The third generation ocean wave models, which integrate the dynamical equations that describe the evolution of a wave field, have been widely used in scientific studies and practical applications. Though input mechanisms, in particular bulk transfer of energy action and momentum to waves, and other source terms are not well established either, the least understood aspect of the physics of wave model is the dissipation terms (Donelan and Yuan, 1994; Young and Babanin, 2006; Babanin, 2011). The dissipation source (sink) terms induced by wave-generated turbulence and random wave-breaking are the subject of the present paper. There are a number of dissipation mechanisms studied previously through either experimental or analytical (or both) approaches, and they have advanced very significantly over the past decades. However, there remains a notable gap in mechanism research pertaining to varied parameterizations and applications in wave models.

Concerning to the wave-breaking dissipation, the most mathematically well-advanced and most frequently utilized whitecap model is that of Hasselmann (1974) which involves the whitecaps are weak-in-the-mean and the dissipation is a linear function of the spectrum, and its parameterization has been proposed and further extended in various WAM-Cycle models by Komen et al. (1984), WAMDI G. (1988), Bidlot et al. (2005), Bidlot (2012), etc. Following the quasi-saturated ideas of Phillips (1985) and numerical modeling framework of Alves and Banner (2003), Ardhuin et al. (2009b, 2010), Filipot and Ardhuin (2012) presented an improved wave-breaking dissipation parameterization as the sum of the saturation-based term and a cumulative breaking term, while the latter represents the smoothing of the surface by big breakers that wipe out smaller waves. The probability model due to wave breaking was proposed by Longuet-Higgins (1969), Yuan et al. (1986) and Hua and Yuan (1992). This kind of wave-breaking dissipation source function was derived from the breaking wave statistics, in which the power of the normalized integral wave steepness parameter is 1/2 and less than that of whitecap model. So the probability model gives lower dissipation value for the high sea state condition (Yuan et al., 1993), and has been applied in the LAGFD (Laboratory of Geophysical Fluid Dynamics)-WAM wave model and MASNUM (Key Laboratory of MArine Science and NUmerical Modeling) wave model (Yuan et al., 1991; Yang et al., 2005). Though conceptually attractive, its practical parameterization fitted to the whitecap model is deprived of its own discretionary estimates of the dissipation rate. Based on the random phase spectral action density balance equation for wavenumber-direction spectra and evolved from earlier WW1 and WW2 (WAVEWATCH I & II) model packages (Tolman, 1991, 1992), the comprehensive source terms were incorporated and employed in WW3 ((WAVEWATCH III) wave model by Tolman and Chalikov (1996), Chalikov and Belevich (1993), Chalikov (1995) , Tolman (2002), etc. Young and Babanin (2006) summarized series of previous experimental attempts to obtain an experimental spectral dissipation function, and based on direct estimates of wave energy loss due to dominant breaking through their comprehensive measurement records at Lake George, they proposed a new spectral dissipation source term due to wave breaking. This experimental parameterization was improved and employed in WW3 (WAVEWATCH III) wave model (Babanin et al., 2007, 2010; Babanin, 2009; Rogers et al., 2012; Liu et al., 2019; WW3DG, 2019). The quantitative match is dubious between the latter applications and the former experimental estimates, despite only a single field record was analyzed but verified approximately by turbulent kinetic energy (TKE) dissipation rates which were retrieved from synchronously ADV measured turbulence spectra (Young and Babanin, 2006). In addition, the physical meaning of wave-breaking dissipation rate (time derivative of wave energy spectrum), as stated in previous studies, remains fuzzy and vague because the wave-breaking process is not continuous but very short and intermittent, which stimulate us to propose an improved analytical postbreaking spectrum expression, from the point of view of the probability theory, for practical implementation. This constitutes one of our primary focal points of this study.

Polnikov (1994, 2005, 2010) and Polnikov and Tkalich (2006) argued that the mechanism of wave energy dissipation is completely conditioned by the interaction between wave motions and the turbulence of the upper water layer, and the latter is generated by a great number of physical processes, including different kinds of hydrodynamic instabilities of mechanical motions near the interface. They solved Reynolds' equation where the Reynolds' stress was expanded into a series with respect to velocity components and their spatial derivatives. The Prandtl mixing-length hypothesis was used to close the turbulent terms in these series. Their physical treatment is attractive, but the theory needs further development (Young and Babanin, 2006), even though it was further constructed by means of the phenomenological similarity method and spectrally justified in the frame of the proposed eddy viscosity model (Polnikov, 2012). Tolman and Chalikov (1996) also suggested a turbulent dissipation analogy for tunable closure modeling. For dominant low-frequency dissipation, if wave motion and turbulence are not correlated, their interaction can be accounted for by introducing an effective, although weak, turbulent viscosity coefficient in the oceanic boundary layer. But for poorly understood high-frequency dissipation, a diagnostic high-frequency dissipation was defined and designed to result in a consistent source term balance. And the total dissipation source term was defined as a linear combination of the above high and low frequency constituents. Their separation parameterization for the dissipation term is physically meaningful and the heuristic arguments have been employed in WW3 ST2 package and widely applied in many practical numerical models. In fact, during the past two decades, a series of studies concerned with mechanisms of nonbreaking wave-generated turbulence and its mixing effects on upper ocean layers have been achieved significantly and their conclusions have been confirmed more exhaustively by experiments. The theoretical wave-generated turbulence theory can be established by two different but relatively consistent approaches: the first approach involves the use of a parameterization form similar to the classical Prandtl mixing-length theory, and the second approach involves the use of the equilibrium solutions of a high-deterministic second-order turbulence closure model between the wave motion shear instability generations and the TKE dissipations (Baumert et al., 2005), then the analytical mixing coefficients were proposed to elaborate the dominant mixing intensity induced by wave-generated turbulence in the upper ocean (Yuan et al., 1999; Yuan et al., 2013). Qualitative and quantitative validations by field measurements and improvements of wave-current coupled modelling indicate the key mixing role in the formation of upper mixed layers (Qiao et al., 2004, 2010; Xia et al., 2004, 2006; Shu et al., 2011; Shi et al., 2016, 2019; Yu et al., 2020; Zhuang et al., 2020, 2021, 2022; Yang et al., 2003, 2004, 2019, 2022). But how the reaction of wave-generated turbulence on ocean waves was disregarded in their studies, which is still lack and needs to investigate further. Babanin (2006), Babanin and Haus (2009), Dai et al. (2010) tested and confirmed the nonbreaking wave-generated turbulence through mechanically generated laboratory wave experiments. Babanin and Chalikov (2012) also presented that the vorticity and turbulence usually occur in vicinity of wave crests and then spread over upwind slope and downward through a numerical wave-turbulence model. Based on their experimental approximations, a parameterization of swell dissipation rate was proposed, verified through altimeter observation data and employed in WW3 ST6 package (Babanin, 2011, 2012; Young et al., 2013; Zieger et al., 2015; WW3DG, 2019). In fact, an analytical parameterization for wave energy dissipation can be effectively deduced through the equilibrium solutions of wave-generated turbulence, which will be described in detail and implemented in numerical experiments below. It constitutes the other focal point of this study to illustrate the important role of dissipation induced by wave-generated turbulence, which is definitely the feedback of imparting of wave shear instability generations on turbulence.

There are also a number of other dissipation mechanisms which are certainly not negligible in the wave system. Prominent negative input source term for swell was introduced by Chalikov and Belevich (1993), Chalikov (1995), Tolman and Chalikov (1996), Tolman (2002) and Chalikov and Babanin (2019) where the phase velocity of waves is larger than the wind velocity, which means that the dynamic pressure of the wind on the forward face of the wave component exceeds the pressure on the backward face and waves accelerate wind, resulting in the momentum and energy fluxes from the waves to the wind. Based on the direct measurements of turbulent air-sea fluxes obtained during several sea expeditions, Grachev and Fairall (2001) verified that long ocean waves (swell) traveling faster than local wind and in the same direction cause upward momentum transport, implying a negative drag coefficient. A weak damping of swells was also introduced by Janssen (2004), who proposed an asymptotic linearization of the small effects of air turbulent eddies. Ardhuin et al. (2009a, 2010), Collard et al. (2009) proposed a nonlinear swell dissipation parameterization, which is related to a laminar-to-turbulent transition of the oscillatory boundary layer over swells, using the spaceborne SAR observed swell fields from the European Space Agency's (ESA) ENVISAT satellite. Interaction of ocean waves and upper ocean turbulence, while the latter is induced by Stokes drift shears, accounts for a significant fraction of the energy losses of the wave field (McWilliams et al., 1997; Teixeira and Belcher, 2002; Ardhuin and Jenkins, 2006; Guo and Shen, 2013, 2014). Evaluations indicated it is much weaker than other dissipations (Ardhuin et al., 2010), despite this its formula is employed in WW3 ST4 package. Model results showed that its effects significantly improve simulations of turbulence characteristics and upper ocean thermal structure (Huang and Qiao, 2010; Huang et al., 2011). Sea bottom-wave interactions and ice-wave interactions have been studied in great detail, which are out of scope of present study and not discussed here.

The objectives of this paper are to explore the dissipation effects of wave-generated turbulence reacting on ocean waves, and to investigate the role of wave-breaking dissipation via an improved postbreaking spectrum expression based on the breaking wave statistical method. The remainder of this paper is organized as follows: Sect. 2 describes the analytical approaches for wave energy dissipation induced by wave-generated turbulence and by random wave-breaking, introduces scale detection comparing to wind input and provides verifications with laboratory observations or comprehensive measurements; Sect. 3 presents application implementations on simple duration-limited growth and decay experiments; Sect. 4 addresses discussions and issues which need complex insights, and some conclusions and suggestions for future research are summarized in Sect. 5.

There remain two undetermined parameters αwt,ρ in the preceding section, which will be discussed further in Sect. 4. Both of them are chosen as tunable parameters in the duration-limited growth and decay experiments undertook below. The numerical experiments were carried out with the MASNUM wave model (Yuan et al., 1991; Yang et al., 2005), and implemented according to the studies of Janssen et al. (1994). We ran the model for seven days, the first two days with a wind speed of 18.45 m s⁻¹, after which the wind dropped to a value of 5 m s⁻¹. The model integration time step was 30 s, so the growth limiter can be switched off and its impacts need not be considered here.

For simplicity, the parameterization of linear growth in spectral density proposed by Komen et al. (1984) and the quasi-linear one by Janssen (1991) for wind input source function Sin are used respectively. As stated above, coefficient αwt should be tuned to 0.02–0.2. Wang et al. (2017, 2018) validated the sea surface whitecap courage obtained from the statistical wave-breaking model with the satellite-derived data and proposed that the range of ρ is 0.53–0.59, and this referenced span is used in the following experiments.

To evaluate the scaling behavior of the new dissipation formulations due to wave-generated turbulence and wave-breaking proposed in Sect. 2, as well as their different effects, several numerical experiments are carried out (Table 2). In the original 3rd generation MASNUM wave model described here (Experiment 1), a parameterization proposed by Yuan et al. (1986),Yuan et al. (1993), Donelan and Yuan (1994) is adopted for wave-breaking dissipation source function Sds, i.e., 28Sds=-d1ω^ωω^2α^α^PM1/2exp⁡-d2(1-εsp2)α^PMα^E(k1,k2) where ω^=μ-1/μ0-1,α^=μ0ω^4g-2, α^PM is the value of α^ for a PM spectrum (α^PM=3.02×10-3). Two critical coefficients d1 and d2 (d1=1.32×10-4,d2=2.61) were retrieved through fitting algorithm with the dimensional expression proposed by Komen et al. (1984). The corresponding dissipation term in the WAM-Cycle models is given as. 29Sds=-Cdsω^ωω^2α^α^PM2E(k1,k2) where Cds is a constant (Cds=2.36×10-5).

In Experiment 2, instead of this source function Sds, Eqs. (20) and (26) are used to calculate the postbreaking spectrum, while other unchanged source functions Sin,Snl, Sbo and Scu are integrated to obtain the incipient-breaking spectrum. Then the dissipation source function induced by wave-generated turbulence Stid is further considered in Experiment 3. In the above Experiments 2 and 3, ρ=0.59 is chosen, while ρ=0.53 in the supplemental Experiment 2S for further comparison.

Figures 7 and 8 show the time evolutions of wave height and peak frequency, where the wind input source function in MASNUM wave model was adopted from Komen et al. (1984). The wave height grows during the first two days, then decreases significantly after the wind drops two days later. Results of Janssen et al. (1994) with circle symbols are listed for comparison with Experiment 1–3. Results of the original MASNUM wave model (Experiment 1) are consistent with that of Janssen et al. (1994), but the deviation of wave height may be noted 96h later due to different power of normalized wave slope in the wave-breaking dissipation source function (Yuan et al., 1991; Donelan and Yuan, 1994). The difference of wave height between Experiment 2 and others, both the maximum quantity and swell decay, can be distinguished apparently. Especially the wave height in Experiment 2 hardly changes during the swell decay process, because the mean swell steepness becomes small gradually and αb′∼1.0 in Eqs. (20) and (26). This indicates that the wave energy loss induced by wave-breaking is inadequate, and the role of prior proposed parameterizations for wave-breaking dissipation may be overestimated in the previous studies. Besides the effect of postbreaking wave spectrum, the dissipation source function induced by wave-generated turbulence Stid is incorporated in Experiment 3. The corresponding modeled wave height and peak frequency are also listed in Figs. 7 and 8, in which different αwt= 0.05 or 0.06 is selected to highlight its effect. Discrepancies for the maximum of wave height and swell decay are reduced much than that of Experiment 2, and its variation has an analogous trend with that of Janssen et al. (1994).

Figure 9 shows the time evolution of wave height, where the wind input source function in MASNUM wave model was adopted from Janssen (1991). Similar interpretation stated above can be obtained, besides that the effect of postbreaking wave spectrum is still inadequate even for that in the supplemental Experiment 2S where ρ=0.53.

The analytical approaches and the corresponding comparisons to laboratory or in-lake site measurements improve further understandings of wave energy dissipation due to wave-breaking and wave-generated turbulence. This study still exhibits some deficiencies and needs to be addressed on comparative assessments and metrics to previous formulations, as well as evaluations of scaling behavior of the new model, etc. Model validation is tentative and requires future enhanced observations correspondingly. Moreover, there remain problems to be addressed that ρ,αwt are chosen as constants in this study. Parameter ρ is associated with the spectrum width εsp. And according to the statistical theory of breaking waves (Cartwright and Longuet-Higgins, 1956; Longuet-Higgins, 1957; Yuan et al., 1986, 2009), both parameters ρ,εsp should be referred to those of incipient-breaking spectrum, which is obtained as an intermediate variable in our model where all source functions are integrated except for the energy loss induced by wave-breaking. In fact, the observed or model outputted wave spectrum would respond to the postbreaking wave spectrum (Donelan and Yuan, 1994). So the prior-to-breaking parameter ρ or εsp is currently still poorly understood. Moreover, Lamarre and Melville (1991), Melville et al. (1992) showed that 30 %–50 % of energy lost by breaking waves is expended on entraining bubbles into the water against buoyancy forces, and the residue contributes to the turbulence generation. The interaction mechanism of the breaking-induced turbulence with the progressive waves is also unknown. It should also be mentioned that the whitecap model originally proposed by Hasselmann (1974) is an after-breaking class model, and its inherent assumptions need experimental verification (Young and Babanin, 2006). The whitecaps are situated on the forward faces of the waves, exert a downward pressure on the upward moving water, but the direct and precise estimates of their negative work on the waves need further studies. Given the intricate interactions state above, in our model proposed here, the choice of ρ as a tuned parameter is tentative for numerical implementation.

The constant αwt is chosen in such a way as a tuned coefficient that it implies the quasi-equilibrium level of wave-generated turbulence, and physically it depends on the normalized shear instability strength of wave motion and further relates to the normalized higher-order moment of wave spectrum (see also Eqs. 7–8). Therefore it is certainly not constant at any instant t in real-world scenarios and its reasonable parameterization should be further studied in the future. This idea is quite similar to the wave-amplitude-based Reynolds number proposed by Babanin (2006), its critical wave Reynolds number for wave-induced turbulence can reach down to a lower margin, Re_wave ∼ 1000, from a series of laboratory experiments (Babanin and Haus, 2009; Dai, et al., 2010). But the appropriate dissipation rate for model implements cannot be inferred from the wave Reynolds number alone and was approached by experimental means (Babanin, 2011; Zieger et al., 2015; Liu, et al., 2019). The gradient Richardson number Rg may be a more appropriate dependency factor for the coefficient αwt, which needs further perspectives. If the gradient Richardson number Rg is smaller than a critical value Rgc, instability of the flow occurs, i.e., Rg=N^32∂uSMα∂x32<Rgc,α=1,2 where N^32=-g∂∂x3ρ^ρ0, the Brunt–Väisälä frequency. Below we let N^3= const ≈0.01 rad s⁻¹ in the upper layers and the critical value Rgc≈1/2 (Baumert and Peters, 2000; Baumert et al., 2005). For a monochromatic non-breaking wave ζ=ηcos⁡(kαxα-ωt), which is deemed as a base motion in the framework of the ocean dynamic system, the instability criterion is simplified as N^32<Rgcω2AK2exp⁡(2Kx3)cos⁡2(kαxα-ωt) We roughly estimate the critical instability depth induced by the monochromatic waves (Table 3), Fig. 10 shows the dependence of the gradient Richardson number versus layer depth for Case 1 in Table 3. Below the wave crests and troughs, there exist bowl-shaped instability regions, which agree to the experimentally observed instantaneous turbulence incurred intermittently at the rear face of the progressive wave profile, but the breaking-in-progress turbulence develops at the front face (Babanin and Haus, 2009; Babanin, 2011). The coefficient αwt is qualitatively related to the two-dimensional instability area proportion as shown in Fig. 10 or the instability volume proportion beneath the wave surface in real scenarios, which requires further research.

Finally, some limitations of underlying assumptions in our theoretical arguments require further discussions. In the k-ε type eddy viscosity model of wave-generated turbulence, the turbulence timescale is assumed to be shorter than wave period. Therefore, the interaction of turbulence at other scales with ocean waves, which also accounts for a significant fraction of the energy losses of the wave field, is not included in our practical numerical wave model. The quasi-equilibrium assumption of wave-generated turbulence poses the problem of an undetermined coefficient, which needs further elaborations especially essential precise measurements. The linear non-breaking wave theory applied to construct analytical solutions for the turbulence generation is an acceptable assumption in the general case of weakly nonlinear situations, and the scale estimation via the shear instability of wave orbital motions only represents a major portion of the turbulent production in the upper ocean. As stated by Yuan et al. (2009), since the wave spectrum in real-world scenarios is not actually narrow, the breaking wave statistical method under a narrow spectrum assumption is not precise enough to estimate the attenuation coefficient for the postbreaking wave spectrum.

The ocean wave energy dissipation is the least understood of the major source terms, previous approaches to estimate the dissipation source function depended on an incomplete description of the physics of the processes including wave-breaking and wave-turbulence interaction. The latest observational efforts offer a possible approach to explore the underlying comprehensive mechanisms.

In the present paper, we attempted to explore the dissipation effects of wave-generated turbulence reacting on ocean waves, and to estimate the energy loss due to wave-breaking via an improved postbreaking spectrum expression based on the breaking wave statistical method. Two new source functions for the above two dissipation processes are proposed and compared respectively to the laboratory or in-lake observations tentatively in Sect. 2, and their different dissipation effects are experimentally analyzed in Sect. 3.

The main conclusion of the study is that we propose an analytical dissipation source function induced by wave-generated turbulence Stid formulated by Eq. (15), together with an improved postbreaking spectrum expression Eb(k1,k2) by Eqs. (20) and (26). The former dissipation term represents the feedback of imparting of wave shear instability generations on turbulence, and the latter expression depicts the intermittent wave-breaking events. Sum of both contributions play critical role of wave energy dissipation.

Here in this paper we roughly estimate the effects of the above two dissipation mechanisms in simple duration-limited growth and decay experiments. Calibration and verification against a series of academic and realistic simulations, including the fetch/duration-limited cases, turning wind (e.g. cold waves or monsoon)/rotatory wind (e.g. extratropical or tropical cyclone) conditions, numerical hindcast and operational forecast in regional and global oceans, will be pursued in our future project, together with considering other concerned dissipation mechanisms.