The phase-averaged wave action equation (WAE) is extended to include the effect of submerged porous media in this paper. The wave dissipation coefficient for submerged permeable media is incorporated in the WAE of wind wave model. The coefficient of the turbulent flow resistance is obtained based on Lan et al. (2016). An estimation method for the characteristic parameter of turbulent frictional resistance in permeable media is proposed in this study. Reasonable comparisons with experiments and numerical results show the present model can be applied to propagating waves over porous beds, reefs and vegetation.
Wave model simulation is one of the main ways for the marine and coastal engineering practitioners to obtain the design wave conditions. Wave generation, transformation and dissipation are influenced by many complex physical mechanisms (wind forcing, wave-currents and wave-wave interactions, shoaling, breaking, bottom friction, porous flow inside the seabed and through reefs, vegetation etc.). The combined effect of the wave refraction and diffraction on the wave transformation can be accounted for by using mild-slope equation (MSE) approaches, Boussinesq-type equation (BE) models or wind wave models (WWM). The MSE and BE are phase-resolving wave models to account for nearshore wave processes (Rojanakamthorn et al., 1989; Gobbi et al., 2000; Behera et al., 2015; Su et al., 2015). On the other hand, the phase-averaged models are used for the simulation of the variation of wave spectra for random short-crested waves in large-scale oceanic deep water and small-scale shallow water regions. Typical examples of commonly used models are WAM (WAMDI Group, 1988), SWAN (Booij et al., 1999a, 1999b), STWAVE (Smith et al. 2001), TOMOWAE (Marcos, 2003) and WWM (Hsu et al. 2005; Liau et al., 2011; Lan et al., 2015, 2016). The phase-averaged wave action balance equation was formulated by Holthuijsen et al. (2003) and Liau et al. (2011) to include wave refraction- and diffraction-induced directional turning rate of the components. The processes of wave generation, dissipation and nonlinear wave-wave interactions are well accounted for in these models based on WAE. The main effects taken into account include wind-wave generation (Komen et al., 1984; WAMDI Group, 1988), nonlinear wave-wave interactions (Hasselmann et al., 1985; Eldeberky and Battjes, 1995; Eldeberky, 1996), whitecapping (Hasselmann, 1974; Komen et al, 1984), bottom friction (Young and Gorman, 1995), and wave breaking (Battjes and Stive, 1985; Eldeberky and Battjes, 1995). Toledo et al. (2012) constructed an extended WAE in terms of linear wave theory of MSE that has an improved behavior for rapid spatial bottom changes as well as ambient current changes. Lan et al. (2015) proposed a theory of WAE for spectral evolution over porous bottom (e.g. sand beds) in the presence of currents, by taking into account the energy dissipation effect induced by porous bottom media. This model was validated through comparisons with the experimental data for wave propagation over porous beds with low permeability. Lan et al. (2016) further proposed a extended WAE with the effect of submerged permeable reefs. However, in terms of the effect of high permeability, they did not make an appropriate method for estimating turbulent friction parameter but followed the past relevant literatures.
