ABNORMAL WAVES IN WAVE FIELD WITH MODULATION INSTABILITY

Abstract

The paper discusses the development of the modulation instability of high-amplitude waves. The behavior of wave field intensity envelope is described by the Lighthill equation complemented by an absorption and an external source. The second model, presented in the paper, describes the appearance of waves of abnormal amplitude on the ocean surface in presence of disturbances of finite amplitude. The results of simplified description based on a so-called modified S-theory are discussed. This theory takes into account the interaction of spectrum modes, which wave vectors are arranged symmetrically with respect to the wave vector of principal wave. The advantage of the modified S-theory is the ability to distinguish clearly the mechanisms of modulation instability and explain the nature of appearance of abnormal amplitude waves with short lifetime. It is also presented the result of calculations without the use of simplifications, that makes possible a verification of the approach based on S-theory. It has been shown that many of the characteristics of both descriptions are close enough at least for initial stage of the nonlinear instability. Maximum amplitude of the envelope as well as the maximum amplitudes of individual waves, and the characteristic time of their appearance and their lifetimes are also similar for both models. It is noted that the envelope of wave field at beginning of nonlinear regime of instability in the Lighthill model exceeds almost three times the average amplitude of waves. It is shown that the occurrence frequency of abnormal waves (calculated in statistics over ensemble and over time) is almost identical for both models. At the initial stage of nonlinear regime of the instability the waves and bursts of envelope with very large amplitude may occur significantly more often than it would be expected from statistically meaningful estimates.