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).
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