Time scales of a low order harmonic resonance of short-crested gravity waves on deep water

Short-crested water waves (SCWs) are the genuine three-dimensional (3D) ocean waves. They host the phenomenon of harmonic resonances (HRs). The existence of HRs depends on their timescales, on whether or not they actually have time to develop. They are associated to superharmonic instabilities that are due to nonlinear quartet interactions. The low order HR(2,6) was chosen to match previous studies. Their multi-branch solutions and their normal forms are computed. Then, their conditions of occurrence, growth rate (inverse timescale) and persistence are discussed. It is shown that at incidence angles for which HR (2,6) occurs, its associated growth may be larger than, or at least of the same order as, those of the well-known modulational and 3D 'horse-shoe' pattern instabilities, which are the primary processes involved in a surface water wave field. Thus HRs seem likely to appear in a SCW field although other processes, that could inhibit their growth, are suggested.