We describe an investigation of the occurrence, statistics, and generation mechanisms of rogue wave in the open sea using direct three-dimensional phase-resolved nonlinear wavefield simulations. To achieve this we develop an efficient nonlinear wavefield simulation capability based on the high-order spectrum method which solves the primitive phase-resolved Euler equations. The simulations account for nonlinear wave-wave interactions up to an arbitrary high order in the wave steepness and are capable of accounting for effects of bottom bathymetry, variable current, and direct physics-based models for wind input and wave breaking dissipation. We apply direct large-scale simulations to obtain a large number of phase-resolved nonlinear wavefields, initially specified by directional wave spectra. The typical spatial-temporal domain size of such numerical nonlinear wavefields is O(103 km2) over evolution time of O(hr). These spatial and temporal scales account for quartet resonant interactions and partially for quintet resonant interactions among wave components in the wavefield. From the simulated nonlinear wavefields, rogue wave events are identified and their occurrence statistics are studied. It is shown that the classic linear theory (i.e. Rayleigh distribution) significantly underestimates the rogue wave occurrence. Second-order theory improves the Rayleigh prediction, but still underestimates the rogue wave occurrence in wavefields with moderately large wave steepness and relatively narrow directional spreading and spectrum bandwidth. The influence of key wave spectrum parameters (such as significant wave height, directional spreading, effective steepness, and spectrum bandwidth) on the rogue wave occurrence is analyzed. The classification of rogue waves according to their configuration is also obtained. The key characteristics of a rogue wave or rogue wave group in terms of kinematics and surface structure are analyzed and quantified. The nonlinear wave simulations, which provide full three-dimensional kinematics and dynamics of rogue wave events, provide a powerful tool for understanding the underlying mechanisms of their generation. They are elucidated by specific examples.
Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity
