M. Serio

Freak Waves in Random Oceanic Sea States

Freak waves are very large, rare events in a random ocean wave train. Here we study their generation in a random sea state characterized by the Joint North Sea Wave Project spectrum. We assume, to cubic order in nonlinearity, that the wave dynamics are governed by the nonlinear Schrödinger (NLS) equation. We show from extensive numerical simulations of the NLS equation how freak waves in a random sea state are more likely to occur for large values of the Phillips parameter alpha and the enhancement coefficient gamma. Comparison with linear simulations is also reported.

The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains

Abstract Rogue waves are rare “giant”, “freak”, “monster” or “steep wave” events in nonlinear deep water gravity waves which occasionally rise up to surprising heights above the background wave field. Holes are deep troughs which occur before and/or after the largest rogue crests. The dynamical behavior of these giant waves is here addressed as solutions of the nonlinear Schrodinger equation in both 1+1 and 2+1 dimensions. We discuss analytical results for 1+1 dimensions and demonstrate numerically, for certain sets of initial conditions, the ubiquitous occurrence of rogue waves and holes in 2+1 spatial dimensions. A typical wave field evidently consists of a background of stable wave modes punctuated by the intermittent upthrusting of unstable rogue waves.

Extreme wave events in directional, random oceanic sea states

We discuss the effect of the directional spreading on the occurrence of extreme wave events. We numerically integrate the envelope equation recently proposed by Trulsen et al. [Phys. Fluids 12, 2432 (2000)] as a weakly nonlinear model for realistic oceanic gravity waves. Initial conditions for numerical simulations are characterized by the spatial JONSWAP power spectrum for several values of the significant wave height, steepness, and directional spreading. We show that by increasing the directionality of the initial spectrum the appearance of extreme events is reduced.

Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a three-dimensional wave basin

A wave basin experiment has been performed in the MARINTEK laboratories, in one of the largest existing three-dimensional wave tanks in the world. The aim of the experiment is to investigate the effects of directional energy distribution on the statistical properties of surface gravity waves. Different degrees of directionality have been considered, starting from long-crested waves up to directional distributions with a spread of ±30° at the spectral peak. Particular attention is given to the tails of the distribution function of the surface elevation, wave heights and wave crests. Comparison with a simplified model based on second-order theory is reported. The results show that for long-crested, steep and narrow-banded waves, the second-order theory underestimates the probability of occurrence of large waves. As directional effects are included, the departure from second-order theory becomes less accentuated and the surface elevation is characterized by weak deviations from Gaussian statistics.

Modulational instability and non-Gaussian statistics in experimental random water-wave trains

We study random, long-crested surface gravity waves in the laboratory environment. Starting with wave spectra characterized by random phases we consider the development of the modulational instability and the consequent formation of large amplitude waves. We address both dynamical and statistical interpretations of the experimental data. While it is well known that the Stokes wave nonlinearity leads to non-Gaussian statistics, we also find that the presence of the modulational instability is responsible for the departure from a Gaussian behavior, indicating that, for particular parameters in the wave spectrum, coherent unstable modes are quite prevalent, leading to the occurrence of what might be called a “rogue sea.” Statistical results are also compared with ensemble numerical simulations of the Dysthe equation.

Solitons, cnoidal waves and nonlinear interactions in shallow-water ocean surface waves

Abstract We analyze shallow-water surface wave data from the Adriatic Sea using a nonlinear generalization of Fourier analysis based upon the periodic inverse scattering transform in the θ-function representation for the Korteweg-de Vries (KdV) equation. While linear Fourier analysis superposes sine waves, the nonlinear Fourier approach superposes cnoidal waves (the travelling wave solution to KdV) plus their mutual, nonlinear interactions . A new procedure is presented for the nonlinear low-pass and band-pass filtering of measured wave trains. We apply the approach to a measured time series and discuss the dynamics of solitons and the physics of the nonlinear interactions in terms of global, spatio-temporal phase shifts amongst the cnoidal waves.

Interaction of two quasi--monochromatic waves in shallow water

We study the nonlinear interaction of waves propagating in the same direction in shallow water characterized by a double-peaked power spectrum. The starting point is the prototypical equation for weakly nonlinear unidirectional waves in shallow water, i.e., the Korteweg–de Vries equation. In the framework of envelope equations, using a multiple-scale technique and under the hypothesis of narrow-banded spectra, a system of two coupled nonlinear Schrodinger equations is derived. The validity of the resulting model and the stability of their plane wave solutions is discussed. We show that when retaining higher order dispersive terms in the system, plane wave solutions become modulationally unstable.

Correlation dimension of underground muon time series

This paper presents results obtained for the correlation dimension of cosmic ray muon time series detected at a depth underground of 570 hg/cm² over a period of 8 years. The corresponding galactic primaries have rigidities of the order of a few teravolts and are therefore at least partially under the influence of solar modulation effects. The principal indication is that the physical mechanisms which shape the muon time series in the frequency range below 2 × 10−5 Hz are entirely stochastic in behavior and have a monofractal structure. The correlation dimension is finite and shows an inverse dependence on the level of solar activity, oscillating between 4.5 ± 0.4 at solar maximum and 6.2 ± 0.6 at solar minimum.

FINITE CORRELATION DIMENSION AND POSITIVE LYAPUNOV EXPONENTS FOR SURFACE WAVE DATA IN THE ADRIATIC SEA NEAR VENICE

We study wind-driven surface wave data taken on an offshore platform in 16 m of water, about 20 km from Venice in the Northern Adriatic Sea. The data are investigated for the effects of chaos and to this end they are subjected to a variety of time series analysis techniques from the field of dynamical systems theory. For certain data sets we find a finite value for the correlation dimension (~7) and a positive value for the largest Lyapunov exponent (~1.5×10−3 bit/sec). In spite of the fact that these results suggest the possibility of chaotic behavior in the data, the correct interpretation is that the data are essentially stochastic, and that the correlation dimensions and Lyapunov exponents result from the anomalous statistical behavior of certain near-Gaussian random processes whose properties we discuss.

The impact of background noise on the determination of the fractal and statistical properties of cosmic ray time series

This paper investigates the influence of background white noise on the determination of the colored random noise properties of cosmic ray experimental signals. We consider fractal methods (Grassberger and Procaccia method, scaling exponent method and fractal length method) and statistical methods (comparison with the Gaussian and Rice curves, and the multivariate scaling analysis technique). The investigation is carried out using two experimental time series with different color (spectral indices α = 1.6 and α = 1.2) and, as a reference, several computer simulations of pure noise-free colored random noise signals with 0 ≤ α ≤ 3 with the same number of points. The results of the investigation is that the presence of noise in a experimental signal “deceives” the algorithms employed in the various analyses causing them to sample noiselike properties (e.g., an infinite or very large fractal dimension and Gaussian or quasi-Gaussian behavior), irrespective of the physical characteristics of the particular data set.