Introduction

The breaker depth index is commonly used to define the wave height at breaking (Horikawa, 1988, pp. 79–88; Van Rijn, 1990, pp. 298–305).

A second parameter, the breaker height index Ω_b = H_b/H_∞ is also widely used (Smith and Kraus, 1990). However, according to the present authors, it induces a greater uncertainty in the prediction of H_b. Indeed, for deep-water wave steepnesses up to 0.02, this parameter is less sensitive than Equation (2) because predictions by existing formulas are approximately between 1 and 1.1 (cf. Rattanapitikon, Vivattanasirisak, and Shibayama, 2003; Smith and Kraus,1990).

Many different formulas for γ_b= have been proposed on the basis of monochromatic wave experiments. Rattanapitikon and Shibayama (2000) and Smith and Kraus (1990) compared some of these formulas with data. It appears that the main governing factors for γ_b are, as for the breaker type, the bottom slope m and the deep-water steepness λ_∞. According to the results of Rattanapitikon and Shibayama (2000), six formulas are interesting to compare.

Weggel (1972)

Battjes (1974)

Ostendorf and Madsen (1979)

Singamsetti and Wind (1980)

Smith and Kraus (1990)

Goda (1970) modified by Rattanapitikon and Shibayama (2000)

It should be noted that Equations (3), (5), and (8) require an iterative technique, whereas Equations (4), (6), and (7) allow for direct computation. The range of validity for these formulas is implicitly assumed to be the range of the data used for their calibration. Both the formulas by Singamsetti and Wind (1980) and Smith and Kraus (1990) were calibrated with their own data set only (cf. Table 1). The formulas by Battjes (1974), Ostendorf and Madsen (1979), and Weggel (1972) are mainly based on the Galvin (1968, 1969) and Iversen (1952) data set and thus have a similar range of validity (0.02 < m ≤ 0.2). Finally, the Goda (1970) formula modified by Rattanapitikon and Shibayama (2000) was calibrated with a similar data set to that compiled for this paper (i.e., 0.00 < m ≤ 0.5).

Comparison with Compiled Data Set

Table 2 summarizes the errors in the predictions of γ_b for the six studied formulas and for the newly developed formula discussed later, including error estimates for all the data as well as for the cases with a plane-sloping beach, a step beach, and a barred beach. The quantities P10 and P20 denote the percentage of values obtained with an error of less than 10% and 20%, respectively, whereas E_rms is the root mean square relative error (%), defined as

The mean value and the standard deviation (SD) of the difference Δ_b = γ_b,pred − γ_b,exp is also included (computed for all the data and the data for which m > 0). The figures in Appendix A show the comparison between predicted and observed breaker depth index obtained for the six formulas with the use of all the experimental data.

The best results for the existing formulas and the use of all the data are obtained with the Battjes (1974), Goda (1970), and the modified Ostendorf and Madsen (1979) formulas. The latter formula also yields the best results for the barred beach data. Finally, the Weggel (1972) formula shows the best results for the step beach experiments, although it gives a constant value γ_b = 0.78 independent of the deep-water wave conditions. Some formulas can reach an agreement with the data of close to 80% for a permitted error of 20%, whereas none reaches 50% for a permitted error of 10%. This deviation can partly be explained by the errors that come from the experimental measurements since the breaker height is typically visually estimated. Also, the break point can be defined in several ways. Singamsetti and Wind (1980) listed seven possible definitions, which could induce additional errors. Closer inspection of the graphs (see Appendix A) allows for the following comments.

Dependence of Breaker Depth Index on Deep-Water Wave Steepness and Beach Slope

With regard to the dependence of γ_b on the deep-water wave steepness, the behavior of each of the studied formulas is not entirely satisfactory. The graphs in Appendix B.1 present the results of comparing data and formulas for varying λ_∞ under fixed bed slope. Although it is quite difficult to observe any trend in the experimental data of Horikawa and Kuo (1966) (m = 0, m = 0.0125; see graphs in Appendix B.1a and 1b), because significant dispersion exists in the data, a decrease in γ_b with an increase in λ_∞ remains the principal tendency in the other data sets. This decrease becomes clearer and quite pronounced in the data from Iwagaki et al. (1974), Singamsetti and Wind (1980), and Smith and Kraus (1990) for mild slopes (see Appendix B.1c–f). Only the Smith and Kraus (1990) formula displays a correct behavior in accordance with this observation. The five other formulas seem insensitive for high values on λ_∞. For larger values of m (see Appendix B.1g), this sensitivity is not so apparent, mainly because of the lack of data. Finally, for smaller values of λ_∞, most of the data indicate a value of γ_b that is nearly independent of λ_∞, with a decrease in λ_∞ (see Appendix B.1c–e with the data of Galvin [1968, 1969], Iversen [1952], Van Dorn [1978] and Walker [1974]). Both the Battjes (1974) and the Singamsetti and Wind (1980) formulas show unsatisfactory predictions for those specific cases. To summarize the influence of the deep-water wave steepness on the breaker depth index, the best observed behavior is displayed by the Smith and Kraus (1990) formula for mild slopes and by the modified Goda (1970) formula for slopes up to 10%.

The beach slope, m, has often been the only parameter used for calibration of semiempirical formulas, mainly on the basis of Miche's (1944) results. Thus, good agreement is observed for most of the formulas for mild slopes (i.e., γ_b displays an exponential dependence on m; see graphs in Appendix B.2). According to the data (except for the step beach experiment), the Smith and Kraus (1990) formula yields the best results for slopes below 0.1. However, for larger slopes, the tendency changes, and γ_b becomes a decreasing function of m (see graphs B.2 with Smith and Kraus' data) after having reached a maximum for m ≈ 0.2. Above this value, the Battjes (1974) and the Singamsetti and Wind (1980) formulas display incorrect behavior. The Ostendorf and Madsen (1979), Smith and Kraus (1990), and Weggel (1972) formulas, which all assume a fixed value for m > 0.1, give better agreement with data, but it is not satisfactory. Only the Goda (1970) formula modified by Rattanapitikon and Shi-bayama (2000) shows a correct behavior for beach slopes up to 0.2. For lower values on the deep-water wave steepness (λ_∞ < 0.03), it seems like the maximum γ_b is reached for a milder slope (m ≈ 0.1), and the observed decrease with larger slopes is smoother.

A New Formula For Breaker Depth Index

One aim of this study was to propose a new formula for γ_b that exhibits the best possible behavior with respect to the dependence on the mean slope m and the offshore wave steepness γ_∞. The new formula is based on Miche's (1944) results similar to the Ostendorf and Madsen (1979) formula, but it employs the offshore wave steepness instead of the ratio h_b/L_b to allow for a direct computation of the breaker depth index.

The Miche (1944) formula can be written as

Assuming shoaling follows the first-order Stokes theory, the wavelength can be expressed as a function of its deep-water value L_∞ = gT²_w/(2π) and the water depth h.

(if kh « 1, k = ). Because h_b = α_bH₃/γ_b, where α_b >1 is the shoaling coefficient at the break point.

With Equation (12), Equation (10) can be rewritten in an explicit form.

Following the method proposed by Rattanapitikon and Shibayama (2000), an empirical function can be fit to the data to obtain the function f_* (m, λ_∞)

Table 3 presents a comparison between the results from the iterative method (Equation [10]) and the explicit method (Equation [13]) with a special fitting of f_*(m, λ_∞), where P20 is the percentage of data points in agreement within a factor of 1.2. If f_* = 1, Equation (13) yields a formula quite close to Miche's (1944) formula. Assuming that f(m, λ_∞) = f(m), Ostendorf and Madsen (1979) fit a straight line using data in the range of 0.1 < m < 0.1. Rattanapitikon and Shibayama (2000) fit a polynomial function of m using a similar data set as the one compiled in this study. The fourth column presents a fit with the use of the present data set and Equation (13). Similar results are found for the formulas. The differences appear to be mainly due to the different data sets used by the authors for the fitting (P20 and E_rms not improved). It appears that employing Equation (13) does not reduce the predictive skill of the formula compared with the iterative relationships. More than 95% of the predicted data points are in agreement within a factor of 1.2. Some differences occur for the small values of γ_b predicted by the iterative formula, which explains the apparent dispersion of the results (E_rms > 2).

These expressions for f (and f_*) are only dependent on the mean slope, although a clear influence from the deep-water wave steepness can be observed in the graphs (Appendix C). Moreover, the polynomial function proposed by Rattanapitikon and Shibayama (2000) diverges rapidly for large slopes. Instead, a sinusoidal function of the beach slope is introduced.

The graphs in Appendix C present f_* as a function of m together with the experimental data for three different intervals of λ_∞ The best polynomial fit (second order) and the final relationship obtained by the authors are included. A maximum of f_* is clearly observed and seems to be a linear function of λ_∞. The following relationship is proposed.

In the same way, the coefficient α, which represents the “curvature” of the sinusoidal function (for m ≤m_max and m >m_max), can be related to the deep-water wave steepness according to the formulas in Equation (17).

Finally, the best fit is obtained (Equations [18] and [19]).

Figure 2 presents the result when plotting predictions with the proposed formula against all data, showing an improvement compared with the previous relationships.

Errors obtained with the new formula are summarized in Table 2. It is interesting to see that only 15% of the results for all beach types are found to have an error greater than 20%, and less than 50% of the results for an error greater than 10%. Predictions for plane, barred, and step beaches are the best among the studied formulas, but, as explained in the previous section, experimental results for step beaches (experiment of Horikawa and Kuo, 1966) can imply an overestimation of γ_b because of possible reflection. However, it can be noted that (as for the Battjes [1974], Goda [1970], Ostenford and Madsen [1979], and Smith and Kraus [1990] formulas) the maximum predicted value by Equation (15) is 1.15, although some observed values reach 1.6. It is a result of the formula being fit to a large amount of scattered data. Finally, because of the errors and uncertainties in the experimental data, it seems that the quadratic error of less than 1.9% is difficult to achieve. One of the great improvements with the new formula is that a correct behavior with regard to beach slope and deep-water wave steepness is obtained for most of the cases (unlike previous formulas).

The distribution of the difference Δ_γb = γ_b,pred − γ_b,exp (cf. Figure 3) follows a Gaussian probability density function (pdf), which implies that errors correspond mainly to the experimental uncertainties or randomness in the breaking process itself. Because Equations (13) and (15) yield a prediction of the mean value of the observed experimental data, they cannot predict extreme observed values. Looking at the figures in Appendix B, it can be observed that, for similar conditions (same slope and offshore wave steepness), a large scatter appears depending on the experimental data. Thus, in an application of Equations (13) and (15), a random variation might be employed to achieve more realistic predictions of γ_b.

Application to Random Wave Model

Modeling random waves in the surface zone follow basically two different approaches:

The calculation of the wave transformation is determined by applying an energy flux balance. For shore-parallel depth contours, it reads,

In case of approach 1 described above, H = H_rms (root mean square wave height) and T = T_p (peak wave period). Many of the random wave models are based on the approach of Battjes and Janssen (1978). The wave heights in the surf zone are characterized by a Rayleigh distribution, which is assumed to be valid for both broken and unbroken waves. The usual criterion employed to truncate the distribution is that a wave is breaking when its height H_i exceeds some fraction of the water depth. So, to a fixed water depth h, there corresponds a critical wave height H_ib over which all the waves are broken (see Figure 4). The quantity Q_b is found by integrating the Rayleigh distribution over all waves for which H_i > H_ib (Equation [22]).

The height H_ib is generally evaluated from calibration of the random wave model with data. A common expression developed by Battjes and Stive (1985) is based on the Miche (1944) formula.

More recently, Ruessink, Walstra, and Southgate (2003) found a linear relationship with the product of the local wave number and the water depth.

As Ruessink, Walstra, and Southgate (2003) discussed, these relationships include not only the physics of the breakers but also encapsulate structural errors in the model, which might explain the different behavior of these calibration parameters compared with “measured” parameters. However, because the definition of H_ib corresponds exactly to the breaker depth index for a monochromatic wave, Equations (13) and (15) can be used (i.e., H_ib = γ_b h), which gives acceptable results.

In the case of the second approach H = H_i (individual wave height from the pdf) and a breaker depth index formula is directly applicable because waves are modeled individually. So, Q_b = 0 if the individual wave is not broken H_i/h ≤ γ_b and Q_b = 1 if the individual wave is broken.

An example is given in Figure 5 of the calculated evolution of random waves according to both approaches, and two different offshore wave conditions and beach slopes are shown. For the first approach, H_ib was estimated by Equations (13) and (15) or Equation (23) with Equations (24) or (25). For the second approach, only Equations (13) and (15) were used.

In Figure 5, ξ_∞ = 0.125 for both cases; however in Figure 5a, λ_∞ is one-fourth of that in Figure 5b, and the beach slope m is half. In Figure 5a, it appears that the wave-by-wave approach of Equation (15) gives similar results to the parametric approach with a calibrated coefficient. On the other hand, in Figure 5b, the parametric model gives similar results independently of the formula used to calculate H_ib. The wave-by-wave approach yields a smaller wave height inside the surf zone. Comparing the wave-by-wave and parametric approaches with Equation (15), it appears that the wave height in the surf zone is always larger with the parametric method. If a coefficient 0.7 (∼1/ ) is added for the calculation of H_ib with Equation (15) similar results are obtained for all conditions. Therefore, assuming that the wave-by-wave approach produces results closer to reality, Equation (15) can be used in a parametric model after introducing a coefficient of 0.7.

Breaker Type

Assuming that an accurate estimation of the breaker depth index can be made for all cases, it is interesting to see how this index is related to the breaker type. Galvin (1968, 1972) defined the following breaker types:

Breaker type is controlled by the bottom slope m and the deep-water steepness λ_∞. Battjes (1974) used the Irribaren number (or surf similarity parameter) to describe breaker type on the basis of previous results of Galvin (1968, 1972):

Relationship Between Breaker Depth Index and Irribaren Number

Figure 6 shows how measured γ_b varies with ξ_∞. It is clearly seen that γ_b is an increasing function of ξ_∞. The relationship proposed by Battjes (1974) is a good approximation on the basis of comparison with all the data. However, marked scatter exists, which is the weakness of this formula, as was observed in the previous section.

In Figure 7, a sensitivity study of the new formula (Equation [15]) is made for γ_b against ξ_∞ employing various beach slopes m (Figure 7a) or various deep-water wave steepnesses λ_∞ (Figure 7b). Curves are limited by imposed constraints, i.e., the deep-water wave steepness cannot be greater than 0.14 and the maximum beach slope is fixed as the maximum value from the experimental data (m ≤ 0.5).

The most interesting result to emerge from these graphs is that they show two different types of behavior of the breaker depth index depending on the breaker type. Indeed, for ξ_∞ < ξ_{∞, cr} ≈ 0.5 (spilling waves, cr is critical value), γ_b is mainly sensitive to the offshore wave steepness λ_∞, and for a fixed ξ_∞, γ_b is nearly independent of the slope m. On the contrary, for ξ_∞ < ξ_{∞, cr} (plunging and surging waves), γ_b is mainly sensitive to the beach slope m, and for a fixed ξ_∞, γ_b is nearly independent of λ_∞. Moreover, the maximum values on the breaker depth index (see Figure 7b) seem to correspond to plunging breakers (according to values of λ_∞, this maximum appears for 0.8 < ξ_∞ < 4). Thus, surging and collapsing breakers only occur for conditions corresponding to the decreasing part of the computed curves. The physical mechanisms controlling these types of breakers do not allow high values of γ_b, even if they were observed for lower values of λ_∞ (≈0.01) and steep beaches (m > 0.2). In the same way, a separation point between spilling and plunging waves could more or less be the inflection point of the increasing part of the computed curves (see Figure 7b), which explains the critical ξ_∞ from Galvin (1968, 1972) varying between 0.2 and 0.5, which Smith and Kraus (1990) also noticed.

Figure 8 presents the final result concerning breaker type prediction on the basis of the Irribaren number and the breaker depth index with the use of data in which complete information is available. The data from Okazaki and Sunamura (1989) were also added because they performed an interesting study on step beaches. Unfortunately, the experimental results for the breaker depth index were not presented in their article. Thus, γ_b has been estimated by Equation (15). The symbols used for this data set have been made smaller.

The lower curve (separating of the nonbreaking wave region) corresponds to Equation (15) with λ_∞ = 0.14. The upper curve (separating of the unstable wave region) corresponds to the maximum values obtained with this equation with a factor of 1.25. The factor 1.25 was employed because the new equation does not compute the maximum values observed from the experimental data. Finally, the three other curves are empirically determined to separate the spilling breaker, plunging breaker, collapsing breaker, and surging breaker regions following the ideas presented in the previous paragraph. Large uncertainties still exist, but they seem to be mainly due to the uncertainties in the available experimental data. This graph provides an improvement for breaker type prediction compared with Galvin's (1968, 1972) relationships (Equations [26]). The separation lines in Figure 8 are defined by the formulas in Equations (27).