RESUMO
We study the limiting behavior of large-amplitude standing waves on deep water using high-resolution numerical simulations in double and quadruple precision. While periodic traveling waves approach Stokes's sharply crested extreme wave in an asymptotically self-similar manner, we find that standing waves behave differently. Instead of sharpening to a corner or cusp as previously conjectured, the crest tip develops a variety of oscillatory structures. This causes the bifurcation curve that parametrizes these waves to fragment into disjoint branches corresponding to the different oscillation patterns that occur. In many cases, a vertical jet of fluid pushes these structures upward, leading to wave profiles commonly seen in wave tank experiments. Thus, we observe a rich array of dynamic behavior at small length scales in a regime previously thought to be self-similar.
RESUMO
A numerical method is introduced for the computation of time-periodic vortex sheets with surface tension separating two immiscible, irrotational, two-dimensional ideal fluids of equal density. The approach is based on minimizing a nonlinear functional of the initial conditions and supposed period that is positive unless the solution is periodic, in which case it is zero. An adjoint-based optimal control technique is used to efficiently compute the gradient of this functional. Special care is required to handle singular integrals in the adjoint formulation. Starting with a solution of the linearized problem about the flat rest state, a family of smooth, symmetric breathers is found that, at quarter-period time intervals, alternately pass through a flat state of maximal kinetic energy, and a rest state in which all the energy is stored as potential energy in the interface. In some cases, the interface overturns before returning to the initial, flat configuration. It is found that the bifurcation diagram describing these solutions contains several disjoint curves separated by near-bifurcation events.
