1. Introduction




The 3-yr field experiment carried out in Lake George, New South Wales, Australia, in 1997–2000 (Young et al. 2005; Donelan et al. 2005) revealed various novel features of wave dynamics. For the wind input Sin, the main novel features (Donelan et al. 2006) are the following: 1) Sin is a nonlinear function of the wave spectrum because the growth rate γ depends on wave steepness and hence on the spectrum itself; 2) the growth rate γ slows down in extreme conditions because of the flow separation (in relative terms; the growth still increases as the wind increases, but not as fast as one would expect by extrapolating the measurement in moderate wind-forcing conditions); and 3) wind input doubles over a breaking wave and hence can increase if the breaking rates are substantial (Babanin et al. 2007). For the whitecapping dissipation term Sds, the novel features are as follows: 1) the threshold for inherent wave breaking demonstrates its existence in terms of (significant) wave steepness (Banner et al. 2000; Babanin et al. 2001), and Babanin and Young (2005) established dimensionless value for such threshold across the entire spectrum; 2) the two-phase behavior of Sds is noteworthy: at any frequency the breaking can happen due to inherent reasons, but above the spectral peak the breaking is also enhanced due to the influence of longer waves on shorter ones (Babanin and Young 2005; Young and Babanin 2006); 3) the direct dependence of Sds on the wind speed when
These Lake George observations resulted in a new set of source functions for wind input Sin (Donelan et al. 2006; Babanin et al. 2007) and whitecapping dissipation Sds (Babanin and Young 2005; Young and Babanin 2006), which were tested in academic models (Tsagareli et al. 2010; Babanin et al. 2010) and subsequently implemented in Simulating Waves Nearshore (SWAN; Booij et al. 1999) and WAVEWATCH III (WW3; Tolman 1991) by Rogers et al. (2012, hereafter RBW12) and Zieger et al. (2015, hereafter ZBRY15), respectively. Practical modeling also required introduction of further observation-based physics such as swell dissipation Sswl (Babanin 2006, 2011; Young et al. 2013) and negative wind input (ZBRY15; Aijaz et al. 2016; Liu et al. 2017). Once the waves stop breaking, the dissipation continues, but due to a different reason: turbulence production by wave orbital motion (i.e., the so-called swell decay Sswl) (Babanin 2006; Babanin and Haus 2009; Young et al. 2013). Note that other mechanisms responsible for Sswl based on the interaction of ocean waves and upper ocean turbulence or air turbulence are also available in the literature (e.g., Teixeira and Belcher 2002; Ardhuin and Jenkins 2006; Ardhuin et al. 2009). This complete set of new physics ready for practical forecast and hindcast received the name of ST61 in 2014 and 2016 public releases of WW3 (WAVEWATCH III Development Group 2016, hereafter T16). Besides, ST6 is now formally part of the SWAN model as well (SWAN Team 2018, version 41.20A). Further academic developments related to ST6 included a new nonlinear interaction term based on the general kinetic equation (Gramstad and Babanin 2016), modules for wave–current interactions (Rapizo et al. 2017), infragravity waves (Nose et al. 2017), and wave–ice interactions (to be released in the 2019 version of WW3).
Since its implementation in SWAN and WW3, this unique source term package, ST6, of
As will be shown in this paper, this shortcoming can be solved by increasing the wind input Sin slightly and then recalibrating tunable parameters of Sds [i.e.,
The Discrete Interaction Approximation of Snl (DIA; Hasselmann et al. 1985) is the crucial component permitting routine application of third-generation wave models (e.g., Hasselmann et al. 1988; Tolman 1991). It however also has some well-known shortcomings as an approximation (see an extended discussion about this issue in section 2b). To investigate the errors in spectral wave models attributable to the DIA, we first specifically optimize another more accurate nonlinear solver, that is, the Generalized Multiple DIA (GMD; Tolman 2013) for ST6, and then conduct a thorough comparison of model simulations with these two different nonlinear solvers. The most prominent advantage of the GMD-based model over the DIA-based model, as later illustrated in this paper, is that the former shows a much higher accuracy in simulating the energy of long-period waves (
This paper is organized as follows. Section 2 provides a brief overview of ST6 source functions (Sin + Sds + Sswl) and the four-wave resonant interactions Snl. Section 3 describes the updates of ST6 over its predecessor, particularly focusing on the retuning procedure. Section 4 presents a detailed analysis of modeled wave spectra from duration-limited simulations, followed by a thorough validation of model performance with a 1-yr global hindcast in section 5. Conclusions in section 6 finalize this paper.
2. Parameterizations
a. ST6 source term package
A brief overview of the ST6 source terms is given here for completeness. The reader is referred to RBW12 and ZBRY15 and references therein for more details.
1) Wind input 
























2) Wave breaking 
















3) Swell dissipation 






b. Nonlinear wave–wave interactions 





Nonetheless, the shortcomings of DIA have also been extensively discussed. First, the two lobes of
















Interaction diagram for
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
3. Model calibrations
Herein ST6 is applied with different parameterizations of Snl. To distinguish these model configurations, we will refer to ST6 + DIA as ST6D, ST6 + GMD as ST6G, and ST6 + WRT as ST6W. When necessary, the combination of the ST4 source terms (Ardhuin et al. 2010) and DIA (hereafter ST4D), which is used for the operational forecasting in NOAA’s National Centers for Environmental Prediction (NCEP; Alves et al. 2015), is also included for comparisons. The wind growth parameter
a. Calibration of ST6D
As mentioned in section 1, our (re)calibration of ST6D is conducted by increasing the (positive) wind input term slightly, and then finding the new tunable parameters existing in the wave breaking and swell decay terms (i.e.,












(a) Dimensionless energy
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
We first made an attempt to determine (
- ST6D-simulated dimensionless energy
and peak frequency should match the KC92 growth curves in (16) reasonably well. Specifically, , , and , where and are the normalized bias and root-mean-square error (RMSE; appendix A). The positiveness of is imposed due to the exclusion of the negative and terms at this stage (i.e., ).5 - For fully developed waves, for example, waves at the 30th point of the coarsest grid where
km (black arrows in Fig. 2), the two-phase wave breaking term in (8) should satisfy (RBW12).
Through our tuning exercises, we found that the two restrictions described above yield a narrow corridor in the
- 3) For a realistic 75-day wave hindcast in Lake Michigan,6 ST6D should predict both
and mean square slope quite accurately (e.g., the RMSE of and are less than 0.2 m and 10−3, respectively), as compared with measurements from a single buoy 45007 (see section 2 of the SOM).
Waves prevailing in Lake Michigan are generally young and free of wind-swell interactions (Rogers and van Vledder 2013). Accordingly, the deactivation of Sswl and negative Sin in this hindcast experiment is still physically sound.
After the optimal
Parameter setting for ST6D, ST6G, and ST6W, including five parameters

























b. Calibration of ST6G
For the GMD parameterization of
Using the same
Since it is extremely expensive, if not impossible, to run ST6W in realistic large-scale applications, we did not try a further calibration of ST6W. For simplicity, ST6W directly inherits all the parameters (i.e.,
c. Fetch geometry
The last topic we would like to discuss in this section is the effect of fetch geometry on wave growth (Young 1999, p. 109). Field studies (Pettersson and Kahma 2005; Ataktürk and Katsaros 1999) suggested that for the same dimensionless fetch χ, the dimensionless energy ε values of mature waves were remarkably lower for the narrow fetch than for the broad fetch. The dimensionless frequency ν was also affected to some extent, but was clearly less sensitive than ε. It was believed that the narrow geometry constrained the development of waves propagating along directions oblique to the long axis of narrow fetches (bays or lakes). As demonstrated in Rogers and Wang (2006, their Fig. 15) and Ataktürk and Katsaros (1999, p. 643), the third-generation wave models are able to provide qualitatively consistent behavior. An interesting detail, which we think is worth mentioning and which we found from our simulations, is that the DIA-based results are more sensitive to the fetch geometry than those from the GMD and WRT.













Fetch-limited simulations from the third grid only (
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
Relative differences between the asymptotic values of the dimensionless energy and frequency (

4. Duration-limited wave growth
The previous section focuses on integrated wave parameters (e.g., significant wave height
- to what extent modeled spectra reflect measured properties of ocean waves, and
- how well the GMD configuration presented in the previous section represents the exact solutions of
(i.e., WRT).
A single-grid-point, duration-limited wave growth experiment is selected here, mainly due to its computational efficiency and its reduced sensitivity to numerical errors (e.g., RBW12). The model setup is the same as the one used in fetch-limited simulations, except that the directional grid is refined from
a. Equilibrium and saturation ranges



















The evolution of (left) omnidirectional frequency spectrum
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
The saturation spectra














Toba’s parameter
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As in Fig. 4, but for
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b. Transition frequency



















The (left) high-frequency energy level α and (right) peak enhancement factor γ vs the dimensionless frequency
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
In Fig. 6 we replotted
c. Spectral peakedness



Both of these forms of (28) with













The evolution of (a) spectral width
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
Therefore, a consistent finding from our Figs. 7 and 8 is that the WRT- and GMD-based models (i.e., ST6W and ST6G), contrary to our expectation, noticeably overestimate the peakedness/narrowness (or alternatively, underestimate the width) of wave spectra. Being an approximation to the WRT, the DIA provides a slightly improved, but still problematic in general, estimation of the spectral peakedness due to its inherent tendency to unrealistically broaden the exact solutions in frequency space. Although counterintuitive, this finding is remarkably supported by a recent numerical study by Annenkov and Shrira (2018). The authors showed (their Figs. 6 and 11) that relative to the WRT results based on the Hasselmann kinetic equation (Hasselmann 1962), their direct numerical simulations based on the Zakharov integrodifferential equation (Zakharov 1968) predict “considerably wider frequency spectra with much less pronounced peaks.” Considering that the Zakharov equation is “the primitive equation for a weakly nonlinear wave field” and “does not employ any statistical assumptions,” and considering that the Hasselmann kinetic equation can be derived from the Zakharov equation “by applying standard closure hypothesis,” Annenkov and Shrira (2018) argued these systematic mismatches “call for revision of the fundamentals” of the Hasselmann kinetic equation. Besides, the poor performance of wave models in simulating the spectral peak implies a difficulty in predicting the occurrence of freak waves.
d. Directional properties


Figure 9 illustrates

Directional spreading
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Figure 10 presents

The
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5. Global hindcast
After a thorough analysis of model skills in specifying the wave spectrum in academic tests, it is necessary to verify performances of ST6D and ST6G in realistic global hindcasts. ST6W is excluded from the analysis here because of its computational infeasibility at the global scale.
SABZ16 conducted a comprehensive study on the comparison and assessment of different source term packages (Tolman and Chalikov 1996; Janssen 2004; Ardhuin et al. 2010; ZBRY15) available in WW3 with a global hindcast of the year 2011. For easy intercomparison with other packages evaluated in SABZ16, we selected the same year (2011) in our simulations. The global model domain is bounded within 78°S–78°N, with a resolution of 1/2° × 1/2°. The resolution of the spectral grid is
a. Comparison against altimeters
Figure 11 presents comparisons of the simulated wave height

Comparison of
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
The spatial distributions of the normalized bias

Error metrics of
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The geophysical mismatches presented in Fig. 12 can be partially, but not fully, explained by the errors of wind forcing (Fig. B1). For example, the overestimation of
b. Comparison against NDBC buoys
The validation of model simulations against in situ buoys was also conducted using observations from a total of 21 stations (Fig. 13) maintained by the U.S. National Data Buoy Center (NDBC; http://www.ndbc.noaa.gov). Closely following SABZ16, we only selected wave buoys that provide two-dimensional wave spectra and that are located in deep water (depth

A total of 21 NDBC buoys (filled circles) used in the model–buoy comparison.
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1





















Taylor diagram summarizing the statistical comparison between NDBC buoys and wave models: (a) ST6D, (b) ST6G, and (c) ST4D. The wave parameters represented by different colored-markers are interpreted in figure legends. Values in parentheses identify the normalized bias
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
Our Fig. 14 suggests that wave height
The directional spreading
The most noticeable advantage of GMD over DIA is seen in comparisons of long-period wave energy
To highlight improvements in simulating high-frequency energy brought about by the recalibrated ST6D over its predecessor (ZBRY15), and improvements in modeling low-frequency energy brought about by use of the GMD over the DIA, we replotted the mean square slope

Averaged mean square slope
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1

Comparison of partial wave heights for (top)
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One detail that needs to be further clarified is that the swell decay coefficient














As in Fig. 14, but for partitioned wave parameters from (a) ST6D, (b) ST6G, and (c) ST4D. The last subscripts w and s of variables printed in figure legends denote parameters for wind sea (markers with black outlines) and swell (markers without black outlines), respectively.
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
Statistical comparison (as represented by the normalized RMSE

The last problem to be addressed here is the computational efficiency of different wave models. Table 4 shows the normalized model run times obtained with wave models using different nonlinear solvers and different prognostic frequency ranges. The high-frequency limit
The normalized model run time for 1-month global hindcast using the model setup summarized in section 5. The high-frequency limit

6. Concluding remarks
In this paper, the observation-based source term package in WW3, that is, the ST6 for Sin + Sds + Swl (RBW12; ZBRY15), is recalibrated and verified through a series of academic and realistic simulations, including the fetch/duration-limited test, a Lake Michigan (no swell) hindcast, and a 1-yr global hindcast. In addition to the traditional bulk (integral) criteria, we introduced spectral metrics for model validations. We also specifically optimized the GMD nonlinear solver for ST6 (Fig. 1) in order to investigate the DIA-induced uncertainties in spectral wave modeling. Key findings are summarized below:
- The updated ST6D (i.e., ST6 + DIA) and ST6G (i.e., ST6 + GMD) not only are skillful in simulating commonly used bulk wave parameters (e.g.,
and wave periods) but also accurately represent the high-frequency wave spectrum [in terms of the saturation spectrum and mean square slope ]. The overestimation of high-frequency wave energy by ST6’s predecessor (e.g., ZBRY15) is resolved by the recalibration described here (e.g., Fig. 15). - In the duration-limited test,
simulated by ST6 models [ST6D, ST6G, and ST6W (i.e., ST6 + WRT); Figs. 4a,c,e] shows a clear transition behavior from the power law of approximately to the power law of approximately . The modeled energy level of the equilibrium range [ ], as represented by the wave age-dependent Toba’s parameter , is in good agreement with field measurements from Hwang et al. (2000a) and Resio et al. (2004) (Fig. 5). The saturation level yielded by these three ST6 models is also consistent with observations from Babanin and Soloviev (1998a), RM10, and Lenain and Melville (2017) (Figs. 4b,d,f). In addition, the ST6-predicted transition frequency from an to is comparable to buoy data from Forristall (1981). - The wave spectra from ST6G are in excellent agreement with those from ST6W, particularly in the frequency space (Figs. 4–9), illustrating the high accuracy of the GMD approach in reproducing exact solutions of Snl from WRT. In the global hindcast, ST6G exhibits a much better performance in predicting low-frequency wave energy. The normalized biases of
(wave period s) given by ST6D and ST4D are 90% and 69%, respectively, whereas such model errors are significantly reduced by ST6G ( ) (Figs. 14 and 16), which is analogous to the findings from Rogers and van Vledder (2013). Nonetheless, contrary to our expectation, ST6G only provides marginal improvement in characterizing different wave systems (i.e., wind sea and swell; Fig. 17). The GMD configuration used here is ~5 times more expensive than the DIA (Table 4) and therefore might not be economically feasible for operational forecasting. - When we fit the generalized JONSWAP spectrum (28) to the modeled
from the duration-limited case, the simulated high-frequency α is generally consistent with previous field studies (e.g., Hasselmann et al. 1976; Donelan et al. 1985). The simulated spectral peakedness, in terms of γ [(28)] or width [(29)], however, generally deviates from field observations (e.g., Donelan et al. 1985; Krivinskii 1991; Babanin and Soloviev 1998a) (Figs. 7 and 8). In particular, the peak of from ST6G and ST6w appears too narrow, consistent with the finding of the recent numerical study by Annenkov and Shrira (2018). - A few problems presented here still remain unsolved, including that 1) the spectral narrowness
and directional spreading are quite poorly resolved in the global simulations, as shown in Figs. 14 and 17; 2) wave models are able, to some degree, to present bimodal structure of short waves (Fig. 10); the lobe ratio , however, is considerably underestimated (SOM); and 3) the model bias in the Southern Ocean is still relatively high ( ). All these issues are left for future research.
- ST6D and ST4D provide good, and very close, performance in estimating the commonly used integral wave parameters (significant wave height
, wave periods, mean square slope etc.), and therefore either of the two is applicable to the operational wave forecasting and hindcasting. Perrie et al. (2018) demonstrated that in their high-resolution wave forecast model systems, ST4D outperformed the physics package originally designed for the WAM model (Hasselmann et al. 1988) but at the expense of at least 50% more CPU time. Considering the slightly higher computational efficiency of ST6D in the 1-yr global hindcast (Table 4), we expect that ST6D may save noticeable computational costs in such refined, high-resolution applications. - The wave spectrum of short gravity waves is crucial to estimate the wave-induced momentum flux from wind (e.g., Chalikov and Rainchik 2011). Since ST6 yields an improved high-frequency tail, it is recommended to further test/verify this package in the fully coupled atmosphere–wave–ocean models (e.g., Fan et al. 2009a; Warner et al. 2010; Chen et al. 2013).
- The ST6G model configuration (five quadruplets) increases computational costs by a factor of about 6, restricting its applicability to research purposes only, at least at this stage. For academic studies particularly concerned with low-frequency wave energy, ST6G is preferred over ST6D.
- Only deep-water waves are considered in our present study. A thorough validation of the updated ST6 configurations in the finite-deep and shallow waters, similar to the work conducted by Aijaz et al. (2016) and van Vledder et al. (2016), is recommended for further analyses.
The authors are grateful to Dr. Kevin Ewans from MetOcean Research Ltd, New Zealand, for discussion about various aspects of wave spectra. The distribution and maintenance of WW3 and GMD codes by NOAA/NCEP is gratefully acknowledged. We appreciate Dr. David W. Wang from the U.S. Naval Research Laboratory for providing his code of the Maximum Entropy Method (MEM). QL, AVB, and IRY acknowledge the support from the DISI Australia–China Centre through Grant ACSRF48199. AVB also appreciates the financial support by ARC Discovery DP170101328. FQ is supported by the international cooperation project on the China–Australia Research Centre for Maritime Engineering of Ministry of Science and Technology, China under Grant 2016YFE0101400. We thank Dr. G. P. van Vledder and another reviewer for their detailed comments and suggestions that have improved our manuscript a lot.
APPENDIX A
Integral Parameters from the Wave Spectrum





















APPENDIX B
The Performance of the CFSv2 Winds
The performance of the CFSv2 winds of 2011 was carefully checked by SABZ16 (their section 2.3), using

As in Fig. 12, but for (a) the normalized bias
Citation: Journal of Physical Oceanography 49, 2; 10.1175/JPO-D-18-0137.1
REFERENCES
Aijaz, S., W. E. Rogers, and A. V. Babanin, 2016: Wave spectral response to sudden changes in wind direction in finite-depth waters. Ocean Modell., 103, 98–117, https://doi.org/10.1016/j.ocemod.2015.11.006.
Alves, J. H. G. M., and M. L. Banner, 2003: Performance of a saturation-based dissipation-rate source term in modeling the fetch-limited evolution of wind waves. J. Phys. Oceanogr., 33, 1274–1298, https://doi.org/10.1175/1520-0485(2003)033<1274:POASDS>2.0.CO;2.
Alves, J. H. G. M., M. L. Banner, and I. R. Young, 2003: Revisiting the Pierson–Moskowitz asymptotic limits for fully developed wind waves. J. Phys. Oceanogr., 33, 1301–1323, https://doi.org/10.1175/1520-0485(2003)033<1301:RTPALF>2.0.CO;2.
Alves, J. H. G. M., S. Stripling, A. Chawla, H. L. Tolman, and A. J. van der Westhuysen, 2015: Operational wave guidance at the U.S. National Weather Service during Tropical/post–Tropical Storm Sandy, October 2012. Mon. Wea. Rev., 143, 1687–1702, https://doi.org/10.1175/MWR-D-14-00143.1.
Annenkov, S. Y., and V. I. Shrira, 2018: Spectral evolution of weakly nonlinear random waves: Kinetic description versus direct numerical simulations. J. Fluid Mech., 844, 766–795, https://doi.org/10.1017/jfm.2018.185.
Ardhuin, F., and A. D. Jenkins, 2006: On the interaction of surface waves and upper ocean turbulence. J. Phys. Oceanogr., 36, 551–557, https://doi.org/10.1175/JPO2862.1.
Ardhuin, F., B. Chapron, and F. Collard, 2009: Observation of swell dissipation across oceans. Geophys. Res. Lett., 36, L06607, https://doi.org/10.1029/2008GL037030.
Ardhuin, F., and Coauthors, 2010: Semiempirical dissipation source functions for ocean waves. Part I: Definition, calibration, and validation. J. Phys. Oceanogr., 40, 1917–1941, https://doi.org/10.1175/2010JPO4324.1.
Ardhuin, F., J. Tournadre, P. Queffeulou, F. Girard-Ardhuin, and F. Collard, 2011: Observation and parameterization of small icebergs: Drifting breakwaters in the Southern Ocean. Ocean Modell., 39, 405–410, https://doi.org/10.1016/j.ocemod.2011.03.004.
Ataktürk, S. S., and K. B. Katsaros, 1999: Wind stress and surface waves observed on Lake Washington. J. Phys. Oceanogr., 29, 633–650, https://doi.org/10.1175/1520-0485(1999)029<0633:WSASWO>2.0.CO;2.
Babanin, A. V., 2006: On a wave-induced turbulence and a wave-mixed upper ocean layer. Geophys. Res. Lett., 33, L20605, https://doi.org/10.1029/2006GL027308.
Babanin, A. V., 2010: Wind input, nonlinear interactions and wave breaking at the spectrum tail of wind-generated waves; transition from f−4 to f−5 behaviour. Ecological Safety of Coastal and Shelf Zones and Comprehensive Use of Shelf Resources, Vol. 21, Marine Hydrophysical Institute, 173–187.
Babanin, A. V., 2011: Breaking and Dissipation of Ocean Surface Waves. Cambridge University Press, 480 pp., https://doi.org/10.1017/CBO9780511736162.
Babanin, A. V., and Y. P. Soloviev, 1998a: Field investigation of transformation of the wind wave frequency spectrum with fetch and the stage of development. J. Phys. Oceanogr., 28, 563–576, https://doi.org/10.1175/1520-0485(1998)028<0563:FIOTOT>2.0.CO;2.
Babanin, A. V., and Y. P. Soloviev, 1998b: Variability of directional spectra of wind-generated waves, studied by means of wave staff arrays. Mar. Freshwater Res., 49, 89–101, https://doi.org/10.1071/MF96126.
Babanin, A. V., and I. R. Young, 2005: Two-phase behaviour of the spectral dissipation of wind waves. Fifth Int. Symp. on Ocean Wave Measurements and Analysis (WAVES2005), Madrid, Spain, CEDEX, 51.
Babanin, A. V., and B. K. Haus, 2009: On the existence of water turbulence induced by nonbreaking surface waves. J. Phys. Oceanogr., 39, 2675–2679, https://doi.org/10.1175/2009JPO4202.1.
Babanin, A. V., I. R. Young, and M. L. Banner, 2001: Breaking probabilities for dominant surface waves on water of finite constant depth. J. Geophys. Res., 106, 11 659–11 676, https://doi.org/10.1029/2000JC000215.
Babanin, A. V., M. L. Banner, I. R. Young, and M. A. Donelan, 2007: Wave-follower field measurements of the wind-input spectral function. Part III: Parameterization of the wind-input enhancement due to wave breaking. J. Phys. Oceanogr., 37, 2764–2775, https://doi.org/10.1175/2007JPO3757.1.
Babanin, A. V., K. N. Tsagareli, I. R. Young, and D. J. Walker, 2010: Numerical investigation of spectral evolution of wind waves. Part II: Dissipation term and evolution tests. J. Phys. Oceanogr., 40, 667–683, https://doi.org/10.1175/2009JPO4370.1.
Banner, M. L., 1990: Equilibrium spectra of wind waves. J. Phys. Oceanogr., 20, 966–984, https://doi.org/10.1175/1520-0485(1990)020<0966:ESOWW>2.0.CO;2.
Banner, M. L., and I. R. Young, 1994: Modeling spectral dissipation in the evolution of wind waves. Part I: Assessment of existing model performance. J. Phys. Oceanogr., 24, 1550–1571, https://doi.org/10.1175/1520-0485(1994)024<1550:MSDITE>2.0.CO;2.
Banner, M. L., A. V. Babanin, and I. R. Young, 2000: Breaking probability for dominant waves on the sea surface. J. Phys. Oceanogr., 30, 3145–3160, https://doi.org/10.1175/1520-0485(2000)030<3145:BPFDWO>2.0.CO;2.
Battjes, J. A., T. J. Zitman, and L. H. Holthuusen, 1987: A reanalysis of the spectra observed in JONSWAP. J. Phys. Oceanogr., 17, 1288–1295, https://doi.org/10.1175/1520-0485(1987)017<1288:AROTSO>2.0.CO;2.
Booij, N., R. C. Ris, and L. H. Holthuijsen, 1999: A third-generation wave model for coastal regions: 1. Model description and validation. J. Geophys. Res., 104, 7649–7666, https://doi.org/10.1029/98JC02622.
Cavaleri, L., and Coauthors, 2007: Wave modelling—The state of the art. Prog. Oceanogr., 75, 603–674, https://doi.org/10.1016/j.pocean.2007.05.005.
Cavaleri, L., and Coauthors, 2018: Wave modelling in coastal and inner seas. Prog. Oceanogr., 167, 164–233, https://doi.org/10.1016/j.pocean.2018.03.010.
Chalikov, D., and S. Rainchik, 2011: Coupled numerical modelling of wind and waves and the theory of the wave boundary layer. Bound.-Layer Meteor., 138, 1–41, https://doi.org/10.1007/s10546-010-9543-7.
Chen, S. S., W. Zhao, M. A. Donelan, and H. L. Tolman, 2013: Directional wind–wave coupling in fully coupled atmosphere–wave–ocean models: Results from CBLAST-Hurricane. J. Atmos. Sci., 70, 3198–3215, https://doi.org/10.1175/JAS-D-12-0157.1.
Delpey, M. T., F. Ardhuin, F. Collard, and B. Chapron, 2010: Space–time structure of long ocean swell fields. J. Geophys. Res., 115, C12037, https://doi.org/10.1029/2009JC005885.