6.1. Surfzone κxx Comparisons
 Previous surfzone field experiments have used the alongshore distribution of point-released dye at the shoreline to estimate κyy, but lack the cross-shore tracer measurements required to estimate κxx quantitatively. Detailed surfzone tracer κxx comparisons are therefore not possible, but the κxx estimated here (Table 3) are within the range of previous κ values [Inman et al., 1971; Clarke et al., 2007]. GPS-tracked drifters, designed to duck under breaking waves and avoid surfing onshore, have been used to estimate surfzone cross-shore diffusivities κxx(d) with alongshore uniform [Spydell et al., 2007, 2009] and rip channel [e.g., Johnson and Pattiaratchi, 2004; Brown et al., 2009] bathymetries. During the HB06 experiment, drifter-based surfzone κxx(d) were estimated [Spydell et al., 2009], but on different days than dye. Observed dye and asymptotic (long-time) drifter κxx have similar magnitudes (around 1 m2 s−1).
 The HB06 drifter-derived κxx(d) were time-dependent. At times less than the drifter Lagrangian time-scale Txx of O(100 s), the drifter κxx(d) increase quasi-ballistically (σ2 ∼ t2 or κxx ∼ t) towards a peak value [Spydell et al., 2009]. In contrast, tracer-derived surfzone κxx are roughly constant in time and σsurf2 ∼ t (Figure 11), indicating Brownian diffusion. However the first dye transects occur near Tp = 100 s where the drifter ballistic regime generally ends [Spydell et al., 2009], and unobserved ballistic tracer dispersion may have occurred between the first transects and the dye source (where Tp ≲ Txx).
 For t > Txx, drifter κxx(d) gradually decreased [Spydell et al., 2009], possibly because drifters sampled the lower diffusivity seaward of the surfzone. Recent dye dispersion studies seaward of the surfzone in ∼10 m water depth [Fong and Stacey, 2003; Jones et al., 2008], with similar plume widths to those observed here, found absolute diffusivities roughly 10 times smaller than the surfzone κxx here. Note that σsurf2 is surfzone integrated, and therefore not an appropriate variable to examine seaward κxx.
6.2. Surfzone Saturation and Diffusion Seaward of the Surfzone
 The (x, yj) profiles far-downstream (largest y) have roughly constant magnitude (i.e., are saturated) across the surfzone for releases R3, R5, and R6 (Figures 7c, 7e, and 7f), and the far-downstream R3 and R5 transects have sharp (x, yj) gradients at the seaward edge of the surfzone (e.g., Figures 7e and 8). These (x, yj) profiles are consistent with a larger surfzone κxx smoothing dye gradients inside the surfzone and a smaller κxx slowly mixing dye farther seaward. In contrast, the two farthest downstream R1 transects have significant amounts of dye outside the surfzone (Figure 7a), but the dye plume continues to spread. The continued dispersion seaward of the surfzone may result from absolute averages over meandering of the non-shoreline attached plume, but could also result from rip currents that transport dye well beyond the seaward edge of the surfzone.
 Although σsurf2 excludes data (and dispersion) seaward of the surfzone, constant σsurf versus Tp does indicate surfzone saturation. The σsurf(yj) for R6 initially grow inside the surfzone, but become constant for Tp > 2000 s in agreement with saturated profiles (Figure 7f). In addition, the nearly constant σsurf in the farthest downstream transects of R4 (Figure 11d) suggest surfzone saturation that is not visually apparent in the (x, yj) profiles (Figure 7d).
6.3. Parameterizing κxx
 Previous dye dispersion studies [e.g., Harris et al., 1963; Inman et al., 1971] parameterized diffusivity with
where Hb is the wave height at the breakpoint, and T is a wave period. With planar bathymetry and constant γ = H/h, these two parameterizations (21, 22) are essentially equivalent. Although previous work found agreement between surfzone diffusivity variability and the parameterizations above [e.g., Bowen and Inman, 1974], the physical mechanism driving cross-shore diffusion was unclear.
 Mechanisms for cross-shore surfzone diffusion investigated here include bore-mixing, shear dispersion, and horizontal vortical-flow. Multiple cross-shore propagating bores with turbulent front faces (a high diffusivity region) can result in net cross-shore diffusion [Feddersen, 2007]. The non-dimensional bore-induced average diffusivity [Feddersen, 2007; Henderson, 2007] is
where ĉ and are the non-dimensional phase speed and wave period, respectively. A dimensional mid-surfzone xx can be derived from the scalings of Feddersen 
 Assuming a self-similar surfzone (H/h = γ) and a mid-surfzone water depth (h = hb/2) then
 With γ = 0.6, the slope between κxx and Hb2T−1 would be near 2.
 Here the incident (measured at frame 7, Figure 1) Hs and mean period Tm (Table 1) are used in the bore induced κxx scaling (25). Although observed κxx generally increase with Hs2Tm−1 (Figure 13a), the correlation is low (r2 = 0.32), and the best-fit slope of 11.7 is a factor 6 larger than expected for bore-induced dispersion (25). The observed cross-shore dye dispersion is probably not dominated by bore-mixing. However, the range of Hs and Tm are small (Table 1) and the κxx error bars (Figure 13a) often overlap, indicating the need for more observations.
 In model simulations [Spydell and Feddersen, 2009], horizontal rotational velocities (i.e., vortical flow) generated by finite crest length breaking [Peregrine, 1998] or shear instabilities of the alongshore current [e.g., Oltman-Shay et al., 1989] were found to be a primary mixing mechanism. Here, a mixing-length scaling, i.e., a velocity scale times a length scale [e.g., Tennekes and Lumley, 1972], is examined using a surfzone width Lx length-scale and a surfzone-averaged low-frequency horizontal rotational velocity scale rot (i.e., cross-shore averaged rot (x) (1) between the shoreline and x = −Lx, Table 1)
where α is a non-dimensional constant. In analogy with Von Kármán's constant of 0.4 in wall-bounded shear flow, or the factor of 0.57 [e.g., Rodi, 1987] in 2-equation (i.e., k − ) models relating diffusivity to a length- and velocity scale product, α is expected to be <1 but still O(1). Surfzone rot (x) includes horizontal rotational flow driven by instabilities in the alongshore current [e.g., Oltman-Shay et al., 1989], finite crest-length wave-breaking [Peregrine, 1998; Spydell and Feddersen, 2009], and wave groups [e.g., Reniers et al., 2004]. The surfzone averaged rot ranges between 0.036–0.09 ms−1 (Table 1).
 The surfzone tracer xx increase with rotLx (Figure 13b) and the linear best-fit gives r2 = 0.59, slope of 0.2, and near-zero y-intercept. The high r2 and an expected slope <1 (for a mixing-length scaling) indicate that rotational velocities (surf-zone eddies) play an important role in cross-shore surfzone tracer mixing. However, similar to Hs and Tm, the range of rot and Lx are relatively small (Table 1), and additional observations of surfzone tracer xx are required to fully test this parameterization (26). A related mixing-length scaling, using instead of rot as the velocity scale, was correlated with alongshore drifter diffusivity [Spydell et al., 2009], and is consistent with the present result because ��rot (x) and V are correlated [Noyes et al., 2004].
 As suggested by Pearson et al. , another possible mechanism for cross-shore surfzone tracer mixing is shear dispersion [e.g., Taylor, 1954] driven by vertical variation of the cross-shore mean velocity (i.e., undertow). The idealized expression, assuming a step function velocity profile, for the shear dispersion driven κxx(sd) [Fischer, 1978] used by Pearson et al. 
where h is the water depth, κzz is the surfzone vertical diffusivity, U+ and U− are the cross-shore velocities in the surface (onshore) and return (offshore) layers with the transition at h/2. Other plausible velocity profiles (e.g., linear) have different functional forms for κxx(sd) [Fischer, 1978], but give similar results when the on-offshore transports are matched between profiles. Using (27) and empirical relationships for κzz and U+, and assuming U− = −U+, Pearson et al.  found good agreement between a laboratory estimated κxx and the corresponding scaled κxx(sd) for shore-normal monochromatic waves.
 The cross-shore shear dispersion scaling (27) is examined with field data derived from the instrumented frames. During each release, U− is given by mid-surfzone cross-shore velocities, measured at the instrumented frames (Figure 1) roughly 0.4 m above the bed in 1–2 m water depth. The maximum U− is −0.07 ms−1, and analogous to Pearson et al. , U+ = −U− is assumed. The vertical cross-shore velocity profile is unknown, however the step function profile assumed in (27) is used for comparison to previous work [Pearson et al., 2009]. At the same locations the estimated surfzone turbulent dissipation rate ε ≈ 4 × 10−4 m2 s−3 (F. Feddersen, Quality controlling surfzone acoustic Doppler velocimeter observations to estimate the turbulent dissipation rate, submitted to Journal of Atmospheric and Oceanic Technology, 2010). Assuming a turbulent length-scale of half the water depth, the resulting κzz derived from a k − ε closure scheme [e.g., Rodi, 1987] are typically κzz ≈ 4 × 10−2 m2 s−1. A linear best-fit of κxx to κxx(sd) (Figure 13c) results in high correlation (r2 = 0.94), but a large slope of 30. The κxx(sd) are expected to be O(1) estimates of cross-shore shear dispersion, but ranged from 35–125 times smaller than the observed κxx (Figure 13c). If vertical tracer gradients exist (section 6.4), the κxx(sd) may be underestimated, however this is unlikely to account for the large differences in magnitude. Although correlations are high, undertow driven cross-shore shear dispersion is apparently not a dominant tracer dispersal mechanism in the observed natural surfzone. In the laboratory, with monochromatic, shore-normal waves [Pearson et al., 2009], horizontal rotational velocities are reduced or absent and the undertow driven shear dispersion mechanism may be dominant.
6.4. Potential Causes for Reduced Downstream M(yj) Relative to Dye Pump Estimates
 Tracer transports at the source M(y = 0 m), estimated using the dye pump rate, are larger than at downstream transects M(y > 0 m), estimated with the observed (yj), V(x), and h(x) (Figure 9). The reasons for the initial M(yj) decrease are unknown, but possible causes, and the implications of those causes on tracer analysis, are explored. One possibility is that pump rates were overestimated by using water (lower viscosity than dye) from a bucket (not the dye tank). However testing on a similar pump system (the original was no longer available) did not support this hypothesis. Pump rate errors would not effect cross-shore moments or xx, but would affect the predicted tracer maxima max(p)(19) used for model data comparison (Figure 12a).
 Increased near-bed dye concentration (where the jet ski does not sample) relative to the surface may be a cause of the reduced downstream M(yj) relative to the pumped M(y = 0 m). The injected dye, with concentration 2.1 × 108 ppb, has a specific gravity of 1.2. In a coastal or open-ocean environment, weak vertical mixing requires density adjustment of the dye to prevent it from sinking towards the bottom [e.g., Ledwell et al., 2004]. In contrast, the surfzone is a region of vigorous vertical mixing, where sand (2.65 specific gravity) is frequently lifted off of the bed and suspended at sediment-water densities >1.001 ρ (where ρ is the density of seawater) [e.g., Beach and Sternberg, 1996] despite grain settling velocities of roughly 0.03 ms−1 [e.g., Hallermeier, 1981]. Maximum tracer concentrations 1 m from the source are estimated at 104 ppb with a density of 1.0001 ρ, based upon the conservative assumptions of a constant 0.1 m vertical dye layer (no vertical mixing), advected by = 0.1 ms−1 (Table 1) and a small-scale cross-shore diffusivity of 0.01 m2 s−1 (from turbulent dissipation, section 6.3). Thus potential tracer induced stratification is considered negligible. With the conservative vertical diffusivity estimate κzz = 10−2 m2 s−1, mid-water column released dye in h = 2 m depth has a surface value >90% of the mid-depth maximum, for tp > 40 s, and is consistent with the visual observations of rapid vertical mixing. Thus, dye tracer is expected to be vertically well mixed at downstream transect locations.
 The region between xin (Table 2) and the x = 0 shoreline (≈10 m wide) was not sampled by the jet ski or included in M(yj), and the excluded near-shoreline tracer transport is a potential cause of the low biased M(y > 0 m) relative to the pump estimated M(y = 0 m). The non-shoreline attached R1, with low shoreline dye concentrations, is not expected to have significant near-shoreline transport, and indeed the M(yj) are roughly conserved from the release point to farther downstream (Figure 9a). Near-shoreline tracer transports are unknown, but qualitative estimates (not shown) are made assuming constant and V between xin and the shoreline. For the two R2 transects closest to the release location, the qualitative near-shoreline estimates are consistent with the correction required to match transect M(y > 0 m) with pump rate M(y = 0 m). For R3, R4 and R6 transects with y < 200 m, the near-shoreline estimates are between 20–33% of the correction required to match M(y > 0 m) and M(y = 0 m), and farther downstream the estimates are negligible. Thus, dye flux inshore of xin may be significant at times, but does not fully explain the generally high bias of pump M(y = 0 m). Using the shoreline bounded analytic solution (13), and neglecting the near-shoreline region (i.e., integrating from xin instead of x = 0 m), increases κxx roughly 14–20%. Thus, the κxx bias for excluding near shoreline tracer is generally low compared with other uncertainties (error bars in Figure 13).
 Other factors also induce M(yj) errors not accounted for in the estimated M(yj) uncertainties (error bars in Figure 9). The bathymetry and alongshore currents V(x) are assumed perfectly alongshore uniform, and alongshore variations would increase M(yj) uncertainties. However, it is not clear that these assumptions can induce a bias.