Hasselmann (1974) expressed the effect of white capping — the sporadic overturning and breaking of steep wave crests — on the spectral energy balance of ocean waves as an equivalent ensemble of random pressure pulses, and showed that such a weak-in-the-mean, locally strong process yields a quasi-linear dissipation source function whose damping coefficient scales with the square of the wave frequency, provided wave scales are large compared with white-cap dimensions. The absolute dissipation level, encoded in a free coefficient η\eta (with higher-order coefficients η1,η2,…\eta_{1},\eta_{2},\ldots), could not be derived because no fluid-dynamical model of the white-cap process existed; it was fixed only indirectly through the spectral energy and momentum balance, whose closure rested on an assumed linear wind-input source function awaiting direct measurement. Subsequent work established that the wind-input measurement program was carried out (Snyder et al. 1981; Janssen's quasi-linear theory 1991), and that the frequency-squared dissipation shape was superseded by calibrated, observation-based parameterizations (Komen et al. 1984; van der Westhuysen et al. 2007; Ardhuin et al. 2010) whose coefficients are still tuned by trial and error, while phase-resolving simulations, which evolve the instantaneous sea surface rather than a statistical spectrum (Korotkevich et al. 2019), suggest these tuned levels may err by an order of magnitude. The surviving open problem is the physical, calibration-free determination of the whitecapping dissipation source function and its coefficients from the fluid dynamics of wave breaking, jointly with an independently determined input, so that the wind-wave energy and momentum balance closes without indirect tuning.

Background

Wind-generated ocean waves are described statistically by a directional spectrum F(σ,θ)F(\sigma,\theta), the distribution of wave energy over (radian) frequency σ\sigma and propagation direction θ\theta, with the deep-water dispersion relation σ2=gk\sigma^{2}=gk linking σ\sigma to wavenumber kk (gg is gravitational acceleration). Numerical wave prediction rests on an energy-balance (radiative-transfer) equation: along the path of each wave component, the local rate of change of the spectrum equals a sum of source functions — energy input from the wind, SinS_{\mathrm{in}}; redistribution among components by resonant four-wave (weakly nonlinear) interactions, SnlS_{\mathrm{nl}}; and dissipation, SdsS_{\mathrm{ds}}. Under a steady, spatially uniform wind the spectrum grows with fetch (the distance over which the wind has blown) toward a fully developed state, classically represented by the Pierson–Moskowitz spectrum, the benchmark observed spectrum of such a sea. The 1973 JONSWAP field experiment (Joint North Sea Wave Project) established fetch-limited growth laws, a power-law equilibrium range F(σ)=αg2σ−5F(\sigma)=\alpha g^{2}\sigma^{-5} at high frequencies, and that this range is governed by a non-local balance involving all three source terms rather than by breaking alone.

White capping — the intermittent overturning and breaking of steep crests, visible as foam — is the dominant sink of gravity-wave energy at moderate and high winds. It is localized, violently nonlinear, and intermittent, so it defeats the weak-interaction perturbation methods that work for SnlS_{\mathrm{nl}}. Hasselmann (1974) nevertheless gave it a statistical treatment: representing white caps as an equivalent ensemble of random pressure pulses acting on the sea surface, he proved that any interaction process that is weak in the mean, though locally strong (he called such processes non-expansible, meaning they admit no convergent expansion in wave slope), produces to lowest order a quasi-linear source function — linear in the wave spectrum, written in the 1974 notation as E(k)E(\mathbf{k}) (the same directional spectrum expressed as a function of the wavenumber vector k\mathbf{k}). For white capping this yields a damping

Swb=−γE(k),γ=η σ2,S_{\mathrm{wb}}=-\gamma E(\mathbf{k}),\qquad \gamma=\eta\,\sigma^{2},

valid when the scales of the waves are large compared with the dimensions of the white caps. The coefficient η\eta — and the coefficients η1,η2,…\eta_{1},\eta_{2},\ldots entering the higher-order terms of the pulse-moment expansion — could not be computed: no fluid-dynamical model of an individual white cap existed from which the equivalent pressure field could be derived. The absolute dissipation level was therefore fixed indirectly, by requiring the energy and momentum balance evaluated with JONSWAP-type spectra to close. That closure in turn assumed a wind input linear in the spectrum, with a Miles-type parameterization (after the shear-flow wave-generation theory of Miles) Sin=ϱaϱw(σcos⁡θ−g/U) E(k)S_{\mathrm{in}}=\frac{\varrho_{a}}{\varrho_{w}}(\sigma\cos\theta-g/U)\,E(\mathbf{k}), nonzero only for σcos⁡θ≥g/U\sigma\cos\theta\ge g/U (ϱa,ϱw\varrho_{a},\varrho_{w} are air and water densities, UU the wind speed, θ\theta measured from the wind direction) — a form resting on surface-pressure measurements that disagreed among themselves. The model implied that nearly all momentum crossing the air–sea interface enters the wave field, and an approximately constant drag coefficient for stationary uniform winds.

The paper itself flagged what remained open: the fluid-dynamical structure of the white-cap process and the coefficient η\eta; the independent determination of SinS_{\mathrm{in}}, for which it called for direct cross-correlation measurements of surface pressure and wave height, including a test of the extrapolation to the fully developed sea; and possible missing processes — laminar viscous dissipation that may dominate at light winds and very small fetches near the onset of white capping (where the required high-frequency spectral levels were themselves uncertain), and non-conservative short-wave interactions carrying energy and momentum into the gravity-capillary range (waves of centimeter scale, restored by both gravity and surface tension), which available data could not quantify. The unresolved target is therefore a physically closed source-term set for the spectral energy balance, not merely a numerical value for one coefficient.

Problem Statement

The problem left open by Hasselmann (1974) is the physical closure of the wind-wave spectral energy balance, in three parts that are jointly required for completion — a single source-term set in which no dissipation coefficient is fixed by tuning:

(1) Construct an explicit fluid-dynamical model of the white-cap process and derive from it the equivalent pressure-pulse ensemble and the coefficients η,η1,η2,…\eta,\eta_{1},\eta_{2},\ldots of the quasi-linear whitecapping dissipation source function, so that the damping γ\gamma and the absolute level of spectral dissipation follow from the physics of breaking rather than from indirect closure of the energy and momentum balance. A justified alternative to the quasi-linear form is admissible if it is derived from the white-cap dynamics rather than fitted to integral balance requirements.

(2) Determine the wind-input source function independently of the closure, so that the linear form Sin=βE(k)S_{\mathrm{in}}=\beta E(\mathbf{k}), its parameterization, the implied partition of air–sea momentum into the wave field, and the extrapolation to the fully developed sea can be tested rather than assumed. The open target here is the non-circular joint closure with the derived dissipation, together with the still-untested validity of the linear input form and of its extrapolation to the fully developed sea.

(3) Establish the regimes and additional processes in which the balance so closed requires modification: the possible dominance of laminar viscous dissipation at light winds and very small fetches near the onset of white capping, including the uncertain high-frequency spectral levels this involves, and the non-conservative interactions that transfer energy and momentum into the gravity-capillary range.

The common completion boundary is one closed energy and momentum balance for the wind-wave spectrum across fetch, wind speed, and development stage, with the additional processes named in part (3) entering as terms of that same closed set wherever they are non-negligible rather than as exceptions to it. Completion is reached when the dissipation level predicted from the physics of breaking and the input independently determined together form one fixed set of source-term coefficients that reproduces the observed spectral growth, the equilibrium range, and the interfacial momentum partition across that whole range without any dissipation or input-function coefficient being adjusted to make the balance close; the absence of tuning is shown by that single coefficient set sufficing over the range rather than by coefficients refitted case by case.

References

  1. Hasselmann, K. (1974). On the spectral dissipation of ocean waves due to white capping. Boundary-Layer Meteorology 6, 107–127. DOI 10.1007/BF00232479. https://doi.org/10.1007/BF00232479
  2. Snyder, R. L., Dobson, F. W., Elliott, J. A., Long, R. B. (1981). Array measurements of atmospheric pressure fluctuations above surface gravity waves. Journal of Fluid Mechanics 102, 1–59. DOI 10.1017/S0022112081002528. https://doi.org/10.1017/S0022112081002528
  3. Komen, G. J., Hasselmann, S., Hasselmann, K. (1984). On the existence of a fully developed wind-sea spectrum. Journal of Physical Oceanography 14, 1271–1285. DOI 10.1175/1520-0485(1984)014<1271:OTEOAF>2.0.CO;2. https://doi.org/10.1175/1520-0485(1984)014%3C1271:OTEOAF%3E2.0.CO%3B2
  4. Janssen, P. A. E. M. (1991). Quasi-linear theory of wind-wave generation applied to wave forecasting. Journal of Physical Oceanography 21, 1631–1642. DOI 10.1175/1520-0485(1991)021<1631:QLTOWW>2.0.CO;2. https://doi.org/10.1175/1520-0485(1991)021%3C1631:QLTOWW%3E2.0.CO%3B2
  5. Banner, M. L., Peregrine, D. H. (1993). Wave breaking in deep water. Annual Review of Fluid Mechanics 25, 373–397. DOI 10.1146/annurev.fl.25.010193.002105. https://doi.org/10.1146/annurev.fl.25.010193.002105
  6. Tolman, H. L., Chalikov, D. V. (1996). Source terms in a third-generation wind wave model. Journal of Physical Oceanography 26, 2497–2518. DOI 10.1175/1520-0485(1996)026<2497:STIATG>2.0.CO;2. https://doi.org/10.1175/1520-0485(1996)026%3C2497:STIATG%3E2.0.CO%3B2
  7. Young, I. R., Babanin, A. V. (2006). Spectral distribution of energy dissipation of wind-generated waves due to dominant wave breaking. Journal of Physical Oceanography 36, 376–394. DOI 10.1175/JPO2859.1. https://doi.org/10.1175/JPO2859.1
  8. van der Westhuysen, A. J., Zijlema, M., Battjes, J. A. (2007). Nonlinear saturation-based whitecapping dissipation in SWAN for deep and shallow water. Coastal Engineering 54, 151–170. DOI 10.1016/j.coastaleng.2006.08.006. https://doi.org/10.1016/j.coastaleng.2006.08.006
  9. Ardhuin, F., Rogers, E., Babanin, A. V., Filipot, J.-F., Magne, R., Roland, A., van der Westhuysen, A., Queffeulou, P., Lefevre, J.-M., Aouf, L., Collard, F. (2010). Semiempirical dissipation source functions for ocean waves. Part I: Definition, calibration, and validation. Journal of Physical Oceanography 40, 1917–1941. DOI 10.1175/2010JPO4324.1. https://doi.org/10.1175/2010JPO4324.1
  10. Babanin, A. V. (2011). Breaking and Dissipation of Ocean Surface Waves. Cambridge University Press. DOI 10.1017/CBO9780511736162. https://doi.org/10.1017/CBO9780511736162
  11. Chalikov, D., Babanin, A. V. (2012). Simulation of wave breaking in one-dimensional spectral environment. Journal of Physical Oceanography 42, 1745–1761. DOI 10.1175/JPO-D-11-0128.1. https://doi.org/10.1175/JPO-D-11-0128.1
  12. Scanlon, B., Breivik, Ø., Bidlot, J.-R., Janssen, P. A. E. M., Callaghan, A. H., Ward, B. (2016). Modeling whitecap fraction with a wave model. Journal of Physical Oceanography 46, 887–894. DOI 10.1175/JPO-D-15-0158.1. https://doi.org/10.1175/JPO-D-15-0158.1
  13. Korotkevich, A. O., Prokofiev, A. O., Zakharov, V. E. (2019). On the dissipation rate of ocean waves due to white capping. JETP Letters 109, 309–315. DOI 10.1134/S0021364019050035. https://doi.org/10.1134/S0021364019050035
  14. Romero, L. (2019). Distribution of surface wave breaking fronts. Geophysical Research Letters 46, 10463–10474. DOI 10.1029/2019GL083408. https://doi.org/10.1029/2019GL083408
  15. Du, J., Bolaños, R., Larsén, X. G., Kelly, M. (2019). Wave boundary layer model in SWAN revisited. Ocean Science 15, 361–377. DOI 10.5194/os-15-361-2019. https://doi.org/10.5194/os-15-361-2019
  16. Lei, Z., Wu, W., Gu, Y., Zhai, F., Li, P. (2023). A general method to determine the optimal whitecapping dissipation coefficient in the SWAN model. Frontiers in Marine Science 10, 1298727. DOI 10.3389/fmars.2023.1298727. https://doi.org/10.3389/fmars.2023.1298727
  17. Ayala, F. F., Montoya, R. D., Osorio, A. F., Babanin, A. V. (2025). On the directional distributions of the observation-based wind input and dissipation terms in a third-generation wave model. Ocean Modelling 196, 102542. DOI 10.1016/j.ocemod.2025.102542. https://doi.org/10.1016/j.ocemod.2025.102542
  18. Ferdaus, M. M., Cooper, N. A., Schmidt, A. B., Pokhrel, P., Ioup, E., Abdelguerfi, M., Simeonov, J. (2025). A comprehensive review of phase-averaged and phase-resolving wave models for coastal modeling applications. arXiv:2511.21856. DOI 10.48550/arXiv.2511.21856. https://arxiv.org/abs/2511.21856
  19. Herman, A., Dragan-Górska, A., Markuszewski, P. (2026). Close relationship between whitecapping-related wave energy dissipation and ambient sound in the open ocean. Geophysical Research Letters 53, e2025GL120342. DOI 10.1029/2025GL120342. https://doi.org/10.1029/2025GL120342