## 1. Introduction

[2] 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,*R*_{s}^{(ss)}) and infragravity (0.004–0.04 Hz, *R*_{s}^{(ig)}) frequency bands, with significant runup elevation *R*_{s} defined as 4*σ*, where *σ*^{2} is the vertical runup variance in that band. During energetic wave events, *R*_{s}^{(ig)} dominates runup on dissipative beaches [*Guza and Thornton*, 1982]. Thus accurately parameterizing *R*_{s}^{(ig)}is critical to predicting wave-driven coastal flooding and its impacts on coastal ecosystems and communities.

[3] Many empirical parameterizations relate , *R*_{s}^{(ss)}, or *R*_{s}^{(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 (*H*_{s,0}*L*_{0})^{1/2} or *β*(*H*_{s,0}*L*_{0})^{1/2} where *L*_{0} is the incident deep water wavelength based on peak (or mean) frequency *f*_{p} (*L*_{0} = (2*πg*)*f*_{p}^{−2}), and *H*_{s,0}is 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 *R*_{s}^{(ig)} using (*H*_{s,0}*L*_{0})^{1/2} have significant skill (*r*^{2} ≈ 0.65 for *R*_{s}^{(ig)} ∝ (*H*_{s,0}*L*_{0})^{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 on*H*_{s,0} and *f*_{p}, but also has frequency spread (*f*_{s}) about *f*_{p} and directional spread (*σ*_{θ}) [*Kuik et al.*, 1988] about the mean angle (assumed zero here), The effect of variable *f*_{d} and *σ*_{θ} on infragravity runup *R*_{s}^{(ig)} is explored here using the Boussinesq wave model funwaveC on a planar beach.

[4] The model (section 2) is shown to reproduce the dependence on (*H*_{s,0}*L*_{0})^{1/2} and *β*(*H*_{s,0}*L*_{0})^{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 *R*_{s}^{(ig)}/(*H*_{s,0}*L*_{0})^{1/2} parameterization depends on *f*_{s} and *σ*_{θ} (section 3.2). A simple non-dimensional parameter (*f*_{p} / *f*_{s})*σ*_{θ,0} (where *σ*_{θ,0}is the deep-water*σ*_{θ}), based on a nonlinear infragravity wave coupling coefficient, collapses the scatter, suggesting a new *R*_{s}^{(ig)} parameterization (section 4).