Observations of drifter dispersion in the surfzone: The effect of sheared alongshore currents

Authors


Abstract

[1] Surfzone dispersion is characterized with single-particle Lagrangian statistics of GPS-tracked drifters deployed on 5 days at Huntington Beach, California. Incident wave heights varied weakly between days, and stationary rip currents did not occur. Generally, the time-dependent bulk surfzone cross-shore diffusivity equation imagexx was similar on all days, reaching a local maxima of approximately 1.5 m2 s−1 between 160 and 310 s, before decreasing to about 1 m2 s−1 at 1000 s. The alongshore diffusivity equation image increased monotonically to 1000 s and was variable between the 5 days. For times greater than 30 s, the alongshore diffusivity is greater than the cross-shore diffusivity, consistent with previous observations. The observed diffusivities are fit to analytic functional forms, from which asymptotic diffusivities and Lagrangian timescales are determined. The asymptotic alongshore diffusivity equation imageequation image varies between 4 and 19 m2 s−1, and this variation is related to the variation in the maximum of the mean alongshore current equation imagem, broadly consistent with a shear dispersion scaling equation imageequation imageequation image. Cross-shore variation in dispersion processes, lumped together in the bulk κ, is apparent in the non-Gaussian probability distribution function of drifter displacements at intermediate times (30 s). Both biased and unbiased diffusivity sampling errors depend on the number and length of drifter trajectories and limit aspects of the analysis.

1. Introduction

[2] Polluted water sickens beachgoers and significantly impacts coastal United States economies [Dorfman and Rosselot, 2008]. Polluted surfzone waters often have high levels of fecal indicator bacteria [Reeves et al., 2004] and human viruses [Jiang and Chu, 2004]. Dilution and diffusion between the surfzone and offshore waters are believed to be the primary cause of (fecal indicator) Enteroccucus bacteria inactivation [Boehm et al., 2005]. Horizontal diffusion and dispersion must be understood to predict the fate of surfzone tracers, including pollution, plankton, and larvae.

[3] Tracer dispersion can be estimated from Lagrangian drifter data. The theory for single-particle (absolute) dispersion in homogeneous turbulence relates Lagrangian velocity statistics to the diffusivity [Taylor, 1921]. Diffusion is ballistic (i.e., linear diffusivity growth with time) at short times, and Brownian (i.e., constant diffusivity) at long times relative to the Lagrangian timescale. Davis [1987, 1991] developed the methodology for studying oceanic absolute diffusion including the effects of inhomogeneity. Surface and subsurface drifters have been used to directly estimate the large-scale diffusivity of the oceanic general circulation [e.g., Lumpkin et al., 2002], the California Current [Swenson and Niller, 1996], and continental shelf regions [e.g., Dever et al., 1998]. Lacasce [2008] provides an excellent review.

[4] A goal of surfzone mixing research is to estimate the surfzone eddy diffusivity, which could be used in a Fickian diffusion equation for a surfzone tracer, and to determine the diffusivity dependence upon surfzone parameters such as wave height and mean currents. Surfzone diffusivity was first estimated by measuring the alongshore spreading rate of fluorescent dye tracer at the shoreline [Harris et al., 1963; Inman et al., 1971; Grant et al., 2005; Clarke et al., 2007]. Surfzone eddy diffusivity estimates varied considerably, in part because the single realization of a the observed tracer patch precluded the averaging necessary for statistically stable diffusivity estimates. More recently, GPS-tracked surfzone drifters [Schmidt et al., 2003] have been used to study surfzone circulation and diffusion in the field. Drifters have been used to estimate absolute and relative diffusivities in rip-current dominated surfzone circulations [Johnson and Pattiaratchi, 2004; Brown et al., 2009], and to observe rip currents and surfzone eddies on irregular bathymetry [Schmidt et al., 2005]. Surfzone drifters have been included in wave-resolving numerical models of transient rip currents [Johnson and Pattiaratchi, 2006].

[5] Two days of drifter observations at Torrey Pines CA in 2004 (TP04 experiment) were used to estimate time-dependent absolute diffusivities [Spydell et al., 2007]. On day one, wave heights were small and mean currents were weak, whereas on day two larger obliquely incident waves drove a strong alongshore current. On both days, initially the cross-shore diffusivity is larger than the alongshore diffusivity (κxx > κyy) but, after many wave periods (≈100 s), κyy > κxx [Spydell et al., 2007]. That is, after initially more rapid cross-shore spreading, alongshore diffusion is faster than cross-shore diffusion. At the longest times studied (≈600 s), diffusivities on day 1 (κxx ≈ 0.75 m2 s−1, and κyy ≈ 2 m2 s−1) were smaller than on day 2 (κxx ≈ 1.25 m2 s−1, and κyy ≈ 4 m2 s−1). However, as discussed by Spydell et al. [2007], this study was limited by the relatively short trajectory lengths on day two (on average ≈500 s), the day with large waves and strong alongshore currents, prompting the use of a biased Lagrangian velocity autocovariance estimator.

[6] The processes leading to time-dependent surfzone diffusivities (and hence dispersion) are not clearly understood. The TP04 day one (small waves) observed drifter dispersion was well modeled with numerical drifters seeded into a Boussinesq wave and current model [Spydell and Feddersen, 2009]. The dominant dispersion mechanism was surfzone macro vortices forced by finite-crest length breaking [e.g., Peregrine, 1998]. Irrotational surface gravity waves (sea swell or infragravity) motions had negligible dispersive capacity [Spydell et al., 2007; Spydell and Feddersen, 2009]. Shear wave generated eddies [e.g., Oltman-Shay et al., 1989] and shear dispersion [e.g., Taylor, 1953] may have contributed to the TP04 day two elevated alongshore diffusivity [Spydell et al., 2007].

[7] Surfzone drifter observations and estimates of absolute diffusivities are still scarce, particularly on beaches without bathymetric controls on the circulation. GPS-tracked Lagrangian surfzone drifter data was collected at Huntington Beach CA on an alongshore uniform beach for five days with moderate waves and varying alongshore currents (section 2). Drifters were released mostly within the surfzone and drifters typically stayed within the surfzone with trajectory lengths between 15 and 30 min. Relative to prior work [Spydell et al., 2007], longer trajectories allow for longer and more stable diffusivity estimates. The observed Lagrangian statistics are presented in section 3. Unbiased Lagrangian velocity autocovariance functions are used to estimate diffusivity and dispersion (section 3.1). The observed diffusivities are fit to analytic functional forms from which asymptotic values and Lagrangian timescales are determined (section 3.2). Analogous to the open ocean [Gille and Llewellyn Smith, 2000; LaCasce, 2005], the nondimensional probability distribution function (pdf) of Lagrangian displacements is estimated and the degree to which the pdf is non-Gaussian is assessed with the Kolmogoroff-Smirnoff (K-S) test (section 3.3).

[8] Aspects of the Lagrangian statistics presented in section 3 are discussed in section 4. The asymptotic surfzone diffusivity κ dependence on surfzone conditions is explored (section 4.1). A previously proposed surfzone cross-shore diffusivity parameterization [e.g., Inman et al., 1971] involving significant wave height and period does not reproduce the observed asymptotic cross-shore diffusivity. The asymptotic alongshore diffusivity variations correspond to variations in the surfzone mean alongshore current maximum, consistent with a mixing-length model and shear dispersion [e.g., Taylor, 1953]. The observed non-Gaussian displacement pdfs at intermediate times are consistent with the observed cross-shore variation of Lagrangian statistics, reinforcing the “bulk” nature of the diffusivity estimates (section 4.2). Here unbiased diffusivity estimates are used, whereas previously [Spydell et al., 2007] biased estimates were used. The sampling errors of the unbiased and biased diffusivity estimates are compared and depend on the number of trajectories, trajectory lengths, and the Lagrangian timescale (section 4.3). Results are summarized in section 5.

2. Observations

[9] Surfzone observations were collected near Huntington Beach CA as part of the Fall 2006 HB06 experiment. The cross-shore coordinate x increases negatively offshore (x = 0 m is at the mean shoreline) and the alongshore coordinate y increases upcoast. The bathymetry was approximately alongshore uniform (Figure 1) and large rip channels were absent during the experiment. In particular, over the region that the drifters sample, the bathymetric nonuniformity statistic χ2 < 0.01 was below the value found to induce circulation nonuniformities [Ruessink et al., 2001]. Seven instrumented tripods were deployed on a cross-shore transect (at y = 0 m) extending 160 m from near the shoreline to 4-m mean water depth (Figure 1, gray dots). Each tripod held a pressure sensor and a downward looking Acoustic Doppler Velocimeter (ADV) from which hourly wave (e.g., significant wave height Hs), and velocity statistics (mean and standard deviation) were calculated.

Figure 1.

Drifter trajectories (black lines) on 3 days. White dots are drifter release locations, and the thick gray curve near x = 0 m is the approximate waterline. Dashed gray line labeled Lsz indicates the outer edge of the surfzone. The instrumented cross-shore transect is indicated by dark gray dots at y = 0 m. Bathymetry contours (thin gray), based on 43 approximately 25-m spaced (in the alongshore) cross-shore transects, are at 1-m intervals.

[10] Ten 50-cm tall, surfzone GPS-tracked drifters [Schmidt et al., 2003], were deployed on five days (17 September and 2, 3, 14, and 15 October; Figure 1) with variable incident wave and mean current conditions (Table 1). Drifter data was collected for 5–6 hours beginning at 1000 local time. Drifters were released repeatedly (Figure 1, open white circles), within or near the surfzone, and allowed to drift freely for 15–30 minutes before being collected and re-released. Drifter tracks suggest advection by alongshore currents and the presence of low-frequency eddies (Figure 1). Drifters rarely advected offshore of the deepest instrumented tripod at x = −160 m and drifters that came too close to shore and touched the bottom were collected and re-released farther offshore. Each drifter release and collection results in a separate drifter track. The number of tracks n0 varied between 59 and 70 with mean trajectory lengths T between 877 and 1376 s (Table 2, rows 1 and 2), for a daily total of 17–27 hours of drifter data.

Table 1. Eulerian Wave and Current Observations on the 5 Days of Drifter Releasesa
Day09/1710/0210/0310/1410/15
  • a

    Statistics from the most offshore frame include the incident significant wave height Hs, mean frequency equation image, mean direction equation image, and directional spread σθ. The maximum mean alongshore velocity equation imagem excludes sensors 1 and 2, which often were close to the shoreline or out of the water. The surfzone width Lsz is obtained from energy fluxes.

Hs (m)0.830.680.650.690.68
equation image (cps)0.090.100.090.090.10
equation image (deg)4.4−4.0−2.07.92.6
equation image (deg)1921221520
equation image (m s−1)0.27−0.13−0.170.350.25
Lsz (m)9974797979
Table 2. Lagrangian Drifter Statistics on Each Daya
Day09/1710/0210/0310/1410/15
n06670635966
T (s)1110 ± 4301376 ± 865877 ± 7221177 ± 8231067 ± 701
  • a

    The first two rows are total number of drifter trajectories n0 and the mean trajectory length T (±1 std) The coefficients of the fitted LVAF (9) and (7) and asymptotic quantities derived therefrom (e.g., equation image) follow. Fit errors (Appendix C) are indicated as ±'s. On 10/15 the t0 error is not calculated and the asymptotic errors are calculated assuming t0equation image. The range of the asymptotic equation imageequation image (and equation imageequation image) indicates likely values (the approximately 68% range) of equation imageequation image that would result with repeated experiments, i. e., the range is the asymptotic diffusivity sampling error (Appendix D).

Axx (m2 s−2)0.019 ± 0.0010.032 ± 0.0020.031 ± 0.0020.020 ± 0.0010.014 ± 0.001
t0 (s)312 ± 12163 ± 10172 ± 10294 ± 91.82 × 106
τxx (s)191 ± 13125 ± 8152 ± 11339 ± 22116 ± 8
Txx(L) (s)74 ± 329 ± 217 ± 2−52 ± 18116 ± 8
equation imageequation image (m2 s−1)1.38 ± 0.050.93 ± 0.040.53 ± 0.07−1.03 ± 0.311.64 ± 0.02
Range equation imageequation image (m2 s−1)0.9–1.90.7–1.20.1–0.9−1.4–(−0.7)1.3–2.0
Ayy (m2 s−1)0.029 ± 0.0010.027 ± 0.0020.034 ± 0.0020.048 ± 0.0010.029 ± 0.001
τyy (s)342 ± 11190 ± 13118 ± 7390 ± 9419 ± 33
equation imageequation image (m2 s−1)10.0 ± 0.125.15 ± 0.093.94 ± 0.0518.6 ± 0.1912.0 ± 0.43
Range equation imageequation image (m2 s−1)7.5–12.74.2–6.13.0–5.014.0–23.78.9–15.7

[11] Each track consists of cross- and alongshore position time series X(t) = (X(t), Y(t)), where t is time, sampled at 1 Hz. Absolute position errors are approximately ±2 m. However, relative position errors, which induce velocity errors, are small and uncorrelated. Specifically, when drifters are at rest, the velocity variances (zero-lagged autocovariance) and diffusivities are three and four orders of magnitude, respectively, smaller than those observed for deployed drifters. Details of the data processing methods appears in the work of Spydell et al. [2007]. For each drifter track, time-collocated position and velocity data (X(t), Y(t), U(t), V(t)) is calculated from the original positions via (for cross-shore position and velocity)

equation image

where dt = 1 s, resulting in drifter positions and (2nd order accurate) velocities that are on the same time grid. Wave-averaged positions and velocities are obtained by smoothing X(t) and Y(t) with a Gaussian filter with a low-pass frequency cutoff of 0.033 Hz. Wave averaged quantities are denoted by tildes (e.g., equation image(t), equation image(t))) as are any statistics derived from them.

[12] The daily averaged incident wave heights, averaged over the drifter deployment on each day, spanned a relatively small range (between 0.65 and 0.83 m). The incident mean wave frequency equation image, directional spread equation image, and surfzone width Lsz (Figure 1 and Table 1) were also approximately constant. However, mean wave direction equation image, and the associated mean alongshore currents, varied significantly over the five days (Table 1). Consistent with the sign of the bulk incident wave angles, the maximum alongshore current equation imagem was positive on 09/17, 10/14, and 10/15 due to the predominant south swell, and negative on 10/02, and 10/03 as a result of westerly wind swell. Although the alongshore current was generally weak, the equation imagem magnitude varied between 0.13 and 0.35 m s−1, almost a factor of three. Wave conditions on each day did not change significantly over the 5–6 hours of drifter releases. The maximum variation in incident Hs was 0.05 m.

[13] Eulerian mean and standard deviation (std) velocities estimated from cross-shore binned (12–19 m bin size) drifter velocities are usually similar to values from the instrumented tripods (Figure 2). For example, the mean alongshore current equation image (Figure 2a) and cross-shore velocity standard deviation std(u) (Figure 2b) compare well. On all days, offshore of the surfzone std(u) ≈ 0.2 m s−1 and increases due to wave shoaling shoreward to a maximum at x ≈ −75 m followed by a shoreward decrease owing to wave breaking. Alongshore velocity standard deviation std(v) also are similar on all days with std(v) ≈ 0.1 m s−1 offshore of wave breaking and increasing shoreward (Figure 2c). However, for unknown reasons the drifter derived std(v) is larger than the ADV observed within the inner surfzone (x > −75 m). Drifter sampling was usually most concentrated (approximately 3 drifter hours per day) in the midouter surfzone (x ≈ −90 m), and was more evenly distributed on 10/14 (Figure 2d). The difference between observed between drifter- and ADV observed equation image is increased close to the shoreline (Figure 2a) at least in part by the relative paucity of near shoreline drifter sampling (Figure 2d) and the increased alongshore velocity variability within the surfzone (Figure 2c).

Figure 2.

Binned drifter (solid lines) and fixed instrument (dashed lines with dots) Eulerian velocity statistics versus the cross-shore coordinate x on each day (colors): (a) mean alongshore currents (equation image), (b) standard deviation of cross-shore velocities (std(u)), (c) standard deviation of alongshore velocities (std(v)), (d) drifter hours in each bin, and (e) mean depth. Cross-shore bin width varies between 12 and 19 m, depending on the day.

3. Single-Particle Lagrangian Statistics

[14] The mean Lagrangian displacement is defined as

equation image

where the average “〈·〉” is over all drifter tracks and all possible t = 0 along a drifter track (Appendix A). In statistically stationary flows, the release time t = 0 is arbitrary along the drifter track, and averaging over all t = 0 (i.e., all possible t lags) is possible. Thus single-particle statistics could be calculated from one drifter track. Anomalous displacement (rx, ry) are defined as

equation image

and anomalous velocities (u(t), v(t)) are defined similarly.

[15] Tracer evolution, in both homogeneous and inhomogeneous flows, can be modeled by a Fickian diffusion equation with the diffusivity obtained from single-particle (or absolute) Lagrangian statistics [Davis, 1987]. Three key related Lagrangian statistics are the Lagrangian velocity autocovariance function (LVAF)

equation image

the absolute diffusivity, or ensemble tracer patch spreading rate,

equation image

and the absolute dispersion D2, or ensemble tracer patch size squared,

equation image

The dispersion [D2(t)]1/2 can be interpreted as the half-width of the ensemble-averaged tracer patch at time t originally released as a delta function. Subscripts denote tensor components, with the yy component calculated analogously (with xy and uv in equations (4), (5), and (6)). Only the diagonal tensor components are analyzed here. The LVAF C(t) is estimated directly from anomalous drifter velocities with the diffusivity κ(t) and dispersion [D2(t)]1/2 derived from C(t) [e.g., Davis, 1987; Spydell et al., 2007].

3.1. Observed Lagrangian Velocity Autocovariance, Diffusivitiy, and Dispersion

[16] Full (unaveraged) and wave-averaged drifter velocities are used in equation (4) to calculate the Lagrangian velocity autocovariances C(t) and equation image (t), respectively (Figure 3). Oscillations in Cxx(t) from cross-shore orbital wave velocities are evident for t < 30 s and decay after many incident wave periods (Figure 3a), i.e., Cxx(t) ≈ equation imagexx(t) for t > 100 s. For t > 150 s, both Cxx and equation imagexx are negative, reaching a minimum near t ≈ 300 s (Figure 3a). For about t ≥ 1000 s, large errors in both cross- and alongshore C(t) estimates result from relatively few observations and cause large C(t) oscillations, limiting useful diffusivity and dispersion estimates to t < 1000 s. Sampling errors are discussed in section 4.3.2.

Figure 3.

Lagrangian velocity autocovariance functions (LVAFs) on 10/03 versus time: (a) cross-shore Cxx and (b) alongshore Cyy. Thin, thick-dashed, and thin-dashed curves are LVAFs derived from velocities with waves C, wave-averaged velocities equation image, and the best fit analytic function equation image, respectively. Error bars (Appendix B) or 68% confidence on equation image are indicated by gray shading.

[17] Large oscillations at short times are not present in Cyy (Figure 3b) as alongshore orbital wave velocity motions are weak. The wave-averaged equation imageyy closely follows Cyy for t > 10 s (Figure 3, compare thin- and thick-dashed curves). After t > 20 s, both Cyy and equation imageyy decrease exponentially. Unlike equation imagexx, equation imageyy is (within 68% confidence limits) positive for t < 1000 s.

[18] Bulk surfzone absolute diffusivities κ (Figure 4), calculated using the non-wave-averaged LVAF C(t) in equation (5), are representative of drifters deployed in, and remaining in, the surfzone for t < 1000 s (Figure 1). With longer deployments, a fraction of drifters presumably would eventually leave the surfzone and be subject to inner-shelf processes resulting in different Lagrangian statistics (i.e., diffusivities).

Figure 4.

Single-particle diffusivity equation image versus time for each day (indicated by color): (a) cross-shore κxx(t) and (b) alongshore κyy(t). Diffusivities are derived directly from the data with waves (colored curves) with sampling error (68% confidence limit, light-colored shading) and from the best fit analytic equation image(t) (dashed black line).

[19] Generally (except on 10/15), the cross-shore diffusivity κxx(t) reaches a maximum around 1.5 m2 s−1 between t = 160–300 s, before slowly decreasing. On all days, the long-time cross-shore diffusivity, κxx(t) at t = 1000 s, varies between 0.5 and 2 m2 s−1. However, the κxx error bars (Appendix B) often overlap for t > 600 s (shaded regions in Figure 4), so long-time κxx values are only marginally statistically different. On all days, the alongshore diffusivity κyy(t) monotonically increases in time, with the most rapid increases at short time (Figure 4b). On 10/02 and 10/03, κyy are approximately constant for t > 500 s. At longer times (t > 200 s), κyy > κxx (Figure 4).

[20] Similar to previous observations [Spydell et al., 2007], at short times the patch-size cross shore [Dxx2]1/2 is larger than alongshore [Dyy2]1/2 (compare dashed-colored with solid-colored curves for t < 20 s in Figure 5b). At longer times, in accordance with the larger alongshore diffusivities, patches become alongshore elongated with [Dyy2]1/2 > [Dxx2]1/2 (Figure 5, compare solid with dashed curves).

Figure 5.

The cross-shore [Dxx2]1/2 (dashed) and alongshore [Dyy2]1/2 (solid) dispersion versus time (color corresponds to day). Figure 5a has a linear-linear axis. The thick-dashed and thick-solid curves are the fitted 09/17 cross- and alongshore dispersion [equation image]1/2, respectively. The 09/17 Brownian (long time, t1/2 growth) regime for the cross shore and alongshore (equations (14) and (15)) is given by the thin-dashed and thin-solid black curves, respectively. The cross-shore Brownian curve follows the equation image1/2 (thick- and thin-dashed curves overlap). The alongshore Brownian curve, labeled with a t1/2, does not follow equation image1/2 (thin- and thick-solid curves do not overlap). Figure 5b has a log-log axis for 1 < t < 200 s. The 09/17 cross- and alongshore (short-time) ballistic regimes (equation (13)) are indicated as thin-dashed and thin-solid curves, respectively.

[21] On all days, the cross-shore patch half-widths [Dxx2]1/2 at longer times (t > 500 s) are similar, with [Dxx2]1/2 ≈ 50 m (Figure 5a, dashed lines). As typical surfzone widths are Lsz ≈ 100 m, it takes approximately 500 s for the ensemble-averaged patch, released in the center of the surfzone, to spread across the surfzone. In contrast, the alongshore patch half-width [Dyy2]1/2 varies considerably at long times, between approximately 80–150 m at t = 1000 s (Figure 5a, solid lines).

3.2. Analytic Forms: Asymptotic Diffusivities and Lagrangian Timescales

[22] Functional forms for the LVAF equation image, diffusivity equation image, and dispersion equation image facilitate calculation and interpretation of single-particle statistics (e.g., asymptotic diffusivities), and simplify estimation of sampling errors. The autocovariance for a first order autoregressive process [e.g., LaCasce, 2008] has the form ∼ exp(−t) as does the Lagrangian velocity autocovariance for modeled turbulent flow [e.g., Yeung and Pope, 1989; Mordant et al., 2003]. Therefore the following functional form for the alongshore LVAF is used,

equation image

where Ayy is the zero-lag Lagrangian velocity autocovariance (i.e., the variance) and τyy is the alongshore Lagrangian timescale. Using this equation image, the analytic alongshore diffusivity is

equation image

For t > 200 s, the observed negative Cxx (Figure 3a) is captured with a modified functional form for equation image,

equation image

The factor (1 − ∣t∣/t0) in equation (9) makes equation image(t) < 0 for t > t0 similar to the observed Cxx (Figure 3). The analytic cross-shore diffusivity is then

equation image

resulting in a equation imagexx maximum similar to that observed (Figure 4).

[23] The parameters β = [Axx, t0, τxx] ([Ayy, τyy] for κyy) are found by minimizing the squared misfit between observed and fitted κ(t), i.e.,

equation image

is minimized for the cross-shore diffusivity. The integral upper limit Tm = 1000 s avoids the large and rapidly growing sampling errors at longer times. Fit parameters are given in Table 2.

[24] The observed equation image and fitted equation image are similar (with fit skill >0.98) in both directions on all 5 days (Figure 4, compare colored with dashed black curves). Similarly, fit equation imageyy is similar to the observed Cyy for t > 10 s (Figure 3b) but in the cross shore it is the fit equation image and the wave-averaged velocity derived equation image that are similar (Figure 3a). Thus consistent with previous field [Spydell et al., 2007] and numerical [Spydell and Feddersen, 2009] surfzone drifter studies, the observed diffusivity κ is due to wave-averaged processes. Surface gravity wave orbital velocities are merely noise in the context of surfzone drifter dispersion.

[25] As tequation imageτxx, the fit equation image (t) becomes the fit asymptotic cross-shore and alongshore diffusivities equation image, i.e., in the cross shore

equation image

and in the alongshore

equation image

which is the classic asymptotic diffusivity expression [Taylor, 1921]. This extrapolation of equation image(t) to long times assumes that the analytic LVAF equation (9) is valid for t > 1000 s.

[26] If τxx < t0, the equation imageequation image are positive indicating a diffusive processes. On all days except 10/14, equation imageequation image is positive and is between 0.7 and 1.7 m2 s−1. On 10/14, t0 > τxx and equation imageequation image is negative (Table 2) due to drifter convergence close to shore at y ≥ 200 m for long times (Figure 1c). Throughout the day, drifters converged near the shore, with only 3 of 32 tracks having cross-shore positions < −100 m when alongshore positions are >200 m. Hence this negative κxx is possibly due to an underlying convergent mean flow (potentially bathymetrically controlled) which is not a diffusive process.

[27] The asymptotic cross-shore diffusivity equation imageequation image fit is usually good as fit errors (representing goodness of fit, Appendix C) range between 1.2% and 30% (number after the ± in Table 2). Only 10/14 has a fit error larger than 14%. However, the equation imageequation image sampling errors (Appendix D), representing the RMS equation image range that would be obtained in a different realizations of an identical experiment, are larger 20–80% (see “range equation imageequation image” in Table 2). On the five days, the alongshore asymptotic diffusivity equation imageequation image spans a much broader range (4 < equation imageequation image < 19 m2 s−1) than equation imageequation image (Table 2) with equation imageequation image fit and sampling errors smaller than those for equation imageequation image (1–3.6% and 18–31% respectively, Table 2).

[28] Theoretically, at short times, tequation imageτ, Lagrangian velocities are correlated and dispersion is ballistic (D2t2) whereas for long times, Lagrangian velocities are uncorrelated and dispersion is Brownian (D2t). The Lagrangian timescale, defined as T(L) = equation image/equation image (t = 0), characterizes the transition between ballistic (tT(L)) and Brownian (tT(L)) dispersion regimes. The alongshore Lagrangian timescale Tyy(L) = τyy, ranges from 118 to 419 s (Table 2) and the cross-shore Lagrangian timescale, Txx(L) = τxx(1 − τxx/t0) is <τxx and varies between 17 and 116 s, except on 10/14 where it is negative (Table 2).

[29] From the analytic LVAF equations (7) and (9), the ballistic (short time) regime (tequation imageT(L)) is,

equation image

with xx replaced by yy for the alongshore. For short times (t < 20 s), the ballistic scaling (13) underpredicts both the cross- and alongshore dispersion (Figure 5), because analytic LVAFs do not include surface gravity wave contributions, only important at these times, particularly in the cross shore (see Figure 3a). Thus the surface gravity wave motions that result in large differences at short times between Cxx(t) and equation image(t) (Figure 3), only cause significant differences in the observed (with waves) dispersion [D2]1/2 and fitted dispersion [equation image]1/2 for t ≲ 20 s (Figure 5b). However, at these times the patch sizes are small ([Dxx2]1/2 and [Dyy2]1/2 are <3 m). The similarity between the analytic LVAF fit equation image and κ and therefore between equation image and [D2]1/2 for t > 20 s demonstrate that surfzone drifter dispersion is due to motions with frequencies below surface gravity wave frequencies. Unlike [Dxx2]1/2 which never closely follows a ballistic scaling (Figure 5b, compare dashed colored and dashed black curves), the alongshore dispersion [Dyy2]1/2 is ballistic for 30 s < t < τyy (Figure 5b, black solid curve).

[30] From the analytic LVAF, the Brownian regime (tT(L)) is

equation image
equation image

Because Txx(L) < Tyy(L), the observed cross-shore dispersion [Dxx2]1/2 is within 90% of Brownian [equation image]1/2 for t ≳ 200 s whereas the observed equation image1/2 does not reach Brownian scaling at t = 1000 s (Figure 5a, thin dashed and thin solid curves are 09/17 Brownian scalings). In particular, according to the fit LVAF parameters, [Dyy2]1/2 would be within 90% of the Brownian scaling for t ≥ 5.25 τyy or for times >2000 s. Thus the HB06 drifter trajectories are too short to observe alongshore Brownian motion.

3.3. Drifter Displacements

[31] From the probability distribution function (pdf) of displacements, aspects of the mixing processes can be inferred. In particular, Gaussian pdfs are expected for homogeneous mixing while non-Gaussian pdfs result from inhomogeneous mixing or coherent structures present in the flow [e.g., Pasquero et al., 2001]. The pdf of cross-shore displacements P(rx), and alongshore displacements P(ry), is calculated on all days for all t displacements. The pdfs are normalized to zero mean and unit standard deviation (t = 1,30, and 500 s for 10/02 are shown in Figure 6).

Figure 6.

The observed normalized probability density function versus normalized displacements at the times t = 1, 30, and 500 s (colors) on 10/02: (a) cross-shore displacements and (b) alongshore displacements. The dashed line is a Gaussian.

[32] Displacement pdfs generally fall into three categories: (1) Gaussian-like, (2) peakier than Gaussian, and (3) “noisy”. Gaussian-like cross- and alongshore displacement pdfs are found for small t (t = 1 s, Figures 6a and 6b, blue lines), and peakier than Gaussian pdfs are often found for intermediate t (e.g., t = 30 s, Figure 6a, green line). As t increases, there are less observations and pdfs become noisy (e.g., t = 500 s, Figures 6a and 6b, red lines). Given finite observations, the degree to which these pdfs truly are or are not Gaussian is unclear. Previously, normalized displacement pdfs were inferred to be largely Gaussian in the surfzone, however, data plotted with a logarithmic ordinate obscured departures from Gaussian and no quantitative tests were applied [Spydell and Feddersen, 2009].

[33] The likelihood that displacement pdfs are Gaussian is determined from a Kolmolgorov-Smirnov (K-S) test, which tests the null hypothesis: “the data is standard normal at the α significance level”. The test statistic d is the maximum absolute difference between the observed normalized-displacement cumulative distribution function (cdf) and a standard normal cdf. The K-S test inputs are the test statistic d and the number of independent observations NI and the K-S test returns the probability p (P value) of obtaining a value of d or larger by chance given NI. The null hypothesis is rejected at the α significance level if p < α. Thus displacement pdfs are more likely Gaussian for larger p. However, even at lower values of p, there is still a reasonable (e.g., for p = 0.5 a 50%) likelihood that the observed pdf is actually Gaussian. Thus the pdf is not Gaussian with confidence unless p is very small (<0.05). Furthermore, as p ∼ 2exp(−2d2NI) for large NI (>O(102)), larger samples are less likely to be Gaussian for the same d. This test is applied on all days for t < 1000 s in both directions giving cross- and alongshore P values px and py, respectively. However, the number of independent displacements NI at each t first must be determined.

[34] The total number of independent displacements NI(t) (Figure 7, dashed lines, right axis) is the sum over the number of independent displacements in each track

equation image

where n(t) is the number of tracks longer than t, Tj is the length of the jth track, τ is either τxx or τyy depending on the direction, and ceil rounds up to the nearest integer. Except on 10/03 and 10/14 where τxx > τyy, the cross-shore NI is larger than alongshore NI for all t (Figure 7, right axis) leading to noisier alongshore pdfs than cross-shore pdfs (Figures 6a and 6b, compare red lines). The estimate of NI neglects spatial correlation between drifters, resulting in NI overestimates and p underestimates.

Figure 7.

Komolgorov-Smirnov (K-S) test p values (left axis, solid curves: px, black; py, gray) and the number of independent displacements NI (right axis, dashed curves: x, black; y, gray) versus time t for (a) 09/17, (b) 10/02, (c) 10/03, (d) 10/14, and (e) 10/15.

[35] The likelihood as measured by p that the displacements are Gaussian varies considerably in day, time, and direction (Figure 7, left axis). On most days, except 10/15, for short times (t < 15 s), px is near one (Figure 7, black solid curves) indicating that displacements are probably Gaussian, consistent with the similarity between the observed pdf and the Gaussian (t = 1 s, Figure 6a, blue curve). At intermediate times, px decreases and reaches a minimum near t = 75 s. On 09/17, 10/02, and 10/14, px generally increases for 200 < t < 1000 s, indicating that the displacement pdfs are more likely Gaussian at longer times. On 10/03, cross-shore displacement pdfs are probably Gaussian for all t > 100 s. The alongshore py is more variable than px in time and across days (Figure 7, compare gray to dark solid curves). Alongshore displacements are most likely to be Gaussian only on 10/03 for t > 400 s, on 09/17 for t close to zero and for 200 < t < 400 s, and on 10/15 for 200 < t < 500 s.

[36] At intermediate times 20 < t < 200 s, the consistently low px values from the K-S test indicate that cross-shore displacement pdfs are probably not Gaussian, but are more likely Gaussian at very short and longer times. Thus the t = 30 s peakier than Gaussian pdf (Figure 6a, green line) appears real and is not an artifact of undersampling. A potential mechanism to explain this is discussed in section 4.2.

4. Discussion

4.1. Parameterizing the Asymptotic Diffusivity

[37] The effect of varying surfzone conditions on fit asymptotic diffusivities equation imageequation image and equation imageequation image is now examined. Inman et al. [1971] link equation imagexx to the incident significant wave height Hs and mean frequency equation image (Table 1) via

equation image

without any explicit diffusivity time dependence. Recently, a similar relationship was obtained with a simple model of surfzone cross-shore tracer diffusion by bores [Feddersen, 2007; Henderson, 2007] where κxx in equation (17) is the wave-averaged cross-shore tracer diffusivity due to bores. For HB06 conditions, the scaling in Feddersen [2007] predicts α ≈ 1.25. However, note that by design surfzone drifters duck under and are not entrained in or dispersed by bores [Schmidt et al., 2003].

[38] Using the incident Hs and equation image (Table 1) and the 4 days with positive equation imageequation image, the fit to equation (17), constrained to go through the origin, results in α = 20 with low skill (0.20) (Figure 8). Fitting to the maximum equation image instead of equation imageequation image results in a similarly poor skill. However, the parameterization (17) cannot be verified or dismissed by the present observations for the following reasons: there are only four HB06 data points, the range of equation image is small (1–1.5 m−2 s−1), and the equation imageequation image sampling error (Figure 8, vertical bars) overlap such that the equation imageequation image are not distinctly different. The present observations do not conclusively test the parameterization (17).

Figure 8.

The HB06 asymptotic cross-shore diffusivity equation imageequation image versus equation image (circles) with best fit α = 20.9 ± 5 and fit skill of 0.2. Wave height Hs and mean frequency equation image are estimated from the most offshore frame. Vertical lines indicated equation imageequation image error bars (Appendix D). The negative equation imageequation image on 10/14 is not shown and is excluded from the linear best fit (dashed line). Also shown (but not included in the fit) are TP04 [Spydell et al., 2007] data points. Due to data limitations, TP04 error bars could not be calculated.

[39] Two days of surfzone Lagrangian drifter data [Spydell et al., 2007] were also collected in 2004 at Torrey Pines Beach CA (TP04). The data were reprocessed with unbiased autocovariances and best fit to the analytic LVAF for consistency with the HB06 data. The TP04 day one with small waves (Hs = 0.5 m) is consistent with the HB06 data and agrees reasonably with equation (17) and α ≈ 20 whereas TP04 day two with large Hs = 1.35 m does not (see Figure 8, squares).

[40] Although the fit skill to equation (17) is poor, the best fit α ≈ 20 is significantly larger than expected for bore-induced dispersion (α ≈ 1.25). Thus equation imageequation image is larger than that expected for tracer mixing by idealized periodic bores. Moreover, the bore-induced κxx timescale is expected to be a few wave periods whereas here τxx ≈ 150 s, consistent with long-time drifter dispersion caused by low-frequency vortical motions [Spydell and Feddersen, 2009]. Thus for long times, cross-shore dispersion induced by vortical motions appears to dominate over breaking wave (bore) induced dispersion.

[41] For the asymptotic alongshore diffusivity equation imageequation image, two scalings are investigated: one based on dimensional considerations and mixing-length arguments [Tennekes and Lumley, 1972] and another related to shear dispersion in a pipe [Taylor, 1953]. The mixing-length equation imageequation image scaling uses the mean alongshore current maximum equation image (Table 1) for the velocity scale as equation image is related to the fluctuating (shear wave) velocity [Noyes et al., 2004]. Using the surfzone width Lsz as a length-scale (see Table 1) yields

equation image

with γ a nondimensional constant of proportionality. Fitting the HB06 observations to equation (18) results in best fit γ = 0.52 ± 0.08 with skill of 0.68 (Figure 9a). The surfzone width Lsz varied little thus fit skill with constant length scale is also similar. TP04 day one equation image follows the scaling (18), whereas day two with the larger mean current does not.

Figure 9.

The HB06 asymptotic alongshore diffusivity equation imageequation image (circles) versus (a) γequation imagexb and (b) equation imageT0. The best fit, constrained to go through the origin (dashed lines), results in (a) γ = 0.52 ± 0.08 with a 0.68 skill and (b) T0 = 154 ± 13 s with a 0.91 skill. The maximum alongshore current equation image excludes sensors 1 and 2, which are close to the shoreline and/or out of the water. Vertical lines indicate equation imageequation image error bars (Appendix D). Also shown (but not included in the fit) are TP04 [Spydell et al., 2007] data points. Due to data limitations, TP04 error bars could not be calculated.

[42] Shear dispersion in a pipe (three dimensional [Taylor, 1953]), adapted to a simple two-dimensional parabolic alongshore current [Spydell et al., 2007], yields

equation image

where equation image is the constant cross-shore pipe diffusivity and equation image = 0 at x = 0, L is assumed. Defining a cross-pipe diffusive timescale T0 = L2/(480 equation image), equation (19) becomes

equation image

As Lsz and equation imageequation image were relatively constant on the four days with equation imageequation image >0, T0 is assumed constant. Fitting to equation (20) yields T0 = 154 ± 13 s with a fit skill of 0.91 (Figure 9b). Note that for days with equation imageequation image >0, the values of T0 and τxx are similar (Table 2). Using a value of equation imageequation image = 1 m2 s−1 for equation image and L = 150 m (where equation image ≈ 0 m s−1, in Figure 2) results in T0 = 46 s, 1/3 of the best fit value. Using daily values of equation imageequation image and Lsz for κxxpipe and L, respectively in equation (19) and allowing for a fit coefficient, results in slightly less skill than with constant T0equation (20). This all indicates that the observed equation imageequation image is largely consistent with the shear dispersion model (19). Differences are potentially due to the violation of shear dispersion scaling assumptions including a constant in time κxx and uniform cross-shore drifter sampling. In summary, the alongshore diffusivities are consistent with both the mixing length scaling and the shear dispersion scaling.

[43] The shear dispersion scaling (19) has equation imageequation imageequation image, indicating that strong alongshore currents result in large alongshore diffusivity. However, the TP04 day 2 (with large equation image) equation imageequation image is not consistent with either the shear dispersion scaling (20) nor the mixing-length scaling (18) found for HB06 (Figure 9). This inconsistency is perhaps due to relatively poor Lagrangian sampling on TP04 day two which had about half the mean trajectory length and total drifter data of that on each HB06 day. Short drifter trajectories and sparse observations result in large sampling errors (section 4.3.2). It is also possible that the equation imageequation image scalings (18) and (19) do not apply at Torrey Pines. Additional observations, on beaches without bathymetric controls on the circulation, are needed to test the generality of these equation imageequation image scalings.

4.2. Displacements

[44] With homogeneous turbulence, the diffusivity κ does not depend on position and displacement pdfs are Gaussian, i.e., the diffusion equation has Gaussian solutions. However, for position dependent diffusivity, Lagrangian statistics are inhomogeneous and displacement pdfs are non-Gaussian in a manner similar to particle separation pdfs in turbulent flows [Richardson, 1926]. Non-Gaussian pdfs may indicate that dispersion is better represented with a spatially dependent diffusivity, than with a single bulk κ as estimated here. The peakier than Gaussian displacement pdfs (Figure 6a at t = 30 s) that correspond to low p values (Figure 7) may result from drifters sampling regions of cross-shore inhomogeneous statistics. This phenomena has been observed for open ocean studies of velocity pdfs [Gille and Llewellyn Smith, 2000; LaCasce, 2005].

[45] The HB06 drifter trajectories clearly sample regions with cross-shore varying statistics. The standard deviation of 1 s rx and ry displacements (proportional to std(u) and std(v) in Figure 2) vary across the surfzone by about a factor of 2.5. Intermediate-time (30 s) displacements have even more cross-shore variation. For example, consider the standard deviation of 30 s displacements σ(xi) binned by the cross-shore midpoint of the displacement. On 10/15, σ(xi) increases toward the shore and becomes constant in the inner surfzone (x > −75 m, Figure 10a, shaded region), varying from offshore to onshore by a factor of six (Figure 10a, circles).

Figure 10.

(a) On 10/15, standard deviation σ(xi) (solid curve and circles) of binned 30 s cross-shore displacements rx(30 s) and fractional number of displacements in each bin w(xi) (dashed curve and squares) versus the cross-shore position x. The inner surfzone (x > −75 m) is indicated by shading. (b) The observed normalized probability distribution function of rx(30 s) over all cross-shore bins (solid line) and limited to inner surfzone (denoted by “i-s”) bins (gray thick line). Also shown are equation image (equation (21), dashed-dot curve) and a normal distribution (dashed curve). To calculate equation image, N(x) = 13 and ntot = 70,157. The inner surfzone ntot = 22,272.

[46] This cross-shore variation in σ(xi) can result in non-Gaussian displacement pdfs. Assume that in the ith bin, there are ni displacements with Gaussian pdf and variance σi2. The average pdf of all rx displacements is given by the weighted sum of the Gaussian pdfs over all the bins,

equation image

where N(x) is the total number of bins and the weight wi = ni/ntot is the fraction of displacements in the ith bin (Figure 10a, squares).

[47] Using the 30 s displacements standard deviations σ(xi) and associated weights w(xi) in equation (21) results in a peakier than a Gaussian pdf equation image that is similar to the observed 30 s displacement pdf P(rx) (Figure 10b, compare solid and dash-dotted), and is clearly different from Gaussian (Figure 10b). Quantitatively, the K-S test p value between the 30 s displacement pdf and equation image is 0.97 whereas compared to a Gaussian it is 0.12. For inner surfzone displacements (x > −75 m), 30 s displacement standard deviations σ(xi) are constant and the inner surfzone (i−s) 30 s displacement pdf Pis is approximately Gaussian (Figure 10b, compare gray and dashed curves, p value of 0.54). Thus displacements in each cross-shore bin appear approximately Gaussian, but when all displacements are lumped into a single bin containing variable statistics, the resulting pdf is non-Gaussian. According to the central limit theorem, the large t displacement pdfs should be Gaussian as many random displacements that span the entire cross-shore region (with differing statistics) are combined during large t displacements. For example, all cross-shore displacement pdfs (except 10/15) become more Gaussian for larger times (Figure 7).

[48] For times where the displacements are non-Gaussian, κxx(t) (and κyy(t), not shown) should depend on both cross-shore location and time. However, the present observations cannot resolve such cross-shore variation. Thus the κ(t) reported here is a bulk value representative of the dispersion in the entire surfzone, and should be used cautiously in a Fickian diffusion equation.

4.3. Estimating the Diffusivity: Biases and Sampling Errors

[49] For the TP04 data, biased LVAFs were used to estimate single particle diffusivities (i.e., κyy(B)(t)) since the number of drifter trajectories was small and the drifter trajectory lengths were short [Spydell et al., 2007]. As the alongshore direction is unbounded, κyy is expected to monotonically increase and eventually asymptote. The relative paucity of TP04 day two data yielded noisy, nonmonotonic unbiased κyy(t), resulting in unexpectedly small long-time κyy. In contrast, the biased κyy(B)(t) monotonically increased. Thus, Spydell et al. [2007] reported biased LVAF based κyy(B)(t).

[50] The pros and cons of using a biased LVAF based equation image rather than an unbiased LVAF based equation image are illustrated with the following example (Figure 11). Realizations of TP04 day two unbiased κyy(t) and biased κyy(B)(t) were calculated from simulated drifter trajectories from a first-order autoregressive process with κequation image= 6 m2 s−1 and τyy = 115 s. A single realization is constructed from n0 = 72 trajectories with mean length (±standard deviation) T = 565 (±186) s, giving a nondimensional mean trajectory length equation image = T/τyy ≈ 5. Each realization represents the κyy(t) and κyy(B)(t) that would be estimated from a realization of drifter releases.

Figure 11.

Realizations of simulated TP04 day 2 alongshore diffusivity versus time: (a) unbiased equation imageyy and (b) biased equation image. There are 36 individual realizations (thin solid lines). The expected diffusivity (equation image, equation image) is the thick dashed line with error bars (68% confidence) given by gray shading.

[51] Due to short trajectories relative to τyy (equation image ≈ 5) and small n0, the unbiased equation image realizations have significant sampling error and are considerably spread about the expected (true) equation image(t), particularly at t >2τyy (Figure 11, compare solid thin and dashed thick curves). Increasing n0 or T reduces the scatter in the equation imageyy realizations. Corresponding biased κyy(B) realizations have a mean error and underpredict the expected equation image(t). However, they are more stable and have less scatter about the expected biased value equation image (Figure 11b, compare solid thin lines and solid thick line). Due to sampling error, some long-time unbiased κyy realizations are smaller than all biased κyy(B) realizations. Thus at times approaching the trajectory length, uncertainties in the long-time unbiased κyy may warrant use of the biased diffusivities. The tradeoffs of using an unbiased (larger sampling error) or biased (larger mean error) κ are considered.

4.3.1. Biased Diffusivity Mean Error

[52] The difference between expected unbiased equation image (t) and biased equation image(t) can be significant (Figures 11a and 11b, compare dashed thick lines) where equation image has a mean error and underestimates the true expected equation image. The mean error magnitude is a function of trajectory length. From trajectories of equal length T, the unbiased and biased alongshore LVAFs are

equation image

respectively. The Cyy(B) denominator uses the full trajectory length T whereas Cyy uses Tt, the number of observations at each t, which decrease with t. Using (equation (7)), the corresponding nondimensionalized analytic LVAFs are

equation image

where equation image = t/τyy and equation image = T/τyy are nondimensional time and trajectory length, respectively. The biased equation image has an error of −equation image exp(−equation image)/equation image. Nondimensional expected unbiased and biased diffusivities are

equation image

Expected unbiased and biased equation image differences are largest for equation image (Figure 12). At equation image, the expected biased diffusivity equation image(equation image) has asymptoted to a maximum. The dimensional equation image(equation image) underestimates κequation image by

equation image

This mean error is largest for short equation image (Figure 12, circles, compare the trajectory end points).

Figure 12.

Expected biased to asymptotic alongshore diffusivity ratio equation image/equation imageequation image versus normalized time equation image = t/τyy for varying trajectory lengths equation image (see line labels). The dashed curve represents the expected unbiased to asymptotic alongshore diffusivity equation image(equation image)/equation imageequation image.

4.3.2. Sampling Errors

[53] Unbiased equation image sampling errors can obscure the desired long-time diffusivity (e.g., Figure 11a). As the number of observations at each t decrease with t, (e.g., for n0 = 1, there is only one observation at t = T). The unbiased κyy sampling error increases rapidly with time as equation image. To estimate the increased sampling error versus increased mean error tradeoff between an unbiased versus biased κyy, the unbiased and biased κyy sampling error dependence upon t and T is now examined.

[54] The unbiased equation image sampling error is

equation image

where E is the expectation operator over many realizations such that equation image(t) = E[equation image(t)]. For the analytic LVAF (7), equation image(equation image) = 1 − exp(−∣equation image∣). The equation image estimation method for trajectories varying in length T is complex (Appendix B). However, for n0 equal length trajectories, the analytic LVAF (7), and nondimensionalizing by equation image = t/τyy and equation image = T/τyy, the sampling error simplifies to

equation image

where

equation image

with Γ the incomplete gamma function. The diffusivity sampling error dependence upon time equation image and trajectory length equation image is examined for the full estimate (equations (B3) and (B4)), and various limits of equation (22) (Figure 13).

Figure 13.

Normalized diffusivity sampling error equation image/equation imageequation image (from equation (B3)) for one trajectory (n0 = 1) of varying lengths equation image (see legend) versus (a) equation image = t/τyy and (b) equation image. The scalings derived from E1(t) (equation (23)) are shown as gray curves: the short-time scaling equation imageequation image (thin-dashed), the long equation image scaling equation image ∼ {ln [equation image/(equation imageequation image)]}1/2 (thick-dashed), and the intermediate scaling equation imageequation image (thin-solid). The rapidly growing portion of the log scaling is only evident for equation image → 1.

[55] For times much shorter than the Lagrangian timescale (i.e., equation imageequation image 1),

equation image

and the error grows linearly in time (Figure 13). For trajectory lengths equation imageequation image 5, error growth is approximately linear for all t (see equation image = 1,5, Figures 13a and 13b). For long trajectory lengths with equation imageequation image 1,

equation image

and grows rapidly as equation image due to the decreasing number of observations (Figure 13b, thick-dashed gray curve). The singularity at equation image predicted by equation (25) is not in the full solution (23) which has a small boundary layer correction of unit thickness at equation image. For equation image ≪ 1, equation (25) reduces to

equation image

i.e., the t1/2 growth given by Davis [1991]. For diffusivities based on unbiased LVAFs, the equation image error growth (equation (26)) only applies for equation image < 0.3 (see Figure 13b). This long-time duration of rapid error growth (equation (25)) is obscured by the logarithmic abscissa in Figures 13a and 13b.

[56] The HB06 drifter mean nondimensional trajectory lengths are equation image ≈ 4–8 resulting in approximately linear (24) sampling error growth. Using the observed mean trajectory length T and n0 in equation (24) results in approximately the full κyy sampling error (equations (B2)(B4)) shown in Figure 4b.

[57] Turning now to the biased diffusivity, the variance about the expected biased diffusivity equation image is

equation image

equation image is the shading in Figure 11b). The small time behavior of equation image (equation image) is the same as equation image (equation image) given by equation (24). For long trajectories equation image,

equation image

which is equivalent to equation (26) for small equation image. Thus both equation image and equation image increase like equation image1/2 for dimensional times much longer than τyy but shorter than the trajectory length T. The most striking difference between equation image and equation image is that equation image grows slower than equation image1/2 and approaches a constant as equation image. This makes individual equation image realizations more stable at long times and is the main reason that biased diffusivity estimates might be preferred to unbiased. However, the choice of a biased or unbiased diffusivity depends not on equation image (t) but on the full biased diffusivity sampling error equation image, which includes contributions %This error includes contributions from the variance and the mean error, i.e.,

equation image

where the mean error is

equation image

Both equation image and equation image parametrically depend upon n0 and T, whereas Δ depends only upon T.

4.3.3. Comparing Biased and Unbiased Diffusivity Sampling Error

[58] Whether to use the biased or unbiased diffusivity estimates ultimately depends upon the ratioequation image. The time-dependence of equation image as a function of n0 and T is examined. With a constant n0 = 72, equation image at all times for short trajectories equation image ≤ 4 (Figure 14a), due to large mean error Δ. However, for equation image ≥8, equation image for equation image >5 due to smaller Δ, quickly growing equation image and relatively constant equation image as equation image (Figure 14a). With a constant trajectory length of equation image = 8, equation image for all equation image for a small number of trajectories (n0 16), whereas for more trajectories equation image only for equation image (Figure 14b). In other words, for sufficient trajectories longer than τyy, so that equation imageequation image is likely approached, equation image and the unbiased diffusivity estimate is better than the biased except for times approaching the trajectory length (tT). Given a priori knowledge of the Lagrangian timescale, the number of drifters (or trajectories), and the acceptable level of sampling error, drifter deployment schemes can be designed to meet these criteria.

Figure 14.

The ratio of the unbiased to biased diffusivity sampling error equation image/equation image versus nondimensional time equation image. (a) The number of drifter trajectories is fixed at n = 72, and drifter trajectory length is varied equation image = [1, 2, 4, 8, 16, 32, 64, 128] indicated by line thickness: the thickest line is the shortest trajectory. (b) Drifter trajectory length is fixed equation image = 8, and the number of trajectories is varied n0 = [4, 8, 16, 32, 64, 128], indicated by line thickness: the thickest line is the most trajectories. Note, both axes limits are different in Figures 14a and 14b.

5. Summary

[59] Surfzone dispersion is described with single-particle Lagrangian statistics of GPS-tracked drifters deployed at Huntington Beach Ca over five days with small variation in incident wave height. On each day, ten drifters were repeatedly deployed in the surfzone for 15–30 min. Drifter tracks revealed the presence of alongshore currents (up to 0.35) and low frequency eddies.

[60] Bulk (representative of entire surfzone) Lagrangian velocity autocovariance functions (LVAFs) were used to estimate diffusivities κ (the integral of the LVAF) and dispersions D2 (the integral of κ) on each day. The time-dependent surfzone cross-shore diffusivity κxx(t) was similar on all days, reaching a local maxima of about 1.5 m2 s−1 at times 160–300 s before slowly decreasing to about 1 m2 s−1. The alongshore diffusivity κyy(t) increases monotonically for all time t, following a ballistic scaling at short times. Trajectories were not long enough to observe alongshore Brownian dispersion. For t > 50 s, the alongshore diffusivity κyy > κxx, consistent with previous observations [Spydell et al., 2007]. Drifters allowed to drift much longer than the present O(1000) s would eventually be subject to inner-shelf or oceanic processes with different Lagrangian statistics.

[61] The observed diffusivities are well fit by analytic functions, from which asymptotic diffusivities and Lagrangian timescales, representative of the entire surfzone, are determined. The cross-shore asymptotic diffusivity ranged from 0.53 ≤ equation imageequation image ≤ 1.64 m2 s−1. %for the four days with positive values. The asymptotic alongshore diffusivities were 4 ≤ equation imageequation image ≤ 19 m2 s−1, a much larger range than previously observed [Spydell et al., 2007]. The analytic LVAF e-folding time τ is generally O(100 s) with the alongshore τyy greater than the cross-shore τxx. The cross-shore Lagrangian timescale is shorter than τxx due to the nonmonotonic equation image time dependence. Thus asymptotic diffusion is apparent sooner in the cross-shore than in the alongshore. The asymptotic cross-shore diffusivity equation imageequation image was not well fit by a previously proposed parameterization based upon the incident wave height and wave period, although the wave height and equation imageequation imagevariability was weak. The asymptotic alongshore diffusivity equation imageequation image is related to the maximum mean alongshore current equation image in a manner consistent with both a mixing-length (∼equation image) and a shear dispersion based (∼equation image) scaling.

[62] The Kolmogorov-Smirnov test shows that the probability density function (pdf) of short-time displacements (≲20 s) is nearly Gaussian. Displacement pdfs then become peakier than Gaussian around t ≈ 30 s, often followed by a return to Gaussian for long time. This pdf peakiness results from cross-shore variability in displacement statistics and is an indication of cross-shore-dependent diffusivity. Cross-shore diffusivity variation is not resolvable with the present data set. Thus the dispersion statistics presented are representative of the entire surfzone and should be used cautiously in surfzone Fickian diffusion equations.

[63] Differences in unbiased and biased diffusivity estimates using the analytic LVAFs were investigated. The biased diffusivity mean error depends upon the ratio of the trajectory length to the Lagrangian timescale. Both the unbiased diffusivity sampling error εκ(t), and the standard deviation of biased diffusivity estimates σκ(B)(t), depend upon the number of trajectories and the trajectory length. For trajectories of varying lengths the formulae are complicated. However, for equal length trajectories, asymptotic regimes were identified to aid error analysis. For trajectories short relative to the Lagrangian timescale, i.e., for the HB06 deployments, the unbiased sampling error is mostly linear with time. For times approaching the trajectory length, the unbiased sampling error grows rapidly due to the decreasing number of observations. For these long times, the biased diffusivity standard deviation grows much more slowly eventually approaching a constant. The biased diffusivity sampling error combines the biased diffusivity standard deviation and the mean error. For many trajectories short relative to the Lagrangian timescale, the biased error is larger than the unbiased due to large biased mean errors. For many trajectories longer than the Lagrangian timescale, the unbiased diffusivity estimate is preferred except at times approaching the trajectory length. However, the biased diffusivity may be preferred if there are few but long (compared to the Lagrangian timescale) trajectories.

Appendix A:: Averaging: Using the Entire Trajectory

[64] To calculate statistics of Lagrangian quantities, the averaging method, denoted by equation image·equation image in equations such as equations (2) and (4), uses all possible t separated observations (velocities or positions) along each trajectory. Although the data are discrete, continuous data is assumed for clarity of presentation. Converting to discrete data is straightforward. For drifter trajectories with varying lengths Ti, the amount (in units of time) of t separated observations is

equation image

where n(t) is the number of trajectories greater than or equal to t in length. When discretized, equation image(t) is the number of observations separated by t. To illustrate, consider two trajectories, T1 = 100 s and T2 = 200 s. For 0 ≤ t ≤ 100 s, n(t) = 2 and equation image(t) = 300 − 2t, while for 100 < t ≤ 200 s, n(t) = 1 and equation image(t) = 200 − t. Using Cxx as an example, averages are given by

equation image

The estimate (A2) is unbiased because the denominator equation image(t) uses the actual number of observations at each t, whereas a biased estimator uses equation image(t = 0) at every t (see section 4.3.1).

Appendix B:: Sampling Errors of Lagrangian Statistics

[65] Sampling errors for the LVAF and absolute diffusivity are defined. For some quantity ζ(t), the sampling error is

equation image

where E[] is the expectation operator and equation image = E[ζ]. Substituting the definition of the statistic ζ into equation (B1), ζ = 1/(Tt)∫?0Ttu′(a)u′(a + t) da for the LVAF (unbiased), and after taking expectations, leads to the LVAF sampling error

equation image

where n(t) is the number of trajectories greater than or equal to t in length and equation image is given in equation (A1). For each trajectory

equation image

where C is the expected LVAF and the sum is over all i trajectories longer than t. Following Spydell et al. [2007], but using the unbiased definition of C, the diffusivity sampling error squared equation image(t) is

equation image

with

equation image

for each drifter trajectory. When calculating the sampling error for the observed LVAF ɛC(t) and diffusivity ɛκ(t) (shading in Figures 3 and 4, respectively), the analytic cross- and alongshore LVAF equation image is used in equations (B2) and (B3) for efficiency as the integrals in equation imagei,C and equation imagei,κ can be analytically determined.

Appendix C:: Nonlinear Least Squares Fit for Diffusivity

[66] The analytic LVAF functions (7) and (9) are found by minimizing the squared misfit of diffusivity residuals

equation image

integrated over time (see equation (11)). The fit is nonlinear in the best fit coefficients β where the number of coefficients is nβ: 3 for the cross shore and 2 for the alongshore. The sampling error covariance matrix for β0 is

equation image

where the ijth component of Q is

equation image

G0 is the minimum value of G, and m is the number of effective samples used in the fit, estimated with m = Tm/Tz where Tz is the first zero (∼100 s) of the biased autocorrelation function of residuals R(t). The square root (the standard deviation) of the diagonal elements of equation (C1) is the error in the fit coefficients (Table 2).

[67] By Monte Carlo simulation, best fit coefficients (e.g., Axx, t0, and τxx) are used to calculate error bars on asymptotic quantities (e.g., equation imageequation image in equation (12)) derived from them. Best fit coefficients are assumed to be Gaussian random variables with means equal to the best fit values and covariances Mβ (Table 2). For example, consider the asymptotic cross-shore diffusivity equation imageequation image. Sets of randomly generated (Axx, t0, τxx) are used to calculate equation imageequation image values using equation (12). The mean is equation imageequation image and the standard deviation is the number following the ± in Table 2. Fit errors for Txx(L) and equation imageequation image are calculated similarly.

Appendix D:: Asymptotic Diffusivity Sampling Error

[68] The asymptotic diffusivity equation image sampling error is the range of fit equation image derived from different realizations of surfzone drifter releases with the same statistics. This range is estimated by performing best fits equation (11) for t ≤ 1000 s to equation image(t) + equation image(t) and κxx(t) − equation image(t) resulting in best fit coefficients (Axx(+), t0(+), τxx(+)) and (Axx(−), t0(−), τxx(−)), respectively. These fit coefficients yield the upper and lower limits of the asymptotic diffusivity sampling error, i.e.,

equation image

where

equation image

with approximately 68% probability. The alongshore asymptotic diffusivity sampling error is calculated similarly. The equation image ranges are given in Table 2 and shown in Figures 8 and 9 as vertical lines.

Acknowledgments

[69] CA Coastal Conservancy, NOAA, NSF, ONR, and CA Sea grant. We thank the staff and students from the Integrative Oceanography Division (B. Woodward, B. Boyd, K. Smith, D. Darnell, I. Nagy, D. Clark, M. Omand, M. Yates, M. McKenna, M. Rippy, S. Henderson) for acquiring the field observations for this research.

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