Spectral and statistical analysis of the internal bore evolution

Undular internal bores appear in oceans as an intermediate stage of internal tidal wave evolution. We study nonlinear disintegration of long sine wave in the framework of the Gardner equation (extended version of the Korteweg – de Vries equation with both quadratic and cubic nonlinear terms) which is actively applied in physical oceanography. We carried out numerical modeling of long sine wave evolution for different signs of the cubic nonlinear term and different initial amplitudes to demonstrate its principal features. The focus of the study is made on spectral and statistical analysis of generated wave field. If cubic nonlinearity is positive and amplitude of sine wave is large enough, soliton-like impulses of both polarities are generated and their interactions may result in the formation of extreme amplitude waves. Statistical analysis of wave heights in time shows permanent exceedance somewhere in the wave field of the level of two significant wave heights (criterion for freak wave appearance). Key-Words: Undular bore, Gardner equation, significant wave height, extreme amplitude waves, solitons, exceedance probability distribution, skewness, kurtosis