Breaker Index for Random Waves

4.5.1. Incipient breaking index of significant wave

Equations (4.5) and (4.6) are examples of the breaker index for regular waves.

There are some people who try to apply such breaker index formulas to coastal

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone 95

waves or random waves, but such application does not yield correct answers. The breaker index for regular waves may be utilized for the highest wave in an irregular wave train, but it cannot be applied for the significant wave, the root-mean-square wave, or any other characteristic wave. When random waves approach the shore, breaking of individual waves occurs gradually with large waves first at the far dis-tance, medium waves next at some distance from the shore, and small waves near the shoreline. The variation of significant wave height from the offshore toward the shore is so gradual that we cannot employ the concept of wave-breaking line, which is so obvious in the case of regular waves.

Against such difficulty of defining the breaking point of significant wave, Kamphuis⁹ introduced the definition of “incipient wave breaking.” He measured cross-shore variations of significant wave height beyond and across the surf zone, drew a curve of wave shoaling trend in the outside of the surf zone and a curve of wave height decay within the surf zone, and called the condition at the cross-point of the two curves as the incipient wave breaking. By using the data of the signif-icant wave height at incipient breaking, he calibrated 11 breaker index formulas and determined the best-fitting proportionality coefficient. For the formula of Eq. (4.5), he obtained the proportionality coefficient of A = 0.12 for significant wave height.

Li et al.²³ have also presented a data of the breaker index of (H_1/3/h)_b on the slope of 1/200, which is fitted to Eq. (4.5) with a modified constant value of 0.12 for the initial stage of breaking. Their breaking condition was some observation of large individual breaking waves in an irregular wave train. Goda²⁴ has prepared a set of diagrams depicting variations of significant wave height across the surf zone (reproduced as Figs. 3.29 to 3.32 in Ref. 25). The boundary lines of 2% decay in these diagrams approximately correspond to the breaker index with A = 0.11, and the water depth (h_1/3)_peak at which the significant wave takes a peak value within the surf zone (Fig. 3.34 in Ref. 25) corresponds to A = 0.11–0.13. Therefore, the incipient breaker index of the significant wave can be expressed with the following formula:

H1/3, b

h_b



incipient

= 0.12 h_b/L0

1− exp



−1.5π(h_b)_incipient L0



1 + 11 s^4/3

. (4.7)

Thus, the incipient breaker index of significant wave is about 30% lower than that of regular waves. The incipient breaking of significant wave corresponds to the condition that the high waves of upper several percent among individual waves have begun to break.

4.5.2. Laboratory data of breaker index of random waves

After incipient breaking, the percentage of wave breaking increases as waves proceed across the surf zone. The ratio of the significant wave height to the water depth gradually increases toward the shoreline. Ting^26,27 made detailed laboratory inves-tigations of random wave deformations on a uniform slope of 1/35, using frequency spectra of broad- and narrow-band with the peak enhancement factor of 3.3 and 100, respectively. Waves had the significant height of H_s= 0.15 m and the spectral

96 Y. Goda

0.10 1.0

0.01 0.1

Broad spec. H1/3/h Narrow spec. H1/3/h Breaker envlp. H1/3/h Broad spec. H^rms/h Narrow spec. H^rms/h Breaker envlp. Hrms/h

Breaker indices,H1/3/h& Hrms/h

Relative depth, h/L0

0.8 0.6

0.4 0.3

0.2

0.03 0.3

Slope = 1/35

Fig. 4.2. Breaker indices forH1/3andH^rmsons = 1/35 with the data by Ting.^26,27

peak period of T_p= 2.0 s. He recorded wave profiles at an offshore station with the depth of 0.457 m and at six stations on the slope with the depth of 0.27–0.0625 m.

Waves at the six stations on the slope had the percentage of breaking ranging from 5% to 94% (the case of broad-band spectrum).

Wave records were analyzed by the zero-downcrossing method, and calculated results of characteristic wave heights and periods are presented in tabular forms.

From these results, the ratios of H_1/3 and H_rms to the local depth (inclusive of mean water level change) are calculated and plotted against the relative water depth h/L0, as shown in Fig. 4.2.

The curves denoted as breaker envelopes are calculated by Eq. (4.6) for s = 1/35 with the proportionality coefficient of A = 0.145 (85% of regular waves) for the significant height H_1/3 and to A = 0.111 (65%) for the root-mean-square height Hrms. Because the percentage of breaking waves is high in these data, an A value higher than that for incipient breaking fits to the data.

The breaker index data for Hrms by Tick is higher than the value proposed by Sallenger and Holman,²⁸ who gave an expression of Hrms/h = 3.2s + 0.32 without inclusion of the relative depth (h/L0) term. They converted the orbital velocity spectra to the surface wave spectra with the transfer function based on the linear theory, and estimated the energy-based Hrms, which must have been smaller than the statistical Hrms value based on direct measurement data of surface profiles.

4.5.3. Description of field data employed for analysis

In the field wave observation at a fixed station, it is not feasible to judge whether individual waves are at the stage of breaking or not, unless simultaneous measure-ments with video cameras are taken. However, we may find out an upper limit of significant wave height for a given water depth by taking an envelope of many data at different relative water depths. For this type of analysis, stationary wave

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone 97

Table 4.3. Summary of stationary coastal wave data employed in the present analysis.

Type of Sampling Significant Significant

Observation wave Water interval height period No.

station gauge depth (m) ∆t (s) H1/3(m) T_1/3(s) of data

Rumoi Port Step-resistance −11.5 0.5 2.2–7.1 5.9–11.7 44

Yamase-Domari Port

” −12.7 0.5 1.9–6.2 7.7–15.6 9

Tomakomai Port ” −10.8 1.0 2.9–5.8 7.7–10.9 9

” −13.8 0.5 2.6–2.8 6.7 – 7.5 2

Ultrasonic −20 0.5 2.4–2.5 6.9–7.4 2

Kanazawa Port ” −20 1.0 1.0–6.8 1.0–6.8 13

Caldera Port, Costa Rica

” −18 0.5 1.5–3.6 14.2–18.4 50

Sakata Port Pressure −14.5 0.5 1.7–9.7 6.3–13.4 123

−10.5 0.5 1.7–6.1 6.5–15.0 123

(Source: Goda and Nagai,²⁹Goda^24,30).

observation data analyzed by Goda and Nagai²⁹and the data of long-traveled swell recorded with an ultrasonic wave sensor reported by Goda³⁰were utilized. Table 4.3 lists the characteristics of these field data. Waves were recorded by means of either step-resistance gauges or ultrasonic wave sensors so that they were reliable regis-tration of surface wave profiles. The data at Tomakomai and Kanazawa as well as Caldera were measured with ultrasonic wave sensors. They are not analyzed for breaker limits but for wave nonlinearity effects to be described in Sec. 4.6.

Table 4.3 also lists the wave records at Sakata Port measured by means of pressure gauges, which were utilized by Goda²⁴ for calibration of his random wave-breaking model. Although there remains a problem of pressure conversion to surface profiles, the conversion error would have been small because of the relatively shallow water depth at Sakata stations (10.5 and 14.5 m). They were included in the present analysis to increase the size of database.

Other sources of nearshore waves are the photogrametric measurement data by Hotta and Mizuguchi^31,32as well as by Ebersole and Hughes.³³Hotta and Mizuguchi mobilized 11 motion-picture cameras set on top of a coastal observation pier at Ajigaura Beach, Ibaragi, Japan. They took film pictures of instantaneous water surfaces simultaneously at some 60 surveyor’s poles erected in the nearshore waters on a line perpendicular to the shoreline stretched over a distance of about 120 m.

Films of surface wave records were taken every 0.2 s for an effective duration of 760 s.

The beach profile in September 1978 had a trough at the distance of 25 m from the shoreline and the slope of about 1/60 offshore of the trough. The beach profile in December 1978 was somewhat uniform without any bar or trough, and the slope was about 1/70. The water depth inclusive of tides at the poles varied from 0.1 to 2.7 m.

Photogrametric measurements of nearshore waves were also executed by Ebersole and Hughes³³ during the DUCK85 campaign in Duck, North Carolina, USA with the cooperation of Dr. Hotta who brought twelve cameras with him and took charge of filming. They referred to this technique as “the photopole method”;

this terminology is employed in the present chapter. Over the distance of 64 m,

98 Y. Goda

Table 4.4. Summary of photopole measurements.

Name Date No. of data h (m) H⁰ (m) T1/3(s) H⁰/L⁰

Ajigaura 1978/09/05 54 0.6–2.7 0.7 8.4 0.0064

1978/12/13–14 175 0.1–1.8 0.5–0.7 7.2–7.9 0.0059–0.0081 DUCK85 1985/09/04–05 99 0.5–2.4 0.3–0.5 10.3–11.1 0.0017–0.0028 SUPERDUCK 1986/09/11–19 140 0.4–3.7 0.5–1.1 5.4–11.5 0.0027–0.0254

12 poles were erected in the initial depth ranging from 0.4 to 1.9 m. Measurements were taken for nine runs during the fourth and fifth days of September 1985. With the variation of the tide level, the actual water depth varied from 0.5 to 2.4 m. The effective duration of wave recording was about 650 s, judging from the number of waves and average periods listed by Ebersole and Hughes.³³

The beach profile was nearly flat for about 25 m from the shoreline with the depth of about 0.5 m below the mean sea level, and it had the slope of about 1/30 beyond that. Another series of photopole measurements were carried out during the SUPERDUCK campaign in 1986. Dr Hughes kindly supplied the author with the data files of measured wave statistics. The number of poles was increased to 20 and the water depth inclusive of tides varied from 0.4 to 3.7 m. The beach profile during SUPERDUCK is not known, but it would have been nearly the same as DUCK85 because of the same season. All the photopole measurement data were analyzed by the zero-downcrossing method, and various statistical wave characteristics were calculated.

Table 4.4 lists the summary of the photopole wave measurement conditions.

The significant wave period T_1/3 has been converted from the pole-averaged values of either the mean period Tmean or the spectral period T_p by assumption of T_1/3= 1.05Tmeanor T_1/3= 0.95T_p, which would be appropriate for swell of very low steepness. The offshore wave height H0 was converted from the significant height H_1/3measured at the most offshore pole using the shoaling coefficient; no refraction effect was taken into consideration as no information of wave direction was available.

All the waves were swell of very low steepness ranging from 0.0017 to 0.0081, except one case of SUPERDUCK with H0/L0= 0.0254.

4.5.4. Field data of breaker index for energy-based significant waves

Coastal surface wave data, pressure-converted wave data, and three sets of photopole data are plotted together in Fig. 4.3 in the form of H_m0/h versus h/L0. The energy-based significant wave height H_m0defined by Eq. (4.8) is employed here instead of the zero-crossing height H_1/3, because the latter is greatly enhanced over H_m0by strong effects of wave nonlinearity and it is not representative of breaking-dissipated wave energy level; this aspect is discussed in Sec. 4.6.

H_m0= 4.0η_rms= 4.0√m0, (4.8) where m₀ denotes the zeroth moment of frequency spectrum.

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone 99

0.1 1.0

0.001 0.01 0.1 1

Step-resistance ( h=11-14m) Presssure gage (h=10-14m) Photopole Ajigaura (h=0.1 -2.7 m) Photopole DUCK85 (h=0.5-2.4m) Photopole SUPERDUCK (h=0.5-3.2 m) Breaker envelope (80% limit)

Waveheight ratio,Hm0/h

Relative depth, h/L0

0.8 0.6 0.4

0.2

Fig. 4.3. Breaker index forHm0based on the field wave data.

Coastal wave data, on the other hand, were not much affected by wave non-linearity effects, and the zero-crossing significant height H_1/3 was almost the same as H_m0.

Many data points in Fig. 4.3 belong to nonbreaking condition, but what interests us is the upper envelope which provides an estimate of the upper limit of breaking wave height. The curve of dash-dot line in Fig. 4.3 has been calculated by Eq. (4.6) for the slope of s = 0.0143 (1/70) with the coefficient being reduced to A = 0.136 (80%). Similar with the laboratory data shown in Fig. 4.2, the wave height ratio H_m0/h is higher than the incipient breaker index of significant wave expressed by Eq. (4.7). It is because the breaker index increases inside the surf zone as the percentage of breaking waves increase.

It is seen that the energy-based significant wave height H_m0 on gentle slopes does not exceed 0.7 times the local water depth except for the low-steepness swell in very shallow water. For the range of h/L₀> 0.03, the upper limit of significant wave height is about 0.6 times the local water depth. Some data points above the dash-dot curve are those of DUCK85 and SUPERDUCK, which were conducted on the beach steeper than the beach in Ajigaura.