BREAKING WAVES AND THE EQUILIBRIUM RANGE OF WIND-WAVE SPECTRA

When scaled properly, the high-wavenumber and high-frequency parts of windwave spectra collapse onto universal curves. This collapse has bee n attributed to a dynamical balance and so these parts of the spectra have been called the equilibrium range. We develop a model for this eq uilibrium range based on kinematical and dynamical properties of break ing waves. Data suggest that breaking waves have high curvature at the ir crests, and they are modelled here as waves with discontinuous slop e at their crests. Spectra are then dominated by these singularities i n slope. The equilibrium range is assumed to be scale invariant, meani ng that there is no privileged lengthscale. This assumption implies th at: (i) the sharp-crested breaking waves have self-similar shapes, so that large breaking waves are magnified copies of the smaller breaking waves; and (ii) statistical properties of breaking waves, such as the average total length of breaking-wave fronts of a given scale, vary w ith the scale of the breaking waves as a power law, parameterized here with exponent D. The two-dimensional wavenumber spectrum of a scale-i nvariant distribution of such self-similar breaking waves is calculate d and found to vary as Psi(k) similar to k(-5+D) The exponent D is cal culated by assuming a scale-invariant dynamical balance in the equilib rium range. This balance is satisfied only when D = 1, so that Psi(k) similar to k(-4), in agreement with recent data. The frequency spectru m is also calculated and shown to vary as Phi(sigma) similar to sigma( -4), which is also in good agreement with data. The theory also gives statistics for the coverage of the sea surface with breaking waves, an d, when D = 1, the fraction of sea surface covered by breaking waves i s the same for all scales. Hence the equilibrium described by our mode l is a space-filling saturation: equilibrium at a given wavenumber is established when breaking waves of the corresponding scale cover a giv en, wind-dependent, fraction of the sea surface. Although both Psi(k) and Phi(sigma) vary with the same power law, the two power laws arise from quite different physical causes. As the wavenumber, k, increases, Psi(k) receives contributions from smaller and smaller breaking waves . In contrast, Phi(sigma) is dominated by the largest breaking waves t hrough the whole of the equilibrium range and contains no information about the small-scale waves. This deduction from the model suggests a way of using data to distinguish the present theory from previous work .