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 been attributed to a dynamical balance and so these parts of the spectra have been called the equilibrium range. We develop a model for this equilibrium range based on kinematical and dynamical properties of breaking waves. Data suggest that breaking waves have high curvature at their crests, and they are modelled here as waves with discontinuous slope at their crests. Spectra are then dominated by these singularities in slope. The equilibrium range is assumed to be scale invariant, meaning that there is no privileged lengthscale. This assumption implies that: (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 with the scale of the breaking waves as a power law, parameterized here with exponent D. The two-dimensional wavenumber spectrum of a scale-invariant distribution of such self-similar breaking waves is calculated and found to vary as Psi(k) similar to k(-5+D) The exponent D is calculated by assuming a scale-invariant dynamical balance in the equilibrium range. This balance is satisfied only when D = 1, so that Psi(k) similar to k(-4), in agreement with recent data. The frequency spectrum 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, and, when D = 1, the fraction of sea surface covered by breaking waves is the same for all scales. Hence the equilibrium described by our model is a space-filling saturation: equilibrium at a given wavenumber is established when breaking waves of the corresponding scale cover a given, 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 through 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.