  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.
