1. Introduction
Breaking wind waves play an important role in air–sea interaction processes including the exchange of energy, momentum, heat, and gases between the ocean and the atmosphere; mixing of the upper ocean and aerosol production (Melville 1996). Many of the present parameterizations of air–sea fluxes rely on empirical relationships of the fractional whitecap coverage versus the wind speed, typically referenced to 10 m above mean sea level (U10), without information about the wind wave spectrum or the spectral energy dissipation.
Recent advances in image-processing techniques, using digitally acquired images of breaking waves, have allowed the visual detection and quantification of the length and velocity of breaking fronts, providing estimates of Phillips’ (1985) Λ(c), the length of breaking fronts with velocities in the range (c, c + dc) per unit surface area. See, for example, Melville and Matusov (2002, hereafter MM); Gemmrich et al. (2008); Thomson et al. (2009); and more recently, for the Gulf of Tehuantepec Experiment (GOTEX), Kleiss and Melville (2010, 2011). Following Phillips, the first, fourth, and fifth moments of Λ(c) are proportional to the breaking rate, wave momentum flux lost to the upper-ocean surface currents, and energy lost from the wave field due to wave breaking. As shown by Kleiss and Melville (2010), the second moment can be related to the active whitecap coverage, which is the fractional area covered by actively breaking waves. Thus, Λ(c) and its moments provide a framework to relate the energy and momentum balances to the breaking rate and active whitecap coverage, which would allow present numerical wind wave models to predict the wave breaking statistics more accurately than the traditional parameterizations that depend on U10 and, in some cases, air–sea temperature differences.
Laboratory observations of the breaking parameter b as a function of the predicted linear maximum slope S of the focusing wave packet. The data by Banner and Peirson (2007) are shown with circles and pluses. The open diamonds show the data from Melville (1994) using results from earlier experiments at the Massachusetts Institute of Technology. The black and gray triangles (DML) are the data in Drazen et al. (2008) from experiments conducted at Scripps Institution of Oceanography (SIO) and Tainan Hydraulics Laboratory (THL), Taiwan, respectively. The bar shows the average error of the field data for b(k) shown in Fig. 14. The dashed and solid lines are given by Eqs. (23) and (24).
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
In this study a semiempirical model of the spectral energy dissipation is presented. The model follows Phillips (1985), by solving for the spectral dissipation from the energy balance—in this case using novel airborne observations of winds and waves in fetch-limited conditions, collected during GOTEX (Melville et al. 2005; Romero and Melville 2010a), three parameterizations of the wind input, and “exact” computations of the nonlinear energy transfer due to four-wave resonant interactions. The model dissipation and the measurements of Λ(c) (Kleiss and Melville 2010) are used to calculate and model a spectral function b(k) that characterizes the strength of wave breaking across the spectrum. Section 2 provides background information. Section 3 describes the semiempirical model and presents the spectral breaking parameter b(k). In section 4 two models of b(k) are proposed and fitted to the data. In section 5 the results are discussed, and the conclusions are given in section 6.
2. Background
The wind input Sin has been studied extensively both theoretically and empirically from both laboratory and field observations. Miles (1957, 1959) provided a theory describing the generation of waves by wind due to shear-flow instability. The theory predicts a small phase shift between the surface pressure and the surface elevation resulting in a transfer of energy and momentum corresponding to 〈p ∂η/∂x〉 c and 〈p ∂η/∂x〉, respectively. Here the angle brackets correspond to a wave phase average, p is the pressure at the surface η, and x is the wind direction.
The nonlinear energy transfer due to four-wave resonant interactions has been known in analytical form since the work by Hasselmann (1962, 1963) and Zakharov and Filonenko (1967). It is characterized by a direct and an inverse cascade of energy toward both higher and lower wavenumbers, respectively. As shown by Young and van Vledder (1993), the nonlinear energy transfer due to four-wave resonant interactions plays an important role in the evolution of the wind wave spectrum downshifting the peak wavenumber and controlling the directional spreading.
Since the GOTEX observations did not collect surface current data, in this study wave energy advection is approximated by 
3. The semiempirical model
The Gulf of Tehuantepec Experiment in February 2004 collected airborne measurements of waves and wind in strong offshore wind conditions. The instruments included the Airborne Topographic Mapper (ATM), which is a conical scanning lidar to measure the sea surface displacement as a function of two horizontal dimensions and time, a fixed lidar (Riegl) to measure the surface displacement along a single cut through the wave field, video imagery of the sea surface to detect and measure the kinematics and length of breaking fronts (Kleiss and Melville 2010, 2011), and a high-frequency radome pressure sensor array to measure the turbulent atmospheric fluxes (Romero and Melville 2010a).
The ATM measurements provided two-dimensional wavenumber spectra with an upper wavenumber limit km = 0.35 rad m−1, before reaching the noise floor (Romero and Melville 2010a). However, the fixed lidar measurements provided orthogonal one-dimensional wavenumber k1 and k2 spectra with k1 approximately aligned with the local winds, covering a wider range of wavenumbers, with an upper wavenumber limit of 2 rad m−1. The fixed lidar measurements showed that at sufficiently high wavenumbers the directional spectrum is consistent with an isotropic form with 
a. Spectral grid and extrapolation
The algorithm used to calculate the nonlinear energy transfer due to four-wave resonant interactions, developed by Resio and Perrie (1991), requires a polar grid with a constant bandwidth, such that δk/k = const, where δk corresponds to the wavenumber resolution of the spectral grid. In this study, the measured ATM spectra were interpolated on a polar grid with a directional resolution of 4.6° and δk/k = 0.0679 with extrapolation to higher wavenumbers matching both the fixed lidar data at intermediate wavenumbers and the Banner et al. (1989) data at larger wavenumbers.
The measured directional wavenumber spectra were extrapolated toward large wavenumbers with two power laws: k−3.5 and k−4 at intermediate and large wavenumbers, respectively. At intermediate scales, for km < k < kt(θ), 
Figure 2 shows an example of the measured directional wavenumber spectrum F(k, θ) with extrapolation to 20 rad m−1, with the solid black line indicating the upper wavenumber cutoff of the measured ATM directional spectrum. There is a smooth transition in the directional distribution from anisotropy, around the spectral peak, to isotropy at large wavenumbers. The same spectrum when integrated in azimuth ϕ(k) = ∫F(k, θ)k dθ yields the omnidirectional spectrum shown in Fig. 3. The integrated spectrum also shows a smooth transition between the measured ATM directional spectrum and the extrapolation to large wavenumbers, matching the fixed lidar measurements at intermediate wavenumbers, as well as the observations by Banner et al. at very large wavenumbers. The tail of the composite spectrum can be described by two power laws, k−2.5 and k−3 at intermediate and large wavenumbers, respectively.
Sample directional wavenumber spectrum F(k, θ) with extrapolated tail at large wavenumbers and θ = 0° corresponding to the direction of the spectral peak. The data were collected during research flight 7 at 15.93°N, 93.13°W during GOTEX. The local wave age cp/u* = 13. The solid black line indicates the upper wavenumber limit of the ATM data. The black dashed lines show k = 1 × 10−1, 1 × 10°, and 1 × 101 rad m−1.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
The solid black line shows a sample omnidirectional spectrum ϕ(k) = ∫F(k)k dθ with extrapolated tail matching the fixed lidar measurements (dark gray line) and the video stereo observations by Banner et al. (1989) (dark gray dashed line). The light gray dashed line is a reference power law of k−2.5. The fixed lidar data show the mean level and standard deviation of the saturation spectrum approximated by an isotropic assumption as 
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
(a) Azimuth-integrated saturation spectrum B(k) = ∫F(k)k4 dθ. (b) Directional spreading σθ defined in Eq. (8). (c) Normalized saturation 
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
b. Wave energy advection
c. Nonlinear energy fluxes
The nonlinear energy fluxes due to four-wave resonant interactions Snl were computed with an exact method, Webb–Resio–Tracy (WRT) by Tracy and Resio (1982), which is based on the work by Webb (1978). Specifically, Snl was calculated with the subroutines by van Vledder (2006), who rewrote the WRT method and implemented it in various numerical wind wave models [e.g., WAVEWATCH III and Simulating Waves Nearshore (SWAN)].
d. Wind input and stress partition
The wind input function Sin is calculated from the directional spectra and the measured friction velocity u* using the parameterizations by Snyder et al. (1981) and Janssen (1991) and a modification to Janssen’s wind input that, motivated by the work of Chen and Belcher (2000), includes a reduction of the forcing at large wavenumbers due to sheltering induced by the longer waves. As discussed in Romero and Melville (2010a), the calculated profiles of the wind stress reported by Friehe et al. (2006) gave nonzero vertical flux divergence at short fetches, which approximated by a linear relationship suggest an underestimation of the stress by 10%, with an error due to scatter of the data of about 35%. At fetches of 230 km or larger, the data showed negligible vertical flux divergence and scatter of about 1%. Since the data analyzed in this study corresponds to the measurements at short fetches, both the 10-m wind speed and the wind stresses are corrected with a 10% increment in order to account for the observed wind stress divergence.
Janssen’s (1991) wind input formulation from Eqs. (13)–(16) was calculated iteratively for each spectrum as outlined in the following steps. 1) The roughness length is estimated by zo = exp(−U10κ/u*)10, where U10 is the wind speed referenced to 10 m MSL as described in Romero and Melville (2010a); 2) Sin is calculated from the directional spectrum and the corresponding measurement of u* with Eqs. (13) and (14); 3) τw and zo are calculated from Eqs. (16) and (15), respectively; and 4) steps 2 and 3 are repeated up to 10 times to ensure a proper convergence, which was typically achieved in less than four iterations.
Equation (18) was calculated iteratively according to the following steps. 1) It is first assumed that 
Figure 5 shows the dimensionless growth rate γ/f as a function of u*/c, where 
Dimensionless growth rate γ/f as a function of u*/c, where γ = Sin(k, θw)/F(k, θw), with Sin(k, θw) and F(k, θw) corresponding to the component of the wind input and the energy spectrum, respectively, in the direction of the wind θw, and f and c are the wave frequency (Hz) and phase speed according to the linear dispersion relationship. The light gray, gray, and dark gray lines correspond to Snyder et al. (1981), Janssen (1991), and Janssen’s sheltered wind input [Eq. (18)], respectively. The symbols show the gravity wave growth data collated by Plant (1982), where both the circles and squares show the field measurements by Snyder et al., whereas the triangles and crosses are the laboratory observations by Shemdin and Hsu (1967) and Wu et al. (1979, 1977), respectively. For error comparison, the bar shows the average error of the data on the strength of breaking b shown in Fig. 14.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
The parameterization by Janssen gives the lowest forcing for weakly forced waves and the largest growth rate for strongly forced waves. The Snyder forcing is larger than Janssen’s for weakly forced waves but much lower than the available data for strongly forced waves. The modification to Janssen’s wind input with sheltering at large wavenumbers gave nearly identical results to the original formula without sheltering near the peak with a reduction for the strongly forced or short waves.
The fraction of wave-induced momentum flux τw to the total stress τ as a function of the wave age is shown in Fig. 6. The wave-induced momentum flux from the Snyder wind input is in close agreement with the momentum flux due to Janssen’s sheltered wind input, being approximately constant at about 50% of the total wind stress.2 In contrast, the wind input by Janssen (1991) gives larger values of τw and shows a weak reduction of τw/τ with increasing wave wage. This trend contrasts with the model results by Banner and Morison (2010) at lower wind speed (12 m s−1). Their results give a weak increase of τw/τ with increasing wave age, as shown in Fig. 6 with a solid black line.
Ratio of wave-induced momentum flux τw to total wind stress τtot as a function of the local wave age cp/u*. The stars, diamonds, and triangles correspond to Snyder et al. (1981), Janssen (1991), and Janssen’s sheltered wind input [Eq. (18)], respectively. The solid line corresponds to the modeling results by Banner and Morison (2010), with U10 converted into u* using the drag coefficient from Large and Pond (1982).
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
e. Spectral energy dissipation
From the spectral energy balance in Eq. (5) three sets of spectral dissipation were calculated from the data, each corresponding to a different parameterization of Sin. Figure 7 shows three examples of the spectral energy balance near the spectral peak at different stages of development, with Figs. 7a–c corresponding to cp/u* = 11, 14, and 17. The wave energy advection Sad, nonlinear energy fluxes Snl, and wind input Sin are shown with solid black, gray, and light gray curves, respectively, and the dissipation Sds is shown with black dashed lines. The wind input and respective dissipation, shown with thin and thick lines, correspond to the semiempirical model with the wind input by Janssen (1991) and Snyder et al. (1981), respectively. Figure 7 shows that all four terms, Sad, Sin, Sds, and Snl, play significant roles in the energy balance. The nonlinear transfers Snl show the typical three-lobe structure, being negative at intermediate wavenumbers and positive at both low and large wavenumbers.
Sample spectral energy balances at different stages of development. The wave energy advection Sad, nonlinear energy fluxes Snl, and wind input Sin are shown with solid black, gray, light gray curves, respectively, and the dissipation Sds is shown with black dashed lines. The wind input and model dissipation shown by the thin and thick lines correspond to the wind input by Janssen (1991) and Snyder et al. (1981), respectively: (a) cp/u* = 11 in research flight 05, (b) cp/u* = 14 in research flight 07, and (c) cp/u* = 17 in research flight 10.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
The calculated wave energy dissipation is based on the balance of the energy transport equation. It is expected that this will correlate with the rate of viscous energy dissipation underneath the breaking waves, reflected in the inertial subrange of turbulence, but equality would only be a special case (cf. Banner and Morison 2010) in an equilibrium situation where the breaking waves are no longer doing work to accelerate the underlying surface currents. Thus, the total breaking wave dissipation may be proportional, but not necessarily equal, to the dissipation rates of turbulence in the water column near the surface.
Since the GOTEX observations did not collect in situ measurements of the energy dissipation, the calculated energy dissipation is compared against the data reported in Thomson et al. (2009) from measurements collected in winds up to 15 m s−1, slightly below the range of wind speeds reported in this study. Figure 8 shows the total dissipation versus the significant slope. This study’s estimates of the dissipation are consistent in magnitude with and show similar variability to that reported by Thomson et al. (2009).
Energy dissipation rates as function of peak wave steepness 
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
f. Statistics of breaking fronts
Kleiss and Melville (2010) present an analysis of airborne visible video images collected during GOTEX. The video imagery combined with data from the global positioning system (GPS) and an inertial motion unit (IMU) were used to quantify the kinematics and lengths of breakers yielding Λ(cbr), where cbr is the speed of breaking. In Kleiss and Melville (2010) the measurements of Λ(cbr) and its moments were analyzed in detail, including its relationship to environmental parameters such as U10 and u* and wave information such as the wave age and wave slope.
Figure 9a shows the azimuth-integrated Λ(cbr) = ∫Λ(cbr, θ)cbr dθ distributions by Kleiss and Melville (2010) that overlap with the spectra being analyzed in this study. The elemental breaking speed is observed as a function of space and time, with the velocity component normal to the breaking front corrected for the underlying orbital velocity (Kleiss and Melville 2011). The measured Λ(cbr) distributions have a peak at low breaking speeds (2–4 m s−1). At larger values of cbr, after the peak, the Λ(cbr) distributions show a decreasing trend with increasing speed that differs substantially from the power law of c−6 previously suggested by the equilibrium model of Phillips (1985). In Fig. 9b, the breaking speed cbr is normalized by cp and Λ is scaled such that Λ(cbr/cp)d(cbr/cp) = Λ(cbr)dcbr. The distributions of Λ(c/cbr) show a trend with wave age cp/u*, generally showing more breaking near the peak of the energy spectrum of younger seas.
(a) The Λ(cbr) distributions (Kleiss and Melville 2010) color coded according to the wave age cp/u*: the black dashed line is a reference power law of c−6 (Phillips 1985) and the vertical dotted line shows co = 4.5 m s−1, corresponding to the lower limit of the Λ(cbr) data used in this study. (b) The breaking speed cbr is normalized by cp and is scaled by cp such that Λ(cbr/cp)d(cbr/cp) = Λ(cbr)dcbr.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
During the course of this work it became clear that inclusion of the measured Λ(cbr) distributions for cbr near and below their peaks would yield unrealistic values of the breaking function b(k). Moreover, the sensitivity analysis in Kleiss and Melville showed that Λ(cbr) provided the greatest sensitivity to processing method and parameters at lower speeds of breaking, and the Λ(cbr) distributions robustly collapsed for the faster breaking speeds, giving greater confidence in these observations. Thus, the measurements of Λ(cbr) for cbr < co = 4.5 m s−1 were neglected in the calculation of the breaking parameter b(k) described in section 3g. The value of co was chosen as a common speed that is after the peak of all the Λ(cbr) distributions (see Fig. 9). A sensitivity analysis (not shown) revealed that the results of this study are not significantly affected by small changes in the chosen value of co.
g. Dimensionless breaking function: b(k)
(a) The Λ(c) distributions transformed from wave breaking speed cbr to phase speed c according to Eq. (22) with α = 0.9. The data are color coded according to the wave age cp/u*. The solid colored lines are the video observations by Kleiss and Melville (2010). The black dashed and solid lines are reference power laws of c−6 and c−4, respectively. The open triangles correspond to the peak of the Λ(c) distributions by Jessup and Phadnis (2005) from laboratory measurements with a wind speed of 9 m s−1, estimated from two different image processing methods. The thick black and gray lines with bars show the bin-averaged predictions of Λ(c) based on Eq. (20) and the model b2(k) (Table 1) and the spectral energy dissipation with wind inputs by Janssen (1991) and Snyder et al. (1981), respectively; the error bars correspond to one standard deviation. (b) As in (a) but with the vertical axis scaled by c5.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
The sensitivity of the spectral breaking parameter b(k) to small variations in α and the different wind inputs is shown in Fig. 11, with b(k) estimated according to Eq. (20) from model dissipation Sds(k) = ∫ Sds(k)k dθ and the observed Λ(c) distributions (Kleiss and Melville 2010) and converted to the wavenumber domain using the linear dispersion relationship. In Figs. 11a,d,g; Figs. 11b,e,h; and Figs. 11c,f,i, the wind inputs used to calculate the dissipation correspond to Janssen (1991), Janssen (1991) with sheltering, and Snyder et al. (1981), respectively. In Figs. 11a–c, Figs. 11d–f, and Figs. 11g–i, the scaling factor α is 0.8, 0.9, and 1.0, respectively. All nine sets of b(k) show substantial variability, spanning one order of magnitude and, on average, a decreasing trend with increasing wave age. The distributions of b(k) approximately show similar shapes with peaks centered near k/kp = 1. The magnitude of b(k) between the spectral peak and the tail of the distribution varies between 1 × 10−4 and 1 × 10−2, with the wind input by Snyder et al. (1981) giving the largest magnitudes of b(k), with the upper values approaching the average values reported by Thomson et al. (2009) of 1.7 × 10−2 and much larger than those of Gemmrich et al. (2008), b = 7 × 10−5.
Spectral strength of breaking b(k) estimated from the model dissipation Sds(k) = ∫ Sds(k)k dθ and the observed wave breaking statistics Λ(c) from Kleiss and Melville (2010). The wind inputs used to calculate the dissipation correspond to (left to right) Janssen (1991); Janssen (1991) with sheltering; and Snyder et al. (1981). The scaling factor α = cbr/c relating the breaking speed cbr to the phase speed c is (a)–(c) 0.8, (d)–(f) 0.9, and (g)–(i) 1.0. The data are color coded by wave age cp/u*.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
Figure 14 shows the bin-averaged data of b(k), corresponding to the data in Fig. 11. The bin-averaged data retained a weak trend with wave age, with larger values for younger seas. Despite the data averaging, the errors of b(k) are large, occasionally approaching 100% at large wavenumbers. This is mainly due to the large uncertainty of Λ(c) and the reduced number of degrees of freedom at large wavenumbers. The error bars shown correspond to 
4. The model of b(k)
The concept of a wave breaking threshold behavior has been around for many years. Following Longuet-Higgins (1969), Snyder and Kennedy (1983) developed a theoretical model for the formation of whitecaps based on a threshold mechanism of the vertical acceleration. The laboratory experiments by Melville and Rapp (1985) and Rapp and Melville (1990) showed that the loss of excess momentum flux associated with wave breaking exhibits a threshold dependence on the input wave slope akc of the focusing wave group, where a is the wave amplitude and kc is the center wavenumber of the energy spectrum. Banner and Tian (1998) and Song and Banner (2002) studied the onset of breaking from numerical simulations of nonlinear unforced irrotational wave groups and its relationship to the rate of energy convergence at the center of the focusing wave packet. Their results were further confirmed experimentally by Banner and Peirson (2007). The field study by Banner et al. (2000) found a threshold behavior for the probability of breaking depending on the significant spectral peak steepness of the local wind sea Hpkp/2. They concluded that Hpkp/2 is an appropriate parameter for characterizing the nonlinear wave group behavior. Banner et al. (2002) investigated the wave breaking probability for multiple scales using the so-called riding wave removal technique. They reported a threshold behavior of the breaking probability on the spectral saturation B(k) and a common threshold behavior dependent on the normalized saturation 
Data fitting
Visual examination of the data of b(k) in Fig. 11 and the saturation spectra in Figs. 4a,c indicate a net phase shift in the peak of the distributions with both B and 
Figure 12 shows the wavenumber at the peak of B(k) and 
Wavenumber at the peak of B(k) and 
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
To improve the saturation based models b1(k) and b2(k), prior to fitting the free parameters of the models, the wavenumber of b(k) and both saturation functions B(k) and 
Spectral strength of breaking b(k) estimated from the semiempirical dissipation Sds(k) = ∫Sds(k)k dθ and the observed wave breaking statistics Λ(c) from Kleiss and Melville (2010). The horizontal axis is normalized by 
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
Parameters A1 and BT in Eq. (25) were calculated from the data in Figs. 14d,f and Figs. 14g,i with the following iterative procedure:
It is first assumed that BT = 0.
- A1 is calculated bywhere kp is the wavenumber at the peak of the energy spectrum and
ξ1 is calculated from A1 and Eq. (30).
BT is calculated from Eq. (28). Finally steps 2–4 were repeated until reaching convergence, typically in less than 20 iterations. The lower limit in Eq. (31) of 0.85kp provides a sufficiently wide range of wavenumbers while preventing the uncertainties of the data at low wavenumbers. Finally, the algorithm described above was also used to fit the parameters A2 and
Bin-averaged spectral strength of breaking b(k) estimated from the model dissipation Sds(k) = ∫Sds(k)k dθ and the observed wave breaking statistics Λ(c) from Kleiss and Melville (2010). The wind inputs used to calculate the dissipation correspond to Janssen (1991); Janssen (1991) with sheltering; and Snyder et al. (1981). The scaling factor α = cbr/c relating the breaking speed cbr to the phase speed c is (a)–(c) 0.8, (d)–(f) 0.9, and (g)–(i) 1.0. The data are color coded by wave age cp/u*, with the black, dark gray, and light gray lines corresponding to bins centered at cp/u* = 11, 13, and 15, respectively. The bars show the total error of b(k), which is described in section 3g.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
The mean results from the fitting algorithm, which is the average of the three cases at different wave ages, are shown in Table 1. Both scaling factors A1 and A2 are of order unity, varying between 3.6 and 5.0 and between 1.6 and 2.3, respectively. The mean threshold saturation parameters BT and 
Average and standard deviation of the parameters of models b1(k) and b2(k) in Eqs. (25) and (26) fitted against the bin-averaged data of b(k). The mean and standard deviation values given correspond to the mean and standard deviation of the parameters fitted for each bin-averaged distribution of b(k).
Comparisons between the bin-averaged data of b(k) and the models b1 and b2 with parameters in Table 1 are shown in Figs. 15 and 16, with α equal to 0.9 and 1.0, respectively. The black dashed line is the bin-averaged data and the dark and light dashed lines are models b1 and b2, respectively. The data and the model are in good agreement and mostly within error bars, with the model b1 giving the best agreement for α = 0.9 and b2 when α = 1.
Fits of the models b1 and b2, shown in gray and light gray dashed lines, respectively, compared with the bin-averaged data of b(k), shown with dark-dashed lines. The scaling factor α = cbr/c relating the breaking speed cbr to the phase speed c is 0.9.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
Fits of the models b1 and b2, shown in gray and light gray dashed lines, respectively, compared with the bin-averaged data of b(k), shown with dark gray–dashed lines. The scaling factor α = cbr/c relating the breaking speed cbr to the phase speed c is 1.0.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
5. Discussion
The results presented on the measured spectral strength of breaking b(k) in section 3g have large errors, with the binned data having on average an uncertainty of 65%. However, as shown in Fig. 5, a 65% uncertainty is not very large when compared to the scatter of the available wind input data and also the laboratory data of b in Fig. 1. Additionally, the measured data of b(k) are limited to wavenumbers in the range 0.5kp < k < 8kp. This range of wavenumbers was motivated partially by the limitations on the measurements of Λ(c) at low values of c, being greatly affected by the image processing methods used to compute Λ, such as the image brightness threshold. Moreover, the assumption is also justified by the fact that at small scales, O(0.1–1.0) m in length (Jessup et al. 1997), wave breaking does not produce significant numbers of bubbles and consequently Λ(c) cannot be reliably measured with visible imagery. Furthermore, the inclusion of the Λ(cbr) data at low values of cbr would yield unrealistic values of b. However, it is expected that the high wavenumber portion of the spectrum not resolved in the present study (
Assuming that the model b2 is applicable for shorter waves, using the mean parameters in Table 1 combined with the semiempirical dissipation function in Eq. (5), it is possible to estimate the Λ(c) distribution that would close the energy balance in Eq. (19), as shown with thick black and gray lines in Fig. 10, correspond to the model with Janssen’s (1991) and Snyder’s et al. (1981) wind input, respectively. The power-law behavior of the predictions of Λ(c) are consistent with the scaling obtained by balancing the energy wind input to the energy dissipation, assuming a saturated spectrum ϕ ∝ k−3. For a constant value of b(k), Λ(c) would scale as c−4 and c−5, for the wind input by Snyder et al. (1981) and Janssen (1991), respectively.
It is interesting to note that the predictions of Λ(c) at low values of c approximately connect the distributions of visible breakers by Kleiss and Melville (2010) and the available measurements in the literature of microbreaking from laboratory experiments in 9 m s−1 winds by Jessup and Phadnis (2005). This highlights the need for simultaneous field measurements of Λ(c) for both visible and microscale breaking. Figures 17a,b show the spectral breaking strength models b1 and b2, respectively. Both models were calculated using the parameters shown in Table 1 with α = 0.9. The dashed lines are extrapolations of the models b1 and b2 based on B(k) and 
Spectral breaking strength models (a) b1 and (b) b2. Both models were calculated using the parameters shown in Table 1 with α = 0.9, and the Janssen (1991) wind input. The dashed lines are extrapolations of the models b1 and b2 based on B and 
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
Stress partition as function of wind speed referenced at 10 m MSL. The closed diamonds, triangles, and stars show this study’s wave-induced momentum flux normalized by the total wind stress corresponding to the wind input by Janssen (1991), Snyder et al. (1981), and Janssen’s sheltered wind [Eq. (18)], respectively. The wave-induced stress τw, viscous stress τν, and stress due to airflow separation from breaking waves τs from the models by Kudryavtsev and Makin (2007), Mueller and Veron (2009), and Banner and Morison (2010) are shown with black lines, gray lines, and open symbols, respectively.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
Longuet-Higgins [1984, Eq. (4.5)] derived the average number of waves in a group with waves exceeding threshold amplitudes in terms of the spectral bandwidth, which gave good agreement with field observations. The analysis by Longuet-Higgins of temporal field observations gave on average eight waves in a group with wave amplitudes exceeding 2〈η2〉1/2, which corresponds to four waves per group in the spatial domain. This is qualitatively consistent with the values shown in Table 1.
6. Conclusions
A semiempirical model for the energy dissipation due to wave breaking is presented. The model is based on the radiative transport equation for fetch-limited wind-wave conditions. The model dissipation is estimated from the wave energy advection from measured spectra in the Gulf of Tehuantepec (Romero and Melville 2010a), combined with “exact” computations of the nonlinear energy transfer due to four-wave resonant interactions and three different parameterizations of the wind energy input. Following Phillips (1985), the observed statistics for the length and kinematics of breaking fronts (Kleiss and Melville 2010) are combined with the model dissipation to calculate a spectral function b(k) that characterizes the strength of wave breaking. The calculated values of b(k) are within the range of previous measurements in the laboratory, with b(k) mostly between 1 × 10−4 and 1 × 10−2, but having significant variability across the different scales and strong sensitivity to the assumed value of α, which is the ratio of the speed of the breaking wave front to the linear phase speed.
Based on the laboratory measurements by Melville and collaborators (Melville 1994), Banner and Peirson (2007), and Drazen et al. (2008), in addition to an inertial scaling argument, b(k) is parameterized in two ways: as a function of the spectral saturation B(k) and the saturation normalized by the directional spreading 
Acknowledgments
We acknowledge the collaboration of Carl A. Friehe4 and Djamal Khelif at the University of California, Irvine, in planning and conducting the GOTEX experiments and in help with the analysis of the atmospheric boundary layer data. We are grateful to Allen Schanot, Henry Boynton, Lowell Genzlinger, Ed Ringleman, and the support staff at the NCAR Research Aviation Facility. We thank Bill Krabill, Bob Swift, Jim Yungel, John Sonntag, and Robbie Russell at NASA EG&G for access to the ATM, its deployment, and initial data processing. This research was supported by grants to WKM from the National Science Foundation (OCE), the Office of Naval Research (Physical Oceanography).
APPENDIX A
Data Smoothing
The computations of the nonlinear transfer are prone to large errors due to their cubic dependence on the energy spectrum. Thus, it is desirable to have a spectrum with a large number of degrees of freedom (DOF) and thus a small spectral uncertainty. The original wave spectrum has 480 DOF and a spectral uncertainty of 25% of the energy as described in Romero and Melville (2010a). In this study, the spectrum was smoothed, prior to the interpolation on the polar grid, increasing the DOF to 3000, which corresponds to a spectral uncertainty of approximately 5% (Young 1999).
Despite the smoothing of the spectrum prior to the calculation of the various source terms,A1 the calculated directional wavenumber dissipation occasionally gives values greater than zero. However, further analysis has shown that these errors generally occur at large angles relative to the spectral peak, typically near |θ − θp| ≈ π/4, and the data confirmed that about 90% of the total dissipation was contained within |θ − θp| < π/4. Figure A1 shows an example of the source terms corresponding to the spectrum shown in Figs. 2 and 3. Figures A1a–d show Sin, Snl, Sad, and Sds. The gray contours show the area where Sds > 0, which is located at about 45° to the right of the spectral peak, overlapping with an area where Sad > 0, Snl < 0, and Sin is rapidly decreasing. According to Eq. (5), this implies that Sin is not large enough to balance or exceed Sad − Snl. This lack of balance is likely due to the asymmetry in the wavenumber plane of Sad, which in turn is induced by the slow rotation of the spectral peak with increasing fetch in the Gulf of Tehuantepec (Romero and Melville 2010a). The dissipation shown in Fig. A1d was obtained by removing the data where Sds > 0 and then filled by interpolation using Gaussian weights. This procedure was repeated with the rest of the data used for the analysis.
Terms in the radiative transport equation: (a) Sin, (b) Snl, (c) Sad, and (d) Sds. The white contours show the area where Sds > 0. The data shown correspond to the spectrum shown in Figs. 2 and 3.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
APPENDIX B
Error Analysis
The uncertainty of the nonlinear energy fluxes δSnl due to the uncertainty of the spectrum δF was calculated directly from the upper and lower error estimates of the spectrum. As expected from the cubic dependence of Snl on the energy spectrum (Phillips 1985), δSnl is approximately 3δF = 15%: about 14% for the lower and 17% for the upper error bound.
The advective term has two main sources of error, the finite directional resolution and the lack of surface current information. The propagation of the error of the former gives 
In this study, all of the errors due to Sad, Snl, and Sin described above, except the contribution of the airflow separation due to breaking waves, have been computed and added according to Eq. (B1). An example of the directional error functions is shown in Fig. B1, with the error of the wind input (Fig. B1a), nonlinear energy fluxes (Fig. B1b), advection (Fig. B1c), and the energy dissipation (Fig. B1d) normalized by the spectral dissipation. The normalized errors show that the error is largest due to the wind input and that the region of positive dissipation is in regions with large errors in all four source terms.
Uncertainties in the dissipation due to (a) the wind input δSin, (b) the nonlinear energy fluxes δSnl, (c) the advection δSad, and (d) the energy dissipation δSds. The error functions shown have been normalized by energy dissipation Sds. Black contours show the area where Sds > 0. The data shown correspond to the spectrum shown in Figs. 2 and 3 and the source term functions shown in Fig. A1.
Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-072.1
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The one-dimensional wavenumber spectra parallel to wind direction reported in MM, extending to about 3 rad m−1, is in excellent agreement with the GOTEX observations (after correcting a processing error of a factor of 2 in MM; see Romero and Melville 2010a), partially filling the gap between the GOTEX observations and Banner et al. (1989).
The ratio τw / τtot reported in Romero and Melville (2010b, Fig. A5) had a scaling error by a factor of 1.56. After correcting this factor, the numerical simulations with the wind input by Snyder (1981) give values slightly larger than those obtained in this study, while the simulations with Yan’s (1987) wind input give τw / τtot near unity for the younger waves and approaching 0.8 for older seas.
The exponent of 5/2 is consistent with the inertial scaling by Drazen et al. (2008) in Eq. (24).
We dedicate this paper to the memory of our colleague and friend, Carl Friehe, who passed away on 1 September 2011, during revision of this paper for publication. The GOTEX experiment would not have been possible without Carl’s experience and dedication to airborne marine atmospheric boundary layer research.
Following a reviewer’s suggestion, Snl was calculated from the spectrum with and without the additional smoothing. It was determined that Snl was nearly identical between the two sets.
