The wave energy density spectrum provides useful information for both coastal engineering practice and applied sciences. However, it is not always available at desired location which can be far away from the observation stations, and should be achieved using approximated approaches. Nevertheless, the spectrum at the desired location may differ significantly from the approximated one due to nonlinear effects. In this paper, long-duration and large-scale evolutions of the wave spectrum in shallow water is investigated numerically. Direct simulations of random seas are carried out by using the weakly nonlinear KdV equation and the fully nonlinear Enhanced Spectral Boundary Integral (ESBI) model respectively. Due to nonlinear effects, the spectral shape is modified and the energy is redistributed after a long-duration (∼1000 peak periods) and large-scale (∼128 peak wave lengths) evolution. The results obtained by using fully nonlinear ESBI model here demonstrate that the flatness occurs only when both the conditions, i.e., large Ursell number and large wave steepness, are satisfied; It will not happen if the Ursell number is large but the steepness is small. This is different from the existing understanding in literature, i.e. spectra tended to become flat (or nearly uniformly distributed) in low frequency part as long as Ursell number is sufficiently large.

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