a. Phillips’ (1985) spectral breaking wave framework

Recent developments in the spectral characterization of breaking wave properties (Phillips 1985, hereinafter P85) in terms of the spectral density of breaking crest length per unit area Λ(k) [or Λ(c)], and associated spectral breaking strength b(k) [or b(c)], where k is wavenumber and c is phase speed, have added a new theoretical and observational framework that potentially adds reliable breaking wave information to routine wave forecasts (e.g., Banner and Morison 2010; Romero et al. 2012; Kukulka and Hara 2008a,b). With this increased measurement and modeling capability, it is timely to investigate further the quantification of the influence of breaking waves in air–sea interaction, especially its role in gas and sea spray exchange, by highlighting a potentially useful breaking strength parameter, b_eff, to be defined below.

In brief, the directional wave energy spectrum evolves according to the radiative transfer equation (Komen et al. 1994), shown below for deep water and negligible currents:

View Expanded

where Φ = ρgΨ(k) is the directional wave energy spectrum, is the mean square wave height, c_g is the group velocity, ρ is the density of water, and g is the acceleration due to gravity. The total source term S_wave = S_in + S_nl + S_ds, where S_in is the atmospheric input spectral source term, S_nl is the nonlinear spectral transfer source term representing nonlinear wave–wave interactions, and S_ds is the spectral dissipation rate, assumed primarily due to wave breaking. Breaking wave dissipation rates have been shown to be roughly 5–1000 times greater than wall layer scaling (Agrawal et al. 1992; Gemmrich 2010; Sutherland and Melville 2015; Terray et al. 1996). In this context, Sutherland and Melville (2015) note that as the wave age increases, the total turbulent kinetic energy (TKE) dissipation rate within the wave boundary layer greatly exceeds the classical wall layer dissipation rate.

In view of the central role of surface and breaking waves in upper ocean dynamics and air–sea fluxes, wave and breaking measurements are paramount. The dissipation of wave energy in the energetic part of the spectrum has attracted considerable recent interest. Presently, parametric versions of S_ds based on satellite data are used in the context of active whitecap fraction (Anguelova and Hwang 2016), of ocean swell dissipation (Ardhuin et al. 2009), and of the TKE dissipation rate due to breaking (Hwang and Sletten 2008).

Underpinning the P85 breaking wave framework are the assumptions that 1) the velocity of a breaking front is equal to the phase velocity c of the underlying gravity wave that is breaking; 2) there is geometric self-similarity of the breaking zone (i.e., the cross-sectional area of the breaking region scales with c⁴); and 3) the deep water dispersion formula relates the observed breaker front velocity to the underlying wavenumber k of the breaking wave. In this context, generic wave crest slowdown (Banner et al. 2014a) needs to be taken into account in transforming between the c and k domains. Published studies (e.g., Kleiss and Melville 2010; Gemmrich et al. 2013) indicate that Doppler correction by the orbital motion of long waves affecting the short waves produces only modest changes to breaking crest length per unit area [Λ(c)] spectra, as defined in P85. We also note that the linear deep water gravity wave dispersion relation underpins Eq. (6.3) of P85, which was modified by Banner and Morison (2010) to establish the spectral form of the dissipation rate distribution associated with wave breaking, expressed in terms of the breaking wave phase velocity c:

View Expanded

Note that S_ds is used here rather than ε as used in P85. Here, the nondimensional coefficient b(c) is the scale-dependent breaking strength.

The directionally integrated form of Eq. (2) is

View Expanded

which defines the turbulent energy dissipation rate S_ds(c) due to active breaking at scale c and its relationship with Λ(c) and b(c). For narrow-banded wave systems typically used in laboratory dissipation rate determinations for breaking, b is assumed to be independent of scale and is found to be a strong function of the characteristic wave steepness (Banner and Peirson 2007; Drazen et al. 2008; Melville 1994). In ocean wave forecasting, spectral modeling studies provide evidence that b(c) may scale with wave age (Banner and Morison 2010; Romero et al. 2012).

b. The effective breaking strength

The Phillips framework provides a compact parameterization for estimating the whitecap fraction, the momentum flux and the TKE dissipation rate from breaking waves. These quantities can be expressed as various moments of Λ(c). Rather than working with the spectral breaking strength, b(c), which is not presently known from measurements, we modify Eq. (3) by defining the effective breaking strength coefficient

View Expanded

where S_ds(c) also cannot yet be measured directly and presently has only been estimated from models (Morison and Banner 2016, manuscript submitted to Ocean Modell.; Romero et al. 2012). Furthermore, the integral of S_ds(c) has been inferred from turbulence measurements in the wave boundary layer. The parameter b_eff is potentially very useful in a growing number of applications as described in section 4b below. The basis of the b_eff construct rests on the documented dominance of the breaking wave contribution to the TKE dissipation rate in the wave boundary layer (Agrawal et al. 1992; Gemmrich 2010; Sutherland and Melville 2015; Terray et al. 1996).

Following Gemmrich et al. (2013) among others (e.g., Schwendeman et al. 2014; Thomson et al. 2009), b_eff was quantified through the relation between the fifth moment of Λ(c_b) and the depth-integrated, total energy dissipation rate, ε, in the wave boundary layer,

View Expanded

where c_b is defined as the “characteristic” speed of the breaker front (a turbulent layer flow) to distinguish it from the phase speed c of the underlying gravity wave that is breaking. The near-surface vertically integrated TKE dissipation rate, ε, may be decomposed into ε = ε_brk + ε_back, where ε_brk is due to wave breaking (whitecapping and microbreaking) and ε_back is the background TKE dissipation rate associated with wind shear and wave turbulence interaction among other processes (Sullivan and McWilliams 2010; Thais and Magnaudet 1996; Veron and Melville 2001). Note that ε is an independent measure of the mean depth-integrated dissipation rate in the wave boundary layer available from subsurface measurements or determined from modeling or parameterization.

As noted above, the background TKE dissipation rate is a small fraction of the total TKE dissipation rate in the presence of significant wave breaking. On this basis ε_brk dominates ε, and the TKE dissipation rate in the wave boundary includes some background turbulence as a residual. In this way, a reliable estimate for the total KE flux into the water column mediated by wave breaking can be obtained by combining b_eff with the fifth moment of Λ(c_b).

In this note, we focus on the characterization of the effective breaking strength b_eff. We outline the historical laboratory measurements, field measurements, and modeling estimates for determining b_eff. We investigate the behavior of b_eff using field results and modeling estimates to determine the dominant dependence of b_eff with wave age, wave steepness, or wind speed. The sparse observational domain results for b_eff are complemented by modeling results to assess the likely general trend, which is subject to future validation. We develop a parameterization for b_eff as a function of wave age or significant wave steepness for use in operational forecasting. In the discussion section below (section 4), we highlight several geophysical products that would be enabled through the availability of a robust parameterization for b_eff that takes advantage of remotely sensed wave-field parameters to construct regional and global maps of these products.

a. Laboratory measurements

Many measurements of laboratory narrow-banded unidirectional focused breaking wave systems have aimed to quantify the breaking strength in Eq. (4) above. The usefulness of these data has been to demonstrate dependences on wave properties (e.g., wave steepness) under controlled conditions, often without wind forcing (e.g., Banner and Peirson 2007; Drazen et al. 2008; Melville 1994; Perlin et al. 2013; Tian et al. 2010). However, it is a key open question as to whether this class of measurements is representative of breaking conditions in broad-banded directional sea states and hence it is not considered here.

b. Field measurements

Several measurement campaigns have aimed at determining b_eff in open ocean conditions. Phillips et al. (2001) measured “sea spikes” or discrete moving events in a set of X-band radar measurements, backscattered from the sea surface at near-grazing incidence with very high spatial and temporal resolution (30 cm in range and 2000-Hz pulse repetition frequency) in steady moderate wind speeds (9.3 ± 0.3 m s⁻¹) over the open ocean off Kauai, Hawaii. They measured Λ(c_b) and inferred that b_eff according to equilibrium range wave theory spans the range (7–13) × 10⁻⁴ for a wave age, , of 10, where c_p is the phase speed of the waves at the spectral peak and is the atmospheric friction velocity. These authors stressed that their results provided no support for a “Kolmogorov cascade” in wind-generated waves analogous to that in turbulence (i.e., energy input from the wind to large wave scales and dissipation from the waves at small scales). The measurements indicate that, in contrast, dissipation is significant at the largest scales of wave breaking and is distributed widely across that spectrum. That is, there is no evidence that the wave energy dissipation is dominated by small-scale waves.

Thomson et al. (2009) analyzed video observations of breaking waves for wind speeds up to 15 m s⁻¹ taken from R/V Henderson at the north end of Lake Washington in the state of Washington in 12-m water depth and on Puget Sound near Richmond Beach, Washington, in 19-m water depth. Energy dissipation by breaking water waves was quantified indirectly from video imagery using Fourier methods (Thomson and Jessup 2009) and directly from in situ acoustic Doppler velocity profile observations. Energy dissipation rate estimates are in reasonable agreement over two orders of magnitude when tuned by choosing b_eff = 0.017 ± 0.03 (for direct measurements of dissipation) and b_eff = 0.013 ± 0.05 (for indirect measurements of wind input). Peak waves were found to comprise only 10% of the total breaking rate, but they contributed up to 75% of the total dissipation rate. In addition, breaking statistics were found to depend on the peak wave steepness and the energy input by the wind.

Gemmrich et al. (2013) gathered open ocean video and dissipation rate observations from R/P FLIP in the Santa Barbara Channel (SBC) and in the Pacific Ocean (PO) off Hawaii during the Office of Naval Research (ONR)-sponsored Radiance in a Dynamic Ocean (RaDyO) experiments (Zappa et al. 2012). The Santa Barbara Channel observations had developing seas and the Pacific Ocean south of Hawaii had mature seas. During the PO experiment, the data show a distinctive and persistent decrease in the easterly wind speed from 10.2 m s⁻¹ at the beginning of the experiment to 8.5 m s⁻¹ at the end. The video data were analyzed to obtain breaking crest length distributions, Λ(c_b), and the corresponding effective breaking strength during the wind-wave conditions. These are among the first experiments using dissipation rate measurements that probe up into the breaking crest regions, together with simultaneous measurements of breaking crest length distributions. The directly measured effective breaking strength parameter during the PO observations was 4.2 (±1.8) × 10⁻⁵ in mature seas with wave age, , of 40–47.

The integrated dissipation rate scaled by b_eff was consistently higher for mature seas compared to developing seas due to higher energy and momentum fluxes from the wind. In this note, we add the directly measured b_eff for the Santa Barbara Channel observations during RaDyO. The SBC experiment experienced a variety of conditions with generally low winds in the early morning with mean wind speed of 4.8 ± 2.7 m s⁻¹ (all ± bounds refer to a combination of natural variability and measurement uncertainty as expressed by the standard deviation), and strong sea breezes up to 12 m s⁻¹ in the evening with mean wind speeds of 7.1 ± 2.2 m s⁻¹. We determined b_eff to be 2.4 (±1.5) × 10⁻³ in developing seas with wave age, , of 22.3 ± 1.5. Here, the estimate of TKE dissipation rate was obtained at a depth of 0.75 m and following the methodology in Sutherland and Melville (2015) extrapolated to the surface based on z⁻¹ scaling in order to determine ε.

Schwendeman et al. (2014) made coupled in situ and remotely sensed measurements of very strongly forced wind waves at short fetch to assess the role of breaking in a developing young wind sea in the Strait of Juan de Fuca, north of Sequim, Washington, aboard the R/V Robertson for wind speeds that ranged from 9.7 to 18.0 m s⁻¹. In situ measurements of TKE dissipation rate from wave-following SWIFT floats and a tethered acoustic Doppler sonar system are consistent with wave evolution and wind input estimated using the radiative transfer equation. The breaking crest length distribution Λ(c_b) was obtained from stabilized shipboard video recordings and processed using the Fourier-based method of Thomson and Jessup (2009). The effective breaking strength parameter b_eff is calculated by comparing the fifth moment of Λ(c_b) with the measured dissipation rates. The mean b_eff value was determined as 3.2 × 10⁻³, with a standard deviation of 1.5 × 10⁻³.

Sutherland and Melville (2015) investigated turbulent kinetic energy dissipation beneath breaking waves. Wind, wave, and turbulence measurements were made in the North Pacific Ocean aboard R/P FLIP, during the ONR-sponsored High Resolution Air–Sea Interaction (HiRes) and RaDyO experiments as well as off the Southern California coast (SoCal). SoCal 2010 took place over 2 days in the Southern California Bight in low to moderate wind conditions (up to 9 m s⁻¹). HiRes was a 14-day deployment on R/P FLIP moored approximately 25 km off the coast of Northern California in generally strong northwesterly winds (up to 17 m s⁻¹). They estimated the TKE dissipation rate through the entire wave-affected surface layer using a new infrared imaging technique for measuring TKE dissipation at the sea surface combined with traditional acoustic subsurface measurements. Furthermore, the infrared imagery allowed for resolving microbreakers and estimating their contribution to the breaking TKE dissipation rate. They concluded that total integrated TKE dissipation rates in the water column agreed well with TKE dissipation rates from breaking for developing to mature wind seas, 20 < < 50 (where c_m is the spectral mean wave phase speed and is the atmospheric friction velocity and c_p/c_m is ~1.4), and that breaking was the dominant source of turbulence in those conditions. Using their measured results for the integrated water column dissipation [ordinate axis in Fig. 16 of Sutherland and Melville (2015)] and the fifth moment of the measured Λ(c_b) [Fig. 6a in Sutherland and Melville (2015)], we calculated the corresponding b_eff values for the RaDyO Hawaii, HiRes 2010, and SoCal 2010 sea state conditions up to = 51. We note that > 35 is beyond the nominal Pierson–Moskowitz transition from wind seas to swell.

c. Spectral models

Carefully validated spectral models allow merging results from an ensemble of wind speed and wave age case studies to predict global ensemble behavior. This can provide useful insights for unifying the trends of the various individual datasets with their often limited range of environmental conditions and different methodologies and uncertainties. Recent spectral wave models forecast spectral breaking wave properties in addition to the usual wave height spectra (Banner and Morison 2010; Morison and Banner 2016, manuscript submitted to Ocean Modell.; Romero et al. 2012).

Banner and Morison (2010) describe a threshold-based formulation for the breaking component of the dissipation rate source term S_ds within a broad bandwidth spectral wind wave model that significantly refined the formulation of Alves and Banner (2003). For computing the spectral evolution, an “exact” form of the nonlinear source function S_nl was used with a spectral wind input S_in term based on Janssen (1991). After a critical reassessment of the strengths of the S_in and S_ds source terms relative to S_nl, this model was able to reproduce measured dimensionless energy evolution, mean squared slope, directional spreading, wind stress, total water-side dissipation rates, and modeled with observed breaking properties in a field experiment (Jessup et al. 2002) where all of these variables were measured. The breaking wave forecast products comprised the breaker crest length distribution, Λ(c), and spectral peak breaking strength, b(c_p).

Romero et al. (2012) proposed a semiempirical determination of S_ds due to surface-wave breaking and a spectral model for the breaking strength parameter, b(k). Their S_ds was based on closing the radiative transport equation for fetch-limited waves measured in the Gulf Of Tehuantepec Experiment (GOTEX) using the measured fetch evolution of the directional spectra, computations of S_nl and three models of the wind input source function. The form of b(k) was inferred from Kleiss and Melville’s (2010) kinematic breaking measurements and their inferred S_ds term, resulting in b = b(k; ), where is the wave age. A model for b(k; ) is proposed that extrapolates an inertial wave dissipation scaling based on laboratory measurements to the field, using spectral saturation to quantify spectral steepness. This model forecasts Λ(c) and the corresponding spectral breaking strength b(c) for different values (0.8, 0.9, and 1.0) of the breaker speed slowdown parameter, α (Banner et al. 2014a). We determined the modeled b_eff for Romero et al. (2012) using the results in their Figs. 14g–i for α equal to 1.0 above k/k_bp equals 1, where k_bp is the peak in b(k).

Sutherland and Melville (2015) applied the Romero et al. (2012) model for the wind sea conditions appropriate to their observations. Note that the modeled Λ(c) values were not available for this estimation of modeled b_eff. We used their computed results of the energy dissipation by wave breaking [abscissa axis in Fig. 16 in Sutherland and Melville (2015)] and the fifth moment of the measured Λ(c_b) [Fig. 6a in Sutherland and Melville (2015)] to estimate the corresponding modeled b_eff values for the RaDyO Hawaii, HiRes 2010, and SoCal 2010 sea state conditions up to of 51.

Morison and Banner (2016, manuscript submitted to Ocean Modell.) further refine the source terms and extends the Banner and Morison (2010) results to a very broad range of wind speeds and wave ages. The spectral breaking properties are derived from S_ds using a model formulation for b(k) that follows in the spirit of Romero et al. (2012) but uses a different functional form based on measured saturation thresholds determined from field data alone. Validation is achieved by comparing forecast and observed Λ distributions. The model also computes b_eff from Eq. (4). For consistency with Romero et al. (2012), Morison and Banner (2016, manuscript submitted to Ocean Modell.) used a slowdown parameter α = 1.0. Of particular relevance to the present study is their resulting compilation of modeled b_eff against wave age for a wide range of sea state conditions.

a. Dependence of b_eff on wave age during field experiments

Figure 1 shows the effective breaking strength, b_eff, versus wave age () measured for the eight field experiments described in section 2b. The RaDyO experiment in the Pacific Ocean near Hawaii included two specific datasets (Gemmrich et al. 2013; Sutherland and Melville 2015). We collated these nine datasets spanning a wind speed ranging from 2 to 18 m s⁻¹ and a wave age ranging from approximately 10 to 80. We note that some of these experiments included data for older seas that are not shown. The error bars plotted for Sutherland and Melville (2015) and for RaDyO SBC (this study) reflect the standard deviation arising solely from variability in the estimates of the TKE dissipation rate. The key result highlighted in this figure is the modest decrease in b_eff with wave age for the field measurements.

Effective breaking strength parameter b_eff plotted against wave age for each of the experiments in the datasets detailed in the legend. Where available, the error bounds are indicated.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Effective breaking strength parameter b_eff plotted against wave age for each of the experiments in the datasets detailed in the legend. Where available, the error bounds are indicated.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

Effective breaking strength parameter b_eff plotted against wave age for each of the experiments in the datasets detailed in the legend. Where available, the error bounds are indicated.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

Another feature of the field measurements highlighted in Fig. 1 is the order of magnitude smaller b_eff levels reported by Gemmrich et al. (2013) compared to Sutherland and Melville (2015) for the same experiment (RaDyO Pacific Ocean near Hawaii). We have confirmed that the integrated fifth moments of Λ(c_b) determined by (Sutherland and Melville 2013, 2015) and Banner et al. (2014b) are within a factor of 2. Hence the order of magnitude difference in b_eff between Sutherland and Melville (2015) and Gemmrich et al. (2013) is related primarily to the difference in their measured ε levels.

The turbulence measurements used in Gemmrich et al. (2013) were published and analyzed in Vagle et al. (2012). The integrated values of TKE dissipation rate for the specific times during breaking measurements shown in Fig. 11 of Gemmrich et al. (2013) ranged from 1.5 to 5.5 × 10⁻⁵ m³ s⁻³ within the top 1.5 m of the water column. These data from Vagle et al. (2012) show average dissipation rate values during the whole experiment were 6 × 10⁻⁵ to 6 × 10⁻⁶ m² s⁻³ between 0.14- to 0.75-m depths and were 6 × 10⁻⁴ to 6 × 10⁻⁵ m² s⁻³ between 1.04- to 1.76-m depths. Specifically, lower TKE dissipation rates were observed closer to the surface than immediately below. However, Fig. 10 in Sutherland and Melville (2015) shows their RaDyO TKE dissipation rate measurements in the top meter to be on average ~5 × 10⁻⁴ m² s⁻³ during the same time period, decreasing from 8 × 10⁻⁴ to 8 × 10⁻⁵ m² s⁻³ over the depth range from 0.13 to 0.87 m. It is important to note here that the TKE dissipation rate measurements of Vagle et al. (2012) and Gemmrich et al. (2013) were in a wave-following reference frame as were the measurements of Sutherland and Melville (2015). Remarkably, Vagle et al. (2012) report TKE dissipation rates in the same depth range (0.14 to 0.75 m) on the same days to be ~5 × 10⁻⁵ m² s⁻³, about an order of magnitude lower than observed by Sutherland and Melville (2015). Vagle et al. (2012) highlight their finding that the dissipation at greater depths (1.04 to 1.76 m) was greater than near the surface, but this level of variability was not reported by Sutherland and Melville (2015). Overall, the dissipation estimates in the wave boundary layer given by Sutherland and Melville (2015) are a factor of 10 greater than for Vagle et al. (2012), which is the major reason for the difference in the b_eff levels of Gemmrich et al. (2013) and Sutherland and Melville (2015). The source of this discrepancy needs to be better understood through future measurements including more diverse instrument arrays that investigate different turbulence sensors and sampling techniques.

In the context of the other datasets, Gemmrich et al.’s (2013) data appear to be systematically low and until the source of the discrepancy is resolved, we have tentatively chosen to parameterize the data excluding this dataset. Furthermore, Schwendeman et al. (2014) point out that the Thomson et al. (2009) values for b_eff are not valid due to the video missing some of the short whitecaps causing a reduction of Λ(c_b) and an overestimate of b_eff. Data from Phillips et al. (2001) and Gemmrich et al. (2008) are not included subsequently since the TKE dissipation rate in their studies was indirectly estimated from scaling arguments rather than direct measurements. Finally, the Schwendeman et al. (2014) data have been consolidated to a single point since no intrinsic trend in the wave age data is observed and the error bars reflect the standard deviation arising solely from variability in the estimates of b_eff and wave age.

b. Parameterization of b_eff with wave age

On the basis of the results shown in Fig. 1 (with the exclusions noted above), we propose the following linear relationship for b_eff in Fig. 2a for use in applications:

View Expanded

based on the apparent primary dependence on wave age (), where A = 3.482 × 10⁻³ (±6.481 × 10⁻⁴) and B = −4.691 × 10⁻⁵ (±1.935 × 10⁻⁵). The error bounds correspond to the 95% confidence intervals. The R² value is 0.98 and the root-mean-squared error (RMSE) is 1.37 × 10⁻⁴. The t statistic for the slope B was found to be −10.43, which confirms that B is significantly different from zero at the 99.5% confidence level. The F statistic for a linear model versus a constant was found to be 108.79 with a p value of 0.009 07. This further confirms that the linear model in Eq. (6) provides an improved statistical fit compared with a constant at the 99% confidence level. Note that the parameterization in Eq. (6) is valid for wave ages 50 and below for reasons discussed in section 4a.

(top) Effective breaking strength parameter b_eff plotted against wave age for the selected subset of field experiments shown in the upper legend, chosen using the rationale in section 3a of the text. The error bounds are indicated. (bottom) As in (top), but showing modeled results. Morison and Banner’s (2016, manuscript submitted to Ocean Modell.) model results correspond to wind speeds ranging from 6 to 24 m s⁻¹. Romero et al. (2012) show mean and standard deviation for the reported wind input functions. The solid black line shows the least squares best-fit linear correlation [Eq. (6)] to the shown measured b_eff field data points.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

(top) Effective breaking strength parameter b_eff plotted against wave age for the selected subset of field experiments shown in the upper legend, chosen using the rationale in section 3a of the text. The error bounds are indicated. (bottom) As in (top), but showing modeled results. Morison and Banner’s (2016, manuscript submitted to Ocean Modell.) model results correspond to wind speeds ranging from 6 to 24 m s⁻¹. Romero et al. (2012) show mean and standard deviation for the reported wind input functions. The solid black line shows the least squares best-fit linear correlation [Eq. (6)] to the shown measured b_eff field data points.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

(top) Effective breaking strength parameter b_eff plotted against wave age for the selected subset of field experiments shown in the upper legend, chosen using the rationale in section 3a of the text. The error bounds are indicated. (bottom) As in (top), but showing modeled results. Morison and Banner’s (2016, manuscript submitted to Ocean Modell.) model results correspond to wind speeds ranging from 6 to 24 m s⁻¹. Romero et al. (2012) show mean and standard deviation for the reported wind input functions. The solid black line shows the least squares best-fit linear correlation [Eq. (6)] to the shown measured b_eff field data points.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

We used the wave model results to investigate whether a robust dependence could be obtained over a wide range of wind speeds. The model predictions described in section 2c from Romero et al. (2012), Sutherland and Melville (2015), and Morison and Banner (2016, manuscript submitted to Ocean Modell.) have been added to the field data (with the previous exclusions) in Fig. 2b. The Romero et al. (2012) and Sutherland and Melville (2015) model results provide a good fit to the field data, as do the model results of Morison and Banner (2016, manuscript submitted to Ocean Modell.). The latter additionally provide a robust prediction that the observed tight wave age dependence is maintained over a wide range of wind speeds. These results support the adoption of a single parameter relationship between b_eff and wave age for use in global predictive models, as discussed in detail below.

c. Dependence of b_eff on significant wave steepness

An allied result that arose from this study is the possibility of using a wave parameter alone to correlate b_eff. We found that the significant wave steepness H_sk_p/2 of the wind sea provided this possibility, where H_s is significant wave height and k_p is the wavenumber of the waves at the spectral peak. Figure 3 shows the (a) observed and (b) modeled variation of these parameters over a wide range of wind conditions. Note here that the Schwendeman et al. (2014) data have been consolidated to two representative points to reflect the apparent clustering in the significant wave steepness data. It is seen that a strong correlation exists, which could be used in applications where only wave conditions are available. On the basis of the results shown in Fig. 3 (with the previous exclusions), we propose the following linear relationship for b_eff for use in applications:

View Expanded

based on the apparent primary dependence on significant wave steepness (H_sk_p/2), where C = 2.108 × 10⁻⁵ (±2.538 × 10⁻³) and D = 1.534 × 10⁻² (±1.640 × 10⁻²). The error bounds are the 95% confidence intervals. The R² value is 0.75 and the root-mean-squared error is 5.06 × 10⁻⁴. The t statistic for the slope D was found to be 2.98, which confirms D is significantly different from zero at the 95% confidence level. The F statistic for linear model versus a constant was found to be 8.86 with a p value of 0.0588, which confirms that the linear model in Eq. (7) provides an improved statistical fit compared with a constant at the 90% confidence level.

As in Fig. 2, but plotted against significant wave steepness H_sk_p/2. The solid black line shows the least squares best-fit linear correlation [Eq. (7)] to the shown measured b_eff field data points.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

As in Fig. 2, but plotted against significant wave steepness H_sk_p/2. The solid black line shows the least squares best-fit linear correlation [Eq. (7)] to the shown measured b_eff field data points.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

As in Fig. 2, but plotted against significant wave steepness H_sk_p/2. The solid black line shows the least squares best-fit linear correlation [Eq. (7)] to the shown measured b_eff field data points.

Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0227.1

• Download Figure
• Download figure as PowerPoint slide

a. Recommendations

We have shown statistically that b_eff is linearly dependent on both wave age and significant wave steepness. A wave-age-based parameterization has been shown to be the more robust and should be used whenever wave statistics are available. If no wave statistics are available, a constant value for b_eff of 2.082 × 10⁻³ (±8.321 × 10⁻⁴) is recommended.

To be clear, this parameterization for b_eff is only valid when wave breaking is the dominant process contributing to the ε. In Fig. 2b, the impact of breaking on b_eff is strongly reduced for wave age greater than 50 and is more heavily controlled by background processes according to Fig. 16 in Sutherland and Melville (2015). Simultaneously, the dissipation rate in Fig. 13 in Sutherland and Melville (2015) is shown to increase with wave age. These coupled observations are highlighted in Fig. 2b, which shows that the measured b_eff is significantly greater than the modeled value for wave age greater than 50. Note that the modeled value only includes effects from breaking whereas the measured value includes all processes. The combined results of Sutherland and Melville (2013) and Sutherland and Melville (2015) show that the measured integrated water column dissipation in the wave boundary layer is greater than the directly modeled integrated S_ds, which is greater than the dissipation by breaking according to the Phillips framework at high wave age. Therefore, the Phillips framework for b_eff in Eq. (5) and parameterized in Eq. (6) breaks down above wave age 50. Use of b_eff should be restricted to wave ages ~ 50 and below. At larger wave age, the breaking contribution becomes a decreasing fraction of the total dissipation and the use of b_eff and Eq. (5) will overestimate the breaking contribution. The b_eff modeling is consistent with this observation.

The breaking strength parameter has been observed in laboratory wave tank studies of breaking waves to depend on a predicted linear maximum slope of the focused 2D wave packets. For broad-banded ocean wave spectra, the spectral breaking strength, b(k), can then be formulated in terms of spectral saturation of the wave field [as in Romero et al. (2012)]. In fetch-limited cases, the entire form of the spectra can then be related to wave age and peak wave steepness. In those cases, it is reasonable to expect that b_eff [which corresponds to an average of b(k)] could also be related to wave age. In more complicated sea states, the relationship between wave age and the spectral form is less clear. In such cases, the spectral saturation has been proposed as a plausible alternative and is independent of the spectral peak.

It is important to use wind-wave or equilibrium-range statistics wherever possible, especially in complicated sea states. Wind-wave statistics may be obtained by carefully separating the wind-wave signature from 2D spectra (Hanson and Phillips 2001; Portilla et al. 2009). Results presented here are limited by the published datasets and are not exclusively based on pure wind sea statistics. Indeed, the data from Sutherland and Melville (2015) are for more complicated sea states. They used a mean phase speed, c_m, rather than the peak, c_p, and found a relationship between c_m and c_p that we implemented here.

Schwendeman et al. (2014) and Schwendeman and Thomson (2015) suggest that the mean-squared slope (mss) in the equilibrium range is another appropriate wave statistic. However, Schwendeman et al. (2014) fetch-limited data for b_eff show no correlation with significant wave steepness or mss. The more recent work by Schwendeman and Thomson (2015) investigating whitecap fraction shows a very slight improvement in correlation and RMSE with mss over significant wave steepness and the same correlation and RMSE with wave age in order to explain the residual variability from a wind speed parameterization of whitecap fraction. At present it is difficult to gauge the usefulness of mss for predicting b_eff since Schwendeman and Thomson (2015) do not show the relationship between TKE dissipation rate and mss, but only the fact that the TKE dissipation rate is highly correlated with wave age. Nevertheless, the fact that both whitecapping and TKE dissipation rate are predicted by wave age provides confidence in the validity of Eq. (6).

Future studies could include further investigation of the wind-sea only dependent mss (Schwendeman and Thomson 2015; Schwendeman et al. 2014), c_m (Sutherland and Melville 2015), or other appropriate wave statistics. Additionally, Sutherland and Melville (2013) propose scaling relationships for Λ(c) based on dimensional analysis that include combined power laws for wave age, significant wave steepness, and Froude number based on ballistic velocity. Nondimensional fetch may also be important for fetch-limited cases as in Kleiss and Melville (2010). An extension of the work presented here should examine and target these various 2D wave spectral statistics and nondimensional scalings in future validations of b_eff.

b. Factors influencing b_eff determinations

A consensus in the structure of Λ(c) and a reduction in the uncertainty in the measurements appear to be emerging. These have been achieved using high-resolution visible and infrared imaging, more stable and diverse platforms, and more comprehensive fields of view. Substantial progress has been made in quantifying breaking with respect to scale for the kinematics from geometric considerations (Sutherland and Melville 2015). Inferring the turbulent energy dissipation rate from these kinematic estimates of breaking is important for air–sea interaction and still requires further refinement and validation.

The accuracy of b_eff is dependent on the measurement bandwidth, the processing bandwidth [i.e., discrete wave (Gemmrich et al. 2013; Kleiss and Melville 2010, 2011) vs Fourier (Thomson and Jessup 2009) processing] and computational methodology (Banner et al. 2014b). Earlier efforts to capture whitecapping on all scales were compromised due to the lack of image resolution, suggesting that the smallest whitecaps may have been underestimated (Schwendeman et al. 2014). Furthermore, Sutherland and Melville (2015) assert that microbreaking can make important contributions to Λ(c). Not accounting for the smallest whitecaps or microbreaking may cause an underestimate of the fifth moment of Λ(c) and therefore overestimate b_eff.

As noted above, b_eff is applicable to wind-driven seas with active breaking within the spectrum, for which the TKE dissipation rate due to breaking is assumed to dominate over the background. Sutherland and Melville (2015) suggest their measurements indicate that breaking is the main mechanism for the excess of TKE dissipation rate above the wall layer. From our modeling for young wind seas (Morison and Banner 2016, manuscript submitted to Ocean Modell.), the TKE dissipation rate due to breaking is approximately 90% of the total, but this reduces to 50%–60% approaching the swell transition limit (U₁₀/c_p ~0.83; Komen et al. 1984). Beyond this limit, nonlinear hydrodynamic spectral transfer becomes the primary driver for spectral peak evolution rather than wind forcing. On this basis, we suggest that b_eff may have an additional uncertainty of a factor of 2 for older wind seas.

c. Applications for b_eff

The availability of a robust parameterization for b_eff enables a range of geophysical products useful for various air–sea interaction applications. These geophysical products include the total whitecap fraction, W, comprised of bubbles generated during active “stage A” wave breaking as well as postbreaking “stage B” whitecap fraction. The latter includes both residual surface foam and rising bubbles that have surfaced due to buoyancy. Primarily, it is the active portion of the whitecap fraction (W_A) that needs to be considered to deduce dynamical processes such as momentum flux and turbulent mixing associated with breaking waves. The time scale T for stage A whitecapping is on the order of half a wave period (Kleiss and Melville 2011) whereas the time scale for stage B depends on bubble-size distribution, water temperature, ocean stratification, surfactants, and atmospheric stability. The momentum flux and TKE dissipation rate from the waves to the upper ocean are additional geophysical products that are determined from the fourth and fifth moments of Λ(c) weighted by b_eff.

Using the proposed framework, these geophysical products can be determined operationally or for research purposes from remote sensing instruments deployed from satellites, aircraft, or ocean-going/fixed platforms. A possible method to generate global maps of W_A from satellite data has been proposed recently by Anguelova and Hwang (2016). P85 pointed out that combining the first moment of Λ(c) with the duration of the active breaking T(c) at scale c provides the active whitecap cover:

View Expanded

Anguelova and Hwang (2016) derive an approximation to the whitecap cover in Eq. (8) to obtain W_A based on S_ds(c) instead of Λ(c), which intrinsically involves b_eff. This seeks to bypass the need for breaking wave measurements and only requires buoy measurements and a parametric model for the energy dissipation (e.g., Hanson and Phillips 1999; Hwang and Sletten 2008) to determine W_A(ε). Their sensitivity analysis demonstrates that b_eff has the largest impact on the W_A(ε) estimates compared to bubble persistence and threshold breaker speed.

Potential applications include, but are not limited to, the following:

1. In coupled atmosphere–wave–circulation forecast models, remote measurements using the [b_eff, Λ(c)] framework are potentially very valuable for validating ε, especially in the context of coupled wave–circulation modeling, where wave field inputs play a strong role in the upper ocean circulation. Key inputs are the dissipation rate and momentum flux from the breaking waves [respectively, the b_eff -weighted fifth and fourth moments of Λ(c)]. The momentum flux from the breaking waves to the circulation also provides a useful indicator for the atmospheric wind stress level.

2. In studies of air–sea gas exchange, actively breaking crests form the stage A whitecap fraction that is critical for determining the bubble-mediated effects of gas transfer, especially for the gases of varying sparingly solubility. Furthermore, the TKE dissipation rate governs the gas transfer velocity for gases of high solubility and moderately forced wind waves without whitecapping for gases of sparing solubility. Evidence from the Southern Ocean Gas Exchange Experiment (SOGasEx) (Zappa et al. 2016, manuscript submitted to J. Geophys. Res.) suggests strongly that the actively breaking crest region whitecap cover (W_A) of breaking waves is important to air–sea gas fluxes.

3. Total whitecap fraction (W) may be used to determine bubble-mediated sea spray aerosol production and heat exchange (Andreas 1998; de Leeuw et al. 2011).