Self-similarity of wind-driven seas

International audienceThe results of theoretical and numerical study of the Hasselmann kinetic equation for deep water waves in presence of wind input and dissipation are presented. The guideline of the study: nonlinear transfer is the dominating mechanism of wind-wave evolution. In other words, the most important features of wind-driven sea could be understood in a framework of conservative Hasselmann equation while forcing and dissipation determine parameters of a solution of the conservative equation. The conservative Hasselmann equation has a rich family of self-similar solutions for duration-limited and fetch-limited wind-wave growth. These solutions are closely related to classic stationary and homogeneous weak-turbulent Kolmogorov spectra and can be considered as non-stationary and non-homogeneous generalizations of these spectra. It is shown that experimental parameterizations of wind-wave spectra (e.g. JONSWAP spectrum) that imply self-similarity give a solid basis for comparison with theoretical predictions. In particular, the self-similarity analysis predicts correctly the dependence of mean wave energy and mean frequency on wave age Cp / U10. This comparison is detailed in the extensive numerical study of duration-limited growth of wind waves. The study is based on algorithm suggested by Webb (1978) that was first realized as an operating code by Resio and Perrie (1989, 1991). This code is now updated: the new version is up to one order faster than the previous one. The new stable and reliable code makes possible to perform massive numerical simulation of the Hasselmann equation with different models of wind input and dissipation. As a result, a strong tendency of numerical solutions to self-similar behavior is shown for rather wide range of wave generation and dissipation conditions. We found very good quantitative coincidence of these solutions with available results on duration-limited growth, as well as with experimental parametrization of fetch-limited spectra JONSWAP in terms of wind-wave age Cp / U10

Coexistence of Weak and Strong Wave Turbulence in a Swell Propagation

By performing two parallel numerical experiments -- solving the dynamical Hamiltonian equations and solving the Hasselmann kinetic equation -- we examined the applicability of the theory of weak turbulence to the description of the time evolution of an ensemble of free surface waves (a swell) on deep water. We observed qualitative coincidence of the results. To achieve quantitative coincidence, we augmented the kinetic equation by an empirical dissipation term modelling the strongly nonlinear process of white-capping. Fitting the two experiments, we determined the dissipation function due to wave breaking and found that it depends very sharply on the parameter of nonlinearity (the surface steepness). The onset of white-capping can be compared to a second-order phase transition. This result corroborates with experimental observations by Banner, Babanin, Young.Comment: 5 pages, 5 figures, Submitted in Phys. Rev. Letter

Horizon effects with surface waves on moving water

Surface waves on a stationary flow of water are considered, in a linear model that includes the surface tension of the fluid. The resulting gravity-capillary waves experience a rich array of horizon effects when propagating against the flow. In some cases three horizons (points where the group velocity of the wave reverses) exist for waves with a single laboratory frequency. Some of these effects are familiar in fluid mechanics under the name of wave blocking, but other aspects, in particular waves with negative co-moving frequency and the Hawking effect, were overlooked until surface waves were investigated as examples of analogue gravity [Sch\"utzhold R and Unruh W G 2002 Phys. Rev. D 66 044019]. A comprehensive presentation of the various horizon effects for gravity-capillary waves is given, with emphasis on the deep water/short wavelength case kh>>1 where many analytical results can be derived. A similarity of the state space of the waves to that of a thermodynamic system is pointed out.Comment: 30 pages, 15 figures. Minor change

On discriminating swell and wind-driven seas in voluntary observing ship data

[1] The global visual wave observations are reanalyzed within the theoretical concept of self-similar wind-driven seas. The core of the analysis is one-parametric dependencies of wave height on wave period. Theoretically, wind-driven seas are governed by power-like laws with exponents close to Toba's one 3/2 while the corresponding swell exponent (À1/2) has an opposite signature. This simple criterion was used and appeared to be adequate to the problem of swell and wind-driven waves discrimination. This theoretically based discrimination does not follow exactly the Voluntary Observing Ship (VOS) data. This important issue is considered both in the context of methodology of obtaining VOS data and within the physics of wind waves. The results are detailed for global estimates and for analysis of particular areas of the Pacific Ocean. Prospects of further studies are discussed. In particular, satellite data are seen to be promising for tracking ocean swell and for studies of physical mechanisms of its evolution. Citation: Badulin, S. I., and V. G. Grigorieva (2012), On discriminating swell and wind-driven seas in Voluntary Observing Ship data

On experimental justification of weekly turbulent nature of growing wind seas

The causes of breaking of deep-water two-dimensional waves are studied. Evolution of initially monochromatic steep waves to the point of breaking was first investigated by means of the fully nonlinear Chalikov-Sheinin model. Individual wave steepness is found to be the single most important parameter which determines whether the wave will break immediately, never break or take a finite number of wave lengths to break. If the wind forcing is superimposed, it can alter the wave-breaking dependences, but these effects appear to be of secondary importance. The results were subsequently verified and elaborated in a laboratory experiment. The experiment demonstrated good qualitative agreement with the numerical simulations and consequently the breaking dependences were quantified. Since the location of breaking onset, which occurs as a result of the natural evolution of nonlinear wave trains can thus be controlled, properties of incipient breakers were measured. It was found that the breaking will occur once the wave reaches the Stokes limiting steepness. Potential applications to field conditions are also discussed