Breakdown of self-similarity at the crests of large amplitude standing water waves

TL;DR

This study uses high-resolution simulations to reveal that large-amplitude standing water waves do not become self-similar at their crests, instead developing oscillatory structures and complex behaviors previously unanticipated.

Contribution

It demonstrates that standing water waves exhibit non-self-similar crest behavior with oscillations, contrasting with traveling waves, and maps the resulting bifurcation structure.

Findings

Standing waves develop oscillatory crest structures.

Bifurcation curves fragment into disjoint branches.

Vertical jets are observed at wave crests.

Abstract

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.