Ouellette et al. 2006). Thus, the Batchelor time tB(r0)=0.25r0U1 (16) increases with r0 (thin vertical lines in Fig. 3c). Except for the smallest initial separation r0 = 250 m, the Batchelor time tB is consistent with Dr2 changing by approximately 15% for all r0. The decreasing in time t compensated EA perturbation dispersion Dr2(t,r0)/t2, beginning at t = 200 s for r0 = 250 m and t = 4000 s for r0 = 1500 m (Fig. 3c), indicates that the dispersion is weaker than Batchelor (tγ with γ < 2) when transitioning out of Batchelor scaling.

Scaling EA Dr2 for each r0 by U2(r0)t2, and time t by the Batchelor time tB(r0), collapses the perturbation dispersion fairly well overall (Fig. 3d). The collapse is very good for 750 ≤ r0 ≤ 1500 m (cyan to orange curves) for which the scaled EA perturbation dispersions are all generally similar to each whereas for r0 = 250 m and 500 m the collapse is not as good. For r0 ≥ 750 m, tB well predicts the time when Dr2/(U2t2) begins to rapidly drop (when t/tB ≈ 2. Thus, for the r0 ≥ 750 m that show the best Batchelor scaling (Dr2~t2, e.g., Fig. 3c), tB indicates well the duration of Dr2~t2 growth, while for the r0 ≤ 500 m that show weaker than ~t2 growth, tB does not correspond to the duration of the initial growth. The scaled dispersions for r0 = 250 m and 500 m (blue curves) drop less rapidly than for r0 ≥ 750 m. For all r0, the departure from Batchelor scaling, and subsequent Dr2 growth slower than t2, results in perturbation dispersions for long times that are approximately 50% of the dispersion that would result if Batchelor scaling Dr2=U2t2 remained valid for all t, i.e., for all curves, eventually Dr2/(U2t2)0.5 for t/tB > 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 Dr2(t), the EA total dispersion Dr2(t) (8) shows no evidence of a power law scaling for t < 104 s (Fig. 4a). For each r0, EA Dr2(t) increases in time from a constant. Relative to the initial Dr2, the increase in time from the constant r02 is largest (~8 times) for the smallest r0 = 250 m and smallest (~2 times) for the largest r0 = 1500 m. For each r0, the total dispersion Dr2(t,r0) (8) is similar to r02+Dr2(t,r0) (cf. colored and thin black curves in Fig. 4a). The deviation Dr2(t,r0)[r02+Dr2(t,r0)] is quantified by Φ(t, r0), see (10) and (11), hence Φ is estimated as Φ(t,r0)=Dr2(t,r0)r02Dr2(t,r0) (colored curves in Fig. 3b). The deviation Φ (10) is due to the correlation between the initial separation and perturbation separation ⟨r0r′(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 r0, initially Φ is larger than Dr2 (cf. Figs. 3b and 4b) but for larger times Dr2(t)>Φ(t) due to faster growth of Dr2(t)~t2 compared to Φ ~ t. The time tΦ(r0) when Φ(t,r0)=Dr2(t,r0) generally increases with r0 (circles in Fig. 4b). For t < tΦ, Φ initially contributes more to Dr2 than does Dr2 and vice versa. Thus, for t < tΦ the quantity Dr2r02=Dr2+Φ differs considerably from Dr2 and grows as ~t due to the Φ contribution. The total dispersion Dr2 does not follow the alternative scaling law Dr2(t,r0)=[r0+U(r0)t]2 as the cross term 2r0U(r0)tΦ due to r0U(r0)tr0r (not shown). Physically, a Φ > 0 with dΦ/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.

Fig. 4.
Fig. 4.

(a) The experiment averaged dispersion Dr2 (8) and (b) Φ (11) vs time t. In (a), thin black curves are r02+Dr2(t). In (b), colored curves are Φ=Dr2Dr2r02, thin black curves are Φ = 2⟨r0r′⟩, and circles indicate the time tΦ when Φ(t)=Dr2(t). Colors correspond to initial separations 250 ≤ r0 ≤ 1500 m (see legend in Fig. 3). In (a) and (b), thin black curves are on top of colored curves for all r0.

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

The EA perturbation diffusivity Kr(t,r0) is calculated from Dr2(t,r0) using (13). For approximately t < 4000 s, Kr~t (Fig. 5a) consistent with a Batchelor regime. In contrast, the total diffusivity Kr(t, r0) is considerably different (Fig. 5b) than Kr(t,r0). For r0 = 1250, 1500 m, Kr(t) is nearly constant for all t whereas for r0 = 250, 500 m, Kr(t) increases more quickly in time. Specifically, Kr(t) changes by a factor of ≈1.7 times for r0 = 1500 m and by ≈12.5 times for r0 = 250 m. The difference between Kr and Kr is due to dΦ/dt where 0.25dΦ/dt=(KrKr) (14). As Φ increases linearly or more slowly than linearly in time (Fig. 4b), 0.25(dΦ/dt) is constant or slowly decreasing in time for all r0 (Fig. 5c). Owing to the different growth rates for Kr(t) and 0.25(dΦ/dt), for t < t0, where t0 is the time when 0.25(dΦ/dt)=Kr(t) (circles in Fig. 5c), 0.25(dΦ/dt) contributes more to Kr(t) than the perturbation diffusivity Kr. For t > t0, dΦ/dt contributes less to Kr(t) than the perturbation diffusivity Kr. This transition time t0 generally increases with r0 (circles in Fig. 5c). The mean separation velocity υr is found from Φ
υr=12r0ddtΦ(t,r0)
and initially (t ≈ 100 s) increases with r0 from about 0.004 m s−1 to ≈0.03 m s−1 (Fig. 5d). The initially constant υr for each r0 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 r0, υr generally decreases in time until t ≈ 4000 s before generally increasing in time from 4000 to 10 000 s.
Fig. 5.
Fig. 5.

(a) The experiment averaged (EA) perturbation diffusivity Kr (13), (b) the EA total diffusivity Kr (12), (c) the EA difference 1/4(dΦ/dt)=KrKr (14), and (d) the EA separation velocity υr (19) vs time t. In (a), the Batchelor scaling ~t (dashed gray) and the RO scaling ~t2 (solid gray) are shown. In (c), circles indicate the time when Kr=1/4(dΦ/dt). Colors correspond to initial separations 250 ≤ r0 ≤ 1500 m (see legend in Fig. 3).

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

The dependence of the EA diffusivity Kr on scale is now explored. Within the Batchelor regime, Kr~(Dr2)1/2 (Fig. 6a) since Dr2~t2 and Kr~t. The maximum length scale (Dr2)1/2 for which the scaling Kr~(Dr2)1/2 applies increases with r0 consistent with the requirement that Dr2<r02 within the Batchelor regime. For instance, Kr deviates from Kr~(Dr2)1/2 at smaller (Dr2)1/2 for r0 = 250 m than for r0 = 1500 m (cf. blue and orange curves in Fig. 6a). The r0-dependent Kr~Dr Batchelor scaling is contrasted with RO scaling for which Kr~Dr4/3 independent of r0 (thick solid gray line in Fig. 6a). However, the perturbation diffusivity Kr appears to approach a r0-independent scaling when the total separation s=(r02+Dr2)1/2 is used for the scale (thin gray line in Fig. 6b). The approach to s4/3 scaling only occurs after the Batchelor time tB for each r0 (circles in Fig. 6b indicate tB). In Fig. 6b, before tB each r0 curve has vertical tails due to sr0 in the Batchelor regime (Dr2r02). The diffusivity Kr is approaching the same ~s4/3 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 Dr, rather than s, results in the same scale dependence because s2Dr2 (Fig. 4a). Moreover, the dependence of the diffusivity Kr on scale s is approximately the same as Kr versus s except Kr versus s lacks the vertical Batchelor regime tails for each r0 as Kr lacks an initial Batchelor regime. Although each r0 curve appears to approach the same RO s4/3 scaling (thin gray line in Fig. 6b), drifter trajectories were too short to definitively observe RO scaling which would have the different r0 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.

Fig. 6.
Fig. 6.

(a) The EA perturbation diffusivity Kr vs (a) EA perturbation dispersion (Dr2)1/2 and (b) total separation s=(r02+Dr2)1/2. Colors correspond to initial separations 250 ≤ r0 ≤ 1500 m (see legend in Fig. 3). In (a), Batchelor scaling ~Dr (dashed gray) and RO scaling ~Dr4/3 (gray) are shown. In (b), the RO scaling ~s4/3 (thick solid gray) and the enstrophy scaling ~s2 (dash–dotted gray) are shown. Circles in (b) indicate the Batchelor time tB.

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

As indicated by the circles in Fig. 3c, squared Batchelor velocities U2(r0) increase with r0 (colored circles in Fig. 7). Overall, there is no consistent U2(r0) scaling over the entire range of r0 and neither enstrophy cascade (~r02) nor inertial subrange scaling (~r02/3) is indicated (cf. colored dots to solid and dotted gray lines in Fig. 7). The structure function S(r) (black curve in Fig. 7) calculated using (5) for all drifter pair data is ~r2/3. For r ≥ 1000 m, the structure function and squared Batchelor velocities are similar, i.e., SU2. However, for r ≤ 750 m, S(r)U2(r0) despite their theoretical equivalence. Specifically, for r0 ≤ 750 m, S(r)>U2(r0) by less than a factor of 2 and the r0 scaling for U2 is steeper than the r scaling for S. The difference between U2 and S is due to S including all (≈104 s) data for each drifter pair trajectory whereas U2 is based on the only the initial growth which uses the first 5 (600 s) times of each pair trajectory. To verify this, a modified structure function S˜ is calculated using (5) but with only the initial times (t ≤ 600 s) of each drifter pair trajectory. Because U2(r0) is found using only the first 600 s of each pair trajectory, U2(r0) should approximately equal the modified structure function S˜(r) for all r0 and r. Indeed, the modified structure function S˜(r), which is essentially Eulerian as the flow has not yet preferentially placed the drifters in convergence zones, is nearly U2(r0) (cf. “x” marks and colored dots in Fig. 7) with largest deviations for r ≥ 1250 m.

Fig. 7.
Fig. 7.

Drifter pair trajectory derived second-order experiment averaged (EA) structure functions S (thick black) (5) vs r. The EA structure function S˜ based on velocity increments from initial drifter pair trajectory times (see text) vs r are shown as “x” marks. Squared Batchelor velocities U2 (circles) (18) are derived from the Batchelor scaling of Dr2 and colored according to r0 (as in Fig. 3). Energy cascade ~r2/3, enstrophy cascade ~r2, and ~r scalings are shown as solid, dotted gray, dash–dotted lines, respectively.

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

Differences between S(r) and S˜(r) for 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 ||δumn|| used to estimate S(r) and S˜(r) for 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 Dr2(t,r0) (9) and SR total dispersions Dr2(t,r0) (8) are calculated. Note that some releases do not have drifter pairs in every r0 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 r0 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 Np for r0 = 500 m is Np = 27 before dropping dramatically for t ≥ 1.5 × 104 s (red curve, Fig. 8a). Averaging over each release separately greatly reduces Np for each r0 compared to averaging over the entire experiment (i.e., compare this Np to that in Fig. 3a). For this r0 with 27 initial pairs, IP perturbation dispersions r2 are variable in time and between drifter pairs (thin blue lines Fig. 8a). For instance, all r2 initially grows in time reaching a local maximum at 5000–8000 s (depending on which r2), then dropping for 1000–2000 s before increasing again. The variability between r2 is considerable with r2 ranging from ≈0 to 5 × 105 m2 for t ≈ 7000 s.

Fig. 8.
Fig. 8.

Dispersion vs time for the 13 Sep 2017 release: (a),(c) perturbation dispersion Dr2(t), and (b),(d) total dispersion Dr2(t) for (left) r0 = 500 m and (right) r0 = 1000 m. Individual pair (IP) dispersion r2 (6) [in (a) and (c)] and IP r2 [in (b) and (d)] are colored curves. In each panel, the mean of all pairs (thick black), i.e., the single release statistic, and the mean ± one standard deviation (thin black) are shown. The number of drifter pairs Np(t) for r0 = 500 m in (a) and r0 = 1000 m in (b) is indicated by the red curve. Means and IP dispersions are shown as long as the number of drifter pairs is within five of the initial number of pairs, i.e., as long as Np(0) − Np(t) ≤ 5 (≈5 h for r0 = 500 m, ≈4 h for r0 = 1000 m). Thus, five IP dispersions are shorter in duration than the mean.

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

The time variability for this r0 = 500 m is also evident in the single release (SR) perturbation dispersion Dr2(t) (the mean of these r2, thick black curve Fig. 8a) with Dr2 increasing from 0 at t = 0 to 0.2 × 106 m2at t = 7000 s then dropping until t ≈ 9000 s before rising to ≈0.4 × 106 m2(at t = 1.5 × 104 s). Overall, for r0 = 500 m the growth of Dr2(t) appears to be t2 despite the local maximum at t ≈ 7000 s. The variability across individual drifter pairs at time t is quantified by the standard deviation of σDr2(t), e.g., (15) and for this r0 scales directly with Dr2(t) for t ≤ 1.5 × 104 s, i.e., σDr2(t)Dr2(t) (e.g., the thin black curve Dr2σDr2 in Fig. 8a is ≈0 for t ≤ 1.5 × 104 s).

The SR total dispersion Dr2(t) for r0 = 500 m shows similar time dependence to the SR perturbation dispersion Dr2(t) (cf. thick black curves Figs. 8a and 8b) but has larger magnitude (note different y scales between Figs. 8a and 8b). Thus, for this release and r0, Φ ≠ 0. For instance, at the local maximum (t ≈ 7000 s), Φ ≈ 0.2 × 106 m2, a value greater than Dr2. Similar to σDr2, the total dispersion variability across pairs for this r0 = 500 m scales with the total dispersion but offset by the initial total dispersion such that σDr2(t)Dr2(t)Dr2(0) (lower thin black curve in Fig. 8b).

For the 13 September 2017 release, the pair number Np for r0 = 1000 m is initially Np = 22 before dropping when t ≈ 1.5 × 104 s (red curve Fig. 8c). The SR perturbation dispersion Dr2(t) for r0 = 1000 m shows similar time dependence to the r0 = 500 m SR Dr2(t). Overall the growth appears quadratic (~t2) but is approximately 8 times larger in magnitude (thick black curves in Figs. 8a and 8c). A point of difference is that Dr2 for r0 = 1000 m remains constant from 5000 to 10 000 s rather than decreasing like Dr2 for r0 = 500 m. Unlike the variability across pairs for r0 = 500 m in which σDr2Dr2, for r0 = 1000 m the variability is smaller than the dispersion with σDr20.4Dr2 (thin black curves Fig. 8c).

The SR total dispersion Dr2(t) for r0 = 1000 m shows similar time dependence to the other 13 September 2017 dispersions (thick black curve Fig. 8d). Again, the magnitude of Dr2 is larger than Dr2 for r0 = 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 Dr2 and Dr2 are constant in time, Φ ≈ 1.6 × 106 m2 a value more than 2 × Dr2 at this t. Like Dr2 for this r0, the total dispersion variability across pairs σDr2 is a fraction of the total dispersion Dr2. In short, similar to EA dispersion, SR perturbation and total dispersions increase with r0 and the total dispersion is larger than the perturbation dispersion for a given r0. 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 Dr2 for all releases is now examined. As such, thick black curves like those in Figs. 8a and 8b are constructed and examined for each release. Similar to the time dependence of Dr2(t) for both r0 = 500 and 1000 m for the 13 September 2017 release (Figs. 8a,b), for all releases and for these r0, the initial time dependence of Dr2(t) is approximately ~t2 (Figs. 9a,b). Thus, the perturbation dispersion for each release is largely consistent with a Batchelor regime.

Fig. 9.
Fig. 9.

For each release (colors; consistent with Fig. 1), the single release perturbation dispersion Dr2 vs time t for (a) r0 = 500 m and (b) r0 = 1000 m initial separations. Red (blue) colors correspond to drifter releases in the northern (southern) portion of the experimental region off of Oceano (Pt. Sal). In (a) and (b), the gray line is ~t2. Statistics are shown only if the number of drifter pairs Np(t) has not dropped by more than 5 from the initial drifter pair number Np(0). Release 4 is not plotted in (b) because this release did not have any drifter pairs with initial separations r0 ≥ 1000 m.

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

The time when Dr2(t) departs from Batchelor scaling varies with release. For instance, the two releases (7 and 11) with small initial Dr2 for r0 = 500 m (red and dark orange curves in Fig. 9a) both depart and grow more slowly than t2 at t ≈ 1000 s, whereas the departure from t2 growth is much later (t ≈ 10 000 s) for release 12 (light blue curve in Fig. 9a). Unlike the experiment averaged Dr2(t), some releases (2 and 5 for r0 = 500 m, and 2 for r0 = 1000 m) transition to faster than t2 growth rather than slower. For example, release 2 for r0 = 1000 m is faster than t2 at t = 2000 s before transitioning to slower than t2 for t > 4000 s (blue curve in Fig. 9b). Although the experiment averaged Dr2(t) indicates that the Batchelor time tB is associated with departure from t2 growth (e.g., Figs. 3c,d), for the individual release Dr2, a clear pattern of when Dr2 departs from t2 growth is not obvious. For the same r0, the Batchelor time tB=0.25r0U1 is inversely proportional to U. Thus, for r0 = 500 m, release 4 (orange curve in Fig. 9a), with the largest U2, ought to depart from t2 growth before the other releases and release 6 (dark red curve in Fig. 9a), with the smallest U2, ought to depart from t2 growth after the other releases. No tB pattern is apparent likely due to the individual release Dr2 not representing an ensemble mean and sampling error arising from individual release dispersion based on few drifter pairs.

The SR perturbation dispersion Dr2(t) range over the releases is substantial. For instance, initially (t = 150 s) the r0 = 500 m Dr2=12m2 for release 6 while Dr2=363m2 for release 4, a factor of 30 difference (Fig. 9a). Thus, squared Batchelor velocities U2 vary by a factor of 30 between releases for r0 = 500 m. For r0 = 1000 m, the largest initial Dr2 (688 m2) is 36 times larger than the smallest Dr2 (19 m2) (Fig. 9b). For both r0 = 500 and 1000 m, the difference between largest and smallest Dr2 persists for all times with the largest Dr2 approximately 30 times larger than the smallest Dr2 for every t (Figs. 9a,b). Although the initial growth of SR dispersion is generally similar to EA dispersion (~t2), 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 Dr2 and releases to the south (cool colored curves in Figs. 9a,b) generally have the largest Dr2. For r0 = 500 m, release 4 (orange curve with largest Dr2) is an exception as this release was in the northern portion of the domain. However, the initial location of the drifters for this release was in deeper water (≈40 m) than the other releases (20–30-m depths) (see Fig. 1), potentially affecting the dispersion.

3) Scaling the dispersion with U2

The initial perturbation dispersion for each release generally follows Batchelor scaling Dr2(t,r0)=U2(r0)t2 (Figs. 9a,b). Here, we examine how the dispersion for each release transitions out of the Batchelor regime and how the dispersion for t = 104 s (i.e., t > tB) depends on U2 and r0. Squared Batchelor velocities U2(r0) for each release and r0 are determined using (18) as the perturbation dispersion for each release is generally consistent with a Batchelor regime. Thus, for r0 = 500 m, U2 is smallest for release 6 and largest for release 4 (initial stacking of curves in Fig. 9a). As there are few crossings of the colored curves in Fig. 9a, large U2 for r0 = 500 m is generally associated with larger later dispersion and small U2 is generally associated with small later dispersion; this is also similar for r0 = 1000 m.

Directly comparing the perturbation dispersion Dr2(t) at t = 104 s for each r0 and release to the squared Batchelor velocities U2 (Fig. 10) clarifies this relationship. The perturbation dispersion at 104 s is generally associated with U2 and the association is largely geographic: northern releases (warm colored dots) have the smallest U2 and Dr2(t) at t = 104 s and southern releases (cool colored dots) have the largest U2 and Dr2(t) at t = 104 s. Although the perturbation dispersion Dr2(t=104s) scales with U2, it is less than what Batchelor dispersion predicts as the colored dots are under dashed black line in Fig. 10 by a factor of approximately 1/2. Thus, as dispersion transitions out of the Batchelor regime, the SR Dr2(t) generally slows, relative to continuing as Batchelor, similar to EA dispersion (Fig. 3d).

Fig. 10.
Fig. 10.

The single release (SR) perturbation dispersion Dr2(t) at time t = 104 s vs the SR squared Batchelor velocity U2. Colors correspond to latitude of the release consistent with Fig. 1. Symbol size corresponds to increasing initial release separation r0 = 250, 500, 750, …, 1500 m (see legend). The dashed black line corresponds to exact Batchelor scaling (3) Dr2(t)=U2t2 at t = 104 s.

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

The initial separation r0 affects the later perturbation dispersion. Generally, for a specific release, Dr2(t=104s,r0) increases with r0 (larger dots of the same color are above smaller dots in Fig. 10). Thus, the initial separation influences the perturbation dispersion after 104 s indicating that r0-independent RO scaling has not been established. However, the variation in Dr2(t=104s,r0) across r0 values for a given release is smaller than differences between releases for the same r0, compare the spread of Dr2(t=104s) for the same colored dots to the spread of Dr2(t=104s) for the same sized dots in Fig. 10. Thus, the conditions (day of release) and/or specific release locations (e.g., north or south of Point Sal) can influence the dispersion more than the initial separation such that single release dispersion can vary significantly from the EA dispersion.

5. Discussion

a. Influence of internal bores on dispersion

The experiment averaged, and individual release perturbation dispersion Dr2(t), both largely follow Batchelor scaling for t ≲ 5000 s (Figs. 3c and 9). However, some individual releases deviate from Batchelor scaling as seen in Figs. 8a and 8c where Dr2(t) can stop growing (5000 < t < 104 s for r0 = 1000 m, Fig. 8c) and even shrink (6000 < t < 9000 s for r0 = 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).

Fig. 11.
Fig. 11.

(a) East–west velocity u (colors) and temperature (black contours at 13°, 14°, 15°, and 16°C) vs time and depth z at the 30-m Pt. Sal mooring [magenta asterisk in Figs. 1a and 1b at (x, y) ≈ (−2, 0.5) km] during the fifth drifter release on 13 Sep 2017. The tide level is indicated by the black curve at z ≈ 0. (b),(c) Pt. Sal 30-m ADCP near surface (z ≈ −3 m) east–west velocity u vs time (thick gray curves; left y axis). Colored curves (right y axis) are the 15 individual drifter cross-shore position X(t) in (b) and individual drifter pair perturbation separations r2 for r0 = 500 m vs time in (c) (same as blue curves in Fig. 8a). The trajectories X(t) and individual perturbation dispersions r2(t) are shown for the first 5 h, the approximate time when the Np for this release and r0 has decreased from the initial Np(0) by 6 (see Fig. 8a). Colors correspond to the initial cross-shore drifter position in (b) and the initial drifter pair cross-shore midpoint in (c). Reds denote positions closer to shore, and blues denote farther offshore positions. Mooring times are offset by −1 h to better align mooring and drifter times as this mooring was onshore of the drifters.

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

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 r0 = 500 m individual separations r2(t) (blue curves in Fig. 8a reproduced as colored curves versus UTC time in Fig. 11c). The faster than linear initial growth for all r2(t) before 1615 UTC appears Batchelor-like, i.e., ~t2. Although individual perturbation separations r2 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 r2 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 r2 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 Dr2(t) at 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 r2(t) increase and decrease before and after, respectively, NLIW arrival (blue curves Fig. 11c) indicated by local maxima in r2(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 r2(t) to grow before the bore arrives. This results in the largest r2 maximum at ≈1700 UTC for pairs with cross-shore initial separations (orange curves in Fig. 11c). For onshore pairs with alongshore initial separations, r2 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 r2(t) similar to, and an approximate continuation of, the initial growth (Fig. 11c). This is reflected in Fig. 10 where the perturbation dispersion Dr2 at t = 104 s for all r0 for this release (darkest blue dots with U20.02m2s2) is similar to but smaller (by approximately 50%) than the initial squared Batchelor velocity prediction. To summarize, NLIWs act to pulse the perturbation dispersion Dr2 by quickly growing, then stopping and reversing individual separations after which Dr2 grows similarly to its initial growth. Although the effect is large in the short term, overall for times longer than a few hours the effect is fleeting as the dispersion generally returns to the initial Batchelor growth. Thus, this NLIW does not appear to be large contributors to the dispersion on time scales longer than a few hours.

b. Comparison to previous work

For the release-averaged and most individual releases, the perturbation dispersion is consistent with Batchelor scaling Dr2=U2(r0)t2 for t < tB with the Batchelor time tB (16) increasing with initial drifter separation r0 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 (Dr2) for which Batchelor scaling is less clear.

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 Dr2~t2 for 5 < t < 84 h. In this study, a thorough examination of Batchelor scaling was not possible as only one small r0 ≈ 7.5 m was considered. In modeling studies of coastal Southern California, for a single r0 ≈ 500 m and t ≤ 1 day, the dispersion Dr2 is similar to t2 in a 40-m resolution model (Dauhajre et al. 2019) and Dr2~t2 for drifters released within 2 km of the shoreline in a 250-m resolution model (Romero et al. 2013). In a model of the Adriatic Sea, the dispersion is approximately Dr2(t)~t2 for r0 ≈ 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 ⟨r2⟩ − ⟨r2 shows ~t2 growth for t ≲ 3 × 104 s (Roth et al. 2017). In these previous studies, Batchelor scaling was not examined. Some previous studies that have found ballistic Dr2~t2 dispersion (LaCasce and Ohlmann 2003; Roth et al. 2017) suggested that it might be due to horizontal shear dispersion. However, the t2 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) ~ r2 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 Dr2~t3 growth (LaCasce 2008).

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 (tB ≈ 1 h, green + in Fig. 3c) for the r0 ≈ 1 km initial separation considered. In Lumpkin and Elipot (2010), where r0 ≈ 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 r0. In Beron-Vera and LaCasce (2016), where r0 ≈ 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 Dr2(t) was analyzed rather than the perturbation dispersion Dr2(t). Ouellette et al. (2006) discusses how Dr2(t) may hinder the correct dispersive scalings due to including initial separation effects, here quantified as Φ. How Dr2(t) or Dr2(t) can lead to different conclusions is examined by considering the experiment averaged Dr2 and Dr2 for r0 = 1 km (green curve in Fig. 3b and Fig. 4a). For t ≤ 104 s, Dr2(t) is well fit by the exponential r02exp[t/(5h)] (green and dash–dotted curves, respectively, in Fig. 12). Previous oceanographic studies (e.g., Ollitrault et al. 2005; Koszalka et al. 2009; Ohlmann et al. 2012) have also found initial Dr2~exp(t/τ) scaling with various e-folding times τ. For instance, in the Santa Barbara Channel for r0 = 7.5 m, Dr2~exp(t) for t < 5 h with an e-folding time of τ = 0.9 h (Ohlmann et al. 2012) as well as for r0 = 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 Dr2~exp(t) is associated with a k−3 wavenumber spectra (Lin 1972), Dr2~exp(t) suggests that the dispersion may be due to a 2D turbulence enstrophy cascade. According to the theory, e-folding times are independent of r0 in contrast to exponential fits to observed Dr2(t,r0) (colored curves in Fig. 4a) for which τ increases with r0. Rather than exponential growth, for r0 = 1 km the offset perturbation dispersion r02+Dr2(t) is well fit by offset Batchelor scaling r02+U2t2 for 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, r02+Dr2(t) suggests that the dispersion is due to k−5/3 rather than k−3 turbulence as incorrectly suggested by the Dr2(t)=r02exp(t/τ) fit valid up to t ≈ 12 000 s. Here, Dr2(t) is well approximated by an exponential r02exp(t/τ) because initially r02exp(t/τ)r02(1+t/τ) and Dr2(t)=r02+Dr2(t)+Φ(t), see (10) and (11), with Φ ~ t and Φ>Dr2 (green curve in Fig. 4b) for t ≲ 3000 s. To properly estimate the dispersive scaling(s), and therefore correctly infer the background turbulent wavenumber spectra, the perturbation dispersion Dr2(t) must be examined (e.g., Ouellette et al. 2006) as correct dispersive scalings are more easily identified in Dr2(t) rather than Dr2(t) which can have Φ ≠ 0 effects.

Fig. 12.
Fig. 12.

The experiment averaged (EA) total dispersion Dr2(t) (green) and offset EA perturbation dispersion r02+Dr2(t) (red) vs time t for the 13 Sep 2017 release and r0 = 1000 m. Solid and dashed black curves are fits to the offset perturbation dispersion: (solid) Batchelor r02+U2(r0)t2 for t < 5000 s, and (dashed) RO (1.3r0)2 + (2.5 × 10−7 m2 s−3)t3 for 6000 ≤ t ≤ 15 000 s. Dash–dotted curve is the exponential fit r02exp[t/(4.7h)] for t < 104 s to Dr2(t).

Citation: Journal of Physical Oceanography 51, 2; 10.1175/JPO-D-20-0170.1

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 r0 offset since the fit for the perturbation dispersion is Dr2(t)C1r02+C2t3 with C1 ≠ 0 whereas for theoretical RO scaling Dr2 would not have the r0 term and C1 = 0. However, in accordance with inertial subrange theory (Batchelor 1950), for large enough t, the t3 term is eventually much larger than r02 term.

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 Dr2 (8), here, the perturbation dispersion Dr2 (9) was also analyzed. Diffusivities Kr (12) and perturbation diffusivities Kr (13) were additionally calculated and analyzed. Statistics were presented for the entire experiment (EA statistics) and each single release (SR statistics) for t ≲ 4 h and for initial drifter separations 250 ≤ r0 ≤ 1500 m.

The EA perturbation dispersion follows Batchelor scaling Dr2(t,r0)=U2(r0)t2 (3) for 1000–3000 s for r0 ≥ 750 m. Consistent with theory, both the duration of Batchelor scaling tB(r0) and squared Batchelor velocities U2(r0) increase with initial separation r0. EA squared Batchelor velocities U2(r0) for r0 = 1500 m are 5 times greater than U2 for r0 = 250 m. For r0 ≤ 1000 m, EA U2(r0) are nearly equivalent to the unbiased velocity structure function S˜ calculated from initial drifter pair trajectories. In contrast to inertial subrange scaling where U2~r02/3, and enstrophy cascade scaling where U2~r02, here U2(r0) does not have a single power-law dependence for all r0. After Batchelor scaling, i.e., t > tB, scale s=(r02+Dr2)1/2 dependent EA perturbation diffusivities suggest approach to the classic Richardson 4/3-law Kr~s4/3 consistent with RO dispersion (1). For the r0 values considered here, the EA Dr2(t) at times larger than tB is smaller than but correlated with the initial Batchelor scaling. Specifically, the dispersion at t = 104 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 r0, the SR Dr2 generally follows Batchelor scaling. For a given r0 and time t, the SR Dr2(t,r0) variation between releases is significant—varying by almost a factor of 100. This variation is attributed to squared Batchelor velocities U2 that are much larger (>10 times) in the southern region of the domain (off Pt. Sal) compared to the northern region (off Oceano). Each release also slows when transitioning out of the Batchelor regime as the SR perturbation dispersion at t = 104 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 r0 was considered. 3) Most importantly, investigating only the dispersion Dr2(t) that can scale differently than the perturbation dispersion Dr2(t) for which Batchelor scaling is clear. Here, Batchelor scaling is not evident in Dr2(t) as the correlations between initial and later separations Φ(t) scale as ~t for short times. Thus, previous studies investigating Dr2(t) are potentially aliased by initial separation effects that are not present in Dr2(t). Thus, analysis of both Dr2 and Dr2 is critical in order to accurately determine the dispersion power law time dependence and therefore the underlying turbulent velocity wavenumber spectra.

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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