Modelling of Ocean Waves with the Alber Equation:Application to Non-Parametric Spectra and Generalisation to Crossing Seas

The Alber equation is a phase-averaged second-moment model for the statistics of a sea state, which recently has been attracting renewed attention. We extend it in two ways: firstly, we derive a generalized Alber system starting from a system of nonlinear Schr\"odinger equations, which contains the classical Alber equation as a special case but can also describe crossing seas, i.e. two wavesystems with different wavenumbers crossing. (These can be two completely independent wavenumbers, i.e. in general different directions and different moduli.) We also derive the associated 2-dimensional scalar instability condition. This is the first time that a modulation instability condition applicable to crossing seas has been systematically derived for general spectra. Secondly, we use the classical Alber equation and its associated instability condition to quantify how close a given non-parametric spectrum is to being modulationally unstable. We apply this to a dataset of 100 non-parametric spectra provided by the Norwegian Meteorological Institute, and find the vast majority of realistic spectra turn out to be stable, but three extreme sea states are found to be unstable (out of 20 sea states chosen for their severity). Moreover, we introduce a novel "proximity to instability" (PTI) metric, inspired by the stability analysis. This is seen to correlate strongly with the steepness and Benjamin-Feir Index (BFI) for the sea states in our dataset (>85% Spearman rank correlation). Furthermore, upon comparing with phase-resolved broadband Monte Carlo simulations, the kurtosis and probability of rogue waves for each sea state are also seen to correlate well with its PTI (>85% Spearman rank correlation)

Laboratory evidence of freak waves provoked by non-uniform bathymetry

We show experimental evidence that as relatively long unidirectional waves propagate over a sloping bottom, from a deeper to a shallower domain, there can be a local maximum of kurtosis and skewness close to the shallower side of the slope. We also show evidence that the probability of large wave envelope has a local maximum near the shallower side of the slope. We therefore anticipate that the probability of freak waves can have a local maximum near the shallower side of a slope for relatively long unidirectional waves. Copyright 2012 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Phys. Fluids 24, 097101 (2012

Rogue waves: Results of the ExWaMar project

Rogue waves, also known as freak waves, have received much attention in the scientific community as well as in media and the marine industry in the past two decades. Forecasts of extreme weather events have always been welcomed by the marine industry. Therefore, the development of warning criteria for rogue waves, being scientifically challenging forecast products, has also been encouraged. Such criteria would help mariners avoid sea states where rogue waves occur. The paper summarizes the main findings of the Norwegian research project ExWaMar dedicated to development of improved warning criteria for extreme and rogue waves. Three approaches for warning criteria for extreme and rogue waves based on information provided by the weather forecast are proposed. They include use of integrated wave parameters, coupling of a phase-averaged wave spectral model and phase-resolving wave model and application of Machine Learning methodology. Challenges related to development of warning criteria for rogue waves are discussed.acceptedVersio

The Zakharov equation with separate mean flow and mean surface

Using the Hamiltonian approach of Krasitskii (J. Fluid Mech., vol. 272, 1994, pp. 1-20), we derive a variant of the Zakharov equation in which the wave-induced mean surface elevation and the surface potential of the wave-induced mean flow are represented as separate variables governed by separate evolution equations. The kernel function of this new variant is simpler, and in particular also well defined in the uniform-wave-train limit for waves on finite depth. This form of the Zakharov equation may be advantageous in some applications. One example is the derivation of nonlinear Schrodinger equations in the narrow-band limit, where the handling of the mean flow and mean surface is significantly simpler than when starting from the original Zakharov equation. In this paper we have used the alternative form of the Zakharov equation to derive a Hamiltonian nonlinear Schrodinger equation for directional waves on arbitrary depth, valid to one order higher in bandwidth than the Hamiltonian equation recently presented by Craig, Guyenne and Sulem (Wave Motion, vol. 47, 2010, pp. 552-563)

Manifold Based Optimization for Single-Cell 3D Genome Reconstruction

The three-dimensional (3D) structure of the genome is important for orchestration of gene expression and cell differentiation. While mapping genomes in 3D has for a long time been elusive, recent adaptations of high-throughput sequencing to chromosome conformation capture (3C) techniques, allows for genome-wide structural characterization for the first time. However, reconstruction of "consensus" 3D genomes from 3C-based data is a challenging problem, since the data are aggregated over millions of cells. Recent single-cell adaptations to the 3C-technique, however, allow for non-aggregated structural assessment of genome structure, but data suffer from sparse and noisy interaction sampling. We present a manifold based optimization (MBO) approach for the reconstruction of 3D genome structure from chromosomal contact data. We show that MBO is able to reconstruct 3D structures based on the chromosomal contacts, imposing fewer structural violations than comparable methods. Additionally, MBO is suitable for efficient high-throughput reconstruction of large systems, such as entire genomes, allowing for comparative studies of genomic structure across cell-lines and different species

Phase-averaged equation for water waves

We investigate phase-averaged equations describing the spectral evolution of dispersive water waves subject to weakly nonlinear quartet interactions. In contrast to Hasselmann's kinetic equation, we include the effects of near-resonant quartet interaction, leading to spectral evolution on the 'fast' O(ε-2} ) time scale, where ε is the wave steepness. Such a phase-averaged equation was proposed by Annenkov & Shrira (J. Fluid Mech., vol. 561, 2006b, pp. 181-207). In this paper we rederive their equation taking some additional higher-order effects related to the Stokes correction of the frequencies into account. We also derive invariants of motion for the phase-averaged equation. A numerical solver for the phase-averaged equation is developed and successfully tested with respect to convergence and conservation of invariants. Numerical simulations of one-and two-dimensional spectral evolution are performed. It is shown that the phase-averaged equation describes the 'fast' evolution of a spectrum on the O(ε-2 ) time scale well, in good agreement with Monte-Carlo simulations using the Zakharov equation and in qualitative agreement with known features of one-and two-dimensional spectral evolution. We suggest that the phase-averaged equation may be a suitable replacement for the kinetic equation during the initial part of the evolution of a wave field, and in situations where 'fast' field evolution takes place

Variational Boussinesq model for kinematics calculation of surface gravity waves over bathymetry

Many of the widely used models for description of nonlinear surface gravity waves, in deep or shallow water, such as High Order Spectral Method (HOSM) and Boussinesq-type equations, rely on the elimination of the vertical coordinate from the basic three-dimensional Euler equations. From a numerical point of view such models are often computationally efficient, which is one of the main reasons that many such models are frequently used in studies on nonlinear surface waves. While surface-based models provide the time-evolution of surface quantities, typically the surface elevation and velocity potential at the surface , they do not directly provide the water particle kinematics in the fluid interior. However, in many practical applications information about the water-particle kinematics is crucial. The present paper presents a new method for the calculation of water-particle kinematics, from information about surface quantities. The presented methodology is a non-perturbative approach based on the fully nonlinear Variational Boussinesq model, and can be applied to wave propagation over both constant and variable water depth. The proposed method is validated on several cases, including Stokes waves, a solitary wave, and irregular waves over flat bottom. We have carried out new laboratory experiments of regular waves over a shoal with measurements of the horizontal velocity specifically taken for validation of the method. We also employ recent laboratory experiments for validation of statistical properties of wave kinematics of long crested irregular waves propagating over a shoal

The generalized kinetic equation as a model for the nonlinear transfer in third-generation wave models

An alternative model for the nonlinear interaction term Snl in spectral wave models, the so called generalized kinetic equation (Janssen J Phys Oceanogr 33(4):863–884, 2003; Annenkov and Shrira J Fluid Mech 561:181–207, 2006b; Gramstad and Stiassnie J Fluid Mech 718:280–303, 2013), is discussed and implemented in the third generation wave model WAVEWATCH-III. The generalized kinetic equation includes the effects of near-resonant nonlinear interactions, and is therefore able, in theory, to describe faster nonlinear evolution than the existing forms of Snl which are based on the standard Hasselmann kinetic equation (Hasselmann J Fluid Mech 12:481–500, 1962). Numerical simulations with WAVEWATCH have been carried out to thoroughly test the performance of the new form of Snl, and to compare it to the existing models for Snl in WAVEWATCH; the DIA and WRT. Some differences between the different models for Snl are observed. As expected, the DIA is shown to perform less well compared to the exact terms in certain situations, in particular for narrow wave spectra. Also for the case of turning wind significant differences between the different models are observed. Nevertheless, different from the case of unidirectional waves where the generalized kinetic equation represents a obvious improvement to the standard forms of Snl (Gramstad and Stiassnie 2013), the differences seems to be less pronounced for the more realistic cases considered in this paper

Statistical properties of wave kinematics in long-crested irregular waves propagating over non-uniform bathymetry

Experimental and numerical evidence have shown that nonuniform bathymetry may alter significantly the statistical properties of surface elevation in irregular wave fields. The probability of “rogue” waves is increased near the edge of the upslope as long-crested irregular waves propagate into shallower water. The present paper studies the statistics of wave kinematics in long-crested irregular waves propagating over a shoal with a Monte Carlo approach. High order spectral method is employed as wave propagation model, and variational Boussinesq model is employed to calculate wave kinematics. The statistics of horizontal fluid velocity can be different from statistics in surface elevation as the waves propagate over uneven bathymetry. We notice strongly non-Gaussian statistics when the depth changes abruptly in sufficiently shallow water. We find an increase in kurtosis in the horizontal velocity around the downslope area. Furthermore, the effects of the bottom slope with different incoming waves are discussed in terms of kurtosis and skewness. Finally, we investigate the evolution of kurtosis and skewness of the horizontal velocity over a sloping bottom in a deeper regime. The vertical variation of these statistical quantities is also presented