Modelling of nonlinear shoaling based on stochastic
evolution equations
A one-dimensional stochastic model is derived to simulate the
transformation of wave spectra in shallow water including generation of
bound sub- and superharmonics, near-resonant triad wave interaction and
wave breaking. Boussinesq type equations with improved linear dispersion
characteristics are recast into evolution equations for the complex
amplitudes, and serve as the underlying deterministic model. Next a set
of evolution equations for the cumulants is derived. By formally
introducing the well-known Gaussian closure hypothesis, nonlinear
evolution equations for the power spectrum and bispectrum are derived. A
simple description of depth-induced wave breaking is incorporated in the
model equations, assuming that the total rate of dissipation may be
distributed in proportion to the spectral energy density.