The dominant instability of near-extreme Stokes waves

The instability of Stokes waves, steady propagating waves on the surface of an ideal fluid of infinite depth, is a fundamental problem in the field of nonlinear science. The dominant instability of these waves depends on their steepness. For small amplitude waves, it is well known that the Benjamin-Feir or modulational instability dominates the dynamics of a wave train. We demonstrate that for steeper waves, an instability caused by disturbances localized at the wave crest vastly surpasses the growth rate of the modulational instability. These dominant localized disturbances are either coperiodic with the Stokes wave or have twice its period. In either case, the nonlinear evolution of the instability leads to the formation of plunging breakers. This phenomenon explains why long propagating ocean swell consists of small-amplitude waves.