Corresponding author: F. Feddersen, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0209 USA. (firstname.lastname@example.org)
 Wave breaking across the surf zone elevates the mean water level at the shoreline (setup), and drives fluctuations about the mean (runup). Runup often is divided into sea-swell (0.04–0.3 Hz) and lower frequency infragravity (0.00–0.04 Hz) components. With energetic incident waves, runup is dominated by infragravity frequencies, and total water levels (combined setup and runup) can exceed 3 m, significantly contributing to coastal flooding and erosion. Setup and runup observations on sandy beaches are scattered about empirical parameterizations based on near-shoreline beach slope and deep water wave height and wavelength. Accurate parameterizations are needed to determine flooding and erosion risk to coastal ecosystems and communities. Here, numerical simulations with the Boussinesq wave model funwaveC are shown to statistically reproduce typical empirical setup and runup parameterizations. Furthermore, the model infragravity runupRs(ig) strongly depends on the incident wave directional and frequency spread (about the mean direction and peak frequency). Realistic directional spread variations change Rs(ig) equivalent to a factor of two variation in incident wave height. The modeled Rs(ig)is shown to vary systematically with a new, non-dimensional spreading parameter that involves peak frequency, frequency spread, and directional spread. This suggests a new parameterization forRs(ig) potentially useful to predict coastal flooding and erosion.
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 Waves incident to a beach elevate the shoreline mean water level (setup, ) and drive fluctuations about the mean (runup). Energetic waves can elevate total water level (combined setup and runup) by as much as 3 m [Stockdon et al., 2006]. Empirical parameterizations of wave-induced setup (super-elevation of the mean shoreline location) and runup (fluctuations of the waterline about the mean) are often used to predict coastal flooding and erosion [Ruggiero et al., 2001; Anselme et al., 2011; Revell et al., 2011]. Runup often is divided into sea-swell (0.04–0.3 Hz,Rs(ss)) and infragravity (0.004–0.04 Hz, Rs(ig)) frequency bands, with significant runup elevation Rs defined as 4σ, where σ2 is the vertical runup variance in that band. During energetic wave events, Rs(ig) dominates runup on dissipative beaches [Guza and Thornton, 1982]. Thus accurately parameterizing Rs(ig)is critical to predicting wave-driven coastal flooding and its impacts on coastal ecosystems and communities.
 Many empirical parameterizations relate , Rs(ss), or Rs(ig) to incident wave conditions and beach slope β [Holman, 1986; Ruessink et al., 1998; Stockdon et al., 2006; Senechal et al., 2011; and others], typically proportional to (Hs,0L0)1/2 or β(Hs,0L0)1/2 where L0 is the incident deep water wavelength based on peak (or mean) frequency fp (L0 = (2πg)fp−2), and Hs,0is the deep-water significant wave height (i.e., unshoaled to deep water on plane parallel bathymetry). On natural, non-planar beaches,β is often approximated as the linear beach slope near the waterline [e.g., Stockdon et al., 2006]. Although empirical parameterization for Rs(ig) using (Hs,0L0)1/2 have significant skill (r2 ≈ 0.65 for Rs(ig) ∝ (Hs,0L0)1/2) [e.g., Stockdon et al., 2006], scatter can be significant. The incident sea-swell directional spectrum, which nonlinearly forces infragravity waves in shallow water, depends not only onHs,0 and fp, but also has frequency spread (fs) about fp and directional spread (σθ) [Kuik et al., 1988] about the mean angle (assumed zero here), The effect of variable fd and σθ on infragravity runup Rs(ig) is explored here using the Boussinesq wave model funwaveC on a planar beach.
 The model (section 2) is shown to reproduce the dependence on (Hs,0L0)1/2 and β(Hs,0L0)1/2 (section 3.1) of existing setup and runup parameterizations, suggesting that it can be used to model runup quantitatively. The scatter about the Rs(ig)/(Hs,0L0)1/2 parameterization depends on fs and σθ (section 3.2). A simple non-dimensional parameter (fp / fs)σθ,0 (where σθ,0is the deep-waterσθ), based on a nonlinear infragravity wave coupling coefficient, collapses the scatter, suggesting a new Rs(ig) parameterization (section 4).
2. Model Description
 The funwaveC model used here, solving the relatively simple [Nwogu, 1993] equations, reproduces field observations of surfzone waves and currents [Feddersen et al., 2011] and of tracer dispersion driven by low frequency vortical motions [Spydell and Feddersen, 2009; Clark et al., 2011]. Cross- and alongshore grid sizes are 1 m and 1.25 m, respectively. The model bottom stress is quadratic in velocity and for simplicity a spatially uniform drag coefficientcd = 0.002 [Feddersen et al., 2011] is used. Increased cd in the swash zone [Puleo and Holland, 2001], which potentially affects runup, is not included. An eddy viscosity wave breaking method [Lynett, 2006] is used with parameter values similar to previous studies [Feddersen et al., 2011]. Runup is implemented using the “thin-layer” method [Salmon, 2002] that adds an extra pressure to the equations to keep a minimum fluid thickness d0 on a sloping shoreline. The d0 depend on the grid spacing and beach slope, and ranged between 3–7 cm. Although momentum is not strictly conserved, this method is simple and reproduces analytic solutions [Carrier and Greenspan, 1958] for a nonlinear standing wave on a beach [Salmon, 2002].
 The model bathymetry (Figure 1) has an offshore region of constant depth (h0= 9.5 m) adjacent to a planar slope region further onshore. The constant depth region length (230–260 m width) contains the wavemaker and an offshore sponge layer (70–100 m width) that absorbs seaward propagating waves. The total cross-shore domain varies between 563–808 m. The alongshore domainLy, between 1.15–2.25 km, is chosen to allow non-zero incident wave angles as small as 3° (at depthh0and peak-frequencyfp) to satisfy the alongshore periodic boundary condition. Rs(ig) is only weakly sensitive to Ly (see Appendix A).
 A wavemaker [Wei et al., 1999], located immediately onshore of the offshore sponge layer, generates approximately the target spectrum
over the frequency range fp ± fs, where fs is the frequency spread and fp is the peak frequency. At fp, kh < 1, within the valid range of Nwogu . The symmetric, normally incident directional spectrum S(θ) has a Gaussian form with width specified by the directional spread σθ [Feddersen et al., 2011].
 A total of 180 model simulations were performed with independently-varied beach slopeβ (between 0.02–0.04), incident Hs (between 0.4–2.5 m), peak frequency fp (between 0.06–0.14 Hz), frequency spreads fs (between 0.0025–0.02 Hz), and target directional spread σθ(between 5–30°). The wave parameters are not independent in naturally occurring waves; low frequency swell is often narrow in frequency and direction, whereas high frequency seas typically have broad spreads. The wave parameter co-variation depends on location and event. The wavemaker only approximately reproduces the targetE(f, θ), and the four incident wave parameters (Hs, fp, fs, and σθ) are estimated on the flat depth region onshore of the wavemaker with model output. The deep water wave height Hs,0 and wavelength L0 are calculated from Hs and fp. The range of fp, β and Hs,0 correspond to Irrabaren numbers ζ = β(L0/Hs, 0)1/2 generally ζ < 0.4, indicating dissipative conditions [Stockdon et al., 2006]. Simulation were sampled for 2400 s after 200 s of model spinup. Alongshore variations in runup statistics were weak.
 The model runup toe location R(t), defined as the most shoreward location where fluid thickness >4d0, varied between 9–21 cm above the minimum fluid thickness d0. If R(t) is too small (relative to d0), the runup is distorted by the thin film pressure head [Salmon, 2002]. Field observed runup statistics depend on the minimum water elevation chosen for the runup toe. Differences in significant runup, between 5 cm and 15 cm elevations, are between 30% and 15% in the sea-swell and infragravity frequency bands, respectively, on a moderately sloped beach with low energy swell waves [Raubenheimer and Guza, 1996]. Empirical runup parameterizations were derived primarily with video observations, which corresponds most closely to a 5 cm minimum elevation [Holland et al., 1995]. The effects of different runup toe definitions over a range of wave conditions and beach slopes are unknown, and are neglected here. Model results are not sensitive to variations in toe thickness between 3d0–5d0.
 Setup is defined as the time- and alongshore average ofR.The significant sea-swellRs(ss) and infragravity Rs(ig)runup elevations are based on the alongshore-averaged runup spectrum integrated over the sea-swell (0.04–0.3 Hz) and infragravity (0.004–0.04 Hz) frequency bands. The upper limit of the infragravity frequency band is typically between 0.04 Hz [e.g.,Herbers et al., 1995] and 0.05 Hz [e.g., Stockdon et al., 2006]. Here, 0.04 Hz is used to avoid leaking incident (lowest peak frequency fp = 0.06 Hz) energy into the infragravity band. In analysis of field observations [e.g., Senechal et al., 2011], a lower infragravity-band limit of 0.004 Hz is used to exclude low-frequency motions (e.g., tides), and this limit is used here for consistency. Model results are insensitive to this limit. Similar to field observations [Holman, 1986; Stockdon et al., 2006], simulated two-percent runup-exceedences follows where Rsincludes both sea-swell and infragravity contributions.
3.1. Model Comparison With Existing Runup Parameterizations
 Model setup and runup statistics , Rs(ss), and Rs(ig) fit to parameterizations based on (Hs,0L0)1/2 and β(Hs,0L0)1/2yield best-fit slopes and squared correlationsr2 (Table 1 and Figure 2) similar to field based results [Ruessink et al., 1998; Stockdon et al., 2006; Senechal et al., 2011]. As with field observations [Stockdon et al., 2006], the setup and the sea-swell band significant runupRs(ss) had a higher r2 with β(Hs,0L0)1/2 than with (Hs,0L0)1/2. Although natural beach profiles are not well represented by a single β, the model best-fit regression-slopes usingβ(Hs,0L0)1/2 are consistent with those found on natural beaches [Stockdon et al., 2006] (Figures 2a and 2b).
Table 1. Regression Statistics Between Runup Quantities and Non-dimensional Parameters (Hs,0L0)1/2 and β(Hs,0L0)1/2a
rms Error (m)
rms Error (m)
The slope and root-mean-square error (rms error) are for a fit forced to go through the origin. The squared correlation isr2. Narrow frequency spreads fs = 0.0025 Hz, that rarely occur in Southern California, are not includedin Rs(ig) statistics.
 Consistent with prior field observations [Stockdon et al., 2006; Senechal et al., 2011], infragravity-band runupRs(ig) skill using (Hs,0L0)1/2 is higher than with β(Hs,0L0)1/2 (r2 = 0.67 and 0.60, respectively, for cases with fs > 0.0025 Hz, Table 1 and Figure 2c). The smallest fs = 0.0025 Hz, which rarely occur in Southern California, are excluded from this comparison (Figure 2c) to video-based runup parameterizations. The best-fit slope of 0.041 betweenRs(ig) and (Hs,0L0)1/2 (solid black line in Figure 2c) is comparable to the Stockdon et al.  slope of 0.06 (dashed line in Figure 2c) and the Senechal et al. slope of 0.05 (not shown). The model reproduces the field-based parameterizations(1) of , Rs(ss) and Rs(ig), supporting the use of model simulations to explore the neglected effects of frequency (fs) and directional (σθ) spread.
3.2. Frequency and Directional Spread Dependence of Rs(ig)
 The contributions of fs and σθ to the scatter of Rs(ig) about the parameterizations (Figure 2c) is now explored. At fixed β, fp and Hs,0 ≈ 1.25 m, the normalized infragravity runup Rs(ig)/(Hs,0L0)1/2 increases with increasing frequency spread fs and decreases with increasing directional spread σθ (Figure 3a). For Southern California, fs = 0.0025 Hz rarely occurs, and associated Rs(ig)/(Hs,0L0)1/2 ≈ 0.02 are smaller than typically observed. As fs increases to (more typical for Southern California) 0.01–0.02 Hz, Rs(ig)/(Hs,0L0)1/2 ≈ 0.05 and the sensitivity of Rs(ig)/(Hs,0L0)1/2 to fs decreases. With fs ≥ 0.01 Hz, the ratio Rs(ig)/(Hs,0L0)1/2 varies by a factor of 1.5 with realistic σθ variation from 5° to 30° (Figure 3b). With constant σθ, fs ≥ 0.01 Hz, and the limited range of moderate β considered (0.02–0.04), the normalized infragravity runup Rs(ig)/(Hs,0L0)1/2 does not obviously depend on β (lack of color banding in Figure 3b. Only Rs(ig)/(Hs,0L0)1/2 depends strongly on spread, as the normalized ratio and have no trend with fs and σθ (not shown).
4. Discussion: Parameterizing Rs(ig)/(Hs,0L0)1/2
 The separate fs and σθ dependence of Rs(ig)/(Hs,0L0)1/2 (Figure 3) is shown to collapse with a single non-dimensional variable. In shallow and constant depth, two approximately co-linear incident waves with slightly different frequencies Δfare in near-resonance with the infragravity wave of frequency Δf, resulting in infragravity wave growth [Herbers and Burton, 1997]. In intermediate and deep water, this nonlinear interaction forces a small, second order bound infragravity wave at Δf [Longuet-Higgins and Stewart, 1962]. The bound wave solution is singular at the shoreline. However, in the limit of small but finite depth, small beach slope, and weak nonlinearity, the steady near-resonant and bound infragravity wave solutions are equal [Herbers and Burton, 1997]. Although infragravity-band runup may be dominated by resonantly forced waves, this equality motivates use of the bound-wave formalism to guide parameterizing the dependence ofRs(ig)/(Hs,0L0)1/2 on fs and σθ.
 The bound total infragravity energy EIG in shallow depth his elated to the (linearly unshoaled on plane parallel contours) deep-water sea-swell frequency directional spectrumE(f, θ) via [Herbers et al., 1995]
where (fmin, fmax) and (Δfmin, Δfmax) are the frequency ranges of the swell and infragravity waves, respectively), and C2is a coupling coefficient. The bound-wave expression(2), used to model the observed directional properties of free infragravity waves [Herbers et al., 1995], was later shown to be related to resonant free infragravity waves [Herbers and Burton, 1997]. C2is maximum for the special, well-studied case of a wave-flume; normal wave incidence with zero directional spread. In this caseC2 depends weakly on Δf (where Δf = f2 − f1) [e.g., Sand, 1982]. However, for directionally spread waves, C2 is sensitive to both Δθ = θ2 − θ1 and Δf [Sand, 1982]. For small angles and narrow-banded waves,
where Δθ ≪ 1 and Δf/fp ≪ 1. Note that Δθfp/Δfneed not be small. With an artificial top-hatE0(f, θ) with frequency width fsand deep-water directional widthσθ,0, integration of (2) with (3) results in EIG that depends strongly on (fp / fs)σθ,0, where deep water σθ,0 = (c0/c)σθ [Herbers et al., 1999] and c0 and care the deep-water and constant-depth region phase speeds atfp. The modeled Rs(ig)/(Hs,0L0)1/2 decreases with decreasing fp and increasing σθ (Figure 3), consistent with (3). In the limit of fs → 0, a monochromatic wave, infragravity wave energy is zero (3), consistent with the trend in the modeled Rs(ig)/(Hs,0L0)1/2 (Figure 3a). This motivates examining the normalized infragravity runup Rs(ig)/(Hs,0L0)1/2 dependence on (fp/fs)σθ,0. For all simulations, Rs(ig)/(Hs,0L0)1/2 variation is well parameterized (skill r2 = 0.69) using (fp/fs)σθ,0 (Figure 4). The best-fit relationship is
Over the range 0.6 < (fp/fs)σθ,0 < 30, Rs(ig)/(Hs,0L0)1/2 varies a factor of 4. A similar relationship holds (although with smaller r2 = 0.54) using σθ. As before (Figure 3b), Rs(ig)/(Hs,0L0)1/2 does not depend systematically on β over the limited range considered (symbol colors in Figure 4). Modifying the Rs(ig) parameterization from using only (Hs,0L0)1/2 (r2 = 0.67, Figure 2a) to also include (fp / fs)σθ,0(4) improves the skill (r2 = 0.80). This suggests a new Rs(ig) parameterization that may improve predictions of coastal flooding and erosion risk.
 The Boussinesq wave model funwaveC is used to simulate shoreline setup and runup over a range of incident significant wave height, peak period, frequency and directional spread, and beach slope. The model uses a simple planar beach with idealized incident wave spectra. Wave runup is simulated with a “thin-layer” method. The model reproduces the existing empirical parameterizations for setup and runup based on (Hs,0L0)1/2 or β(Hs,0L0)1/2. The focus here is understanding infragravity runup, which in energetic conditions dominates the sea-swell runup. The normalized runupRs(ig)/(Hs,0L0)1/2 is shown to depend on frequency (fs) and directional (σθ) spread of the incident wave spectrum. Motivated by a simple analysis of near-resonant infragravity waves, the scatter about theRs(ig)/(Hs,0L0)1/2parameterization is collapsed by a single non-dimensional variable (fp/fs)σθ,0 (σθ,0is the deep-water directional spread). Although the model incident wave field and bathymetry are idealized, the results suggest that including (fp / fs)σθ,0 in parameterizations could improve predictions of infragravity runup and coastal flooding during energetic wave events.
Appendix A:: Sensitivity to Model Domain and Offshore Sponge Layer Size
 The sensitivity of Rs(ig)to variations in model geometry (alongshore and cross-shore domain size, and offshore sponge layer width) was examined with a subset of simulations. ReducingLy by 40% resulted in small (<10%) changes in Rs(ig), much less than the variation of Rs(ig)/(Hs,0L0)1/2 associated with fs and σθ. Results with the 70 to 100-m wide sponge layer (used in the simulations, the base case,Figure 1) were compared with results from simulations with a 700 m wide sponge layer and with an additional 600-m long constant depth domain before the 100-m wide sponge layer. Relative to the base case, the 700-m long sponge layer simulations reduced infragravity energy reflection at the offshore model boundary, and the 600-m longer domain simulations altered the tank mode frequencies. Although the infragravity runup spectra varied with different cross-shore domain and sponge layer configurations, the normalizedRs(ig)/(Hs,0L0)1/2 varied by <10%.
 Support was provided by US Army Corps of Engineers, the California Department of Boating and Waterways, and NSF. This study was conduced in collaboration with SPAWAR Systems Center Pacific under grant SI-1703 from the Strategic Environmental Research and Development Program (SERDP). The authors contributed equally to this work.
 The Editor thanks two anonymous reviewers for assisting in the evaluation of this paper.