*2.6. Parameterization of the Wave-Breaking Index*

2.6.1. Constant Wave Breaking Criterion (*γ*)

The parameter *γ*, given in Equation (4), is an adjustable coefficient that allows for the effects of the bottom slope on the waves and is one of the important parameters in the energy dissipation formulation. Miche [52] derived the theoretical value of *γ* = 0.88 for a flat bottom. Based on the average of several reasonable wave breaking observations, Battjes and Janssen [48] suggested a constant breaker index of *γ* = 0.8. The values for the breaking index *γ* range from 0.60 to 0.83; however, *γ* = 0.73 has been used in the most operational third-generation wave models, which is taken as the averaged value over a larger data set [9]. In the present study, a value of *γ* = 0.78 was specified for the SWH simulation following the results from Chen et al. [21] and Hsiao et al. [22–24] if the wave-circulation model adopted a constant breaker index.

2.6.2. Wave Breaking Criterion Based on Local Steepness or Peak Steepness

The wave breaking process in shallow nearshore waters is affected by the seafloor profile (i.e., bottom slope) and incident wave steepness. Battjes and Stive [9] found that a hyperbolic tangent function could be used for predicting the SWHs in shallow-water coastal areas and a relationship between the breaking index *γ* and the wave steepness *s*0:

$$
\gamma = 0.5 + 0.4 \tanh(33 \text{s}\_0) \tag{13}
$$

in which *s*0 can be expressed as follows:

$$\text{so} = H\_{\text{rms0}} / \lambda\_0 \tag{14}$$

where *Hrms*0 is the root-mean-square wave height in deep waters, *λ*0 is the wavelength and is defined as *λ*0 = *gd*/ *f* , *f* is the wave period and *g* is the acceleration of gravity. The wave breaking criterion in Equation (13) is regarded as the local steepness or peak steepness based on *f* .
