The instability of near-extreme Stokes waves

We study the stability of Stokes waves on a free surface of an ideal fluid of infinite depth. For small steepness the modulational instability dominates the dynamics, but its growth rate is vastly surpassed for steeper waves by an instability due to disturbances localized at the wave crest, explaining why long propagating ocean swell consists of small-amplitude waves. The dominant localized disturbances are either co-periodic with the Stokes wave, or have twice its period. The nonlinear stage of instability for steep wave evolution reveals the formation of a plunging breaker.