Weakly turbulent laws of wind-wave growth

The theory of weak turbulence developed for wind-driven waves in theoretical works and in recent extensive numerical studies concludes that non-dimensional features of self-similar wave growth (i.e. wave energy and characteristic frequency) have to be scaled by internal wave-field properties (fluxes of energy, momentum or wave action) rather than by external attributes (e.g. wind speed) which have been widely adopted since the 1960s. Based on the hypothesis of dominant nonlinear transfer, an asymptotic weakly turbulent relation for the total energy ε and a characteristic wave frequency ω* was derived ε ω*⁴/g² = α_ss (ω*³ dε/dt/g²)^1/3. The self-similarity parameter α_ss was found in the numerical duration-limited experiments and was shown to be naturally varying in a relatively narrow range, being dependent on the energy growth rate only. In this work, the analytical and numerical conclusions are further verified by means of known field dependencies for wave energy growth and peak frequency downshift. A comprehensive set of more than 20 such dependencies, obtained over almost 50 years of field observations, is analysed. The estimates give α_ss very close to the numerical values. They demonstrate that the weakly turbulent law has a general value and describes the wave evolution well, apart from the earliest and full wave development stages when nonlinear transfer competes with wave input and dissipation.