Surface gravity waves in deep fluid at vertical shear flows

Special features of surface gravity waves in deep fluid flow with constant vertical shear of velocity is studied. It is found that the mean flow velocity shear leads to non-trivial modification of surface gravity wave modes dispersive characteristics. Moreover, the shear induces generation of surface gravity waves by internal vortex mode perturbations. The performed analytical and numerical study provides, that surface gravity waves are effectively generated by the internal perturbations at high shear rates. The generation is different for the waves propagating in the different directions. Generation of surface gravity waves propagating along the main flow considerably exceeds the generation of surface gravity waves in the opposite direction for relatively small shear rates, whereas the later wave is generated more effectively for the high shear rates. From the mathematical point of view the wave generation is caused by non self-adjointness of the linear operators that describe the shear flow.Comment: JETP, accepte

Numerical modeling of surface wave development under the action of wind

The numerical modeling of two-dimensional surface wave development under the action of wind is performed. The model is based on three-dimensional equations of potential motion with a free surface written in a surface-following nonorthogonal curvilinear coordinate system in which depth is counted from a moving surface. A three-dimensional Poisson equation for the velocity potential is solved iteratively. A Fourier transform method, a second-order accuracy approximation of vertical derivatives on a stretched vertical grid and fourth-order Runge–Kutta time stepping are used. Both the input energy to waves and dissipation of wave energy are calculated on the basis of earlier developed and validated algorithms. A one-processor version of the model for PC allows us to simulate an evolution of the wave field with thousands of degrees of freedom over thousands of wave periods. A long-time evolution of a two-dimensional wave structure is illustrated by the spectra of wave surface and the input and output of energy

Statistical properties of nonlinear one-dimensional wave fields

A numerical model for long-term simulation of gravity suface waves is described. The model is designed as a component of a coupled Wave Boundary Layer/Sea Waves model, for investigation of small-scale dynamic and thermodynamic interactions between the ocean and atmosphere. Statistical properties of nonlinear wave fields are investigated on a basis of direct hydrodynamical modeling of 1-D potential periodic surface waves. The method is based on a nonstationary conformal surface-following coordinate transformation; this approach reduces the principal equations of potential waves to two simple evolutionary equations for the elevation and the velocity potential on the surface. The numerical scheme is based on a Fourier transform method. High accuracy was confirmed by validation of the nonstationary model against known solutions, and by comparison between the results obtained with different resolutions in the horizontal. Tne scheme allows reproduction of the propagation of steep Stokes waves for thousands of periods with very high accuracy. The method here developed is applied to simulation of the evolution of wave fields with large number of modes for many periods of dominant waves. Tne statistical characteristics of nonlinear wave fields for waves of different steepness were investigated: spectra, curtosis and skewness, dispersion relation, life time. The prime result is that wave field may be presented as a superposition of linear waves is valid only for small amplitudes. It is shown as well, that nonlinear wave fields are rather a superposition of Stokes waves not linear waves. Potential flow, free surface, conformal mapping, numerical modeling of waves, gravity waves, Stokes waves, breaking waves, freak waves, wind-wave interaction

Freak waves: their occurrence and probability

This paper describes the results of more than 4000 long-term (up to thousands of peak wave periods) numerical simulations of nonlinear gravity surface waves performed for the investigation of properties and estimation of statistics of extreme ('freak') waves. The method of solution of two-dimensional potential wave equations based on conformal mapping is applied to the simulation of wave behavior assigned by different initial conditions, defined by the Joint North Sea Wave Observation Project and Pierson-Moskowitz spectra. It is shown that nonlinear wave evolution sometimes results in the appearance of very big waves. There are no predictors for the appearance of extreme waves; however, the height of dimensional waves is proportional to the significant wave height. The initial generation of extreme waves can occur simply as a result of linear group effects, but in some cases the largest wave suddenly starts to grow. It is followed sometimes by a strong concentration of wave energy around a peak vertical. It takes place typically for one peak wave period. It happens to an individual wave in physical space, with no energy exchange with surrounding waves taking place. A probability function for steep waves has been constructed. Such type of function can be used for the development of operational forecast of freak waves based on a standard forecast provided by a three-dimensional generation wave prediction model (WAVEWATCH or wave modeling)

On the nonlinear energy transfer in the unidirected adiabatic surface waves

The results of numerical simulation of the adiabatic evolution of waves are presented. The model is based on the fully nonlinear 1D equations of potential waves written in conformal coordinates. It is shown that a wave spectrum is subject to strong fluctuations. Most of such fluctuations are reversible, however a residual effect of the fluctuations causes downshifting of the spectrum. The rate of downshifting depends on nonlinearity

Numerical investigation of freak waves

Abstract not available

Numerical simulations of coupled sea waves and boundary layer dynamics

Abstract not available