Cyclic recurrence in nonlinear unidirectional ocean waves

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Abstract
A fully nonlinear model is developed for the unidirectional propagation of periodic gravity wave groups in deep water, in which the shape of the group envelopes changes cyclically. It is intended to describe the slow-time evolution of wave groups on the open ocean surface, and to generalize the cyclic recurrence that can occur during the sideband modulation of Stokes waves and Schrödinger wave groups. The weak nonlinear interactions are shown to concentrate the wave energy at the centre of each group at regular intervals, causing the waves there to be of greater height locally in space and time. This is suggested as one mechanism for the local wave breaking that is observed on the open ocean surface. The cyclically recurring wave groups may be interpreted as the limit-cycle stage in a progression from uniform wave groups to chaos on the forced, damped, ocean surface.