## 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 (κ_{xx} > κ_{yy}) but, after many wave periods (≈100 s), κ_{yy} > κ_{xx} [*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 (κ_{xx} ≈ 0.75 m^{2} s^{−1}, and κ_{yy} ≈ 2 m^{2} s^{−1}) were smaller than on day 2 (κ_{xx} ≈ 1.25 m^{2} s^{−1}, and κ_{yy} ≈ 4 m^{2} 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.