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Modeling surf zone tracer plumes: 2. Transport and dispersion

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Abstract

[1] Five surf zone dye tracer releases from the HB06 experiment are simulated with a tracer advection diffusion model coupled to a Boussinesq surf zone model (funwaveC). Model tracer is transported and stirred by currents and eddies and diffused with a breaking wave eddy diffusivity, set equal to the breaking wave eddy viscosity, and a small (0.01 m2 s−1) background diffusivity. Observed and modeled alongshore parallel tracer plumes, transported by the wave driven alongshore current, have qualitatively similar cross-shore structures. Although the model skill for mean tracer concentration is variable (from negative to 0.73) depending upon release, cross-shore integrated tracer moments (normalized by the cross-shore tracer integral) have consistently high skills (≈0.9). Modeled and observed bulk surf zone cross-shore diffusivity estimates are also similar, with 0.72 squared correlation and skill of 0.4. Similar to the observations, the model bulk (absolute) cross-shore diffusivity is consistent with a mixing length parameterization based on low-frequency (0.001–0.03 Hz) eddies. The model absolute cross-shore dispersion is dominated by stirring from surf zone eddies and does not depend upon the presence of the breaking wave eddy diffusivity. Given only the bathymetry and incident wave field, the coupled Boussinesq-tracer model qualitatively reproduces the observed cross-shore absolute tracer dispersion, suggesting that the model can be used to study surf zone tracer dispersion mechanisms.

1. Introduction

[2] The rates and mechanisms of surf zone horizontal tracer (e.g., pollution, nutrients, sediment, and larvae) dispersion are understood poorly, and numerical models may be useful for investigating the underlying dispersion processes. However, numerical surf zone tracer models have not been validated, a necessary step before investigating dispersion mechanisms.

[3] Surf zone tracer dispersion has been modeled analytically and numerically. Simple Fickian analytic models were used to estimate bulk surf zone diffusivity from field data [Harris et al., 1963; Inman et al., 1971; Clarke et al., 2007; Clark et al., 2010]. Fickian models may be able to predict bulk surf zone tracer dispersion with the appropriate diffusion coefficient. However, surf zone diffusivity values are poorly known, and diffusivity parameterizations have not been validated over a broad range of conditions. Coupled tracer and (wave-averaged) circulation models have been sparingly used to simulate tracer transport in the nearshore and surf zone [Tao and JianHua, 2006; Issa et al., 2010], but comparisons with observations are very limited [Rodriguez et al., 1995].

[4] Scaling arguments [Harris et al., 1963; Inman et al., 1971] and an idealized model [Feddersen, 2007; Henderson, 2007] suggest that the bulk (averaged over many waves) cross-shore tracer diffusivity equation imagexx from turbulent mixing at the front face of broken waves (bores) scales as equation imagexxHs2Tm−1, where Hs and Tm are the incident significant wave height and mean period, respectively. However, this scaling had marginal correlation (r2 = 0.32) when compared with recently observed bulk cross-shore dye diffusivities [Clark et al., 2010]. Stirring due to low-frequency (f < 0.03 Hz) horizontal surf zone eddies may induce a significant amount of cross-shore tracer dispersion. Higher correlation (r2 = 0.59) was found for a surf zone–eddy mixing length scaling κxxequation imagerot(IG)Lx, where Lx is the surf zone width and equation imagerot(IG) is a surf zone (cross-shore) averaged bulk infragravity (0.004–0.03 Hz) eddy velocity, suggesting that low-frequency eddies may be a primary dispersion mechanism [Clark et al., 2010]. An undertow-induced cross-shore shear dispersion scaling [Pearson et al., 2009] was not found to be applicable [Clark et al., 2010]. Overall, the mechanisms of tracer dispersion and their relative importance are not well understood.

[5] Time-dependent wave-resolving surf zone models (most commonly Boussinesq models), include the broad range of processes, from individual breaking waves to low-frequency eddies and mean currents, required for investigating surf zone tracer dispersion mechanisms. Boussinesq surf zone models, solving an extended version of the nonlinear shallow water equations with weak nonlinearity and dispersion [e.g., Peregrine, 1967; Nwogu, 1993; Wei et al., 1995], have been used to examine surf zone drifter dispersion in directionally spread random wave fields [Johnson and Pattiaratchi, 2006; Spydell and Feddersen, 2009; Geiman et al., 2011], but have not been used for surf zone tracer modeling. Finite crest length wave breaking within Boussinesq models provide a (vertical) vorticity source for forcing horizontal eddies [Peregrine, 1998] at a range of length scales, which induced surf zone drifter dispersion at scales between 20 and 200 m [Spydell and Feddersen, 2009]. Surf zone drifters duck under, and are not dispersed by entrainment in, the front face of breaking waves [e.g., Schmidt et al., 2003, 2005]. By resolving individual wave breaking, Boussinesq models also provide a mechanisms for breaking waves to mix tracer. Thus, a depth-averaged tracer advection diffusion equation coupled to a Boussinesq model contains both stirring by the horizontal eddy field (e.g., vertical vorticity) and the breaking wave mixing mechanisms.

[6] Here, five surf zone tracer releases from the HB06 experiment in Huntington Beach, California [Clark et al., 2010] are simulated with the coupled tracer and Boussinesq model funwaveC. The Boussinesq model is described by Feddersen et al. [2011] (hereinafter referred to as Part 1), and compared with Eulerian wave and current observations. The model reproduces the observed significant wave height and (except for one release) alongshore currents. Low-frequency eddies are well modeled in the infragravity frequency (f) band (0.004 < f < 0.03 Hz), but are overpredicted by a factor of 2 in the very low frequency (VLF, 0.001 < f < 0.004 Hz) band. The HB06 tracer experiments and previous results are summarized in section 2. The tracer model and averaging method are described in section 3.

[7] Mean tracer concentrations are well modeled for 3 out of 5 releases (section 4). For all releases, model skills for cross-shore integrated tracer first and second moments are high, and the model reproduces the observed bulk cross-shore surf zone diffusivity (section 5). The causes of model-data mismatch for mean tracer and alongshore tracer transport are discussed in sections 6.1 and 6.2, respectively. The downstream dilution of the modeled mean plume is consistent with a Fickian analytic solution (section 6.3). The effect of the modeled breaking wave eddy diffusivity on cross-shore tracer dispersion is discussed in section 6.4. Mixing length scalings for modeled bulk cross-shore diffusivity κxx, using bulk low-frequency eddy velocities, are examined in section 6.5. The results are summarized in section 7.

2. HB06 Observations and Dye Releases

[8] The predominant south swell during the HB06 experiment drove strong alongshore currents upcoast (toward the northwest). Waves and currents were measured on a 140 m long cross-shore array of 7 bottom mounted tripods, denoted F1–F7 from near the shoreline to roughly 4 m water depth (Part 1, Figure 1). The observations at F2 were often poor quality, and are not included in the subsequent analysis. Hourly significant wave heights Hs ranged from 0.41 to 1.02 m during the tracer releases, with mean wave periods Tm from 9 to 9.9 s, and directional spreads σθ from 15° to 23°. Mean (in time) alongshore currents V(x) (where x is the cross-shore distance from the shoreline) were generally maximum near mid surf zone, except for one release with maximum V(x) near the shoreline. Eulerian wave and current observations are described by Clark et al. [2010] and compared with the funwaveC Boussinesq model in Part 1.

[9] Five continuous dye tracer releases (denoted R1, R2, R3, R4, and R6) were performed on different days [Clark et al., 2010]. Dye tracer was injected 0.5 m above the bed in roughly 1 m water depth (4–54 m from the shoreline), at rates between 1.3–7.1 mL s−1 (263–1489 ppb m3 s−1). The tracer was advected downstream with the mean alongshore current, forming shore parallel plumes, and measured near the surface for between 40 and 121 min (depending on the release) with a jet ski mounted fluorometer system [Clark et al., 2009].

[10] Visual observation indicated rapid vertical tracer mixing (tracer reaching the surface within several meters of the source), and patchy and highly variable tracer plumes. Dye was sampled on repeated cross-shore transects at 3–9 downstream locations, between 16 and 565 m from the tracer source. With increasing downstream distance y from the dye source, the mean cross-shore tracer profile D(x, y) peak concentrations decreased and cross-shore widths increased. The cross-shore profiles were often shoreline attached, roughly resembling a half-Gaussian, with a maxima near the shoreline. Bulk cross-shore surf zone diffusivities κxx were estimated from the downstream evolution of the plume squared cross-shore length scale, and varied between 0.5 and 2.5 m2 s−1 [Clark et al., 2010].

3. Surf Zone Tracer Modeling and Analysis

3.1. Tracer Model Description

[11] The 5 tracer releases analyzed by Clark et al. [2010] are simulated with a time-dependent wave-resolving Boussinesq model (funwaveC, Part 1). The model bathymetry is based on the observed alongshore-averaged survey bathymetry (Figure 1b). Waves matching the observed incident angle, directional spread, and energy spectrum are generated by the model wave maker, and propagate toward the shore where they “break” and dissipate (by the breaking eddy viscosity νbr). Model wave breaking drives alongshore currents and low-frequency (f < 0.03 Hz) surf zone eddies. The observed significant wave height Hs(x), mean alongshore current V(x), and bulk rotational infragravity (IG) velocities equation imagerot(IG)(x) are modeled with high skill. Bulk very low frequency (VLF) rotational velocities were overpredicted by about a factor of 2 (Part 1).

Figure 1.

(a) Plan view of a typical model domain (R4 example). The cross-shore distance from the “shoreline” is x, and Y is the alongshore coordinate. Gray regions indicate sponge layers and the wave maker. The cross-shore tracer domain (dashed lines) is bounded by the offshore wave maker and the onshore sponge layer. Stars indicate release locations for model tracers A (Yrl = 250 m), B (Yrl = 500 m), and C (Yrl = 750 m), and the arrow indicates the direction of the mean alongshore current V. (b) Typical model cross-shore bathymetry h versus x (R4 example), with a flat region at 7 m depth for the offshore sponge layer and wave maker and a 0.3 m depth flat region for the onshore sponge layer.

[12] A depth-averaged tracer module, coupled to the time-dependent Boussinesq model funwaveC, allows for three separate noninteracting tracers (denoted A, B, and C) released at different locations. Each tracer samples a different part of the flow field, increasing the degrees of freedom for quantities averaged over the statistics of all three tracers. Model tracer evolves according to an advection-diffusion equation,

equation image

where d is the tracer concentration (in ppb), h is the still water depth, η is the free surface elevation, equation imagebr is the breaking wave eddy diffusivity, κ0 is the background diffusivity, equation image is the two-dimensional horizontal gradient operator, and u is the model horizontal velocity vector, which for small kh is approximately the depth-averaged velocity. Tracer is injected into the model at (x = xrl, Y = Yrl) with the input flux M0 (δ is the Kronecker delta function).

[13] In (1), κbr is set equal to the breaking wave eddy viscosity νbr (e.g., momentum and tracer are assumed to mix identically), and the background diffusivity κ0 = 0.01 m2 s−1, is two orders of magnitude smaller than the observed bulk κxx. The κbr is nonzero only on the front face of a breaking wave (bore), whereas κ0 is applied everywhere. The inclusion of the breaking eddy viscosity allows the breaking wave mixing mechanism discussed by Feddersen [2007] to be examined relative to other tracer dispersion mechanisms.

[14] The vertically integrated Boussinesq and tracer models lack cross-shore dispersion by vertically sheared currents (i.e., undertow). However, this mechanism was not found to be significant in a natural surf zone with directionally spread waves [Clark et al., 2010], and rapid vertical mixing [Feddersen and Trowbridge, 2005; Ruessink, 2010; Feddersen, 2011] implies little vertical tracer structure within the surf zone. However, vertical structure may be important seaward of the surf zone [Kim and Lynett, 2010].

[15] The cross-shore tracer domain (dashed lines, Figure 1a) is embedded in the full Boussinesq model domain. The offshore tracer boundary (set to d = 0 ppb) is located just onshore of the wave maker region between x = 232 and 260 m from the shoreline, depending upon release. The onshore tracer boundary is typically located ≈5 m onshore of the start of the sponge layer (the depth of the flat region is h0 = 0.2–0.35 m), where a no-flux boundary condition is applied. In contrast to the η and u periodic alongshore boundary conditions, the tracer alongshore boundary conditions (at both ends of the 1500 m alongshore domain) are open, allowing tracer to advect out of the domain (Figure 1a). The alongshore tracer boundary conditions affect tracer concentrations within approximately 25 m of the boundary, and these regions are excluded from the analysis.

[16] The model spins up for 2000 s before starting continuous releases of tracers A, B, and C at alongshore locations Yrl = 250, 500, and 750 m, respectively, from the upstream boundary (Figure 1a). Model and observed cross-shore release locations xrl and tracer injection rates M0 are equal (Table 1). Model instantaneous tracer concentrations d(A,B,C), sea surface elevation η, cross-shore and alongshore currents (u and v), and breaking wave eddy diffusivity κbr are output every 2 s over the entire domain.

Table 1. Model Tracer Release Parameters: Input Tracer Flux M0 and Cross-Shore Release Location xrl
ReleaseM0 (ppb m3 s−1)xrl (m)
R1263−54
R2647−13
R31256−10
R41489−22
R6485−12

3.2. Model Tracer Analysis: Averaging

[17] The model tracer advects downstream with the mean alongshore current forming a shore-parallel plume that widens with downstream distance. Instantaneous d(A) model tracer plumes (Figures 2a, 2c, and 2e) are variable and patchy, with eddy-like tracer structure seaward of the surf zone (x < −100 m). The cross-shore structure of modeled low-frequency rotational motions (i.e., eddies) is discussed in Part 1.

Figure 2.

(a, c, and e) Instantaneous d(A) and (b, d, and f) mean D(A) (time average over 6000–14,000 s after each tracer release begins) modeled tracer A concentration as a function of x, the cross-shore distance from the “shoreline,” and y, the alongshore distance from the dye source, for R1 (Figures 2a and 2b), R4 (Figures 2c and 2d), and R6 (Figures 2e and 2f). In each panel the black star indicates the cross-shore release location (xrl, Table 1).

[18] The D(A)(x, y), D(B)(x, y), and D(C)(x, y) represent mean modeled tracers A, B, and C, time averaged in a fixed reference frame (x = 0 m at the shoreline and y = 0 m at the release location) between 6000 and 14,000 s after the tracer release started. Time averaging begins once the tracer plume has reached quasi-equilibrium (see Figure 3). The averaging times used for the observed means D(obs) are limited by instrument and environmental parameters to between 40 and 120 min. The model averages are over 133 min (8000 s) after tracer is equilibrated (Figure 3). Stability of the numerical results is further increased by averaging statistics over tracers A, B, and C. Averages over one 5600 s wave maker recurrence cycle (Part 1) are nearly identical to the 8000 s averages presented here, suggesting the wave maker recurrence does not effect the tracer results significantly. The observed D(obs) (this notation differs slightly from Clark et al. [2010]) and model D(A,B,C) mean plumes are time averaged in fixed coordinates (i.e., absolute averaged), which includes any plume meandering in the resulting (absolute) diffusivity estimates. Relative averaging (e.g., in center of mass coordinates [Csanady, 1973]), which separates plume meandering from smaller-scale mixing, is not used here because the interpretation of relative averages is unclear near the shoreline boundary [Clark et al., 2010].

Figure 3.

The R4 total tracer A volume T(A) versus time after the tracer release began, where T(A)(t) = ∫∫ (h + n)d(A)(t)dy dx) is integrated over the entire cross-shore tracer domain and from the upstream model boundary to 250 m downstream of the tracer source (where R4 diffusivities are estimated). For t > 3000 s the quasi-steady-state T(A) oscillates about a mean. The R4 T(A) is representative of other tracers and releases.

[19] Mean tracer D(A) plumes (Figures 2b, 2d, and 2f) are much smoother than the instantaneous tracer (Figures 2a, 2c, and 2e). The absolute concentration (in ppb) varies between model releases (relative shades of gray between panels in Figure 2), due to different tracer injection rates (Table 1), different V magnitudes (stronger V decreases tracer concentrations for a given injection rate), and varying amount of cross-shore dispersion.

4. Mean Cross-Shore Tracer Profiles and Alongshore Tracer Transport

4.1. Mean Cross-Shore Tracer Profiles

[20] Model D(A) and observed D(obs) mean tracer profiles at three representative downstream y are shown for all releases in Figure 4. D(A) and D(obs) profiles for R3, R4, and R6 are usually shoreline attached (maxima at or near the shoreline), with decreasing peak concentrations and increasing cross-shore widths with downstream distance y (Figures 4c4e). The mean tracer concentration skill for each transect is estimated by 1 − 〈(D(obs)(x, y) − D(A,B,C)(x, y))2x,y/〈D2(obs)(x, y)〉x,y, where 〈〉x,y is the mean over x and y, for regions where D(obs) > 5 ppb (thus avoiding relatively large instrument noise at low concentrations). Mean R3, R4 and R6 skills, averaged over all transects and the three model tracers in each release, are between 0.5 and 0.73 (Table 2), consistent with the qualitative agreement in Figures 4c4e.

Figure 4.

Modeled D(A) (solid) and observed D(obs) (dashed) mean tracer profiles versus x for (a) R1, (b) R2, (c) R3, (d) R4, and (e) R6, with alongshore distance y from the source indicted by the legend in each panel. Observed transects extend from seaward of the tracer plume to the inner transect edge xin.

Table 2. Mean Tracer Concentration Skilla
R1R2R3R4R6
  • a

    For each release, the mean tracer concentration skill 1 − 〈(D(obs)(x, y) − D(A,B,C) (x, y))2x,y/〈D2(obs)(x, y)〉x,y, averaged over all observed transects where D(obs) > 5 ppb and all three (A, B, and C) model tracers.

−2.70−8.890.700.500.73

[21] For release R1, the magnitudes and shapes of D(A) and D(obs) are roughly similar, and both model and observed mean tracer spread in the cross-shore with downstream distance (Figure 4a). However, at y = 56 and 107 m the D(obs) maximum is farther to the shoreline than for D(A) (Figure 4a), which may be explained by seaward advection of the observed plume [Clark et al., 2010]. This cross-shore displacement between D(A) and D(obs) maxima results in negative skill for R1 (Table 2), despite the similarity in shape.

[22] The R2 D(A) disperses similarly to D(obs), however the D(A) magnitudes are significantly larger than D(obs) (Figure 4b) which results in negative skill (Table 2). The differences in mean tracer magnitude are most pronounced near the shoreline where D(A) are often 2–5 times larger than D(obs) (Figure 4b).

4.2. Alongshore Tracer Transport

[23] Model M(A,B,C)(y) and observed M(obs)(y) alongshore tracer transports are [Clark et al., 2010]

equation image

where xin is the observed inner transect edge (i.e., where observations end near the shoreline, Figure 4), xF7 is the F7 location (the farthest seaward velocity observation), and V(x, y) is the mean alongshore current averaged over the same times as D(x, y). Note that the model M(A,B,C) uses alongshore varying V(x, y) while the observations assume alongshore uniform V(x) as measured on the cross-shore array. The model V(x, y) vary weakly alongshore (Part 1), and using alongshore averaged model V(x) does not change M(A,B,C) significantly. The xin range from −17 to −10 m, and xF7 range from −146 to −162 m from the shoreline. This M(y) estimate excludes the region shoreward of xin and seaward of xF7, and excludes alongshore eddy tracer fluxes by using time averaged V and D.

[24] The M(A,B,C) and M(obs) have roughly similar structure and decrease slightly at large y, except the overestimated M(A,B,C) in R3 (Figure 5). Estimates between individual M(A), M(B), and M(C), sometimes vary by 50%. The model tracer input flux is equal to the observed dye release flux M(obs)(y = 0) (circles in Figure 5), however the M(A,B,C) do not necessarily match at the source and are not conserved downstream. The difference between the observed input flux (y = 0, circles in Figure 5) and observed and modeled downstream transport M(obs)(y > 0) may be due to neglected alongshore eddy fluxes in (2) or to tracer transported onshore of xin (e.g., R3 D(A) at x > − 15 m in Figure 4c) or offshore of xF7. For the model, this is examined in section 6.2.

Figure 5.

Modeled M(A,B,C) (colored curved) and observed M(obs) (open black triangles with error bars) alongshore tracer transport (2) versus y, for releases (a) R1, (b) R2, (c) R3, (d) R4, and (e) R6. The observed dye release rate is estimated by the open black circle at y = 0.

5. Cross-Shore Integrated Tracer Moments and Bulk Surf Zone Diffusivity κxx

5.1. Definitions

[25] Observed and modeled cross-shore tracer plume structures are compared using cross-shore integrated surface tracer moments, which are consistent with a Fickian framework [Clark et al., 2010]. These moments are normalized by the total tracer (cross-shore D integral), and thus independent of absolute concentration. The surface center of mass μ is the D first moment [Clark et al., 2010]

equation image

where xout, the offshore extent of the observed transects, varied from −105 to −298 m over all transects. The jet ski always drove seaward until dye concentrations were not detectable. The model xout is taken at the seaward tracer boundary.

[26] Surf zone bulk cross-shore diffusivity κxx is estimated using the surf zone–specific squared cross-shore length scale σsurf2, a shoreline based second moment [Clark et al., 2010]

equation image

integrated from the seaward extent of the surf zone x = −Lx (at the location of maximum Hs) to x = xin. The Hs were modeled with high skill (Part 1), thus modeled and observed Lx are similar (12 m RMS difference). However, the Hs are observed at discrete locations (roughly 20 m apart) resulting in coarse Lx resolution. For comparisons, the observed Lx are used in (4) to estimate model and observed σsurf2. The shoreline based (i.e., without subtracting μ) moment σsurf2 is appropriate for estimating κxx near a boundary, assuming the alongshore plume axis is parallel to the shoreline, i.e., no large-scale cross-shore advection of the mean plume [Clark et al., 2010].

[27] For each release, a bulk equation imagexx is estimated from transects that are well contained in the surf zone, thus not effected by smaller diffusivities seaward of the surf zone. Transects are defined as well contained in the surf zone when equation image < 0.55, where equation image is the ratio of plume σsurf2 to the σsurf2 for a cross-shore uniform tracer concentration [Clark et al., 2010]. For each release, the bulk equation imagexx is

equation image

where equation imagexx and β are fit constants. The plume alongshore advection time

equation image

is the approximate plume age at a downstream location y, where the overbar represents a surf zone average (cross-shore average over the surf zone). The observed equation image are estimated using the cross-shore array of current meters [Clark et al., 2010], and the model equation image is averaged over the surf zone (−Lx < x < 0) and the alongshore region between the release location y = 0 and the farthest downstream location where κxx is estimated (equation image < 0.55). The Fickian solutions used to derive (5) assume constant depth, however numerical solutions to the depth varying case have similar surface tracer moments (i.e., (3) and (4)) and the resulting κxx are within 10% of the constant depth estimates [Clark et al., 2010]. The observed R1 plume differs from the other releases because the plume moved seaward and did not interact strongly with the shoreline (Figure 4a) [Clark et al., 2010]. Thus, the observed R1 κxx(obs) is estimated from the squared cross-shore length scale σ2, where the cross-shore advection is removed (for details see Clark et al. [2010]).

5.2. Surface Center of Mass μ

[28] For all releases, the observed μ(obs) and modeled μ(A,B,C) generally move seaward at an approximately constant rate with increasing downstream distance y, for y < 300 m (Figure 6). The downstream evolution of μ(obs) and μ(A,B,C) are similar for R2, R3, and R4 (Figures 6b6d). The R1 μ(obs) and μ(A,B,C) are similar at the closest transect to the source (Figure 6a), but μ(obs) magnitudes are slightly larger than μ(A,B,C) for the two farthest downstream transects, consistent with seaward advection of the observed R1 plume (Figure 4a), possibly by unresolved local bathymetric variation [Clark et al., 2010]. The R6 modeled μ(C) closely match the μ(obs), but μ(A) and μ(B) magnitudes are generally larger than the μ(obs), with more alongshore variation in μ(A) and μ(B) than μ(obs). The disparity corresponds with small patches of D(A) (x < − 88 m, Figure 2f) and D(B) seaward of the surf zone. In R4 and R6 where the plume was measured farther downstream (y > 300 m), the rate that μ moves away from the shoreline decreases (Figures 6d and 6e) presumably owing to weaker mixing seaward of the surf zone. The μ(A,B,C) skill, 1 − 〈(μ(obs)(y) − μ(A,B,C)(y))2y/〈μ2(obs)(y)〉y, is estimated for each tracer and release. The mean μ(A,B,C) skill over all releases and tracers is 0.88 indicating good model-data agreement.

Figure 6.

Modeled (colored) and observed (black triangles with error bars) surface center of mass μ versus y for releases (a) R1, (b) R2, (c) R3, (d) R4, and (e) R6. The mean model skill over all releases is 0.88.

5.3. Cross-Shore Dispersion and κxx

[29] The model σsurf2(A,B,C) and observed σsurf2(obs) plume squared cross-shore length scales (4) increase with increasing plume alongshore advection time tp(6), and are qualitatively well modeled for R2, R3, R4, and R6 (Figures 7b7e). The initial increase in σsurf2(A,B,C) and σsurf2(obs) is roughly linear in tp (Figure 7) consistent with Brownian diffusion regimes. The σsurf2(A,B,C) skill, 1 − 〈(σsurf2(obs)(tp) − image averaged over releases R2, R3, R4, and R6 is 0.92. For the purpose of comparison, the R1 σsurf2(obs)(tp) is compared with the modeled σsurf2(A,B,C)(tp) − 〈σsurf2(A,B,C)(tp = 0)〉A,B,C (Figure 7a). The R1 modeled and observed squared cross-shore length scales evolve similarly (Figure 7a), but this comparison is qualitative and skill is not estimated.

Figure 7.

Modeled (color curves) and observed (black or white squares with error bars) squared cross-shore length scale σsurf2 versus plume age tp for releases (b) R2, (c) R3, (d) R4, and (e) R6 and (a) σsurf2(A,B,C) (tp) − equation imageσsurf2(A,B,C)(tp = 0)〉A,B,C (modeled) and σ2 (observed) for release R1. Tracer profiles that are well contained in the surf zone, where κxx is fit, are indicated by black squares (observed) or the region below the dashed gray line (model) with equation image < 0.55. The σsurf2(obs) initial conditions (assuming a δ function at tp = 0) are indicated by the black stars. The mean σsurf2(tp) skill over releases R2, R3, R4, and R6 is 0.92.

[30] Mean modeled cross-shore surf zone diffusivities equation imageκxxA,B,C (averaged across tracers A, B, and C) are estimated by least squares fits (5) where the tracer plumes are surf zone contained (equation image < 0.55, below the dashed gray lines in Figure 7). The R1 κxx(obs) is a special case discussed by Clark et al. [2010]. Linear fits to σsurf2(A,B,C) versus tp have high r2 values, with a mean r2 = 0.87. The 〈κxxA,B,C errors are derived in the same manner as the observations [Clark et al., 2010], and include uncertainties in σsurf2(A,B,C) and variations between σsurf2(A,B,C) best fit slopes [Wunsch, 1996]. The 〈κxxA,B,C range from 0.67 to 2.83 m2 s−1 (Table 3).

Table 3. Mean Model 〈κxxA,B,C Derived From σsurf2 Versus tp (Figure 8)
Release〈κxxA,B,C
R10.73 ± 0.29
R21.02 ± 0.17
R31.49 ± 0.30
R42.83 ± 0.76
R60.67 ± 0.07

[31] Model 〈κxxA,B,C and observed κxx(obs) are similar (Figure 8), with correlation r2 = 0.72. The skill, 1 − 〈(κxx(obs) − 〈κxxA,B,C)2R1−R6 〈(κxx(obs))2R1−R6, is 0.40. Model and observed cross-shore dispersion κxx are qualitatively similar for the given bathymetries and incident wave fields.

Figure 8.

Mean modeled 〈κxxA,B,C versus observed κxx(obs), with a dashed line indicating perfect agreement. The κxx(obs) and 〈κxxA,B,C error bars are estimated from the σsurf2 versus tp fit slope errors as detailed by Clark et al. [2010]. The skill is 0.40.

6. Discussion

6.1. Model-Data Comparison

6.1.1. Mean Plume Concentration D and Alongshore Transport M

[32] The magnitude and cross-shore structure of tracer concentration D(A,B,C) is more difficult to model than cross-shore integrated, normalized moments (μ and σsurf2), because D(A,B,C) depends on the details of V(x), eddy stirring, and the input tracer flux. Model and observed D(A) and D(obs) are similar with good skill (0.5–0.73) for releases R3, R4, and R6 (Figures 4c4e), where the waves, V(x), and eddy velocities were also well modeled (Part 1). However, other releases have significant deviations in plume location (R1, Figure 4a) or tracer magnitude (R2, Figure 4b) leading to low (negative) D(A) skill. The difference in R1 cross-shore plume location likely results from cross-shore advection of the mean D(obs) plume (i.e., the along-plume axis is not parallel to shore). The R2 D(A) are reasonably matched in the outer surf zone (x < − 60 m, Figure 4b), but D(A) magnitudes are often 2–5 times greater than D(obs) near the shoreline, contrasting with the good agreement between R2 M(A,B,C) and M(obs) tracer transports (Figure 5b). Near the shoreline, the R2 model V ≈ 0.05 m s−1 substantially underpredicts the observed V ≈ 0.3 m s−1 (Part 1, Figure 6), but combined with the overpredicted D(A) (Figure 4b), results in good M model-data agreement (Figure 5b).

6.1.2. Cross-Shore Moments μ and σsurf2 and Diffusivity κxx

[33] Although D(A,B,C) skill is variable and sometime negative, the normalized cross-shore integrated moments μ(A,B,C)(3) and σsurf2(A,B,C)(4), representing cross-shore plume structure, have high mean skill (0.88 and 0.92, respectively). For example, despite low D(A,B,C) skill, R2 has high μ(A,B,C) skill (Figure 6b) and the best agreement (highest σsurf2(A,B,C) skill) with the observed cross-shore dispersion (Figure 7b). This is in part due to scaling the model and observations with tp which reduces the sensitivity to R2 V(x) model errors. Thus cross-shore diffusivities κxx may still be accurately modeled when V(x) and D(A,B,C) are not.

[34] Drifter observations indicate ballistic dispersion (σ2t2) for times ≲ 50 s and Brownian dispersion at longer times [Spydell et al., 2009]. Tracer observations were generally at downstream distances corresponding to tp > 100 s [Clark et al., 2010], and the observed and the modeled σsurf2 are consistent with Brownian diffusion (σ2t, Figure 7). At short times (where there are no observations), a ballistic dispersion regime is not apparent in the model σsurf2(A,B,C), potentially because of a mix of background diffusivity, bore mixing, and eddy stirring which all have different time scales.

[35] Model 〈κxxA,B,C and observed κxx(obs) are similar with correlation r2 = 0.72 and moderate 0.40 skill (Figure 8). Thus, given only the bathymetry and incident wave field, the coupled Boussinesq-tracer model qualitatively reproduces the observed cross-shore absolute tracer dispersion and suggests that the model can be used to study the mechanisms of surf zone tracer dispersion.

6.1.3. R6 Dispersion, Seaward of the Surf Zone

[36] The R6 μ(C) closely matches μ(obs), but μ(A) and μ(B) are farther seaward, resulting in the lowest μ(A,B,C) mean skill (0.68) of all releases. Despite the highest D(A,B,C) skill (Table 2), low concentrations of D(A) and D(B) extend much farther seaward than the D(obs) (see Figure 4e for D(A)), thus increasing μ(A) and μ(B) magnitudes. This may indicate model mixing rates seaward of the surf zone are larger, or have different structure, than observed. Seaward of the surf zone, vertical tracer structure, not accounted for here, may also become important [e.g., Kim and Lynett, 2010].

6.1.4. Potential Sources of Error

[37] The variation between modeled individual tracer (A,B,C) statistics (e.g., σsurf2(A,B,C), Figure 7) is due to two factors. First, small alongshore variations in the surf zone eddy field (e.g., Part 1, Figures 14 and 15) result in each tracer experiencing slightly different stirring statistics. Second, the velocity field stirring the tracer has a red spectrum (Part 1, Figure 13) that is intrinsic (as in 2D turbulence) and does not depend on the model wave maker. Thus, averaging for 8000 s may not be sufficient to completely converge tracer statistics from the three release locations. Significantly longer model simulations allow for more lower-frequency energy, possibly negating any reduction in uncertainty provided by longer averages. Statistical stability in 〈κxxA,B,C is increased by averaging over the 3 alongshore separated tracers.

[38] The observed κxx(obs) estimates assume σsurf2(obs)(tp = 0) = xrl2 (tracer δ function at the release location). However, the model σsurf2(A,B,C)(tp = 0) for R2, R3, and R4 are larger (by 153–724 m2) than the assumed xrl2 value for the observations (Figures 7b7d). The elevated model σsurf2(A,B,C) relative to xrl2 is due to intermittent model tracer recirculation upstream of the tracer source (e.g., Figure 2a), and consistent with visual observations. Additional model experiments with nonbreaking waves on a steady current demonstrate that cross-shore tracer dispersion due to orbital wave motions is weak and contributes only a small fraction of the model σsurf2(A,B,C) near y = 0 m. The assumed σsurf2(obs)(tp = 0) = xrl2 likely underestimates the actual value, and the observed fit slopes (5) and κxx(obs) may be slightly overestimated.

6.2. Alongshore Tracer Transport: Eddy Fluxes and Cross-Shore Integration Limits

[39] For equilibrated conditions (t > 6000 s, Figure 3) and conserved tracer, the time averaged alongshore tracer transport is expected to be constant downstream of the source. However, the model and observed M(2) are not conserved, do not match the input flux, and vary downstream by up to 50% (Figure 5). Model and observed tracer transports M are both estimated using time-averaged D and V(2), excluding alongshore eddy fluxes, and neglect the regions onshore of xin and offshore of xF7. The time-averaged total alongshore tracer transport equation image(y) is estimated with

equation image

where v and d are the instantaneous model alongshore velocity and tracer concentration, respectively, and the time-averaged equation image[h + η]vdequation imaget includes both mean and eddy alongshore tracer fluxes. The xin < x < xF7 integral limits are used for comparison with M(A,B,C)(2). The R4 ℳ(A), representative of other tracers and releases, matches the input flux at y = 0 and varies less downstream than M(A) (Figure 9). The ℳ(A) and M(A) have roughly similar magnitudes, indicating small alongshore eddy fluxes, consistent with the assumptions used to derive κxx(5).

Figure 9.

Alongshore tracer transport estimates M(A)(2), equation image(A)(7), and equation imagedomain(A) (see legend) versus y for R4. Note that equation imagedomain(A) is defined similar to equation image(A) but is integrated over the entire cross-shore tracer domain. Mean and eddy fluxes are included in both equation image(A) and equation imagedomain(A). The observed dye release rate is given by the open black circle at y = 0 m.

[40] A domain integrated total transport estimate equation imagedomain(A) is defined similarly to (7) but integrated over the entire cross-shore tracer domain (Figure 1). The equation imagedomain(A) decreases less downstream than M(A) and equation image(A) (Figure 9). The equation image(A) are initially (y < 100 m) smaller than equation imagedomain(A) because equation image(A) excludes tracer shoreward of xin. The equation image(A) are also smaller farther downstream (y > 200 m) because tracer transport seaward of xF7 is excluded. The downstream decrease in equation imagedomain(A) is due to tracer losses at the offshore boundary, indicating that a larger cross-shore domain, in addition to incorporating the effects of vertical variation of tracer and currents and stratification, are needed to study tracer evolution seaward of the surf zone.

6.3. Simple Fickian Equation Comparison: Tracer Maxima

[41] Cross-shore diffusivity κxx is estimated here and by Clark et al. [2010] using a simple Fickian solution, where the tracer cross-shore maxima decrease downstream as Dmaxtp−1/2 [Clark et al., 2010]. The individual Dmax(A,B,C) model tracers are similar, and the mean [〈DA,B,C]max over tracers A, B, and C is compared with the expected tp−1/2 dependence over the surf zone contained region where κxx is estimated.

[42] The R1, R4, and R6 [〈DA,B,C]max decrease similarly to tp−1/2 (Figure 10). The R2 and R3 [〈DA,B,C]max initially (tp < 200 s) decrease similarly to tp−1/2, but decrease more rapidly with tp > 200 s (Figure 10), possibly because tracer is leaking into deeper water seaward of the surf zone. Linear regressions of the form [〈DA,B,C]max = Atγ, with A and γ fit constants, yield γ slightly greater than 0.5. The similarity between [〈DA,B,C]max and tp−1/2 indicates that (5) is appropriate for estimating κxx, and that the modeled absolute diffusion is generally well represented by a simple Fickian equation, when the tracer is well contained within the surf zone. Diffusivities estimated from [〈DA,B,C]max versus tp (not shown) are similar to those estimated from σsurf2(A,B,C), but are much noisier and include uncertainties in the absolute tracer concentration.

Figure 10.

Mean (over tracers A, B, and C) cross-shore tracer maxima [〈DA,B,C]max versus plume age tp, for the downstream region where tracer is well contained within the surf zone (i.e., the region where κxx is estimated). The tp−1/2 slope based on Fickian diffusion is indicated by the dashed black line.

6.4. Tracer Dispersion Induced by Breaking Wave κbr

[43] For time-averaged breaking wave (bore) induced diffusion, scalings similar to κxxHs2Tm−1, were suggested by several previous studies [Harris et al., 1963; Inman et al., 1971; Clarke et al., 2007; Feddersen, 2007; Henderson, 2007], but had lower skill (0.32) than alternate scalings and best fit slope smaller than expected when applied to the HB06 observed κxx [Clark et al., 2010]. The relative importance of simulated bore diffusion is investigated for the modeled HB06 tracer plumes.

[44] Tracer mixing by breaking waves is modeled with a breaking eddy diffusivity κbr (set equal to the local breaking eddy viscosity νbr), which propagates with the front face of a breaking wave (bore) [Feddersen, 2007]. In the absence of other dispersion mechanisms, a tracer patch that is much wider than the cross-shore width of a bore (approximately the water depth) has a bulk cross-shore diffusivity given by the time-averaged breaking diffusivity 〈κbrt [Henderson, 2007].

[45] The R4 〈κbrt increases from zero, far seaward of the surf zone, to a maxima near the outer surf zone (x ≈ − 100 m), and then decreases toward the shoreline (Figure 11). Although R4 has the largest breaking diffusivities of all releases, the maximum 〈κbrt = 0.06 m2 s−1 is much smaller than the O(1) estimates for 〈κxxA,B,C (Figure 8 and Table 3), suggesting the effect of bore mixing [Feddersen, 2007] on absolute averaged tracer properties is weak.

Figure 11.

Time-averaged breaking wave diffusivity 〈κbrt (solid) and background diffusivity κ0 = 0.01 m2 s−1 (dashed) versus x for R4.

[46] The weak effect of breaking wave induced κbr on cross-shore absolute dispersion is demonstrated by an additional R4 simulation with two tracers released at the same location, one with breaking and background diffusivities κbr + κ0 and another with only background κ0 applied to the tracer field. The R4 model σsurf2 with and without κbr are almost identical (Figure 12, other releases are similar), demonstrating that model bore-induced mixing is insignificant to bulk surf zone cross-shore dispersion for the obliquely incident, directionally spread wave conditions modeled here.

Figure 12.

Model σsurf2 versus y for two R4 tracers with identical release location: one with full breaking-induced diffusivity κbr + κ0 (black) and another with background diffusivity κ0 only (gray).

6.5. Model Mixing Length κxx Scalings

[47] A mixing length scaling for the cross-shore diffusivity

equation image

was compared with observed diffusivities, where equation imagerot(IG) is a surf zone–averaged infragravity horizontal rotational velocity (estimated following Lippmann et al. [1999] and discussed in Part 1) and Lx is the surf zone width [Clark et al., 2010]. This scaling (8) was correlated (r2 = 0.59) with observed equation imagexx(obs), and suggested that stirring by infragravity (IG, 0.004 < f < 0.03 Hz) eddies (vortical motions) was a significant cross-shore tracer dispersion mechanism. In addition, note that although equation imagexx(obs) was also correlated with equation imageLx, equation image appeared in the formulation for κxx(obs) and the correlation could be artificially high. The observed IG band rotational velocities &#55349;&#56497;rot(IG)(x) were well reproduced by the model (see Part 1, Figure 14). Here, the mixing length scaling (8) is investigated for the modeled dispersion.

[48] The model equation imagerot(IG) are estimated by cross-shore averaging the model equation imagerot(IG)(x) over the surf zone (−Lx < x < 0). Over the five releases, the model 〈equation imagebrA,B,C and equation imagerot(IG)Lx are related (Figure 13a), with a best fit slope of 0.1. This slope is near the observed best fit slope of 0.2 suggesting that this scaling is also applicable in the model. However, the squared correlation r2 = 0.29 is lower than observed (r2 = 0.59). The scaling is not expected to represent all the dispersive processes in the surf zone, and the diffusivity is expected to be nonzero (and positive) when equation imagerot(IG)Lx = 0. The positive best fit y intercept (Figure 13a) is roughly consistent with this expectation.

Figure 13.

Model equation imageequation imagebrequation imageA,B,C versus (a) equation imagerot(IG)Lx, (b) equation imagerot(IG+VLF)Lx, and (c) equation imageψ(RMS)Lx scalings. The dashed gray line indicates linear fits to each scaling, and r2 correlations are 0.29 (Figure 13a), 0.60 (Figure 13b), and 0.63 (Figure 13c).

[49] Observed and modeled very low frequency (VLF, f < 0.004 Hz) contributions to the bulk rotational velocity estimate (Part 1, Figure 15), not considered by Clark et al. [2010], have similar magnitudes to those in the IG band (Part 1, Figure 14), and may represent a significant contribution to cross-shore mixing. A mixing length scaling equation imagexxequation imagerot(IG+VLF)Lx similar to (8), using a surf zone averaged horizontal rotational velocity equation imagerot(IG+VLF) integrated over both IG and VLF frequency bands (0.001 < f < 0.03 Hz), is tested for the model. The equation imagerot(IG+VLF)Lx scaling has a best fit slope of 0.06, and a higher correlation (r2 = 0.60) with 〈κbrA,B,C (compare Figure 13b with Figure 13a), indicating that VLF motions are likely important to model cross-shore tracer dispersion in the surf zone. The best fit y intercept is near zero, similar to equation imagerot(IG)Lx. The model rotational motions in the VLF frequency band are roughly twice the observed velocities, thus VLF motions may be more important to tracer dispersion in the model than in the field.

[50] Estimates of rotational velocities using a colocated pressure and velocity measurement [Lippmann et al., 1999] are useful for field applications, but involve assumptions about the low-frequency wave field. More accurate and complete rotational velocities are estimated by decomposing the model instantaneous velocity field into rotational uψ and irrotational uϕ components [e.g., Spydell and Feddersen 2009; Part 1], where model vorticity comes entirely from the rotational uψ. The majority of contributions to uψ are in the IG and VLF frequency bands (Part 1, Figure 13).

[51] While equation imagerot(IG+VLF) combines both cross-shore and alongshore rotational motions, only the cross-shore component of the rotational velocity field (i.e., uψ) is expected to mediate cross-shore dispersion. A bulk cross-shore rotational velocity equation imageψ(RMS), estimated from the surf zone averaged RMS model cross-shore rotational velocities uψ, is applied to the scaling, i.e., equation imagexxequation imageequation imageψ(RMS)Lx. This scaling has a squared correlation r2 = 0.63 (Figure 13c), similar to the equation imagerot(IG+VLF) squared correlation, and a best fit slope of 0.1. The best fit y intercept is negative, but close to zero. This scaling again suggests that VLF motions are an important factor in model dispersion.

[52] Unlike the diffusivity scalings and simple Fickian solutions (best fits to κxx and σsurf2, respectively) the Boussinesq model is not tuned to match tracer statistics. Given the similarity between model and observed tracer dispersion, the model can give insight into tracer dispersion mechanisms and improve the skill and reliability (over a range of beach and wave conditions) of diffusivity scalings. Improved scalings may provide the rapid (albeit approximate) estimates needed to predict pollutant dispersal in an emergency.

7. Summary

[53] A time-dependent wave-resolving Boussinesq surf zone model funwaveC, coupled with a tracer advection diffusion equation, is used to simulate 5 tracer releases from the HB06 experiment. The model, using the observed bathymetry and incident wave spectra, reproduces the cross-shore evolution of significant wave height, mean alongshore currents, and low-frequency rotational motions, i.e., eddies (Part 1). Model tracer is transported by currents, stirred by eddies, and mixed with a breaking wave eddy diffusivity κbr, and a small (0.01 m2 s−1) background diffusivity. Three noninteracting model tracers were released 250 m apart in the alongshore at the rates and cross-shore release locations of the observations.

[54] Similar to the observations, the continuously released model tracers form alongshore parallel plumes in the wave-driven alongshore current, with decreasing peak concentrations and increasing cross-shore widths with downstream distance from the source. Modeled D(A,B,C) and observed D(obs) mean tracer profiles are often shoreline attached (near-shoreline maxima). Three releases (R3, R4, and R6) have high D skill (0.5–0.73) with well matched plumes. Two releases (R1, R2) have negative skill, associated with a mismatch in plume cross-shore location (R1), or differences in the modeled and observed mean alongshore current near the shoreline (R2).

[55] The modeled alongshore tracer transport M agrees with the data for most releases, but is overestimated for R3. Small tracer losses at the seaward model boundary do not effect surf zone dispersion results, but indicate a much larger cross-shore domain would be required to examine processes seaward of the surf zone. Alongshore tracer eddy fluxes are small, and in agreement with neglecting alongshore tracer dispersion in cross-shore diffusivity estimates.

[56] The observed and modeled cross-shore integrated moments, normalized to remove the dependence on absolute concentration, agree well for all releases. The model D(A,B,C) surface centers of mass μ(A,B,C) move seaward with downstream distance, and agree well with observations (0.88 skill over all releases). The plume squared cross-shore length scale σsurf2 (second moment) is used to estimate bulk cross-shore diffusivity κxx. The downstream evolution of model and observed σsurf2 is similar, with high skill (0.92).

[57] Mean model equation imageκxxA,B,C are similar to observed κxx(obs), with good correlation (r2 = 0.72) and skill of 0.40. Observed κxx(obs) were correlated with a mixing length scaling based on bulk infragravity (IG) cross-shore rotational velocities equation imagerot(IG), however modeled 〈κxxA,B,C have lower correlation (r2 = 0.29) with this scaling. Alternative mixing length scalings including both IG and very low frequency (VLF, f < 0.004 Hz) rotational motions, have higher r2 = 0.60–0.63 correlations with 〈κbrA,B,C. The mean model wave-breaking eddy diffusivity is small and does not effect the bulk dispersion significantly.

[58] The good overall agreement between model and observed tracer plume properties indicates that, given the bathymetry and incident wave field, coupled time-dependent Boussinesq and tracer models can be used to predict surf zone mean tracer evolution and are appropriate for studying the mechanisms of surf zone tracer dispersion.

Acknowledgments

[59] This research was supported by SCCOOS, 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. Okihiro, M. Omand, M. Yates, M. McKenna, M. Rippy, S. Henderson, and D. Michrokowski) were instrumental in acquiring the field observations. We thank these people and organizations.