Modelling of the temporal and spatial evolutions of weakly nonlinear random directional waves with the modified nonlinear Schrödinger equations

The temporal and spatial evolutions of nonlinear wave group with an initial Gaussian envelope are theoretically studied under the governing of MNLS equations, demonstrating that the temporal and spatial versions of numerical model are not always consistent in the whole evolution process, particularly in the presence of strong nonlinearity. Moreover, a large set of numerical simulations, performed respectively by these two versions of numerical model, are systematically compared to mechanically generated waves with different initial directional spreading and Benjamin-Feir Index, mainly focusing on the evolution properties of surface elevations such as the coefficients of skewness and kurtosis, the probability density function, and the maximal surface elevation. On the whole, it can be argued that the statistical properties of both numerically simulated wave fields are basically consistent with the laboratory observations.