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Shoaling Transformation of Wave Frequency-Directional Spectra T JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. C1, 3013, doi:10.1029/2001JC001304, 2003 Shoaling transformation of wave frequency-directional spectra T. H. C. Herbers and Mark Orzech Department of Oceanography, Naval Postgraduate School, Monterey, California, USA Steve Elgar Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA R. T. Guza Integrative Oceanography Division, Scripps Institution of Oceanography, La Jolla, California, USA Received 15 January 2002; revised 3 July 2002; accepted 24 July 2002; published 21 January 2003. [1] A Boussinesq model for the nonlinear transformation of the frequency-directional spectrum and bispectrum of surface gravity waves propagating over a gently sloping, alongshore uniform beach is compared with field and laboratory observations. Outside the surf zone the model predicts the observed spectral evolution, including energy transfers to harmonic components traveling in the direction of the dominant waves, and the cross- interactions of waves traveling in different directions that transfer energy to components with the vector sum wavenumber. The sea surface elevation skewness and asymmetry, third-order moments believed to be important for sediment transport, also are predicted well. Effects of surf zone wave breaking are incorporated with a heuristic frequency- dependent dissipation term in the spectral energy balance equation and an empirical relaxation of the bispectrum to Gaussian statistics. The associated coefficients are calibrated with observations that span a wide range of surf zone conditions. With calibrated coefficients, the model predicts observed surf zone frequency spectra well and surf zone skewness and asymmetry fairly well. The observed directional spectra inside the surf zone are broader than the predicted spectra, suggesting that neglected scattering effects associated with the random onset of wave breaking or with higher-order nonlinearity may be important. INDEX TERMS: 4560 Oceanography: Physical: Surface waves and tides (1255); 4546 Oceanography: Physical: Nearshore processes; 4263 Oceanography: General: Ocean prediction; 4255 Oceanography: General: Numerical modeling; KEYWORDS: ocean waves, wave spectra, beach, surf Citation: Herbers, T. H. C., M. Orzech, S. Elgar, and R. T. Guza, Shoaling transformation of wave frequency-directional spectra, J. Geophys. Res., 108(C1), 3013, doi:10.1029/2001JC001304, 2003. 1. Introduction where each component obeys the linear gravity wave dispersion relation and the mismatch from resonance Á is [2] Models for the transformation of ocean surface waves across a beach are important to the prediction of nearshore small, transfer energy from the incident wave components to circulation and sediment transport. In addition to the well- higher (e.g., harmonic) and lower (e.g., infragravity) understood linear processes of shoaling and refraction, the frequency components. These interactions not only broaden wave transformation is affected by nonlinear interactions the frequency spectrum in shallow water, but also phase- and wave breaking. Nonlinear interactions between triads of couple the spectral components, causing the characteristic steepening and pitching forward of near-breaking wave wave components with frequencies (w) and cross- (k) and alongshore (l ) wavenumbers satisfying crests [e.g., Freilich and Guza, 1984; Elgar and Guza, 1985]. This nonlinear evolution is described well by depth- integrated Boussinesq equations for weakly nonlinear, w w w 0; 1a 1 þ 2 þ 3 ¼ ð Þ weakly dispersive waves in varying depth [Peregrine, 1967]. Alternative forms of Boussinesq-type equations have k1 þ k2 þ k3 ¼ Á; ð1bÞ been derived to extend their application to deeper water [e.g., Kaihatu and Kirby, 1995] and stronger nonlinearity [e.g., l1 þ l2 þ l3 ¼ 0; ð1cÞ Weietal., 1995]. Recently, fully nonlinear Boussinesq equations [Weietal., 1995] have been used to describe the Copyright 2003 by the American Geophysical Union. detailed time evolution of waves and wave-driven currents 0148-0227/03/2001JC001304$09.00 on complex bathymetry [Chen et al., 2000]. All the above 13 - 1 13 - 2 HERBERS ET AL.: SHOALING TRANSFORMATION OF WAVE SPECTRA models are deterministic. That is, for a given set of time- increase in directional spreading that is not understood dependent boundary conditions, the model yields time series [Herbers et al., 1999]. of fluid velocities and sea-surface elevation at shoreward [6] Here, a numerical implementation and field evalua- locations. Although these time-domain Boussinesq models tion of a stochastic Boussinesq model for directionally can be applied to two-dimensional beaches with arbitrary spread waves propagating over an alongshore uniform incident wave conditions, the computations are numerically beach [Herbers and Burton,1997]arepresented.The expensive for large domains and require detailed boundary model, based on a third-order closure, consists of a coupled conditions that often are not available. set of first-order evolution equations for the wave spectrum [3] Alternatively, the shoaling evolution of random waves E(w, l) and bispectrum B(w1, l1, w2, l2). The two-dimen- on a beach can be predicted with stochastic models that sional spectrum E(w, l) defines the energy density of solve evolution equations for statistically averaged spectral component (w, l), and the four-dimensional (complex) wave properties [e.g., Agnon and Sheremet, 1997; Herbers bispectrum B(w1, l1, w2, l2) defines the average phase and Burton, 1997]. These models are numerically efficient relationship of a triad (Equation (1)) consisting of compo- and can be initialized at the offshore boundary with wave nents (w1, l1), (w2, l2), and (Àw1 Àw2, Àl1, Àl2). A one- spectra obtained from routine directional wave measure- dimensional version of the model for normally incident, ments or regional wave model predictions. However, unlike non-breaking waves was developed and field-tested by deterministic models that solve (approximate) equations of Norheim et al. [1998]. The predicted evolution of wave motion without any assumptions about higher-order statis- frequency spectra agreed well both with observations and tics, stochastic models require a statistical closure that may with predictions of the deterministic Boussinesq model of yield significant errors over long propagation distances and Freilich and Guza [1984]. Here, a full two-dimensional in regions of strong nonlinearity. implementation of the model for directionally spread waves [4] Theories for nonlinear wave-wave interactions are is presented, including heuristic extensions to incorporate rigorous, but surf zone wave breaking is not well under- dissipation associated with wave breaking (following Kai- stood and is modeled heuristically. Scha¨ffer et al. [1993] hatu and Kirby [1995]) and a relaxation to Gaussian include a turbulent surface roller in a time domain Bous- statistics in regions of strong nonlinearity [e.g., Orszag, sinesq model that yields a realistic description of the 1970; Holloway and Hendershott, 1977]. evolution of wave profiles in the surf zone. Most models [7] The model (section 2) is evaluated with data from two for the breaking of random waves are based on the field experiments on an ocean beach near Duck, North analogy of individual wave crests with turbulent bores Carolina (described in section 3) and from a laboratory [Battjes and Janssen, 1978; Thornton and Guza, 1983]. simulation of waves observed on a beach near Santa Barbara, Although these bore models yield robust estimates of bulk California. Model predictions of the evolution of wave dissipation rates in the surf zone, the spectral character- frequency-directional spectra, skewness, and asymmetry istics of the energy losses are not specified, and somewhat are compared with observations of nonbreaking (section 4) arbitrary quasi-linear spectral forms of the dissipation and breaking (section 5) waves, followed by a summary function are used in Boussinesq models [Mase and Kirby, (section 6). 1992; Eldeberky and Battjes, 1996]. Boussinesq model predictions of wave frequency spectra in the surf zone 2. Model Description appear to be insensitive to the precise frequency depend- 2.1. Evolution Equations ence of the dissipation function, but predictions of wave skewness and asymmetry are considerably more accurate if [8] Spectral evolution equations for directionally spread dissipation is weighted toward high-frequency components surface gravity waves propagating over an alongshore uni- of the spectrum [Chen et al., 1997]. Estimates of nonlinear form beach [Herbers and Burton, 1997] are reviewed energy transfers in the surf zone based on bispectral briefly and extended to include heuristic surf zone damping analysis of near-bottom pressure fluctuations confirm the and relaxation terms. The sea-surface elevation h(x, y, t)of dominant role of triad interactions in the spectral energy surface gravity waves propagating over an alongshore uni- balance [Herbers et al., 2000]. The observed decay of the form beach has the general Fourier representation spectral peak is primarily the result of large nonlinear energy transfers to higher frequencies where the energy X1 X1 ÂÃÀÁ presumably is dissipated. hðÞ¼x; y; t Ap;qðÞx exp ilqy À wpt ; ð2Þ [5] Although many studies of the shoaling evolution of p¼À1 q¼À1 wave frequency spectra have been reported, the evolution of the wave directional spectrum has received less attention, where wp = pÁw and lq = qÁl are the frequency and despite
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