a. Previous studies of wave breaking probability

Measurement of the breaking probability of dominant ocean waves is a challenging task, particularly when carried out by unattended instruments in the open ocean. Here, there is no consensus on exactly which properties need to be measured to provide unambiguous results on the occurrence and scale of the breaking waves. Identifying which environmental variables determine the breaking probability has been even more elusive.

While visual recording of wave breaking occurrence is arguably the most reliable method, other methods have been reported in the literature based on measurements of surface elevation, velocities, and accelerations. Various signal-processing techniques have been explored to extract breaking occurrences from such wave records, including Fourier and wavelet analysis. Other detection methods have used measurements of void fraction, whitecap coverage, subsurface turbulence, underwater sound, infrared properties of the surface, and microwave backscatter. Reconciling different sets of observations is complicated by the lack of overlap among the various techniques. Finally, we note that both wave field parameters and wind parameters have been proposed as appropriate independent variables, as described below.

b. Wave-related dependence

Much attention has focused on the joint probability density of individual wave heights and periods. Underlying this approach is the notion of a critical local steepness, or crest acceleration, based on results for limiting steepness of steady irrotational waves by Stokes (1880). These kinematic aspects of breaking onset have been investigated experimentally by Longuet-Higgins (1986), among others, and have been extended by several authors. Snyder and Kennedy (1983) proposed a probability model for breaking in the spectrum based on a spectral interpretation of the crest acceleration threshold, involving a relationship between the breaking probability and the fourth spectral moment (the acceleration spectrum). In Kennedy and Snyder (1983) and Snyder et al. (1983), numerical simulations and a field experiment were undertaken to verify the model, with encouraging initial agreement found between their simulated results, field results, and the model predictions. The acoustic observations of Ding and Farmer (1994) also provided support for this breaking probability model, but with some uncertainty in the underlying model parameter α. This class of probability model was extended by Yuan et al. (1986) and Papadimitrakis et al. (1988), among others. A similarly based parameterization of breaking influence was proposed by Huang (1986), using as the primary independent variable the significant wave steepness parameter S = 〈η²〉/λ₀, where 〈η²〉 is the mean square wave height and λ₀ is the dominant wavelength.

Other observational studies have investigated the use of a local wave steepness threshold. These studies indicate that this approach is not robust when applied to the onset of wave breaking at sea, where a rather diffuse separation between breaking and nonbreaking wave populations is observed on the basis of individual wave steepness. Holthuijsen and Herbers (1986) visually monitored wave breaking events that occurred at a wave-rider (acceleration) buoy in an open ocean field site. They detected wave breaking events over a wide range of length scales, and these were registered synchronously with the buoy signal. The joint probability analysis of heights and periods of breaking and nonbreaking waves showed considerable overlap and did not clearly resolve breakers on the basis of individual wave steepness. Other findings of this study were that the probability of breaking was correlated with wind speed, varying from 0.10 to 0.16 within the observed range of wind speeds of 8–12 m s⁻¹. Also, it was observed that two-thirds of the breaking waves were observed to occur in one-third of the wave groups. Further, these authors reported that breaking occurred most commonly in the center of a group, highlighting the importance of groupiness in the breaking process. This confirmed the earlier findings of Donelan et al. (1972) that breaking was visually correlated strongly with wave group structure.

c. Dependence on wind forcing

Several studies using different breaking detection methods have attempted to establish a connection between breaking probability and various atmospheric forcing parameters. The earliest such studies were of a related quantity W, the percentage whitecap cover. A very strong wind speed correlation W ∼ U^3.75₁₀ was proposed by Wu (1979). It should be noted, however, that W may be rather different from the dominant wave breaking probability because it integrates whitecap contributions of all wave scales and may include residual foam cover.

The field study of Thorpe and Humphries (1980) used a breaker detection scheme based on a jump in the surface slope associated with the leading edge of the spilling region. It is believed that this method primarily registers dominant wave breaking events. These authors showed a clear correlation between U₁₀ and the number of breaking waves per wave period, but did not provide a quantitative relationship for the observed correlation. A similar wave breaking detection method was also used by Longuet-Higgins and Smith (1983). Their instrument operated in wind speeds ranging from 1 to 14 m s⁻¹ and they reported breaking probabilities of up to a few percent. The breaking probability increased with wind speed and inverse wave age.

The laboratory study of Xu et al. (1986) reported that the breaking probability P (ratio between the number of breaking waves and the total number of waves examined) increased rapidly with wind velocity U and followed a power law P = U^2.2, with P reaching 100% at winds near 20 m s⁻¹ in their wind wave tank. Their detection method based on a threshold of the temporal derivative of the wave height signal is considered to be unsuitable for field conditions involving a broad range of breaking wave scales.

Weissman et al. (1984) used the high-frequency spectral band (18–32 Hz) of fine wire wave probe records to quantify breaking wave occurrences for wind waves on Lake Washington. For a wind velocity of about 6 m s⁻¹ at a fetch of 7 km, they found the probability of breaking P was 8.64% with a breaking rate of 17/km. In this approach, breaking of the different scales is not easily separated and the high signal bandwidth is not usually available in ocean wave records.

Katsaros and Atakturk (1992) used continuous video records of breaking event occurrences at their wire wave probe on Lake Washington. They correlated their results for spilling and plunging breakers using five different atmospheric parameters, both individually and in various combinations, for the dependence of breaking probability on wind forcing. They found the best correlation was 0.81 for the following fit to their data:

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where u∗ is the friction velocity and c_p is the phase speed of waves at the spectral peak. In this correlation, the second term describes the influence of the wind stress and the third term reflects the stage of wave development. While there was no explicit information given on the range of scales of breaking waves, we associate the majority of their reported spilling and plunging wave breaking events with the dominant waves, as they reported microscale breaking statistics separately.

d. Other detection methodologies

There have been other closely related studies based on a range of conventional and innovative detection methods (e.g., Stolte 1994; Melville et al. 1992; Ding and Farmer 1994; Su et al. 1996; Bass and Hay 1997; Jessup et al. 1997; Gemmrich and Farmer 1999; among others), but these mostly have not provided breaking occurrence statistics for the dominant waves alone. An exception is the Ding and Farmer (1994) study that used a small array of four omnidirectional hydrophones deployed on a freely drifting float to map the location and movement of breaking waves in the ocean. Their technique detected the breakers and measured their propagation speed and geometrical evolution. Very recently, Gemmrich and Farmer (1999) described measurements of the scale and occurrence of breaking waves at sea for a wide range of conditions, based on void fraction conductivity measurements. They found that breaking probability was not highly correlated with either wind speed or wave age. Interestingly, during one deployment they observed that the breaking probability continued to increase after the wind had dropped from 17 to 7 m s⁻¹. Their observations did show a high correlation, however, when framed in terms of a relative wind input strength parameter that compares the wind input strength to that of a fully developed windsea at that wind speed.

From the literature, it is clear that there is little consensus on the environmental parameter or parameters that control the onset of breaking of the dominant waves. In this paper, we extend the discussion on this fundamental issue through our investigation of a diverse new dataset. Our analysis offers further confirmation that the traditional approach of relating breaking probability to the standard wind forcing parameters provides a relatively poor correlation when results are analysed for the broad range of wave conditions represented in our dataset. We then propose and investigate an alternate primary dependence for breaking probability of the dominant ocean waves based on the results of numerical experiments on nonlinear wave group hydrodynamics. The theoretical background is summarized in the following section. As an extension of this approach, we report preliminary breaking statistics observed for frequency bands above the spectral peak and examine how closely breaking probability for a higher-frequency band conforms to the dependence observed for the dominant waves at the spectral peak.

a. Other possible factors influencing breaking onset

The wind drift, upper ocean shear, wind forcing, breaking-induced turbulence, and three-dimensionality have all been recognized as potentially important factors in influencing the onset of breaking. There is little available direct observational evidence of the importance of these factors, so we sought guidance from numerical simulations of idealized cases where possible. We used the DP code to examine cases of a superimposed linear shear layer considerably stronger than the levels recently reported by Terray et al. (1999) and found only a small (<10%) reduction in the evolution time to breaking. We also found that that the influence of very strong wind forcing was not of primary importance for the evolution time to breaking. In one case, the time to breaking was reduced by 15%, while in another case, the wind forcing delayed the onset of breaking. While these model findings remain to be validated experimentally, they suggest strongly that nonlinear wave group dynamics is the primary mechanism involved. Another related influence is the background turbulence injected by the field of breaking waves. While central to the effective eddy viscosity in the surface layer (e.g., Craig and Banner 1994), its local influence on breaking onset is not known, but we anticipate this to be secondary. A more direct issue is three-dimensionality, as the dominant ocean waves have a small but finite directional spreading distribution and large whitecaps are often short-crested in appearance. A body of literature on three-dimensional effects is beginning to develop and recent laboratory investigations (e.g., Kolaini and Tulin 1995; She et al. 1997; Nepf et al. 1998) have quantified modest differences in breaking-related local steepness and energy losses between these short-crested wave trains and comparable two-dimensional wave trains. Nevertheless, it is apparent in all of these studies that the evolution to breaking is still underpinned by their nonlinear wave group behavior.

On this basis, we propose that the key factor associated with the onset of wave breaking is nonlinear momentum and energy transfer within wave groups with direct wind forcing, shearing current influence, and three-dimensionality of secondary importance. This approach suggests an alternative basis for parameterizing the environmental dependence of the average dominant wave breaking probability in terms of the nonlinearity of the wave field. We look to the available field data described in section 3 for observational support for these ideas and for a suitable environmental parameterization in section 4.

a. Black Sea data

This section reports the results of a comprehensive wave breaking analysis of Black Sea data. The experimental arrangement involved visual surveillance of waves passing over a wave probe, with collocated breaking events labeled electronically by an observer. With this data we were able to investigate the distribution of breaking probability with respect to the distance from the spectral peak. The first 12 records (Table 1) were obtained in 1993 during the 31st cruise of the R/V Professor Kolesnikov (henceforth PK) operated by the Marine Hydrophysical Institute (MHI) in Sebastopol. The last four records are from the 1986 field experiment conducted by the Air–Sea Interaction Department at MHI.

The PK wave data were recorded using an accelerometer buoy, as described by Babanin et al. (1993). Briefly, the buoy diameter was 0.58 m and its operational bandwidth was 0.08–1.0 Hz, which easily covered the wave frequencies of interest. It was deployed around 100 m from the ship to avoid any interference between the ship and the wind and wave fields. Record lengths of 34 min to 68 min were acquired using a sampling frequency of 4 Hz. One of the authors (AVB) observed the buoy from the vessel and triggered an electronic signal to register the passage of a whitecap over the buoy. This signal was recorded synchronously with the buoy data. The observer varied the duration of the electronic label according to the geometrical size of the whitecap, providing an approximate indication of individual breaker dimensions for future analysis. The observer was about 10 m above sea level, allowing a clear view of whitecaps with scales down to the size of the buoy. The environmental data collected simultaneously included 10-min averages of the 10-m wind speed and direction, measured by a standard Russian M63-MP anemometer on the ship’s bow, together with mean water and air temperature data.

The first 10 records were obtained with the ship anchored about 3.2 to 6.4 km offshore on the relatively flat west coast of the Crimea. Relatively young, strongly forced, fetch-limited waves with U₁₀/c_p = 2.7–4.1 were observed. Their spectral peak frequencies f_p ranged from 0.3 to 0.44 Hz. The wind speed U₁₀ ranged from 11.7 to 15.0 m s⁻¹. Records 16 and 18 were gathered from the drifting vessel during the same PK cruise in the Eastern Mediterranean Sea, during which older waves were observed approaching full development, with U₁₀/c_p = 0.9 and 1.15. Their rather low observed peak frequencies of 0.16 and 0.17 Hz corresponded to 8.4 and 11.2 m s⁻¹ wind speeds respectively. The last four wave records were recorded from an oil rig situated 60 km offshore in the northwest region of the Black Sea in horizontally uniform water of 30-m depth. The four-wire (0.3-mm diameter nichrome) directional array wave gauge used to collect the wave data, as well as the platform and its location, are described in detail by Babanin and Soloviev (1998a,b). The wave sensors had a linear regression between readings and water level accurate to within 1% of the greatest deviation and were able to resolve wave frequencies up to approximately 10 Hz. The wave gauge was located about 15 m away from the nearest platform leg and only waves unperturbed by the platform influence were recorded. For records 211 through 244, the fetch exceeded 400 km and was practically unlimited. The records were 12.5 min long and were recorded using a sampling interval of 0.05 min. The observer located 16 m above the wave probe array monitored one of the wave probes and triggered a signal whenever a whitecap of any size occurred at the probe. The wind wave data, obtained with much higher frequency resolution than previous records, embraced moderate to well-developed waves with peak frequencies f_p = 0.16–0.27 Hz, wind speeds U₁₀ = 8.7–10.7 m s⁻¹, and wave development stages U₁₀/c_p = 1.0–1.7. The complete environmental dataset included 1-min averaged wind speeds, 5-min averaged surface current speed, and direction measurements at 3-h intervals (Babanin 1988), as well as water and air temperatures recorded every 3 hours. To avoid possible airflow distortion by the platform, an M63-MP anemometer was deployed on a tower 42 m above sea level, well above the deck buildings. The drag coefficient correlation of Large and Pond (1981) was used to estimate the 10-m wind speed from the observed data.

The surface elevation time series with labeled individual breaking events were analyzed to determine individual breaker properties and breaking statistics. The accelerometer data series from the PK cruise were segmented to reduce the effects of low-frequency trends and integrated twice prior to analysis. With shorter waves riding on longer ones, occasional confusion could occur as to which wave scale was actually breaking but, since dimensional information on the individual whitecaps was available from the label length, we believe that uncertainties of this type were negligible.

b. The Southern Ocean data

This dataset was obtained during the Southern Ocean Waves Experiment (SOWEX) staged in June 1992 (see Banner at al. 1999). Nadir-look video images of the ocean surface were recorded from a research aircraft flown upwind at altitudes of 680 m (10 June) and 250 m (11 June) over the Southern Ocean 225 km southwest of Tasmania. The video frame digitization resolved 320 × 200 pixels, providing spatial resolutions of 1.62 m/pixel and 0.59 m/pixel, respectively. Table 2 provides a summary of the experimental conditions.

A video recorder interfaced to a personal computer was programmed to play back an assigned number of sequential video frames. During the processing, the software digitized these frames and counted the number of pixels in each whitecap. The number of breakers passing through a chosen point on the screen was counted and estimates were made of the distances and travel time of these breakers. After correction for the wind and aircraft velocities, the phase speeds and frequencies of the breakers were calculated. We only considered breakers propagating with speeds close to the phase speed of the spectral peak. The methodology is very time consuming and only one representative breaking probability was obtained for each day. This was the only way to include breaking statistics for the very long Southern Ocean waves, consistent with our goal of including the widest possible range of wavelengths.

The processed video segments were 5 and 9 min long, respectively, consistent with the duration of the steady upwind flight legs at constant aircraft altitudes and representing approximately 30 and 40 min records at a fixed point. The peak frequencies of 0.083 and 0.074 Hz have wavelengths of 220 m and 290 m, respectively. Corresponding to the prevailing 10-m wind speeds of 16 m s⁻¹ and 20 m s⁻¹, these waves were nominally just below and just above the fully developed Pierson–Moskowitz (1964) state. Further details on the SOWEX instrumentation are given in Banner et al. (1999).

c. Lake Washington data

The Lake Washington dataset, described by Katsaros and Atakturk (1992), represents the other extreme of natural deep water wind waves with very short wavelengths. These short fetch-limited waves were generated by light winds. For this dataset, U₁₀ ranged from 3.4 to 6.8 m s⁻¹, with peak wave frequencies f_p = 0.55 to 0.75 Hz and wave development stages of U₁₀/c_p = 1.5 to 2.5. The waves were measured using a wire wave gauge, with breaking events recorded by a video camera observing the wave gauge. A detailed analysis provided the number of plunging, spilling, and microscale breakers for each of the sixty-six 17-min records. In the present paper, we combined the available plunging and spilling breaker statistics to quantify the breaking waves in the spectral peak band.

a. Dependence on the dominant wave steepness

To address the primary goal of this study, a parameter was needed that reflected the fundamental role of nonlinear group dynamics in determining breaking onset. Since the present focus is on mean breaking characteristics, it was very desirable to be able to relate this parameter to standard properties of wave fields, such as wave height spectra.

As discussed earlier in section 2, DP reported the existence of an initial carrier wave steepness threshold that separates wave groups that evolve to breaking from those that undergo recurrence without breaking. Their result, reproduced in Fig. 3a, shows that the breaking/recurrence threshold depends on the number of waves N in the group, although this dependence is seen to be relatively modest beyond N = 3. On the other hand, the Banner–Tian growth rate parameters have a threshold that appears to be less sensitive to N. However, it is not possible to determine these parameters from routinely available wave data.

Based on these considerations, we investigated the potential suitability of the significant wave steepness at the spectral peak as the primary parameter associated with the onset of dominant wave breaking. Using the DP code, we computed T_b, the time to breaking onset from an initial nearly uniform wave group configuration, as a function of initial steepness (ak)₀ for N = 5. This result is shown in Fig. 3b. We concluded that initial steepness (ak)₀ had the strongest influence on T_b, with the inverse time to breaking almost quadratically dependent on (ak)₀. We also found that for a given initial steepness of 0.12, T_b was insensitive to the choice of N for N ≥ 5. As discussed in section 1, the influence on T_b of other possible variables such as the strength of wind forcing and background shear was also found to be secondary. Our computational approach is based on investigating the time to breaking of a quasi-uniform wave group. Clearly, this class of initial condition is not realizable in open ocean situations, so our results on the behavior of T_b are only intended to be suggestive of potential dependences for open ocean correlations.

We conjectured that the stochastic nature of group lengths in ocean waves should average out the variation with N of the hypothetical critical initial steepness suggested by the numerical experiments as exerting an underlying influence on the onset of breaking. This behavior is paralleled in laboratory wave tank experiments, for example, Rapp and Melville (1990). This led us to propose its counterpart for real ocean waves: a significant spectral peak slope threshold based on the observed wave spectrum. We note that Huang (1986) previously used a closely related parameter, the significant wave slope, in his modeling of breaking influence on upper ocean dynamics.

b. Definitions of key breaking wave parameters

The breaking of local wind-generated seas often takes place in the presence of low steepness background swells from distant storms, and we assume here that these low steepness swell waves have little effect on the dynamics of breaking events of the dominant local wind sea. Hanson and Phillips (1999) describe how the swell contributions may be filtered out to provide the significant wave height of the local wind sea.

A significant wave steepness may be derived simply from H_sk_p/2, where H_s = 4m₀ is the significant wave height of the local wind sea and k_p is its spectral peak wavenumber. This steepness measure, however, contains some contribution from the shorter, higher-frequency wind wave components. Thus, for our present purpose, we can conveniently define the significant spectral peak steepness of the local wind sea by

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where

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Here F(f) is the frequency spectrum of the windsea after filtration of any background swell. We propose that ε is an appropriate parameter as it not only reflects the mean steepness of the dominant waves, but also provides a measure of their nonlinearity. Its suitability as a measure of nonlinear energy flux within dominant wave groups and breaking onset is assessed below.

a. Dependence on surface shear

A wind-driven vertical shear current is usually present in the ocean, but its magnitude and depth distribution is a complex research issue (e.g., Craig and Banner 1994). The results of Teles da Silva and Peregrine (1988), Millinazzo and Saffman (1990), Banner and Tian (1998), among others, suggest that a background linear vertical shear current should have a dynamical influence on the evolution to breaking of a deep-water gravity wave train. Banner and Tian (1998) reported that the presence of a superposed linear vertical shear current contributes to breaking onset by marginally reducing the time to breaking. The actual wind-induced surface layer shear is a complex and elusive region for reliable measurements because of the presence of the orbital motions of surface waves as well as the breaking-induced current shear (Terray et al. 1999). For the purpose of the present paper, we seek a surface shear measure that has the correct order of magnitude and is readily estimated from routine measurements. We proceeded from the recent PIV wind wave measurements of Peirson and Banner (1999, manuscript submitted to J. Fluid Mech.), who reported typical values of the surface drift to wind friction velocity u_s/u∗ ∼ 0.03, exclusive of the Stokes drift contribution. Based on typical corresponding U₁₀ levels, we adopted a representative magnitude for the surface layer wind-induced current u_s ∼ 0.01U₁₀. As our indicative [O(0.1)] shear influence parameter for the wave field, we chose the ratio of the wind-induced surface current u_s to maximum orbital velocity u_o of a linear surface gravity wave with height equal to the significant wave height H_s:

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The Black Sea and Lake Washington breaking wave datasets plotted in Figs. 5a,b show that the breaking fraction increases with the shear parameter, although with a far lower visual correlation than seen for the peak steepness. Also, the existence of a shear threshold is not as evident as with the steepness. These conclusions are consistent with the secondary role of vertical shear reported in the numerical computations discussed above. To reflect this possible secondary dependence, we retain Δ as a potentially important variable in the ensuing analysis through an assumed linear perturbation term of the form (1 + Δ).

b. Dependence of the breaking probability on the wind forcing strength

The dependences of breaking probability on wind speed and on wave age alone were shown in Figs. 1 and 2. Our unpublished numerical experiments with wind forcing indicate that it has only a secondary impact on the calculated time to breaking relative to the unforced case when only nonlinear interactions were operative. We therefore chose to incorporate the wind forcing dependence through an assumed linear perturbation term of the form (1 + γ), where γ = U₁₀f_p/g is an inverse wave age parameter with a dynamic range from 0 to ∼0.5 for field conditions. We note, however, that for very strongly forced conditions such as occur in wind wave tanks, this parameter can become significantly larger than 1. For such strongly forced conditions, the fractional rate of addition of momentum from the wind can be substantial and the primary dependence on hydrodynamic forcing may not be valid for such conditions.

c. Overall dependence for the combined datasets

In this section we examine how well the diverse datasets from the three measurement sites conform to the proposed parameterisations.

First, the composite data dependence on the dominant wave steepness ε alone is shown in Fig. 6 using log–log axes to highlight visually any power law behavior. Note that this plotting format suppresses all data points with zero breaking probability. The fit to the diverse dataset is very encouraging and the legend shows the best-fit statistics using a standard analysis with the ordinate data taken as the independent variable. This approach was motivated by our belief that the determination of the number of observed breaking events had intrinsically less error than the spectral peak steepness determination. In any event, this method gave a better visual fit to the data. The overall correlation was found to be close to quadratic, with a correlation coefficient of 0.78.

Second, for the combined dataset, the dependence of the spectral peak breaking probability on the composite steepness/wind-forced shear parameter ε(1 + Δ)(1 + γ) is shown in Fig. 7. This combined parameter incorporates the observed trends of breaking probability primarily on the spectral peak steepness with a secondary dependence on the surface layer shear and wind forcing parameters that vanishes when surface layer shear and wind input are absent. The correlation coefficient is marginally higher at 0.81, with a slightly reduced exponent and reduction of the overall error tolerance on the fitted parameters. This modest improvement is consistent with the notion of a secondary influence on breaking probability due to a superposed vertical shearing current and wind forcing in conjunction with the primary influence due to nonlinear intragroup modulation effects.

d. Note on long-term variability

An interesting result on the sensitivity of the correlation to the averaging time was found during our analysis of the Black Sea data. While 20-min records are usually regarded as sufficient for determining wave spectra, the number of wave groups required to provide stable breaking probabilities is found to be longer. The result of splitting several of the 40-min records into two 20-min records produced significantly more scatter in the dependence of b_T on ε. This scatter is shown in Fig. 8 for the nine cases identified in Table 1. Our limited duration data records and the relatively infrequent onset of dominant wave breaking did not allow us to further identify the source of this variability and future studies involving longer data records are clearly needed.