ON THE DETERMINATION OF THE ONSET OF BREAKING FOR MODULATING SURFACE GRAVITY WATER-WAVES

Determining the onset of wave breaking in unforced nonlinear modulatin g surface gravity wave trains on the basis of a threshold variable has been an elusive problem for many decades. We have approached this pro blem through a detailed numerical study of the fully nonlinear two-dim ensional inviscid problem on a periodic spatial domain. Two different modes of behaviour were observed for the evolution of a sufficiently s teep wave group: either recurrence of the initial state or the rapid o nset of breaking, each of these involving a significant deformation of the wave group geometry. For both of these modes, we determined the b ehaviour of dimensionless growth rates constructed from the rates of c hange of the local mean wave energy and momentum densities of the wave train, averaged over half a wavelength. These growth rates were compu ted for wave groups with three to ten carrier waves in the group and a lso for two modulations with seven carrier waves and three modulations with ten carrier waves. We also investigated the influence of a backg round vertical shear current. Two major findings arose from our calcul ations. First, due to nonlinearity, the crest-trough asymmetry of the carrier wave shape causes the envelope maxima of these local mean wave energy and momentum densities to fluctuate on a fast time scale, resu lting in a substantial dynamic range in their local relative growth ra tes. Secondly, a universal behaviour was found for these local relativ e growth rates that determines whether subsequent breaking will occur.