Superharmonic Instability of Stokes Waves

TL;DR

This paper investigates the superharmonic instability of nearly limiting Stokes waves through numerical simulations, revealing new instability branches, eigenvalue collision mechanisms, and a universal scaling law for eigenvalues as wave steepness increases.

Contribution

It introduces new numerical methods to analyze superharmonic instability branches and uncovers a universal power law scaling for eigenvalues near the limiting wave.

Findings

Eigenvalues collide at the origin leading to instability.

Unstable eigenvalues follow a universal power law.

Surface profiles of unstable modes are provided online.

Abstract

A stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behaviour of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through http://stokeswave.org website.