ABSTRACT:
The ultimate aim of studies of random wind waves is to predict probability density function of wave characteristics, primarily wave height, at any given place and time. We study long-term nonlinear evolution of typical random wind waves which are characterized by broad-banded spectra and quasi-Gaussian statistics. Within the framework of wave turbulence paradigm the evolution of wave spectra is described by the kinetic (Hasselmann) equation derived in the sixties and now routinely employed in operational forecasting. In contrast in present the probability density function is found using some empirical formulae. We find the departure of wave statistics from Gaussianity from first principles using higher-order statistical moments (skewness and kurtosis) as a measure of this departure. Non-zero values of kurtosis mean an increase or decrease of extreme wave probability (compared to that in a Gaussian sea), which is important for assessing the risk of freak waves and other applications. The key point is as follows. For water waves, there are two different contributions to kurtosis. The first one is due to bound harmonics, while the second one is linked to genuine nonlinear wave-wave interactions. The latter contribution requires information on the phases of all interacting waves and, therefore, can be found only with direct numerical simulation. We perform such a simulation of wind-generated random wave fields, using a specially designed algorithm, based on the Zakharov equation for water waves. In generic situations, the contribution to kurtosis due to wave interactions is shown to be small compared to the bound harmonics contribution. This observation enables us to determine higher momenta by calculating the bound harmonics parts directly from spectra using asymptotic expressions. Thus, the departure of evolving wave fields from Gaussianity is explicitly contained in the instantaneous wave spectra. This enables us to broaden significantly the capability of the existing systems for wave forecasting: in addition to simulation of spectra it becomes possible to simulate also higher momenta. We found that the contributions due to bound harmonics to both skewness and kurtosis are significant for oceanic waves, and non-zero kurtosis (typically in the range 0.1-0.3) implies a tangible increase of freak wave probability.
From the analysis of the wave kinetic equation, it is well known that wave spectra of a developing wind wave field and swell often evolve in a self-similar manner. For such self-similar regimes we derive asymptotic formulas for skewness and kurtosis of a random wave field generated by constant wind, and for swell. These asymptotic properties are verified by direct numerical simulation.
ADDITIONAL INFORMATION:
Professor Shrira’s research interests include: general theory of nonlinear waves, wave turbulence, coherent flow patterns, wave processes in environmental fluid mechanics, geophysical fluid dynamics, physical oceanography, wave-flow interactions, extreme waves, wind waves, tsunamis, transitions to turbulence, Lagrangian dynamics, and remote sensing of the sea surface. Since 2000 he is the Professor of Applied Mathematics at Keele University. http://www.keele.ac.uk/scm/staff/professors/victorshrira