Breaking waves and the equilibrium range of wind-wave spectra

Abstract

When scaled properly, the high-wavenumber and high-frequency parts of wind-wave 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.