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.

New conformal mapping for adaptive resolving of the complex singularities of Stokes wave.

A new highly efficient method is developed for computation of travelling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularities above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for the Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows us to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced.