1. Introduction
Surface waves have been described as the “gearbox” between the atmosphere and ocean (Ardhuin et al. 2005). In particular, wave breaking plays an important role in many air–sea exchange and upper-ocean processes. At moderate to high wind speeds the momentum transfer from wind to ocean currents passes through the wave field via wave breaking. The breaking of surface waves is responsible for the dissipation of wave energy, and thus wave breaking is a source of enhanced turbulence kinetic energy (TKE) in the near-surface layer. Breaking waves not only transfer energy, momentum, heat, and gases from the atmosphere to the ocean surface layer, they also foster aerosol generation and latent heat flux due to sea spray. Breaking waves also disperse pollutants and generate underwater sound. Comprehensive overviews of the role of wave-induced turbulence in upper-ocean dynamics and air–sea exchange processes are given by Thorpe (1995), Melville (1996), and Duncan (2001).
Furthermore, wave breaking affects wave development as it dominates the dissipation of wave energy and controls wave growth. To improve our understanding of wave-breaking-related processes a twofold approach is necessary: (i) detailed process studies of energy dissipation rates, mixing, sound generation, etc. and (ii) knowledge of the occurrence and scale of wave breaking. Here we focus on the latter.
Defining multiscale breaking probabilities by Eq. (1) requires knowledge of the total number Nall of wave crests of a certain scale passing a fixed point. However, this measurement is not commonly available. A more practical measure was introduced by Phillips (1985). Realizing that the scale of a breaking wave may be partly represented by the length of the breaking crest and its propagation speed and direction, he defined Λ(c, θ), the spectral density of breaking wave crest length per unit area with speed c and propagation direction θ. Note that this is a measure of the length of the actively breaking crest perpendicular to the wave propagation, and thus does not give the whitecap area, which also depends on the swept extent of the whitecap in the direction of breaking. Throughout this article we will use the notation Λ(c) = ∫π−πΛ(c, θ) dθ, except when resolving the directional distribution of the breaking crests is required explicitly.
Any quantitative assessments of energy dissipation rates and momentum fluxes depend directly on the proportionality factor b, and therefore require a thorough understanding of its value and functional behavior. The original conceptual model assumes that the energy dissipation is caused by a thin whitecap propagating down the forward face of the breaking crest and opposing the underlying wave orbital motion. Here we will introduce a complementary dissipation rate model based on the mass redistribution in a breaker. Both concepts predict b ≪ 1. This will be discussed further in section 4.
Although the theoretical framework of energy dissipation rates based on breaking crest distributions was developed two decades ago, observations of Λ(c) in the ocean are still very limited (Ding and Farmer 1994; Phillips et al. 2001; Melville and Matusov 2002) and show a wide range of variation. We describe observations of breaking crest distributions in the open northeast Pacific at wind speeds of 10–12 m s−1 and examine the implied energy dissipation rate, momentum flux, and breaking frequency. Preliminary results have been reported in Gemmrich (2005).
2. Measurement approach
Video imaging of the ocean surface spans a wide range of applications. For example, this method has been used to analyze modulations of the phase speed of short gravity waves (Gotwols and Irani 1980), small-scale surface renewal processes (Gemmrich and Hasse 1992), total whitecap coverage (Monahan and O’Muirheartaigh 1986), and characteristics of individual breakers (Melville and Matusov 2002). Recent technological developments allow for the automated processing of digital video records.
Observations of the surface wave field were taken during 24 September–10 October 2000 as part of the Fluxes, Air–Sea Interaction and Remote Sensing (FAIRS) experiment aboard Research Platform FLIP in the open ocean 150 km offshore of the central Californian coast (Gemmrich and Farmer 2004). Two analog black/white video cameras were mounted on R/P FLIP: camera 1 was located on the railing at approximately 12 m above still water level and camera 2 on the center boom at about 9-m height. The nominal field of view of camera 1 is 15.4 m × 20.5 m and is slightly distorted due to a small camera inclination. Camera 2 is nadir looking, resulting in a 9 m × 12 m footprint, which partially overlaps the view of camera 1. R/P FLIP drifted freely, orienting itself at about 15° to the wind direction. Thus, the video images are downwind and several meters to the side of FLIP’s hull, and are thought to represent the undisturbed wave field.
A sonic range finder, mounted on a boom next to the center boom, recorded the surface elevation at 4-Hz sampling rate (data courtesy of A. Jessup, APL UW, 2000, personal communication). The directional wave field was monitored with a set of eight 100-kHz side-scanning Doppler sonars mounted on the hull at three different depths (15, 31.5, 91.5 m) oriented in four orthogonal directions. In this way the surface values of the wave orbital motion along the sonar direction are recorded to a range of roughly 300 m. The directional wave spectrum is extracted from the sonar data using the maximum likelihood method (Trevorrow 1995).
a. Processing of video recordings
Our goal in analyzing the video recordings is to extract the length and propagation speed of actively breaking wave crests within a known surface area. As a first step, the analog video signal is converted into the digital MPEG format of 640 × 480 pixel size (Fig. 1, top). The resulting pixel size of 3.2 × 10−2 m (1.9 × 10−2 m for camera 2) resolves even the smallest visible whitecaps without limitations.
In this study we focus on the breaking crest length, not the total whitecap area. The breaking crest length is best extracted by taking the difference between successive images. In these differential images, the forward edge of a breaking crest results in a strong positive signal, stationary features are eliminated, and the rear side of a whitecap leads to a negative signal (Fig. 1, second panel). The theoretical minimum speed that can be resolved is given by the pixel sizes times the frame rate. Thus, for the original frame rate of 30 frames per second, differential signals are limited to propagation speeds >1 m s−1. Therefore, we downsample the sequence of differential images to 10 frames per second. This improves the theoretical resolvable propagation speed to >0.3 m s−1, corresponding to a 1 pixel displacement. Hence, a breaker moving at 1.5 m s−1 travels 5 pixels from one frame to the next. Next, the differential images are converted into binary images. This conversion is based on a threshold that is determined manually by trial and error and remains fixed. Any interconnected positive-valued pixels are considered as individual image objects. One of the advantages of analyzing differential images is that the average value of the differential images depends only marginally on the overall brightness of the raw image, thus justifying the common threshold value approach taken here. Ideally, the binary images would contain only the forward edge of breaking crests (Fig. 1, third panel). However, occasionally the tether lines and the floatation of the deployed instrumentation also remain as positive signals. The properties of these objects are rather different from genuine whitecaps and these signals are removed during the subsequent analysis, as will be discussed below. In the next step, a standard Matlab image processing routine is applied to approximate each object in the binary image by an ellipse. Its centroid location (Xi, Yi), major and minor axis length and orientation, Li, Wi and ϕi, respectively (Fig. 1, bottom), as well as the corner locations and the total area of the object are recorded onto a file. Occasionally the thresholding mechanism splits a single whitecap edge into several segments, as seen in the third image of Fig. 1 (third panel). Therefore, neighboring objects within a distance less than 20 pixels are combined into a single object if the following conditions are met: the major axis orientations of the segments are within 15° of each other and the linear fit through the centroids is aligned to within 15° with the average of the major axes orientations. If these conditions are not met, the segments are recorded as individual objects.
The task of extracting the speed and length of breaking crests consists of (i) tracking recorded objects across subsequent images and (ii) verifying that these objects, indeed, represent whitecaps. For each object of a starting image the corresponding object in the following image is searched for. The search is based on following criteria, which were established by trial and error and confirmed by visual cross checking for individual breaking events:
the propagation direction is within ±90° of the wind direction
the propagation direction is within ±30° of the minor axis orientation
the centroid displacement is limited to Δs < 1.5cp/Fs, where cp is the dominant wave phase speed and Fs is the image frame rate
the changes in object area and the major axis length are less than 25%.
If a corresponding object can be identified, it is flagged as being assigned and then the search is continued in subsequent images until the object propagates outside the image or it ceases to exist; that is, in the raw video image the contrast between the whitecap and the background is below a certain threshold or the whitecap remains stationary. Generally, whitecaps are tracked across a range of N = 5 to 12 images, corresponding to 0.5–1.2 s. Their displacement speed ci = Δsi Fs is estimated for each pair of successive images, where Δsi is the displacement of the object centroid from image i to image i + 1.
The whitecap displacement velocity is the combination of the true whitecap propagation, that is, the nonlinear phase velocity of the breaking wave cbrk, plus any advection owing to underlying long waves with velocity vi so that ci = cbrk + vi. The advection velocity vi depends on the phase of the long wave and is highest at the long-wave crest. Its determination will be described later. Since short waves break near the crest of the long waves, the whitecap displacement speed is often largest at the beginning of the breaking. We assign the whitecap record a single whitecap displacement speed c = c1/3, that is, the mean of the highest one-third of the speeds within the record, a duration T = N/Fs, a breaking crest length L = max(Li), a mean propagation direction θ = tan−1[(YN − Y1)/(XN − X1)], and a time of occurrence t = t1. These whitecap properties are recorded to a file for further processing, including the correction for riding wave advection.
If no corresponding object can be identified in the subsequent image based on the above search criteria, the object is flagged as a nonwhitecap signal and excluded from further processing. Visual tests revealed that this scheme removed more than 90% of nonwhitecap signals, but captured all large-scale breaking waves (Fig. 1) and close to 90% of whitecaps with weaker contrast.
b. Breaker speed extraction
As stated above, the whitecap displacement is due to a combination of the propagation of the breaking crest, which occurs at the phase velocity of the breaking wave, and advection by the orbital motion of underlying longer waves. Therefore, an estimate of the long-wave orbital motion is required to extract the true phase velocity of the breaking wave.
The speed of the orbital motion at the crest of a monochromatic wave is v = aω, where a is the wave amplitude and ω the wave frequency. As described below, the advection speed owing to long waves may be estimated from a time series of the surface elevation.
3. Observations
The FAIRS experiment included wind conditions ranging from almost calm up to 15 m s−1. The resulting wave field ranged from pure swell conditions with significant wave height Hs < 1 m to young wind seas on top of swell with Hs > 4 m. Here we report on four datasets recorded under various wind forcing and wave field conditions (Fig. 2). Significant wave height Hs and peak period Tp are based on the acoustic rangefinder data and are in good agreement with wave property estimates from data of side-scanning sonars mounted on FLIP’s hull. The wind stress estimates τ (provided by J. Edson, WHOI, 2000, personal communication) were obtained by the eddy correlation method.
Dataset I, 0950–1450 UTC 29 September 2000, follows a period of increasing wind speed. For several days prior to this dataset winds were very light (<4 m s−1) and the wave field was dominated by swell with Hs < 1 m. Approximately 11 h before the start of this dataset the wind speed u10 increased steadily for a period of 10 h and peaked at u10 = 12.8 m s−1. Throughout the dataset the wind speed stayed nearly constant at 12 m s−1 with steady direction from west-northwest, creating unlimited fetch conditions. At the beginning of the dataset the significant wave height Hs was 2.8 m and continued to rise to 3.1 m. The dominant wave period was fp = 0.13 Hz and the wave age cp/u* ≃ 26, where u* is the friction velocity in air. This dataset represents a developing sea, based on the fact that the wave height increased during this deployment and despite the relatively large wave age.
Dataset II, 0005–0230 UTC 3 October 2000, occurred after 3 days of sustained wind speed u10 > 10 m s−1, with well-developed windwaves at wave age cp/u* ≃ 33, significant wave height Hs = 3 m, and dominant frequency fp = 0.11 Hz. The wind speed was u10 ≃ 11.5 m s−1 and wind direction from west-northwest. This dataset represents a developed sea.
The third dataset, 2345 UTC 3 October–0245 UTC 4 October 2000, approximates a fully developed sea. The dominant frequency remained unchanged at fp = 0.1 Hz, the wind speed had increased to u10 ≈ 12.5 m s−1 and the significant wave height reached Hs = 3.2 m. The wave age was cp/u* ≃ 33. The wind direction remained west-northwest, that is, an unlimited fetch condition. For this dataset, the data coverage of camera 1 is limited to 0145–0230 UTC 4 October.
Dataset IV, 1100–1430 UTC 10 October 2000, occurred at the end of a rapid increase of wind speed from less than 5 to about 13 m s−1. The wind direction stayed constant at 270°, resulting in unlimited fetch. The significant wave height increased from <2 to >4 m. The dataset itself covers a period of slow increase in wind speed, from 11.8 to 13 m s−1, but with a significant increase in wave height from Hs = 2.5 to Hs = 3.9 m and a reduction in dominant wave frequency from fp = 0.16 to fp = 0.10 Hz. The wave age was cp/u* ≃ 29. This dataset represents a growing sea superimposed upon a significant swell and will be labeled mixed sea. Lighting conditions were poor during this dataset, which likely results in an underestimation of small-scale breakers.
a. Scale of breaking waves
Our analysis extracts the velocity cbr of every breaking event occurring within the video footprint. It has been suggested that due to strong nonlinear effects the phase speed cbr of a breaking wave is less than the phase speed c of a corresponding linear wave, cbr < c (Melville and Matusov 2002; Jessup and Phadnis 2005). Here we take the observed phase speed of breaking waves cbr as a surrogate for the scale of wave breaking, and its normalization with the dominant phase speed cbr/cp yields information on the spectral occurrence of wave breaking.
Wave breaking occurs over a wide range of scales (Fig. 3). However, the breaking scales cover different ranges of the wave spectrum, depending on wave development. In the developing sea case, phase speeds of breaking waves span from approximately 1/10 of the dominant phase speed (i.e., with a wavelength corresponding to 1/100 of the dominant wavelength) up to the dominant wave speed. At the other extreme, in the fully developed sea we observed hardly any breaking at scales corresponding to phase speeds larger than about 0.4 cp. As the wave age increases, the distribution of breaking scales narrows significantly and the peak of the breaker phase speed distribution shifts from about 0.4 cp to 0.2 cp.
For all four datasets the distributions of breaking wave phase speeds obtained from camera 2 and camera 1 (Fig. 3) are in good agreement, despite the distortion introduced by the oblique view of camera 1. The largest scales of wave breaking seem to be not fully resolved by camera 2, resulting in a cutoff of breakers with phase speed greater than approximately 14.5 m s−1. However, according to camera 1 data, this cutoff affects only <1% of breaking events.
b. Breaking crest length distribution Λ(c)
For each breaking event the whitecap propagation speed cbr, the mean propagation direction θbr, the length of the major object axis Lbr, and the event duration tbr are known.
The along-crest length Lbr of the breaking section of wave crests covers more than two orders of magnitude in our datasets, but its distribution is likely a function of the wave field directional spreading. A related, nondimensional property is the breaking aspect ratio Lbr/λbr, where λbr = 2πg−1c2br is the wavelength associated with the breaking crest. The limited field of view of our cameras severely limits the assessment of aspect ratios of intermediate- to large-scale breakers (Fig. 4). Nevertheless, some trends may be seen in our datasets. Based on statistical arguments one can assume that those breaking crests that are only partially included in the camera field of view are, on average, twice as long as the resolved part (assuming one end is visible). Thus, aspect ratios of larger wave scales, say c/cp ≥ 0.5, are most likely not more than twice the directly observed values. We find that aspect ratios are generally smaller than unity, the majority being less than 0.4. Interestingly, in the developing sea most aspect ratios of short breaking waves are greater than 0.4, and thus are much larger than those of intermediate- to large-scale waves. Again, these results are severely limited by the camera field of view and further data are required to test our preliminary findings. However, the subsequent analysis of breaking crest length distributions Λ(c) deals with crest lengths per unit area, and this analysis is not restricted by the camera field of view.
c. The dissipative scale
Phillips’ (1985) concept of spectral wave breaking relates whitecapping to spectral components of the wave field. A natural wave field consists of a continuum of spectral components, and there are different ways to associate an individual whitecap with such components. Here we regard a whitecap as a single entity associated with a single spectral wave component, defined by the propagation speed of the centroid of the breaking crest. A similar approach is taken by Ding and Farmer (1994) who tracked the source of ambient noise associated with wave breaking events. Phillips et al. (2001) identified breaking crests in radar backscatter signals. Since their data do not include azimuthal information, the crest is essentially represented by a single point and therefore is associated with a single speed and wave component.
Recently, Jessup and Phadnis (2005) presented two methods, based on particle imaging velocimetry (PIV) algorithms, to track microscale breakers. In one case they identified breaking crests by a threshold applied to an IR image. The propagation of the crest is then tracked by PIV and the mean velocity is assigned to define the spectral component of the breaker. In the second method, breaking events are identified as regions of increased velocities and the breaker is associated with the mean speed within these regions. Although these methods differ quantitatively, they agree fundamentally in that they each associate a breaking event with a single spectral wave component.
Melville and Matusov (2002) take an entirely different approach and treat a whitecap as an aggregation of breaking crest segments of different spectral wave components. Their video data, recorded from an airplane at 450-m altitude, has a sampling rate of 5 Hz and a pixel resolution of roughly 0.5 m. Based on a brightness threshold, they identify the perimeter of a whitecap and by PIV track the velocity of each pixel of the perimeter. The mean velocity of the rearward pixels defines the whitecap advection, which is then subtracted from the velocity of the forward perimeter pixels. A running average is applied to the velocity calculations, which results in approximately 0.5-m spacing of velocity estimates along the forward edge of a whitecap. This method differs in two ways from the methods described above: (i) only actively expanding whitecaps are included in the analysis and (ii) individual whitecaps contribute to a range of different spectral wave scales, many of them corresponding to phase speeds much slower than the mean advancement speed of the breaking crest.
Generally, the breaking crest length of individual events increases with wave scale (Fig. 4). Therefore, the breaking crest length distribution Λ(c) shows a slightly different behavior than the breaking occurrence rates given in Fig. 3. For the following analysis, dataset I, which occurred during a period of rapid wave field development, has been divided into two segments of equal numbers of whitecap events. The four datasets thus form five data segments.
For all five data segments, Λ(c) peaks at intermediate wave scales with phase speeds of 3–4 m s−1, corresponding to c/cp ≈ 0.3 (Fig. 5). Around the peak of the Λ(c) distribution, the values from the different data segments vary by roughly a factor 2, whereas at the smallest and the largest wave scales the different datasets spread more than one order of magnitude. Phillips (1985) predicts Λ(c) ∝ u3*, and Melville and Matusov (2002) found that the scaling factor (10/u10)3, where u10 is the 10-m-height wind speed (m s−1), collapsed their datasets recorded at different wind speeds (7.2, 9.8, and 13.6 m s−1). However, in our four datasets this scaling factor varies by less than 15% and therefore does not significantly reduce the spreading between Λ(c) values of the different datasets.
At scales larger than the peak of the observed Λ(c) distribution, Λ(c) indeed falls off approximately as c−6, consistent with the equilibrium range concept. However, the drop-off of Λ(c) values at wave scales corresponding to c ≤ 4 m s−1 is somewhat unexpected. To test whether the shape of the Λ(c) curve could be caused by systematical biases introduced in the processing scheme, the limitations of its spatial and temporal resolutions are estimated. The phase speeds of the two lowest bins in Fig. 5 are c = 1.5 and c = 3 m s−1, corresponding to wavelengths λ = 1.4 and 5.8 m and wave periods τ = 1 and 2 s. Assuming that the duration of the breaking process is longer than 0.5τ and the breaking crest length L is 0.7 times the downwind extension of the breaking patch (Phillips et al. 2001) yields L > 0.3λ. That means even for the smallest breaker scales resolved we expect a breaking crest length L > 0.5 m, which is easily detected by the video resolution. However, the processing scheme requires a sequence of at least four images to estimate the object propagation speed. The sampling of the differential images is 10 Hz; that is, only breaking events of duration tbr ≥ 0.4 s are resolved. This limitation might reject some of the smallest breakers as it requires tbr ≥ 0.4τ. At the largest breaker-scale bin with c = 13.5 m s−1, the duration threshold yields a centroid propagation distance of 5.4 m, which is equivalent to 60% of the downwind extent of the video image. Thus, breakers with c = 13.5 m s−1 originating closer than 5.4 m to the downwind edge of the video image will be rejected by the automated processing.
To test the severity of these limitations we processed dataset II at the full differential images sampling rate of 15 Hz. The Λ(c) values obtained at the two different sampling rates show the same functional behavior (Fig. 6). Small differences between the two cases give an indication of the accuracy of the processing scheme. The low Λ(c) values at small wave scales cannot be attributed to the video sampling rate. Small-scale breakers are associated with less air entrainment than larger breakers, and it is possible that the visible contrast of some small-scale breakers is not sufficient to be picked up by the automated processing, thus underestimating the breaking crest length in the lower velocity bins.
During the course of the experiment, the wind direction varied only slightly (Fig. 2). Furthermore, in all four datasets swell and wind waves were aligned within 30°, producing nearly unidirectional wave fields. This is also reflected in the directional distribution of the breaking crest length (Fig. 7). Here we present data in the nondimensional form c2pg−1Λ(c/cp, θ), that is, the breaking crest length normalized by the dominant wave scale k−1p (where kp is the dominant wavenumber) as a function of the breaker phase speed normalized by the dominant phase speed. Again, dominant breakers occur only in the younger sea cases (I, IV), whereas in the developed wave fields little breaking is associated with wave scales corresponding to c/cp ≥ 0.6.
The camera orientations are only known within about 15°. Therefore, for the individual datasets the whitecap propagation direction is rotated to a zero median direction, which can be assumed to be the mean wind direction. In terms of directional distribution we do not see any significant differences between the four data sets (Fig. 7). Breaking waves occupy the entire range of positive downwind directions −90° < θ < 90°, particularly at short to intermediate wave scales (c/cp ≤ 0.5). However, the weighted spreading width [∫π−π θ2 Λ(c/cp, θ) dθ/∫π−π Λ(c/cp, θ) dθ]1/2 is much narrower, about 30°, and does not vary significantly with wave scale. This is roughly consistent with Phillip’s (1985) expectation that “a preponderance of breaking-wave events advance in a direction close to that of the wind.”
4. Discussion
The first, fourth, and fifth moments of the breaking length distribution provide information on the kinematics and dynamics of breaking waves. However, only breaking events exceeding a certain visible brightness contrast due to air bubbles are included in the following analysis. This excludes weaker, small-scale breakers with, say, c < 1 m s−1, as well as microscale breakers.
The breaker speed cbr is somewhat less than the linear phase speed of the breaking wave cln. Here we take α = 0.9, based on wave tank results (Banner and Peirson 2007), which is a slightly weaker reduction than α = 0.8 assumed in Melville and Matusov (2002).
a. Total breaking rate
The breaking rate R at a fixed location is equivalent to the fractional surface area turnover rate at an arbitrary time. Thus, R is an important quantity for air–sea exchange processes, for example, relevant for specifying surface renewal and bubble entrainment in air–sea gas flux models. In the framework of breaking crest length distribution, the breaking rate is a direct quantity, independent of empirical constants. Therefore, R can provide a consistency check for the observed Λ(c) distributions.
The breaking rate R ranges between roughly 50 and 120 breaking events per hour (Fig. 8). Previous observations by various investigators using a wide range of observational techniques (for a summary, see Gemmrich and Farmer 1999) report breaking rates R ≃ 0.1/τp to 0.8/τp, where τp is the dominant wave period. For open ocean conditions these rates relate to the same range of values as observed in this study. The video images provide an independent estimate of the breaking rate RI. In an automated but somewhat crude scheme, we calculate the mean brightness of a fixed spot of 10 × 5 pixels. If the mean brightness exceeds a certain threshold, which is determined by the overall image brightness, a breaking event is registered. We find reasonable agreement between the breaking rates based on the breaking crest distributions RΛ and RI from the video recordings (Fig. 8).
On the other hand, breaking rates inferred from breaking crest length distributions reported in Melville and Matusov (2002) are R ≃ 2 h−1 and R ≃ 6 h−1 for u10 = 9.8 and 13.6 m s−1, respectively. These breaking rates are very low compared to previous observations under similar conditions and are likely linked to the limited video resolution and the elemental whitecap analysis applied in their study.
The breaking rate also provides insight to what conditions are favorable to wave breaking. Banner et al. (2002) found the breaking rate at specific scales to depend on the normalized wave saturation σ(ω), defined in (2), for the corresponding frequency band. This indirect breaking criterion is also verified by our current data. Saturation levels during datasets II and III are lower than for the two other datasets. Moreover, at lower frequencies, corresponding to waves with c/cp > 0.6, the saturation in II and III is below the threshold level (Fig. 9) and no breaking occurred at these wave scales (Fig. 3). For the angular spreading D(ω) we chose an empirical expression that closely follows Hwang et al. (2000), but levels off at small wave scales. We chose D = 0.3 for ω/ωp ≤ 1 and D = 0.8 for ω/ωp ≥ 3.5 with a fifth-order polynomial matching in between.
We find the breaking rate R to depend roughly linearly on the mean saturation level σb, with higher saturation levels leading to more frequent breaking (Fig. 10a). Extrapolation of our data yields σb ≃ 4 × 10−3 as a threshold for the onset of breaking; however, the band-averaged saturation, and therefore this threshold value, depends on the chosen bandwidth. Our choice of ωp ≤ ω ≤ 5ωp yields a threshold in good agreement with the threshold behavior of spectral breaking rates reported in Banner et al. (2002). The good agreement in magnitude and functional dependency of breaking rates inferred from the breaking crest length distributions and various different and independent observation techniques is encouraging in suggesting that our analysis captures the majority of whitecap events.
Gemmrich (2005) reports a close-to-linear relation between breaking rates and the fraction P(ak > χ) of wave crests exceeding a certain steepness threshold χ. It turns out that P is a surrogate for the saturation level (Fig. 10b), and it is preferable to relate wave breaking to the dynamically relevant saturation level σb rather than the indirect threshold-dependent quantity P.
b. Momentum flux and energy dissipation rate
Spectrally resolved momentum fluxes and energy dissipation rates are inferred from the fourth and fifth moments of Λ(c), respectively (6), (4). However, the transfer function includes the unknown proportionality factor b, and we present momentum and energy fluxes without the factor b (Fig. 11). In all five data segments the momentum flux due to breaking waves is strongest at wave scales corresponding to phase speeds of about 6–9 m s−1. At wave scales with c < 5 m s−1 the momentum flux supported by whitecapping waves falls off by roughly three orders of magnitude. Large-scale breaking events are more sparse, and therefore the momentum flux supported by larger wave scales fluctuates considerably between the four datasets. Energy dissipation rates due to whitecapping are even more dominated by larger wave scales. The spectral distributions of the energy dissipation rates peak at c = 8 to 10 m s−1.
Budget ratios of the total energy dissipation rate in the surface layer and the total momentum flux in the atmospheric boundary layer yield ceff = O(1 m s−1) (Gemmrich et al. 1994). Terray et al. (1996) analyzed a variety of field observations of direct turbulence measurements and presented the effective phase speed, normalized by the dominant phase speed, as a function of wave age. Applying their results to our datasets results in ceff = (0.06 to 0.17) cp (Fig. 12), corresponding to ceff ≃ 1 to 2 m s−1. Based on the theoretical functional form Λ(c) ∝ c−6, the effective phase speed is ceff = 0.95 m s−1.
The energy input into the water column via skin friction scales as Eskin ∝ u3*. Thus, the ratio ceff/u* describes the enhancement of energy input due to waves. As the wave field develops, this ratio increases initially. However, for more mature seas the reduction of the spectral peak enhancement causes ceff/u* to slowly decrease with wave age (Terray et al. 1996) which is also seen in our data (Fig. 12b).
Both quantities c̃eff and ceff represent the ratios of total energy dissipation rate and total momentum flux supported by a certain wave band; in the case of c̃eff only wave scales associated with whitecaps are included, whereas ceff includes momentum and energy transfer by the entire capillary–gravity wave band. Thus, the difference between c̃eff and ceff helps to address the role of small-scale breaking waves without visible air entrainment.
A first estimate of γ is obtained from model results, estimating the cumulative spectrum of wave-induced stress per wave band (Kudryavtsev et al. 1999), combined with an estimate of the partitioning factor γ̃ [defined in (24)]. Assuming fully developed seas, 10 m s−1 wind speed, and γ̃ = 0.85 (Banner and Peirson 1998), the model predicts that nearly 30% of the total air–sea momentum flux is supported by wave scales corresponding to c ≥ 1.5 m s−1. However, it is likely that γ depends on wave development and wind speed.
c. Proportionality factor b
Both models predict the proportionality factor b to be small. However, it is difficult to estimate on theoretical or empirical grounds the range of values as well as potential dependencies.
In all three models, it is assumed that in nature the physical process of dissipating energy spans the same time as the active breaking of a wave crest is occurring. All three concepts confirm that the energy dissipation rate scales to the fifth power of the wave speed and predict the proportionality factor b to be small. Most likely, this value is not a constant. Dimensional analysis only requires a nondimensional proportionality factor. However, all of the models described here suggest that b may depend on the wave geometry, say b = b(σb). In recent wave tank experiments the value of b varied by an order of magnitude depending on the breaker strength, with the lowest b values found for the weakest breaking (Banner and Peirson 2007).
The numerical value of b is crucial for any quantitative estimates of wave energy dissipation or wave–current momentum transfer, but it is very uncertain, with reported values applicable to the ocean ranging from b ≈ 10−3 (Phillips et al. 2001) to b ≈ 10−2 (Melville and Matusov 2002) and assumed to be scale independent.
From the definition of γ, these estimates are identical, yielding 3.2 × 10−5 ≤ b ≤ 10.1 × 10−5 (Fig. 13). Thus, in our four datasets b varies by a factor of 3 and is one to two orders of magnitude smaller than previous estimates (Phillips et al. 2001; Melville and Matusov 2002). Surprisingly, the apparent dependence of b on the wave development, expressed as wave age cp/u* (Fig. 13) is opposite to our initial expectations; b increases with wave age. For the five data segments, the airside momentum flux τa varied only by 12% (Fig. 2). Thus, variations in b are mainly due to differences in the breaking crest length distributions. However, uncertainties in the empirical values of ceff (Terray et al. 1996) and γ may also play a significant role in this counterintuitive dependence of b on the wave development.
The energy transferred from the wind to the wave field (25) is expected to be roughly twice as large in dataset I as in II (Fig. 12). However, the integrated fifth moments of the breaking crest distribution L5 = ∫c5Λ(c) dc show a ratio close to 6:1, resulting in the about three times larger b values for dataset II. The differences in L5 between datasets are due largely to the presence of dominant breakers in the younger sea states. These large-scale breakers are significantly less frequent than intermediate-scale breakers and their statistics are less robust. The L5 values are also affected by the bandwidth Δc and the video footprint. However, erroneous rejection or inclusion of breaking events modifies L5 only at a roughly linear dependence; that is, a hypothetical identical value of b for datasets I and II would imply that two-thirds of the large-scale breakers (c ≥ 9 m s−1) in dataset I are processing artifacts. Similarly, erroneous classification of 10% of all events into the next higher speed band would increase L5 by less than 7%. Therefore, we conclude that the observed trend of increasing b values with increasing wave development cannot be attributed to the uncertainties in the observed breaking crest length distributions. We take the difference of Λ(c) from the full sampling rate and subsampled differential imagery (Fig. 6) as an indication of the estimate of these uncertainties. The resultant uncertainty in L5 is 6%. Nevertheless, the apparent wave-development dependence of b remains puzzling and counter to the trend from wave tank experiments. This warrants further study. Based on our results here, we suggest that for open ocean conditions b = (7 ± 3) × 10−5.
5. Conclusions
Close range video recording of the ocean surface was used to estimate the along-crest dimension and propagation speed of whitecaps. From these open ocean data, at four different wave development stages, the breaking crest length distribution Λ(c) has been calculated. An important result is the specification of breaker scales in these different conditions. In fully developed sea states (cp/u* = 33) only intermediate- to small-scale breaking waves, corresponding to c/cp ⪅ 0, 6, are observed. On the other hand, in developing seas (cp/u* = 26), all wave scales, including the dominant waves, are part of the breaking spectrum. This has important implications for the shape of the energy dissipation term Sdis used in spectral wave models. The occurrence of wave breaking at a particular scale is linked to the wave saturation. In a previous study (Banner et al. 2002), it was found that the onset of breaking is linked to the wave saturation, normalized by the directional spreading of the wave energy. Breaking occurred only for cases with normalized wave saturation larger than 4.5 × 10−3. This threshold behavior has been confirmed in this study. Also, the mean wave saturation within the frequency band ωp ≤ ω ≤ 5ωp is a good indicator for the breaker passage rate at a fixed location. The breaker passage rate obtained from the first moment of Λ(c) indicated 50 to 100 breakers per hour, in close agreement with independent observations. This provides strong validation for the observed breaking crest length distributions Λ(c).
The fourth and fifth moment of Λ(c) are linked to the momentum flux from the waves to currents and the wave-induced energy dissipation rate, respectively. These fluxes are dominated by intermediate-scale waves and no indication is found of a Kolmogorov-type energy cascade in the wave field, where energy dissipation would be dominated by small-scale breakers. We find the proportionality factor b = (7 ± 3) × 10−5, which is required for a quantitative assessment of wave energy dissipation rate based on breaking crest length distributions. This is much smaller than previously reported (Phillips et al. 2001; Melville and Matusov 2002). In a conceptual model, describing wave breaking as displacement of water parcels from the crest into the trough region, we offer a heuristic explanation why we expect b to vary depending on the wave state. However, a detailed study of the functional dependence of the proportionality factor b is still crucial before implementing breaking crest length distributions into wave energy dissipation functions.
In the case of developed wave fields and 12 m s−1 wind speed, about 11% of the air–sea momentum flux is supported by air-entraining breaking waves. This fraction increases for younger seas.
The Duncan–Phillips concept of estimating wave kinematics and dynamics from breaking crest length distribution was introduced more than two decades ago. Advances in video technology and computational image processing have opened this concept to relatively easy field observations. We expect that an increase in such observational data, covering various wave field conditions, will help to solve the remaining open issues, in particular the functional dependence of the proportionality factor b. Furthermore, this study is based only on wave breaking that generated visible air entrainment, and we recommend that future studies also include smaller-scale breaking waves and microbreakers.
Acknowledgments
This work was made possible with the funding by the Canadian Foundation for Atmospheric and Climate Sciences (CFCAS). Data collection was part of the FAIRS project, supported by the Office of Naval Research. We are indebted to the crew of R/P FLIP and the technical staff of the acoustical oceanography group at the Institute of Ocean Sciences (Sidney, BC). A. Jessup (APL-UW) and J. Edson (WHOI) provided supplementary data. MLB also gratefully acknowledges support during this investigation from the U.S. Office of Naval Research under the CBLAST Departmental Research Initiative. Part of the analysis was done during one author’s (JG) visit to the Institute for Baltic Sea Research Warnemünde, and the hospitality and many insightful discussions with H. Burchard and L. Umlauf are greatly acknowledged. The critical reviews by two anonymous reviewers helped to improve the manuscript.
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Extraction of breaking crest properties: (top) sequence of video frames, (second panel) corresponding differential images, (third panel) forward edges of breaking crest based on threshold filter of differential images, and (bottom) major axis and centroid of ellipse fitted through forward crest edges. All sequences are shown at a 10-Hz frame rate.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Environmental conditions during the FAIRS experiment. Midpoints of individual datasets are marked by I – IV. (top) wind direction dd, (middle) significant wave height Hs and (bottom) wind speed at 10-m height u10.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Distribution of the normalized whitecap propagation speed cbrk/cp as obtained from camera 1 (gray bars) and camera 2 (black bars). Individual panels correspond to the four datasets indicated in Fig. 2. The indicative mean breaker passage rate for these datasets is given in Fig. 10a.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Distribution of the normalized individual breaking crest length Lbr/λbr for different wave bands centered at normalized phase speed c/cp, ranging from 0.1 (black circle), to 0.3 (black triangle-down), 0.5 (gray diamond), 0.7 (light gray triangle-up), and 0.9 (white square). Individual panels correspond to the four datasets indicated in Fig. 2. Note, intermediate- to large-scale waves are likely not fully resolved.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
(top) Breaking crest length distribution Λ(cbr) as function of the breaker phase speed cbr. The symbols (○, ▿, □, ◊, ☆) correspond to datasets Ia, Ib, II, III, and IV, respectively. The dashed line indicates a c−6 dependence, predicted in Phillips (1985). (bottom) Same data transformed to linear wave phase speed cln. (See section 4)
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Breaking crest length distribution Λ(cbr) as function of the breaker speed cbr for dataset II. Sampling rates of differential video frames are 10 Hz (open circle) and 15 Hz (asterisks).
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Nondimensional breaker length distribution c2p g−1Λ as a function of the propagation direction θ and the normalized linear phase speed of breaking waves cln/cp as obtained from the four datasets indicated in Fig. 2. The contour spacing is 0.5 and the thick contour line corresponds to −2.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Breaking rate RΛ, calculated from breaking crest length distributions, compared to breaking rate RI, estimated from the passage rate of bright targets passing a fixed spot in the video frame: symbols as in Fig. 5.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Normalized wave saturation σ [defined in (2)] as function of frequency f (datasets as indicated in top-left corner). The dashed horizontal line represents the threshold for onset of breaking as determined in Banner et al. (2002); the vertical line depicts the dominant wave frequency.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
(a) Breaking rate R as function of mean normalized saturation level σb within the band 1 ≤ ω/ωp ≤ 5. (b) Fraction of wave crests with steepness ak > π/14 vs mean normalized saturation level σb: symbols as in Fig. 5.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Moments of Λ(c) as function of the linear breaking wave phase speed c. Multiplied by the factor b [Eqs. (3), (5)], these data points correspond to the (top) breaking wave–induced momentum flux and (bottom) breaking wave–induced energy dissipation rate: symbols as in Fig. 5.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
The effective phase speed ceff related to the energy input from wind to waves (Gemmrich et al. 1994). (top) Effective phase speed normalized by dominant phase speed, as function of inverse wave age. Data points are matched to the result of Terray et al. (1996), depicted by the dashed line. (bottom) Effective phase speed normalized by friction velocity as function of wave age: symbols as in Fig. 5.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
Proportionality factor b as a function of wave age cp/u*: symbols as in Fig. 5.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3762.1
