NEW BREAKER INDEX FORMULA IN PARAMETRIC WAVE MODEL

INTRODUCTION Phase-averaged parametric wave models have been widely used to predict nearshore wave height transformation. The performance of parametric models depends significantly on the wave breaker index (γ), which controls the amount of breaking energy dissipation. Previous parameterizations improved the model predictability by considering the breaker index as a tunable coefficient, while made less effort to the physical interpretation for the proposed formulas. Indeed, inconsistency from the physical perspectives might exist. For example, several studies proposed γ increases with the increasing offshore wave steepness (s0) or wave height (H0), and increases with the increasing local normalized water depth (kh). However, according to the surf similarity parameter, the spilling breaker (with a smaller wave height-to-depth ratio) more likely occurs far away from the shoreline (with a larger kh), is corresponding to a larger s0. Therefore, the parameterization of γ still requires further investigation by considering the comprehensive influences of the offshore wave parameters and the local water depth, as well as the possible relationships with the breaker type and the surf zone state.


INTRODUCTION
Phase-averaged parametric wave models have been widely used to predict nearshore wave height transformation. The performance of parametric models depends significantly on the wave breaker index (γ), which controls the amount of breaking energy dissipation. Previous parameterizations improved the model predictability by considering the breaker index as a tunable coefficient, while made less effort to the physical interpretation for the proposed formulas. Indeed, inconsistency from the physical perspectives might exist.
For example, several studies proposed γ increases with the increasing offshore wave steepness (s0) or wave height (H0), and increases with the increasing local normalized water depth (kh). However, according to the surf similarity parameter, the spilling breaker (with a smaller wave height-to-depth ratio) more likely occurs far away from the shoreline (with a larger kh), is corresponding to a larger s0. Therefore, the parameterization of γ still requires further investigation by considering the comprehensive influences of the offshore wave parameters and the local water depth, as well as the possible relationships with the breaker type and the surf zone state.

INVERSED MODELLING
For model calibration, we use the 313-h Hrms data collected at 14 cross-shore locations (extending to the water depth of about 4.5 m) during Duck94 experiment from September 21, 1994to October 4, 1994. For model verification, the 529-h data collected in Duck from October 4, 1994to October 26, 1994, 841-h data collected in Egmond from October 15, 1998to November 19, 1998 and 816-h data collected in Terschelling from May 25, 1994 to June 28, 1994 are used.
On the basis of the work by Ruessink et al. (2003), the present study uses the inverse modelling approach to revisit the γ with respect to a wide variety of conditions. Based on field datasets, variation of the γ is inversely calculated using the wave breaking model proposed by Baldock et al. (1998). Detailed description of the inversed modelling approach can be found in Ruessink et al. (2003).
It is noted that Ruessink et al. (2003) only retained the γ values for Db > 15 N/ms to avoid spurious results.
However, we find this treatment is not necessary in the present study as it may cause the miss of some important information. In this study, all γ values for Db > 0 N/ms are retained. It is found that the data of Db > 15 N/ms accounts for only a small portion (18%) of all data, and it mainly represents the wave conditions when s0 is relatively large (approximately > 0.02). Therefore, the calibration of γ under wave conditions with smaller offshore wave steepness is likely not included in Ruessink et al. (2003). Figure 1 presents the relationship between γ and kh. As shown, the γ data for Db >15 N/ms is in good agreement with the formula (γ = 0.76kh + 0.29) of Ruessink et al. (2003), which describes a positive correlation between γ and kh. However, this relationship no longer applies if all γ data is considered. Especially, the γ data seems to follow a different trend under wave conditions with smaller s0 and Db. Moreover, the γ data for Db > 15 N/ms (mainly corresponding to the energetic wave conditions with large s0) is generally higher than other data, implying that the breaker index depends on both the local water depth and the offshore wave parameters. Based on the inverse model results, a composite dependence of γ on both s0 and kh is found. As provided in Figure 2, we reveal a positive γ -kh relationship for larger s0, and a negative γ -kh relationship for smaller s0. Based on such composite relationships, a new γ formula is proposed as, (1)

RESULTS
The model (Hereafter ZL2020) performance in predicting Hrms using the new breaker index formula (Equation 1) together with the breaking dissipation model of Baldock et al. (1998) is quantitatively evaluated with the root-meansquare percentage error (RMSPE).
Using the available field datasets in three coasts, the present model (ZL2020) is compared with seven previous models [R2003 in Ruessink et al. (2003, TG1983 in Thornton and Guza (1983), JB2007 in Janssen and Battjes (2007), B1998 in Baldock et al. (1998), BJ1978 in Battjes and Janssen (1978)], which are widely used in parametric wave modelling. Since Apotsos et al. (2008) have significantly improved the accuracy of TG1983, JB2007, B1998 and BJ1978 by tuning γ for each model based on the similar datasets, here we will employ the newly tuned γ formulas of Apotsos et al. (2008) in these models (hereafter TG1983_t, JB2007_t, B1998_t, BJ1978_t) for a more rigorous inter-comparison. Figure 3 compares the median percentage errors of eight models. All models have the errors between 5% and 20%, consistent with the results of Apotsos et al. (2008). The model error is generally lower at Duck than at two other sites, possibly because most of the models have been calibrated with part of (or full) Duck data. ZL2020 shows lowest errors at Duck and Egmond, and very slightly higher error than TG1983_t but still remarkably lower than other models at Terschelling. The BSS values in Table 1 indicated that using ZL2020 model reduces the errors averagely by 16% at Duck, 19% at Egmond, 21% at Terschelling, and by 22% over R2003, 10% over TG1983_t, 15% over JB2007_t, 17% over B1998_t, 20% 319 over BJ1978_t, 23%.

CONCLUSIONS
A composite dependence of γ on both the offshore wave steepness (s0) and the normalized local water depth (kh) is revealed. A new γ formula is proposed to take such composite relationships into consideration. The model implementing this new γ formula systematically reduces the prediction error of wave height compared with five widely used models in previous literatures.