Spectral and statistical properties of the equilibrium range in wind-generated gravity waves

TLDR

Recent wave spectrum measurements and remote sensing observations show that the 1958 upper‑limit asymptote hypothesis is invalid, and that spectral forms derived by Kitaigorodskii and observed by Toba are consistent with the equilibrium range. The study reexamines the equilibrium range using modern insights into wave‑wave interactions, wind energy input, and wave‑breaking. The authors derive multiple spectra (slope, directional, instantaneous line traverse) and provide expressions for spectral loss rates of action, energy, and momentum due to wave‑breaking, linking these to measured spectral levels and directional distributions. The analysis shows that the equilibrium‑range spectrum follows a p≈½ wavenumber law and a u g σ⁻⁴ frequency law, that wave‑breaking momentum flux asymptotically approaches a large fraction of the wind stress while energy flux to turbulence grows logarithmically with fetch, and that statistical relations for whitecap density, bubble production, and coverage can be derived.

Abstract

Recent measurements of wave spectra and observations by remote sensing of the sea surface indicate that the author's (1958) conception of an upper-limit asymptote to the spectrum, independent of wind stress, is no longer tenable. The nature of the equilibrium range is reexamined, using the dynamical insights into wave–wave interactions, energy input from the wind and wave-breaking that have been developed since 1960. With the assumption that all three of these processes are important in the equilibrium range, the wavenumber spectrum is found to be of the form , where p ∼ ½ and the frequency spectrum is proportional to u*gσ−4. These forms have been found by Kitaigorodskii (1983) on a quite different dynamical basis; the latter is consistent with the form found empirically by Toba (1973) and later workers. Various derived spectra, such as those of the sea-surface slope and of an instantaneous line traverse of the surface, are also given, as well as directional frequency spectra and frequency spectra of slope.The theory also provides expressions for the spectral rates of action, energy and momentum loss from the equilibrium range by wave-breaking and for the spectrally integrated rates across the whole range. These indicate that, as a wave field develops with increasing fetch or duration, the momentum flux to the underlying water by wave-breaking increases asymptotically to a large fraction of the total wind stress and that the energy flux to turbulence in the water, occurring over a wide range of scales, increases logarithmically as the extent of the equilibrium range increases. Interrelationships are pointed out among different sets of measurements such as the various spectral levels, the directional distributions, the total mean-square slope and the ratio of downwind to crosswind mean-square slopes.Finally, some statistical characteristics of the breaking events are deduced, including the expected length of breaking fronts (per unit surface area) with speeds of advance between c and c+dc and the number of such breaking events passing a given point per unit time. These then lead to simple expressions for the density of whitecapping, those breaking events that produce bubbles and trails of foam, the total number of whitecaps passing a given point per unit time and, more tenuously, the whitecap coverage.