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Cross-shore surfzone tracer dispersion in an alongshore current

Authors


Abstract

[1] Cross-shore surfzone tracer dispersion in a wave driven alongshore current is examined over a range of wave and current conditions with 6 continuous dye releases, each roughly 1–2 hours in duration, at Huntington Beach, California. Fluorescent dye tracer released near the shoreline formed shore parallel plumes that were sampled on repeated cross-shore transects with a jet ski mounted fluorometer. Ensemble averaged cross-shore tracer concentration profiles are generally shoreline attached (maximum at or near the shoreline), with increasing cross-shore widths and decreasing peak values with downstream distance. More than a few 100 m from the source, tracer is often well mixed across the surfzone (i.e., saturated) with decreasing tracer concentrations farther seaward. For each release, cross-shore surfzone absolute diffusivities are estimated using a simple Fickian diffusion solution with a no-flux boundary at the shoreline, and range from 0.5–2.5 m2 s−1. Surfzone diffusivity scalings based on cross-shore bore dispersion, surfzone eddy mixing length, and undertow driven shear dispersion are examined. The mixing-length scaling has correlation r2 = 0.59 and the expected best-fit slope <1, indicating that horizontal rotational motions are important for cross-shore tracer dispersion in the surfzone.

1. Introduction

[2] Beaches and the adjacent surfzone are used for recreational and commercial activities, and provide habitat to a variety of fish and benthic species. Beach related tourism provides yearly revenue of about 1 billion dollars in Los Angeles and Orange Counties, California, U.S.A. [Hanemann et al., 2001]. These economic and environmental resources are threatened by polluted terrestrial runoff that frequently drains onto the shoreline where it is entrained and spread in the surfzone [Boehm et al., 2002]. Waterborne pollution threatens public health, causing both gastrointestinal and upper respiratory symptoms in exposed beach goers [Haile et al., 1999], and results in frequent beach closures [Noble et al., 2000]. A model predicting the transport and dilution of surfzone pollutants would improve beach management. However, the processes that mix tracers within the surfzone are understood poorly.

[3] Fluorescent dye tracers have been used to investigate surfzone mixing and transport [Harris et al., 1963; Inman et al., 1971; Grant et al., 2005; Clarke et al., 2007]. Visually observed tracer patches initially dispersed cross-shore until the surfzone was saturated (approximately uniform cross-shore dye concentration), followed by dominant alongshore dispersion [Harris et al., 1963; Inman et al., 1971; Clarke et al., 2007]. After several hours, surfzone tracer patches were observed to stretch 5–8 km alongshore while remaining within a few surfzone widths of the shoreline [Grant et al., 2005]. Seaward of the surfzone, visually slower dispersion suggested that mixing was weaker than within the surfzone.

[4] A wide range of field estimated surfzone diffusivities (κ ∼ 10−3–104 m2 s−1) have been found by fitting dye tracer data to Fickian diffusion solutions assuming constant alongshore currents and depth [Harris et al., 1963; Inman et al., 1971; Clarke et al., 2007]. Harris et al. [1963] estimated alongshore diffusivity κyy by measuring dye concentrations from bottle samples collected at several shoreline locations during both point and continuous tracer releases. Inman et al. [1971] sampled point released dye with bottles at the shoreline and at the visually estimated dye patch center. Clarke et al. [2007] estimated diffusivities by fitting a 2-D advection diffusion solution to point dye releases that were bottle sampled at several shoreline locations.

[5] Cross-shore tracer structure was not observed in previous surfzone field studies [Harris et al., 1963; Inman et al., 1971; Clarke et al., 2007]. Diffusivity estimates were derived from single realizations in space and time, without the ensemble averaging over plume and patch fluctuations needed for stability in diffusivity estimates [e.g., Csanady, 1973]. In addition, local waves and currents were generally not measured, complicating the interpretation of diffusivity parameterizations.

[6] Laboratory experiments using shore-normal monochromatic waves without [Harris et al., 1963] and with [Pearson et al., 2009] an imposed alongshore current have been used to study surfzone tracer dispersion. Harris et al. [1963] estimated a combined cross- and alongshore diffusivity κ, and used turbulent dissipation and an eddy length scale to derive a κ ∼ Hb2Tb−1 scaling, where Hb was the breaking wave height and Tb was the mean breaking wave period. Pearson et al. [2009] estimated a cross-shore diffusivity κxx from mean cross-shore dye profiles at several locations downstream from a continuous dye source, and proposed a cross-shore shear dispersion scaling using the sheared mean cross-shore current (undertow) and a vertical diffusivity.

[7] Using a shoreward propagating region of diffusivity to represent the mixing effects of a broken wave (bore), the effects of single and multiple waves on cross-shore tracer concentrations were investigated using numerical models [Feddersen, 2007]. A non-dimensional cross-shore average diffusivity equation image = equation image/(ĉequation image) was derived where ĉ and equation image are the non-dimensional cross-shore wave speed and wave period [Feddersen, 2007; Henderson, 2007].

[8] Drifters have also been used to estimate surfzone diffusivities. On roughly alongshore uniform beaches, drifter estimated diffusivities were time-dependent with asymptotic (long-time) κxx between 0.5–1.5 m2 s−1 and asymptotic κyy between 2–18 m2 s−1 [Spydell et al., 2007, 2009]. Good agreement was found between the asymptotic κyy and both mixing-length and shear dispersion scalings. At beaches with irregular bathymetry that control circulation (i.e., rip channels), estimated asymptotic diffusivities were κxx = 0.9–2.2 m2 s−1 and κyy = 2.8–3.9 m2 s−1 [Brown et al., 2009], and estimated relative diffusivities were κxx ≈ −0.8–2 m2 s−1 and κyy = 1.8–4.8 m2 s−1 [Johnson and Pattiaratchi, 2004]. Unlike tracers, drifters duck under breaking waves and are not entrained in the front face of a bore. Diffusivities for drifters and tracers may differ.

[9] Here, field observations of continuously released surfzone dye tracer plumes in quasi-steady alongshore currents on generally alongshore uniform bathymetry are presented. Tracer experiments are conducted over a range of wave and current conditions (section 2.1). Dye released into the surfzone (section 2.2) is measured on repeated cross-shore transects by a dye sampling jet ski (section 2.3). Using ensemble (absolute) averaged cross-shore concentration profiles, cross-shore integrated tracer statistics are estimated (section 3). Variation in individual tracer profiles, the structure of mean profiles, and the downstream evolution of tracer profile statistics are described in section 4. A simple Fickian diffusion model for tracer released at the shoreline with a no-flux shoreline boundary (section 5.1) is used to estimate surfzone absolute κxx (section 5.2) from mean dye profiles that are well contained in the surfzone. The Fickian solution is compared with observed tracer moments (section 5.3). The observed surfzone κxx are compared with other surfzone κxx estimates (section 6.1), and inferences are made about the relative strength of mixing seaward of the surfzone (section 6.2). Three κxx scalings and related dispersion mechanisms are discussed (section 6.3), and the possible causes of decreased tracer transport between the dye release pump and downstream transects are examined (section 6.4). Section 7 is a summary.

2. HB06 Experiment

2.1. Field Site, Waves, and Currents

[10] The HB06 experiment took place from September 14th to October 17th, 2006 in Huntington Beach, California located 50 km south of Los Angles. The approximately straight, 1 km long study beach faces 214° southwest. Offshore islands strongly effect the incident waves by blocking shore-normal southwesterly swells, and obliquely incident waves from the west or south often drive strong alongshore currents.

[11] The X and Y coordinates are the cross-shore distance from the mean sea level (MSL) shoreline, and the alongshore distance from the instrumented transect (Figure 1), respectively. Bathymetry (Figure 1a) was surveyed three times on 42 cross-shore transects using a GPS equipped jet ski, ATV, and hand pushed cart [Seymour et al., 2005]. The alongshore and time-averaged bathymetry slope is 0.03 seaward of about 2 m depth, decreases to 0.006 between roughly 0.7 m and 2 m depth, and steepens to 0.075 on the beach face (Figure 1b). Changes in the seaward portion of the bathymetry over time were small. However, a small trough near the shoreline early in the observations subsequently accreted (shaded region between −50 < X < −10 m in Figure 1b). The tidal range is typically less than ±1 m.

Figure 1.

(a) Plan view of HB06 bathymetry (depth) contours versus cross-shore distance X from the MSL shoreline, and alongshore distance Y from the instrumented frames (black crosses). Thin curves are depth contours (labeled in m) and the thick black contour is at mean sea level (MSL). (b) Mean depth versus X, with depth equal to zero at the MSL shoreline (dashed black line). The gray region indicates the bathymetry standard deviation over Y and time, and black crosses indicate approximate vertical instrument locations.

[12] Seven tripod frames with pressure sensors and acoustic Doppler velocimeters (ADVs) were deployed in a 140 m long cross-shore array from near the shoreline to 4 m mean depth (Figure 1a). Frames are numbered from 1 (shallowest) to 7 (deepest). Frame 7 was always seaward of the surfzone. Frame 1 was 11 m from the MSL shoreline, and the ADV was out of the water during low tides.

[13] During the six HB06 dye release experiments (R1 through R6) the dominant south swell drove surfzone alongshore currents and dye in the +Y (up coast) direction. For each dye release, x is the cross-shore distance from the mean shoreline (tide dependent), and y is the alongshore distance from the continuous dye source. Significant wave heights Hs(x), alongshore currents V(x), and horizontal (low-frequency) rotational velocities equation imagerot (x) [Lippmann et al., 1999] were measured at each frame (Figure 1) and averaged over the duration of each release (Figure 2 and Table 1). The Hs(x) are estimated from pressure spectra (depth corrected to the surface) over the sea-swell band. Following Lippmann et al. [1999], low frequency vortical motions equation imagerot (x) are estimated by removing irrotational infragravity wave energy from the observed velocity via

equation image

where ũ, equation image, equation image are the cross-shore, alongshore, and pressure spectra respectively, f is frequency, and the integral is over the infragravity band (0.004 < F < 0.03 Hz). This equation imagerot (x) estimate approximates shear wave velocity variance [Noyes et al., 2002].

Figure 2.

(a) Significant wave height Hs, (b) mean alongshore current V, and (c) horizontal rotational velocities equation imagerot versus cross-shore distance from the shoreline x for each dye release (see legend).

Table 1. Wave and Current Statistics for Each Dye Releasea
ReleaseDateHs (m)Tm (s)θ (deg)σθ (deg)equation image (ms−1)equation imagerot (ms−1)Lx (m)
  • a

    Release number, release date, incident (frame 7) mean significant wave height Hs, mean period Tm, wave angle θ, directional spread σθ, surfzone averaged mean alongshore current equation image, surfzone averaged horizontal rotational velocity equation imagerot, and surfzone width Lx are shown.

R1Sep 180.909.59.814.60.180.075101
R2Sep 221.029.10.923.10.230.082122
R3Sep 280.849.97.817.80.210.073112
R4Sep 29 am0.959.16.518.30.370.088116
R5Sep 29 pm0.939.06.317.80.310.090116
R6Oct 110.419.27.015.30.070.03688

[14] For releases R1 through R5, Hs(x) shoaled to a maximum near x = −110 m then decreased towards the shoreline as broken waves dissipated (Figure 2a). For R1–R5, V(x) and equation imagerot (x) had similar cross-shore structure with mid-surfzone maxima (Figures 2b and 2c). Wave heights during R6 were smaller than the other releases, reaching a maximum closer to the shoreline (x = −88 m), with a weak Hs(x) decay towards the shoreline (Figure 2a). Unlike R1 through R5, R6 also had V(x) and equation imagerot (x) (Figures 2b and 2c) maxima close to the shoreline.

[15] Averaged over each release, the incident (Frame 7) Hs range from 0.41 to 1.02 m, mean wave periods Tm from 9.0 to 9.9 s (from energy weighted pressure spectra over the sea-swell band), incident wave angle (equation image) from 0.9 to 9.8 degrees down coast (after significant shoaling and refraction), and directional spread from 14.6 to 23.1 degrees (Table 1). The surfzone width Lx is between 88 m and 122 m, with the seaward edge of the surfzone x = −Lx defined as the cross-shore location of the Hs maximum (Table 1). Cross-shore averaging over the frames within the surfzone results in surfzone averaged mean alongshore currents (equation image) between 0.07 and 0.37 ms−1, and surfzone averaged equation imagerot between 0.036 and 0.090 ms−1 (Table 1).

2.2. Dye Release Methods

[16] Concentrated Rhodamine-WT dye (21% by weight) was released continuously at 1.3–7.1 mL s−1 into the surfzone during mid- to high tide. The cross-shore dye release location x0 (in about 1 m depth) varied between −4 and −22 m, with one release (R1) much farther offshore at x0 = −54 m (Table 2). A battery powered peristaltic pump mounted on a 2 m tall heavy metal cart forced dye through a small tube to 0.5 meters above the bed, terminating into a small 10 cm long diffuser hose. Rapid vertical mixing was visually observed, and measured surface dye concentrations were reduced to 400 parts per billion (ppb) within a few meters of the source indicating that concentrated dye (1.2 specific gravity) was quickly diluted to a specific gravity near 1. Although dye was not measured near the bed, dye is expected to be vertically well-mixed due to vigorous surfzone mixing. The possibility of vertically varying dye is discussed in section 6.4.

Table 2. Dye Sampling Parameters for Each Releasea
ReleaseDuration (min)x0 (m)xin (m)TransectsNj(j)equation imagej(j)τdecorr (s)
  • a

    Release number, sampling duration, cross-shore dye release location x0, inner transect integration limit xin, number of downstream transect locations, average number of realizations on each transect 〈Nj(j) (where 〈·〉(j) is the average over all transect locations j), average degrees of freedom on each transect 〈equation imagej(j), and the estimated Eulerian decorrelation time τdecorr are shown.

  • b

    For R3 and R6, data to estimate τdecorr were not available so the largest estimate (τdecorr = 135 s) is used.

R166−54−13315.315.341
R240−13−1159.87.564
R3117−10−13611.35.9135b
R464−22−1784.54.479
R566−4−1455.84.9135
R6121−12−1099.28.9135b

2.3. Dye Sampling Methods

[17] Dye concentration D was measured with a flow-through fluorometer mounted on a GPS tracked jet ski [Clark et al., 2009], allowing measurements on cross-shore transects through the surfzone where small boats cannot operate. An onboard position display facilitated repetition of predetermined transects. Water was pumped from an intake 20 cm below the surface into a debubbler, thus reducing the number of large bubbles entering the optical instruments. The water subsequently passed through a turbidity sensor to estimate the remaining bubble interference, and finally through a Rhodamine WT fluorometer. Dye fluorescence measurements are corrected for bubble effects [Clark et al., 2009], with resulting root mean square (rms) errors estimated to be less than 2.7% of D. Mixing within the flow-through system smoothes sharp gradients in dye concentration over time scales less than 2.4 s. The time for water to move through the flow-through system and reach the fluorometer varied by ±0.84 s, resulting in spatial errors (matching dye measurements to GPS positions) of a few meters, dependent upon jet ski speed [Clark et al., 2009].

[18] Dye tracer plumes were sampled for 40 to 121 min durations (Table 2) downstream from the dye source on cross-shore transects (e.g., R3 and R6 examples in Figure 3). The dye plume was allowed to advect past the farthest downstream transect for roughly 20 min prior to sampling, insuring that initial transients had moved beyond the sampling region. Inbound transects were driven from seaward of the dye plume towards the shoreline until the jet ski turned around in roughly 0.5 m water depth (<10 m from the shoreline). Inbound transects were shore normal, uninterrupted, and driven just in front of a broken bore to reduce the number of bubbles entering the flow-through dye sampling system. Outbound transects were not analyzed because large amounts of air was entrained when the jet ski jumped over bores, and transects were often interrupted while avoiding waves. Inbound sampling over the same part of the wave orbital cycle (e.g., just in front of a bore) may bias the cross-shore dye locations by roughly ±1–2 m (using linear theory for typical HB06 surfzone conditions). The alongshore distances between transects varied between roughly 20 and 250 m, and the largest downstream distance was 686 m.

Figure 3.

Jet ski dye measurements D (concentration in color) during releases (a) R3 and (b) R6 versus cross-shore distance from the shoreline x, and alongshore distance y from the dye source (green star). Only inbound (traveling towards the beach) transects are shown. Dashed gray line indicates the seaward edge of the surfzone.

[19] Each transect location was repeated 1–4 times before moving to the next location, and the entire pattern was repeated several times. Each transect through the dye plume yields a realization (or snapshot, denoted with an i) of cross-shore dye concentration Di(x, yj) at alongshore transect location yj. The number of transect locations for each release ranged from three (R1) to nine (R6). The number of realizations on a transect is Nj, and the release averaged realizations per transect 〈Nj(j) (where 〈·〉(j) is the average over all transect locations j in a release) varies between 4.5 and 15.3 (Table 2).

[20] Individual Di(x,yj) realizations include instrument dye measurement errors (with uncertainty ±0.027 D) and errors from the ±0.84 s uncertainty in flow-through system delay time τ [Clark et al., 2009]. The delay time error is assumed to have a Gaussian probability density function (PDF) P(τ) with 0.84 s standard deviation. Total rms dye measurement errors εi (x, yj) are estimated from squared dye variations and squared dye measurement errors integrated over P(τ)

equation image

where c is the roughly 1–5 ms−1 jet ski speed. In general, εi is <20% of Di.

3. Tracer Means and Moments

3.1. Absolute Averages

[21] Turbulent tracer dispersion has time varying structure, and ensemble averages (over realizations in time) are used to describe mean (or bulk) tracer statistics [e.g., Taylor, 1921; Batchelor, 1949; Csanady, 1973]. Absolute averages are taken in a fixed coordinate frame, and include the effects of both meandering (varying advection of the realization center of mass) and relative diffusion about the realization center of mass. Relative averages remove meandering by averaging in a center of mass coordinate system (relative to each realization), and isolate the effects of relative diffusion by smaller spatial and temporal scale processes. Batchelor [1952], Csanady [1973], Fong and Stacey [2003], and many others discuss absolute and relative averaging.

[22] For each release, cross-shore profiles of mean (absolute averaged) concentration equation image(x, yj) at each yj transect location are constructed by averaging Nj transect realizations Di(x, yj) in shoreline coordinates x, i.e.,

equation image

where 〈·〉(i) is the average over all realizations i. Absolute averaging is used for simplicity because the interaction of a tracer plume with a boundary (i.e., the beach) complicates the interpretation of relative averages. For example, Di(x, yj) realizations with shoreline maxima (shoreline attached), equal dye mass, and different cross-shore widths give varying individual centers of mass at a transect location yj (Figure 4). The center of mass variation may imply that the plume is meandering, but could also be explained (e.g., this example) by turbulent fluctuations widening some shoreline attached realizations more than others.

Figure 4.

Schematic cross-shore tracer concentration in (a) shoreline and (b) center of mass coordinates, illustrating the difficulty in estimating relative diffusivity near the shoreline (shoreline coordinate zero). Tracer averages (thick black curves) are from three realizations (gray curves) with varying cross-shore widths. The center of mass for each realization is indicated with a star in the corresponding shade of gray.

[23] Rms errors equation image(x, yj) in the mean equation image(x, yj) are estimated by

equation image

where equation imageequation imagei2(x, yj)equation image(i) is the mean squared dye measurement error, equation image[Di(x, yj) − equation image(x, yj)]2equation image(i) is the dye variance at each x, and &#55349;&#56489;j is the degrees of freedom at a transect location j (Appendix A). The dye variance is usually much larger than the instrumental error. Using &#55349;&#56489;j in (4) accounts for consecutive Di(x, yj) that are not independent. The &#55349;&#56489;j are estimated from the Eulerian tracer decorrelation time (Appendix A), where &#55349;&#56489;j = 1 and &#55349;&#56489;j = Nj correspond to completely dependent and completely independent sampling, respectively (release averaged 〈&#55349;&#56489;j(j) are given in Table 2).

3.2. Mean Profile equation image(x, yj) Cross-Shore Integrated Statistics

[24] Two cross-shore integrated tracer statistics are estimated from mean equation image(x, yj) profiles; the alongshore tracer transport M(yj), and the surface-center of mass (first moment) μ(yj). These statistics are both functions of yj, the alongshore distance from the dye source. The jet ski cross-shore transects are driven as shallow as possible without running aground, and the location of the equation image(x, yj) inner ends vary. To avoid propagating transect end variations into equation image(x, yj) statistics, the shoreward integral limit is at xin (Table 2), the inner equation image(x, yj) end that is farthest from the shoreline (for each release). Taking the integral limit at xin biases mean transect statistics equally, rather than randomly as with variable inner ends. The xin are generally shoreward of the cross-shore dye release location x0, and little data is discarded. The effect of the unsampled region near the shoreline on M(yj) is discussed in section 6.4.

[25] An idealized tracer plume conserves alongshore tracer transport (i.e., flux through the xz plane). Assuming vertically well-mixed tracer, alongshore uniform depth h(x) and alongshore current V(x), negligible tracer-alongshore current covariance, and negligible dye offshore of frame 7 (xF7), the mean alongshore tracer transport M(yj) is defined as

equation image

The transport M at y = 0 is given by the estimated pump flow rate (m3 s−1) times the initial dye concentration (2.1 × 108 ppb).

[26] The tracer plume surface-center of mass μ(yj) is given by the equation image(x, yj) first moment

equation image

where the offshore limit of equation image(x, yj), xout, is always seaward of the tracer plume. For non-shoreline attached tracer plumes (i.e., no plume-shoreline interaction) and no cross-shore advection, μ(yj) is expected to remain constant [e.g., Csanady, 1973]. In contrast, shoreline attached plumes spread (i.e., disperse) away from the shoreline, and μ(yj) magnitudes are expected to increase downstream. Similar to μ(yj), the shoreline also complicates estimates of the plume second moment, as discussed in section 5.

[27] Errors equation imageM (yj) and equation imageμ (yj) in cross-shore tracer statistics M(yj) and μ(yj) are estimated using Monte Carlo simulations. For each transect, 104 simulated equation image(x, yj) are generated from the observed equation image(x, yj) plus Gaussian noise, where the noise variance is equal to equation imageequation image(x, yj) (4). The tracer statistic errors εM(yj) and εμ(yj) are estimated as the standard deviation of simulated M(yj) and μ(yj) calculated from the simulated equation image(x, yj). The errors are dependent on the εequation image decorrelation length-scale, thus the Monte Carlo process is repeated with cross-shore decorrelation length-scales between zero and twice the surfzone width Lx. The maximum εM (yj) and εμ (yj) over the range of decorrelation length-scales, are used.

4. Observations of Surfzone Tracer Plumes

4.1. Tracer Cross-Shore Structure

[28] Continuous surfzone dye releases in an alongshore current form tracer plumes (e.g., Figure 3) similar to a smokestack plume in the wind, with plume axis parallel to the shoreline. The positive alongshore current (Figure 2b) advects dye downstream (+y) from the dye source (green star, Figure 3). The initially concentrated dye dilutes and spreads cross-shore as it is advected downstream. The plumes were visually patchy with adjacent high and low concentration areas at all alongshore distances from the dye source, and the patch length-scale increased with distance. Bores did not “surf” dye to the shoreline, although the plume cross-shore width visually widened with each passing bore [e.g., Feddersen, 2007]. Bore-mixing was most apparent when plume widths were <10 m, and difficult to observe when plume widths were visually >40 m.

[29] At all cross-shore locations, individual Di(x, yj) vary about the mean (e.g., Figure 5). At transect locations close to the source (e.g., Figures 5a and 5d), dye maxima are sometimes seaward of the shoreline, and have “meandering like” variations (roughly ±10 m). Farther downstream from the dye source (Figures 5b, 5c, 5e, and 5f) the Di(x, yj) realizations are more shoreline attached, and may significantly interact with the shoreline boundary.

Figure 5.

Cross-shore dye concentration transects Di(x, yj) versus x for (a–c) R3 and (d–f) R6 at three downstream locations y from the dye source (see top left). Individual realizations Di(x, yj) are in color and the mean equation image is a dashed black curve. R3 (Figures 5a–5c) and R6 (Figures 5d–5f) have different vertical scales.

[30] The mean tracer profile errors equation image (x, yj) (4), indicated by the light-colored regions about the mean in Figure 6, combine variability between realizations (e.g., Figure 5) and dye measurement errors (2). For all realizations, dye inter-realization variance (last term of (4)) dominates equation image (x, yj) and is on average 45 times greater than mean squared dye measurement errors. The equation image(x, yj) increase with increasing equation image(x, yj) (e.g., Figure 6). The R3 equation image (x, yj) are larger than R6 because (in addition to larger equation image(x, yj)) R3 has shorter times between transects, resulting in lower degrees-of-freedom &#55349;&#56489;j (Table 2).

Figure 6.

Mean dye profile curves equation image(x, yj) with lighter regions indicating equation image(x, yj) ± equation image(x, yj), for dye releases (a) R3 and (b) R6 at three alongshore distances y from the dye source (see legend). The dashed gray line indicates the seaward edge of the surfzone. Vertical scales differ.

[31] The mean tracer profiles equation image(x, yj) (Figure 7) average over stirring and meandering, and are smoother than individual profiles Di(x, yj) (e.g., Figure 5). Most mean profiles equation imagei(x, yj) are shoreline attached with maxima at or near the shoreline (Figure 7). The exception, R1, has maxima in the mid- to outer-surfzone, likely because dye was released in the mid- to outer-surfzone (x0 = −54 m), and yj sampling distances are short (Figure 7a). On all releases except R5, the initially narrow equation image(x, yj) profiles disperse across the surfzone and peak concentrations decrease with downstream distance from the source (Figure 7). Release R5 was sampled far downstream of the the dye source (yj > 282 m) where tracer had already spread (i.e., saturated) across the surfzone (Figure 7e), with smaller concentrations seaward of the surfzone. The two farthest downstream equation image(x, yj) profiles of R3 (Figure 7c, with expanded scale in Figure 8) and R6 (Figure 7f) are also surfzone saturated. The strong gradients in equation image profiles seaward of the saturated surfzone are consistent with decreased diffusivity.

Figure 7.

Mean tracer concentration equation image(x, yj) versus x for releases (a) R1, (b) R2, (c) R3, (d) R4, (e) R5, and (f) R6. Colors indicate different downstream alongshore distances y (see legends). The surfzone is between x = 0 m and the vertical dashed gray line. Mean (over all releases) fractional errors equation image (x, yj)/equation image(x, yj) are 0.38 ± 0.16 (for D > 5 ppb), and 0.78 ± 0.29 (for D < 5 ppb). Vertical and horizontal scales vary.

Figure 8.

equation image(x, yj) versus x for the two R3 transects farthest downstream from the dye source (expanded view of Figure 7c). The surfzone is between x = 0 m and the vertical dashed gray line.

4.2. Alongshore Evolution of equation image(x, yj) Statistics

[32] The downstream y evolutions of tracer transport M(yj) (5) and surface-center of mass μ(yj) (6), estimated using equation image(x, yj) mean profiles, are examined. Tracer seaward of frame 7 (x < −150 m from the MSL shoreline) is neglected (5) and reduces some M(yj) at large y. Alongshore variation in V(x) is also neglected, and complete vertical mixing is assumed. Nevertheless, for all releases, the downstream (i.e., not including y = 0 m) tracer transports M(y > 0 m) generally vary by less than a factor of 2 (Figure 9), and (except for R4) are either roughly constant downstream (e.g., R1 and R3) or monotonically decrease (e.g., R2). Thus, a significant amount of the dye measured on the first transect (yj > 0 m) is accounted for farther downstream.

Figure 9.

Tracer alongshore transport M(yj) (5) versus y, with error bars ±εM for releases (a) R1, (b) R2, (c) R3, (d) R4, (e) R5, and (f) R6. The initial condition at y = 0 m is the injected dye transport (concentration times flow-rate).

[33] The reason that the pump-rate calculated M(y = 0 m) are larger than downstream estimated M(y > 0 m) is not known. Consistent fluorometer calibrations over multiple batches of calibration standards indicate that fluorometer instrumentation error is not the cause. Other possible reasons are discussed in section 6.4. All other tracer moments considered here (e.g., μ(6)) are normalized by the cross-shore surface tracer integral on each transect, thus reducing the effect of tracer transport variations.

[34] The equation image(x, yj) (Figure 7) tracer surface-centers of mass μ(yj) (6) are initially grouped (Figure 10) near the cross-shore dye release locations x0 (Table 2). Farther downstream, the shoreline attached R2–R6 μ(yj) generally move seaward, consistent with shoreline attached dye profiles (Figures 7b7f) broadened by cross-shore dispersion (Figure 10, and section 5.3), and does not likely represent cross-shore advection of the mean plume. In contrast, R1 tracer was released mid-surfzone (Table 2) and the μ(yj) appear (Figure 7a) to move seaward by advection of the mean plume, and not plume widening near a boundary. For shoreline attached profiles, surfzone saturation and a lower diffusivity seaward of the surfzone would result in decreased seaward μ(yj) movement at large y, and is apparent in release R6.

Figure 10.

Tracer surface-center of mass μ(yj) (6) versus y. The mean μ(yj) error over all transects and releases is εμ(yj) ≈ 14 m.

5. Dispersive Plume Widening and Surfzone Cross-Shore Diffusivity κxx

5.1. Simple Diffusion Models

[35] Surfzone cross-shore turbulent mixing results from many mechanisms with different time- and length-scales. The appropriate model diffusivity depends on the model dynamics and the scales resolved. For example, a two-dimensional (2D) horizontal eddy-resolving Boussinesq model [e.g., Johnson and Pattiaratchi, 2006; Spydell and Feddersen, 2009] would require a much smaller diffusivity than a simple bulk model averaged over longer time scales that combine eddy stirring into a bulk (Fickian) diffusivity. Here, a simple Fickian diffusion model is presented that provides an analytic method for estimating the bulk diffusivity from observations of equation image(x, yj), and also has solutions relating diffusivity, tracer surface-center of mass, and dilution of maximum tracer concentration.

[36] The time invariant 2-D Fickian advection-diffusion model for vertically-mixed mean tracer concentrations equation image(x, y) with a tracer source Q0 (m3 ppb s−1) at x = x0 and y = 0, and assuming a constant cross-shore diffusivity κxx (Brownian diffusion regime), a long narrow plume (∂2/∂y2 ≪ ∂2/∂x2), and cross-shore variable depth h(x) is

equation image

[37] Assuming constant surfzone averaged depth equation image (neglecting dh/dx) and that tracer is advected downstream by the surfzone averaged mean alongshore current equation image, (7) becomes

equation image

and allows for analytic solutions. For HB06 cross-shore variable bathymetry h(x) (Figure 1b), observed V(x) (Figure 2b), and dye release locations (Table 2), numerical solutions to the constant (8) and cross-shore varying depth and alongshore current (7) equations are similar (particularly for y > 50 m), as is the evolution of cross-shore integrated moments. One reason for the similar solutions may be the relatively flat terraced surfzone bathymetry (see −75 < X < −10 m in Figure 1b). Defining a plume alongshore advection time tp

equation image

where tp is the time for a section of the plume moving with equation image to reach a downstream location y, (8) reduces to the familiar 1-D diffusion equation

equation image

where equation image0 = Q0/(equation image). On an unbounded domain (9) has a Gaussian solution

equation image

where κxx is related to the tracer second moment σ2 [e.g., Csanady, 1973]

equation image

and σ2 is defined as

equation image

where μ(6) is calculated over the ±∞ domain. On a semi-infinite domain (−∞ < x < 0) with a no-flux boundary condition at the x = 0 shoreline

equation image

(9) has the method of images solution

equation image

[38] For the case of dye released at the shoreline (x0 = 0), the solution (13) is a shoreline attached half-Gaussian. The cross-shore diffusivity for (13) is

equation image

where σsl2 is the shoreline based second moment

equation image

[39] Thus, for shoreline attached plumes (R2, R3, R4, and R6) equation imagexx is estimated from (14) and (15), with modification (section 5.2). For plumes that are well separated from the shoreline (i.e., R1), κxx is estimated from (11) and (12). Applying (14) and (15) to numerical solutions of the full advection diffusion equation (7) with observed release parameters (Table 2), and cross-shore varying h(x) (Figure 1) and V(x) (Figure 2b), yields κxx estimates within 10% of the modeled value. Therefore, the cross-shore uniform h and equation image approximations are not expected to bias κxx significantly.

5.2. Estimating Surfzone κxx

[40] Surfzone absolute equation imagexx are estimated by applying (14) and (15) to shoreline attached mean tracer profiles equation image(x, yj), with some adjustments to capture surfzone specific κxx. Because σsl2 is sensitive to tracer seaward of the surfzone, a surfzone-specific 2nd moment σsurf2 is defined, similar to σsl2, but integrated only over the surfzone, i.e.,

equation image

thus excluding dye seaward of the surfzone. With a delta function source at x = x0 (Table 2) and y = 0 (i.e., Tp = 0), σsurf2 (0) = x02.

[41] With a reduced equation imagexx seaward of the surfzone [e.g., Harris et al., 1963], initially rapid surfzone cross-shore dispersion would slow as a tracer spreads offshore. A surfzone saturation ratio equation image is used to select transects with dye well contained in the surfzone, and exclude surfzone saturated transects effected by reduced seaward equation imagexx. For each transect, equation image is the ratio of the measured σsurf(yj) to that of a saturated surfzone (i.e., constant surfzone concentration)

equation image

Unsaturated transects, between the source and the farthest downstream transect satisfying a surfzone saturation ratio criterion equation image < ℛ0, are included in κxx estimates. The cutoff threshold ℛ0 = 0.55 is chosen to include many σsurf2(yj) in κxx fits while not biasing κxx estimates by more than 10% (Appendix B).

[42] For shoreline attached releases (R2, R3, R4, R6), surfzone equation imagexx is estimated from least squares σsurf2 versus Tp fits, i.e.,

equation image

where equation imagexx and β are fit constants, and from (15)β is expected to be close to the initial condition x02. The equation imagexx error εκ is estimated from the fit slope error assuming the variance of the residuals is equal to image [e.g., Wunsch, 1996], where image is estimated with the same Monte Carlo methods as εM and εμ (section 3.2). All transects between the release location and the farthest downstream transect with ℛ ≤ ℛ0 = 0.55 are included in the fit (solid black symbols, Figures 11b, 11c, 11d, and 11f). For shoreline attached releases, estimated κxx ± εκ range from 0.5 ± 0.08 to 2.5 ± 0.62 m2 s−1 with generally high squared correlation coefficients r2 (Table 3). The R5 κxx is not estimated because all downstream transects are surfzone saturated, with ℛ > 0.55 (Figure 11e).

Figure 11.

Plot of σ2 ± equation image (Figure 11a) and σsurf2 ± image (Figures 11b–11f) versus Tp for releases (a) R1, (b) R2, (c) R3, (d) R4, (e) R5, and (f) R6. Black symbols indicate points used in the κxx fits (dashed gray curves) between Tp = 0 and the farthest downstream transect with dye largely confined within the surfzone (ℛ ≤ ℛ0). Errors image and image are estimated in the same manner as εM and εμ (section 3.2).

Table 3. Estimated κxx Fitsa
 2[κxx ± εκ]tp + βr2
  • a

    Estimated κxx from a nonshoreline attached σ2 versus tp fit (R1) and shoreline attached σsort2 versus tp fits (R2, R3, R4, and R6). Squared correlations r2 are given for all releases, with an exception for R1 where a two point fit gives the trivial result r2 = 1.

R12[0.8 ± 0.31]tp + 0-
R22[1.3 ± 0.26]tp + 2510.94
R32[1.6 ± 0.68]tp + 2330.95
R42[2.5 ± 0.62]tp + 5450.56
R62[0.5 ± 0.08]tp + 510.97

[43] For release R1 with mid-surfzone release location, mean tracer profiles equation image(x, yj) (Figure 7a) are not shoreline attached, thus (14) through (17) do not apply. The non-shoreline attached R1 surfzone κxx is estimated using the common definition (11) for absolute dispersion without a boundary [e.g., Fong and Stacey, 2003; Jones et al., 2008], where the cross-shore moments μ and σ2 are integrated from xin to the seaward transect limit xout. Using the initial condition σ2 = 0 (at Tp = 0) and the first downstream σ2 (surfzone contained by inspection, Figure 7), the resulting best-fit is κxx = 0.8 ± 0.31 m2 s−1, and r2 cannot be estimated from the two point fit.

5.3. Half-Gaussian Shoreline-Attached Model Data Comparison

[44] For shoreline attached R2, R3, R4, and R6 releases, the observed downstream evolution of equation image is similar to the half-Gaussian solution (13) with x0 = 0 m. For example, within the surfzone the observed σsurf2 increase linearly with tp (black symbols in Figures 11b, 11c, 11d, and 11f) with generally high r2 (Table 3) as is expected for (13) and (14), and is consistent with assumption (section 5.1) of constant (in time and space) surfzone equation imagexx. This model also predicts the downstream evolution of the maximum tracer equation imagemax(p)

equation image

and surface-center of mass μ(p)

equation image

where G(x0 = 0) is the shoreline-attached half-Gaussian solution (13) with dye released at x0 = 0. Note that μ(p) moves offshore owing to the presence of the shoreline, not from advection. Both predictions are now compared with observations using the estimated surfzone κxx.

[45] For the R2, R3, R4, and R6 transects used in equation imagexx estimation (solid symbols in Figure 11b, 11c, 11d, and 11f), and representative of surfzone mixing, the observed equation imagemax and predicted equation imagemax(p)(19) are consistent (Figure 12a). The predictions are slightly larger than the observations, and may result from using pump rate estimated Q0 = M(y = 0 m) (larger than transect estimates, Figure 9) in equation image0(19) or higher tracer concentrations near the shoreline (x > xin, Table 2) where the jet ski does not sample. Although the R3 and R6 equation imagemax(p) have large errors at the first downstream transect, the skill (defined as 1 − 〈(equation imagemax(p)equation imagemax)2〉/〈equation imagemax2〉 over all releases) of 0.76 is high. The observed μ and predicted μ(p) are also consistent (Figure 12b) with skill (defined similarly to equation imagemax skill) of 0.90. The shoreward bias of μ(p) relative to μ (Figure 12b), may result from assuming a shoreline release (x0 = 0 in (20)) in μ(p). In addition μ estimates may be biased seaward by neglecting the near-shoreline region between xin (Table 2) and the x = 0 m shoreline, where the jet ski does not sample. The seaward μ(yj) movement for R2–R6 can be explained as dispersive widening of the shoreline attached plume near a boundary (e.g., Figures 7b7f). The linear σsurf2 growth with tp, the predicted decrease in normalized maxima, and the correspondence of μ and μ(p), all indicate (13) well describes the downstream evolution of surfzone contained tracer released near the shoreline.

Figure 12.

(a) Predicted tracer maxima equation imagemax(p) versus observed equation imagemax, and (b) predicted μ(p) versus observed μ, for surfzone-contained shoreline attached profiles used in equation imagexx fits (releases R2, R3, R4, and R6). The predicted equation imagemax(p) = 2equation image0/equation image(19) and μ(p) = −2equation image(20) use the observed best-fit equation imagexx (Figures 11b, 11c, 11d, and 11f). The dashed line indicates perfect agreement. The skill in Figure 12a is 0.76, and the skill in Figure 12b is 0.90.

[46] The shoreline attached moment σsurf2 (yj) (16) and the half-Gaussian solution (13) assume that the equation imagemax cross-shore locations remain at the shoreline, however the observed locations vary slightly (Figure 7). Consistent with the assumed shoreline maxima, the “Péclet numbers” (LUxx) for shoreline attached releases are small (<0.12), where U is the mean cross-shore velocity of the tracer maxima (for profiles used in κxx fits), L = U tmax, and tmax is the maximum tp included in κxx fits. The small “Péclet numbers” and the agreement between μ and μ(p) are consistent with neglecting cross-shore advection for shoreline attached profiles. In contrast, the “Péclet number” for the non-shoreline attached R1 is 3.9 and the cross-shore advection is accounted for in (11) and (12).

6. Discussion

6.1. Surfzone κxx Comparisons

[47] 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).

[48] 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 (σ2t2 or κxxt) towards a peak value [Spydell et al., 2009]. In contrast, tracer-derived surfzone κxx are roughly constant in time and σsurf2t (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 TpTxx).

[49] 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

[50] The equation image(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 equation image(x, yj) gradients at the seaward edge of the surfzone (e.g., Figures 7e and 8). These equation image(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.

[51] 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 equation image(x, yj) profiles (Figure 7d).

6.3. Parameterizing κxx

[52] Previous dye dispersion studies [e.g., Harris et al., 1963; Inman et al., 1971] parameterized diffusivity with

equation image

or

equation image

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.

[53] 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 equation image [Feddersen, 2007; Henderson, 2007] is

equation image

where ĉ and equation image are the non-dimensional phase speed and wave period, respectively. A dimensional mid-surfzone equation imagexx can be derived from the scalings of Feddersen [2007]

equation image

[54] Assuming a self-similar surfzone (H/h = γ) and a mid-surfzone water depth (h = hb/2) then

equation image

[55] With γ = 0.6, the slope between κxx and Hb2T−1 would be near 2.

[56] 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.

Figure 13.

Estimated surfzone cross-shore diffusivity equation imagexx ± equation imageκ versus (a) Hs2Tm−1, (b) equation imagerotLx, and (c) κxx(sd). The fit slopes are 11.7 and 0.2, and r2 correlations are 0.32 and 0.59 for Hs2Tm−1 (Figure 13a) and equation imagerotLx (Figure 13b), respectively. In Figure 13c, the r2 = 0.94 correlation is high, but κxx(sd) magnitudes are much smaller than the observed κxx.

[57] 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 equation imagerot (i.e., cross-shore averaged equation imagerot (x) (1) between the shoreline and x = −Lx, Table 1)

equation image

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., kequation image) models relating diffusivity to a length- and velocity scale product, α is expected to be <1 but still O(1). Surfzone equation imagerot (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 equation imagerot ranges between 0.036–0.09 ms−1 (Table 1).

[58] The surfzone tracer equation imagexx increase with equation imagerotLx (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 equation imagerot and Lx are relatively small (Table 1), and additional observations of surfzone tracer equation imagexx are required to fully test this parameterization (26). A related mixing-length scaling, using equation image instead of equation imagerot as the velocity scale, was correlated with alongshore drifter diffusivity [Spydell et al., 2009], and is consistent with the present result because &#55349;&#56497;rot (x) and V are correlated [Noyes et al., 2004].

[59] As suggested by Pearson et al. [2009], 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. [2009]

equation image

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. [2009] found good agreement between a laboratory estimated κxx and the corresponding scaled κxx(sd) for shore-normal monochromatic waves.

[60] 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. [2009], 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

[61] 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 equation image (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 equation imagexx, but would affect the predicted tracer maxima equation imagemax(p)(19) used for model data comparison (Figure 12a).

[62] 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 equation image = 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.

[63] 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 equation image 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).

[64] 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.

7. Summary

[65] The cross-shore surfzone dispersion of a continuously released dye tracer in an alongshore current was observed during six dye releases. Tracer concentrations were measured on repeated cross-shore transects, at various alongshore distances from the dye source, with a unique GPS-tracked jet ski dye sampling platform. Tracer is advected with the mean alongshore current (i.e., downstream) forming plumes that become wider and more diluted with distance downstream. Mean cross-shore profiles equation image(x, yj) often have concentration maxima at or near the shoreline (shoreline attached) with decreasing concentration offshore, qualitatively consistent with a half-Gaussian shape. At large downstream distances from the source, equation image(x, yj) is approximately constant across the surfzone with decreasing concentrations farther seaward, consistent with much lower diffusivity seaward of, than within, the surfzone.

[66] Tracer alongshore transport M(yj) and surface-center of mass μ(yj) are estimated from the equation image(x, yj). The mean alongshore M(yj) is roughly conserved downstream of the dye source, and is typically a factor of 2 smaller than the injected dye flux. For shoreline attached profiles the μ(yj) move offshore with downstream distance. For shoreline attached profiles (R2–R6) the offshore μ movement with increasing y is associated with plume widening and not seaward advection of the mean plume.

[67] Surfzone cross-shore absolute diffusivities (equation imagexx = 0.5–2.5 m2 s−1), based upon a simple Fickian diffusion model near a boundary, are estimated from mean equation image(x, yj) profiles. To estimate surfzone diffusivity, only mean tracer transects where tracer is surfzone-contained are included in κxx fits. For shoreline attached profiles, the estimated diffusivities, the observed tracer surface-center of mass, and the observed tracer maxima are all consistent with the Fickian modeled half-Gaussian solution.

[68] Three potential mechanisms for cross-shore tracer dispersion in the surfzone are examined by testing cross-shore diffusivity parameterizations. A breaking-wave induced κxx parameterization has low correlation with observed κxx (r2 = 0.32), and the best-fit slope is larger than expected. Undertow driven shear dispersion estimates have high correlation (r2 = 0.94), but significantly under-predict the observed κxx, indicating that this mechanism is not a dominant term in cross-shore surfzone tracer dispersion. A mixing-length parameterization based on 2D horizontal rotational velocities (surfzone eddies) with length-scales of the surfzone width or less has good correlation (r2 = 0.59) and a best-fit slope <1 (as expected). This suggests that the observed tracer dispersion is primarily due to surfzone eddies forced either by shear instabilities (shear waves) or by finite-crest-length wave breaking.

[69] The reasons for the decreased alongshore tracer transport M(yj), relative to pump rate estimates, are unknown and possible causes are examined. Tracer induced stratification is estimated to be negligible, and unlikely to explain the M(yj) decrease. Tracer transport in the the neglected near-shoreline region (where the jet ski does not sample) is generally not large enough to account for the M(yj) decrease. Neglecting tracer near the shoreline may bias κxx estimates up to 20%.

Appendix A:: Degrees of Freedom in Estimating equation image(x, yj)

[70] For each release, the degrees of freedom equation imagej at each yj are estimated from the Eulerian decorrelation time τdecorr and the times between transect realizations. Surfzone dye concentration time series (not shown), measured by fluorometers [Clark et al., 2009] mounted on the instrumented frames (Figure 1), are used to estimate τdecorr = A(0)−1 image [e.g., Emery and Thomson, 2001], where A(τ) is the lagged (τ) dye concentration autocorrelation function and τmax is the maximum lag (roughly the duration of each dye release). Sequential Di realizations separated by times greater than τdecorr are assumed independent and add one to equation imagej. A group of realizations separated by times less than τdecorr are assumed fractionally independent and add 1 + (equation imagebequation imagea)τdecorr−1 to equation imagej, where equation image and equation image are the mean times of the first and last realizations in the group. The resulting equation imagej is between 1 and number of realizations Nj (Table 2).

Appendix B:: Surfzone Saturation Ratio for Estimating κxx

[71] Estimates of σsurf2 (yj) are only included in cross-shore surfzone diffusivity equation imagexx fits (14) if the mean tracer is surfzone contained, so that the fit equation imagexx represents surfzone diffusivity rather than a combination of the surfzone and the region seaward. To quantify which transects are well contained in the surfzone, a surfzone saturation ratio ℛ (17) is defined as the ratio of σsurf2 (yj), to the σsurf2 value for uniform tracer across the surfzone (i.e., saturated). The threshold ℛ0 for determining which yj locations to include in κxx fits is developed for shoreline attached profiles (the majority of observations) by modeled tracer diffusion.

[72] The surfzone is likely a region of high diffusivity with lower diffusivity seaward. The transition between these two regions is not understood. Thus, two possible extremes for tracer diffusion are considered to determine the equation image threshold. The first is constant diffusivity on a semi-infinite domain with a shoreline no-flux boundary. The second is constant diffusivity within the surfzone (width Lx) with zero diffusivity seaward. This is modeled as a closed domain with no-flux boundaries at the shoreline and the seaward surfzone edge. Diffusivity along the seaward edge of the surfzone is somewhere in between these two extremes.

[73] Non-dimensional variables are introduced

equation image

and result in the non-dimensional diffusion equation with a delta function source at the shoreline

equation image

solved on the semi-infinite and closed domains described above. equation image profiles (Figure B1a) are initially Gaussian until equation image reaches equation image = 1 and either moves beyond the surfzone (semi-infinite domain) or interacts with the surfzone boundary (closed domain). The closed domain increases dye concentrations in the outer surfzone (Figure B1a), resulting in larger closed domain equation imagesurf2 relative to the semi-infinite domain (Figure B1b). The equation imagesurf2 are linear with respect to equation image for equation image < 0.05, but asymptotically approach the surfzone saturation limit [equation imagesurf]2 = 1/3 for large equation image (Figure B1b). Fitting equation imagexx to equation imagesurf2 for equation image < 0.05, where equation imagesurf2 growth is linear (Figure B1b), produces the correct equation imagexx = 1 (Figure B1c). Including data with equation image > 0.05, where the equation imagesurf2 growth rate decreases, reduces the fit equation imagexx from the true value (Figure B1c).

Figure B1.

(a) Modeled non-dimensional dye concentration equation image versus non-dimensional equation image at three times (equation image = 0.05, 0.15, 0.25), for (black curves) diffusion on a semi-infinite domain (no-flux boundary at equation image = 0) where the equation image are truncated at equation image = 1, and (dashed grey curves) diffusion on a closed 0 < equation image < 1 domain with no-flux boundaries. (b) Non-dimensionalized equation imagesurf2 versus non-dimensionalized equation image, with saturated equation imagesurf2 = 1/3 (dotted curve) for reference. (c) Non-dimensional fit equation imagexx (using equation imagesurf2 with 0 < equation image < equation image0) versus equation image0 and, (dot-dashed curve) the equation imagexx = 0.9 threshold used to determine the equation image0 cutoff.

[74] The greatest possible number of field σsurf2 should be used to estimate equation imagexx without significantly biasing equation imagexx from the surfzone value. Requiring that the fit equation imagexx ≥ 0.9 gives the threshold equation image0 = 0.48 and equation image0 = 0.62 for the semi-infinite and closed domains, respectively (Figure B1c), with average equation image0 = 0.55. Only transects between the dye source (y = 0 m) and the farthest downstream transect where ℛ < ℛ0 are included in κxx fits (black symbols, Figures 11b11f).

Acknowledgments

[75] This research was supported by CA Coastal Conservancy, NOAA, NSF, ONR, and CA Sea Grant. Staff, students, and volunteers from the Integrative Oceanography Division (B. Woodward, B. Boyd, K. Smith, D. Darnell, I. Nagy, M. Omand, M. Yates, M. McKenna, M. Rippy, S. Henderson, D. Michrokowski) were instrumental in acquiring these field observations. We thank these people and organizations.

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