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 xx 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 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 varies between 4 and 19 m2 s−1, and this variation is related to the variation in the maximum of the mean alongshore current m, broadly consistent with a shear dispersion scaling ∼ . 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.
 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.
 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  provides an excellent review.
 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].
 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. , 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.
 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].
 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).
 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.
 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.
 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
Statistics from the most offshore frame include the incident significant wave height Hs, mean frequency , mean direction , and directional spread σθ. The maximum mean alongshore velocity m 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.
Table 2. Lagrangian Drifter Statistics on Each Daya
1110 ± 430
1376 ± 865
877 ± 722
1177 ± 823
1067 ± 701
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., ) 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 t0 → . The range of the asymptotic (and ) indicates likely values (the approximately 68% range) of that would result with repeated experiments, i. e., the range is the asymptotic diffusivity sampling error (Appendix D).
Axx (m2 s−2)
0.019 ± 0.001
0.032 ± 0.002
0.031 ± 0.002
0.020 ± 0.001
0.014 ± 0.001
312 ± 12
163 ± 10
172 ± 10
294 ± 9
1.82 × 106
191 ± 13
125 ± 8
152 ± 11
339 ± 22
116 ± 8
74 ± 3
29 ± 2
17 ± 2
−52 ± 18
116 ± 8
1.38 ± 0.05
0.93 ± 0.04
0.53 ± 0.07
−1.03 ± 0.31
1.64 ± 0.02
Range (m2 s−1)
Ayy (m2 s−1)
0.029 ± 0.001
0.027 ± 0.002
0.034 ± 0.002
0.048 ± 0.001
0.029 ± 0.001
342 ± 11
190 ± 13
118 ± 7
390 ± 9
419 ± 33
10.0 ± 0.12
5.15 ± 0.09
3.94 ± 0.05
18.6 ± 0.19
12.0 ± 0.43
Range (m2 s−1)
 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. . 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)
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., (t), (t))) as are any statistics derived from them.
 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 , directional spread , and surfzone width Lsz (Figure 1 and Table 1) were also approximately constant. However, mean wave direction , 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 m 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 m 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.
 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 (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 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).
3. Single-Particle Lagrangian Statistics
 The mean Lagrangian displacement is defined as
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
and anomalous velocities (u(t), v(t)) are defined similarly.
 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)
the absolute diffusivity, or ensemble tracer patch spreading rate,
and the absolute dispersion D2, or ensemble tracer patch size squared,
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 x → y and u→ v 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
 Full (unaveraged) and wave-averaged drifter velocities are used in equation (4) to calculate the Lagrangian velocity autocovariances C(t) and (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) ≈ xx(t) for t > 100 s. For t > 150 s, both Cxx and xx 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.
 Large oscillations at short times are not present in Cyy (Figure 3b) as alongshore orbital wave velocity motions are weak. The wave-averaged yy closely follows Cyy for t > 10 s (Figure 3, compare thin- and thick-dashed curves). After t > 20 s, both Cyy and yy decrease exponentially. Unlike xx, yy is (within 68% confidence limits) positive for t < 1000 s.
 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).
 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).
 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).
 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
 Functional forms for the LVAF , diffusivity , and dispersion 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,
where Ayy is the zero-lag Lagrangian velocity autocovariance (i.e., the variance) and τyy is the alongshore Lagrangian timescale. Using this , the analytic alongshore diffusivity is
For t > 200 s, the observed negative Cxx (Figure 3a) is captured with a modified functional form for ,
The factor (1 − ∣t∣/t0) in equation (9) makes (t) < 0 for t > t0 similar to the observed Cxx (Figure 3). The analytic cross-shore diffusivity is then
resulting in a xx maximum similar to that observed (Figure 4).
 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.,
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.
 The observed and fitted are similar (with fit skill >0.98) in both directions on all 5 days (Figure 4, compare colored with dashed black curves). Similarly, fit yy is similar to the observed Cyy for t > 10 s (Figure 3b) but in the cross shore it is the fit and the wave-averaged velocity derived 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.
 As tτxx, the fit (t) becomes the fit asymptotic cross-shore and alongshore diffusivities , i.e., in the cross shore
and in the alongshore
which is the classic asymptotic diffusivity expression [Taylor, 1921]. This extrapolation of (t) to long times assumes that the analytic LVAF equation (9) is valid for t > 1000 s.
 If τxx < t0, the are positive indicating a diffusive processes. On all days except 10/14, is positive and is between 0.7 and 1.7 m2 s−1. On 10/14, t0 > τxx and 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.
 The asymptotic cross-shore diffusivity 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 sampling errors (Appendix D), representing the RMS range that would be obtained in a different realizations of an identical experiment, are larger 20–80% (see “range ” in Table 2). On the five days, the alongshore asymptotic diffusivity spans a much broader range (4 < < 19 m2 s−1) than (Table 2) with fit and sampling errors smaller than those for (1–3.6% and 18–31% respectively, Table 2).
 Theoretically, at short times, tτ, Lagrangian velocities are correlated and dispersion is ballistic (D2 ∼ t2) whereas for long times, Lagrangian velocities are uncorrelated and dispersion is Brownian (D2 ∼ t). The Lagrangian timescale, defined as T(L) = / (t = 0), characterizes the transition between ballistic (t ≪ T(L)) and Brownian (t ≫ T(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).
 From the analytic LVAF equations (7) and (9), the ballistic (short time) regime (tT(L)) is,
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 (t) (Figure 3), only cause significant differences in the observed (with waves) dispersion [D2]1/2 and fitted dispersion 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 and κ and therefore between 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).
 From the analytic LVAF, the Brownian regime (t ≫ T(L)) is
Because Txx(L) < Tyy(L), the observed cross-shore dispersion [Dxx2]1/2 is within 90% of Brownian 1/2 for t ≳ 200 s whereas the observed 1/2 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
 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).
 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].
 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.
 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
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.
 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.
 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.
4.1. Parameterizing the Asymptotic Diffusivity
 The effect of varying surfzone conditions on fit asymptotic diffusivities and is now examined. Inman et al.  link xx to the incident significant wave height Hs and mean frequency (Table 1) via
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  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].
 Using the incident Hs and (Table 1) and the 4 days with positive , 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 instead of 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 is small (1–1.5 m−2 s−1), and the sampling error (Figure 8, vertical bars) overlap such that the are not distinctly different. The present observations do not conclusively test the parameterization (17).
 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).
 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 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.
 For the asymptotic alongshore diffusivity , 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 scaling uses the mean alongshore current maximum (Table 1) for the velocity scale as 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
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 follows the scaling (18), whereas day two with the larger mean current does not.
 Shear dispersion in a pipe (three dimensional [Taylor, 1953]), adapted to a simple two-dimensional parabolic alongshore current [Spydell et al., 2007], yields
where is the constant cross-shore pipe diffusivity and = 0 at x = 0, L is assumed. Defining a cross-pipe diffusive timescale T0 = L2/(480 ), equation (19) becomes
As Lsz and were relatively constant on the four days with >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 >0, the values of T0 and τxx are similar (Table 2). Using a value of = 1 m2 s−1 for and L = 150 m (where ≈ 0 m s−1, in Figure 2) results in T0 = 46 s, 1/3 of the best fit value. Using daily values of 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 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.
 The shear dispersion scaling (19) has ∼ , indicating that strong alongshore currents result in large alongshore diffusivity. However, the TP04 day 2 (with large ) 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 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 scalings.
 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].
 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).
 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,
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).
 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 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 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 Pi−s 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).
 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
 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.  reported biased LVAF based κyy(B)(t).
 The pros and cons of using a biased LVAF based rather than an unbiased LVAF based 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 κ= 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 = T/τyy ≈ 5. Each realization represents the κyy(t) and κyy(B)(t) that would be estimated from a realization of drifter releases.
 Due to short trajectories relative to τyy ( ≈ 5) and small n0, the unbiased realizations have significant sampling error and are considerably spread about the expected (true) (t), particularly at t >2τyy (Figure 11, compare solid thin and dashed thick curves). Increasing n0 or T reduces the scatter in the yy realizations. Corresponding biased κyy(B) realizations have a mean error and underpredict the expected (t). However, they are more stable and have less scatter about the expected biased value (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
 The difference between expected unbiased (t) and biased (t) can be significant (Figures 11a and 11b, compare dashed thick lines) where has a mean error and underestimates the true expected . The mean error magnitude is a function of trajectory length. From trajectories of equal length T, the unbiased and biased alongshore LVAFs are
respectively. The Cyy(B) denominator uses the full trajectory length T whereas Cyy uses T − t, the number of observations at each t, which decrease with t. Using (equation (7)), the corresponding nondimensionalized analytic LVAFs are
where = t/τyy and = T/τyy are nondimensional time and trajectory length, respectively. The biased has an error of − exp(−)/. Nondimensional expected unbiased and biased diffusivities are
Expected unbiased and biased differences are largest for (Figure 12). At , the expected biased diffusivity () has asymptoted to a maximum. The dimensional () underestimates κ by
This mean error is largest for short (Figure 12, circles, compare the trajectory end points).
4.3.2. Sampling Errors
 Unbiased 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 . 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.
 The unbiased sampling error is
where E is the expectation operator over many realizations such that (t) = E[(t)]. For the analytic LVAF (7), () = 1 − exp(−∣∣). The 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 = t/τyy and = T/τyy, the sampling error simplifies to
with Γ the incomplete gamma function. The diffusivity sampling error dependence upon time and trajectory length is examined for the full estimate (equations (B3) and (B4)), and various limits of equation (22) (Figure 13).
 For times much shorter than the Lagrangian timescale (i.e., 1),
and the error grows linearly in time (Figure 13). For trajectory lengths 5, error growth is approximately linear for all t (see = 1,5, Figures 13a and 13b). For long trajectory lengths with 1,
and grows rapidly as due to the decreasing number of observations (Figure 13b, thick-dashed gray curve). The singularity at predicted by equation (25) is not in the full solution (23) which has a small boundary layer correction of unit thickness at . For ≪ 1, equation (25) reduces to
 The HB06 drifter mean nondimensional trajectory lengths are ≈ 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.
 Turning now to the biased diffusivity, the variance about the expected biased diffusivity is
(± is the shading in Figure 11b). The small time behavior of () is the same as () given by equation (24). For long trajectories ,
which is equivalent to equation (26) for small . Thus both and increase like 1/2 for dimensional times much longer than τyy but shorter than the trajectory length T. The most striking difference between and is that grows slower than 1/2 and approaches a constant as . This makes individual 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 (t) but on the full biased diffusivity sampling error , which includes contributions %This error includes contributions from the variance and the mean error, i.e.,
where the mean error is
Both and parametrically depend upon n0 and T, whereas Δ depends only upon T.
4.3.3. Comparing Biased and Unbiased Diffusivity Sampling Error
 Whether to use the biased or unbiased diffusivity estimates ultimately depends upon the ratio. The time-dependence of as a function of n0 and T is examined. With a constant n0 = 72, at all times for short trajectories ≤ 4 (Figure 14a), due to large mean error Δ. However, for ≥8, for >5 due to smaller Δ, quickly growing and relatively constant as (Figure 14a). With a constant trajectory length of = 8, for all for a small number of trajectories (n0 16), whereas for more trajectories only for (Figure 14b). In other words, for sufficient trajectories longer than τyy, so that is likely approached, and the unbiased diffusivity estimate is better than the biased except for times approaching the trajectory length (t → T). 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.
 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.
 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.
 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 ≤ ≤ 1.64 m2 s−1. %for the four days with positive values. The asymptotic alongshore diffusivities were 4 ≤ ≤ 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 time dependence. Thus asymptotic diffusion is apparent sooner in the cross-shore than in the alongshore. The asymptotic cross-shore diffusivity was not well fit by a previously proposed parameterization based upon the incident wave height and wave period, although the wave height and variability was weak. The asymptotic alongshore diffusivity is related to the maximum mean alongshore current in a manner consistent with both a mixing-length (∼) and a shear dispersion based (∼) scaling.
 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.
 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
 To calculate statistics of Lagrangian quantities, the averaging method, denoted by · 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
where n(t) is the number of trajectories greater than or equal to t in length. When discretized, (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 (t) = 300 − 2t, while for 100 < t ≤ 200 s, n(t) = 1 and (t) = 200 − t. Using Cxx as an example, averages are given by
The estimate (A2) is unbiased because the denominator (t) uses the actual number of observations at each t, whereas a biased estimator uses (t = 0) at every t (see section 4.3.1).
Appendix B:: Sampling Errors of Lagrangian Statistics
 Sampling errors for the LVAF and absolute diffusivity are defined. For some quantity ζ(t), the sampling error is
where E is the expectation operator and = E[ζ]. Substituting the definition of the statistic ζ into equation (B1), ζ = 1/(T − t)∫?0T−tu′(a)u′(a + t) da for the LVAF (unbiased), and after taking expectations, leads to the LVAF sampling error
where n(t) is the number of trajectories greater than or equal to t in length and is given in equation (A1). For each trajectory
where C is the expected LVAF and the sum is over all i trajectories longer than t. Following Spydell et al. , but using the unbiased definition of C, the diffusivity sampling error squared (t) is
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 is used in equations (B2) and (B3) for efficiency as the integrals in i,C and i,κ can be analytically determined.
Appendix C:: Nonlinear Least Squares Fit for Diffusivity
 The analytic LVAF functions (7) and (9) are found by minimizing the squared misfit of diffusivity residuals
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
where the ijth component of Q is
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).
 By Monte Carlo simulation, best fit coefficients (e.g., Axx, t0, and τxx) are used to calculate error bars on asymptotic quantities (e.g., 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 . Sets of randomly generated (Axx, t0, τxx) are used to calculate values using equation (12). The mean is and the standard deviation is the number following the ± in Table 2. Fit errors for Txx(L) and are calculated similarly.
 The asymptotic diffusivity sampling error is the range of fit 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 (t) + (t) and κxx(t) − (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.,
with approximately 68% probability. The alongshore asymptotic diffusivity sampling error is calculated similarly. The ranges are given in Table 2 and shown in Figures 8 and 9 as vertical lines.
 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.