On the determination of the onset of breaking for modulating surface gravity water waves

Abstract

Determining the onset of wave breaking in unforced nonlinear modulating surface gravity wave trains on the basis of a threshold variable has been an elusive problem for many decades. We have approached this problem through a detailed numerical study of the fully nonlinear two-dimensional inviscid problem on a periodic spatial domain. Two different modes of behaviour were observed for the evolution of a sufficiently steep wave group: either recurrence of the initial state or the rapid onset of breaking, each of these involving a significant deformation of the wave group geometry. For both of these modes, we determined the behaviour of dimensionless growth rates constructed from the rates of change of the local mean wave energy and momentum densities of the wave train, averaged over half a wavelength. These growth rates were computed for wave groups with three to ten carrier waves in the group and also for two modulations with seven carrier waves and three modulations with ten carrier waves. We also investigated the influence of a background vertical shear current.Two major findings arose from our calculations. 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, resulting in a substantial dynamic range in their local relative growth rates. Secondly, a universal behaviour was found for these local relative growth rates that determines whether subsequent breaking will occur.