**Appendix A**

The basic wave propagation balance equations in the WAVEWATCH-III (WW3) model can be briefly described as follows:

$$\frac{\mathbf{D}}{\mathbf{D}t}(\mathbf{N}(\mathbf{k},\theta;\mathbf{x},\mathbf{t})) = \frac{\mathbf{S}(\mathbf{k},\theta;\mathbf{x},\mathbf{t})}{\sigma} \text{ and} \tag{A1}$$

$$\mathbf{S} = \mathbf{S}\_{\text{in}} + \mathbf{S}\_{\text{nl}} + \mathbf{S}\_{\text{ds}} + \mathbf{S}\_{\text{bot}} + \mathbf{S}\_{\text{db}\prime} \tag{A2}$$

where the wavenumber-direction spectrum N is the basic spectrum of WW3 in terms of wavenumber k, wave propagation direction θ, space dimension x, and time dimension t; N is the wave action density spectrum; σ is the intrinsic frequency; and S(k,θ;x,t) describes the net of sources and sinks from the wavenumber-direction spectrum. The sink term S(k,θ;x,t) contains the impacts of linear and nonlinear wave propagation energy, including wind energy input Sin, a wave–wave interaction term Snl, dissipation Sds, and the empirical parameterizations of wave–bottom friction Sbot and depth-induced breaking Sdb. The parameterizations of these terms are conveniently provided by the WW3 model, as described in the technical manual. The four wave-induced effects were calculated based on several parameters. Specifically, the parametrization of input/dissipation terms is referred to switch ST2+STAB2 [13,57], and the packages for processing nonlinear terms on quadruple wave–wave interactions is referred to switch DIA [5,58].
