Breakdown of Self-Similarity at the Crests of Large-Amplitude Standing Water Waves
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.
- Publication:
-
Physical Review Letters
- Pub Date:
- October 2011
- DOI:
- 10.1103/PhysRevLett.107.184501
- arXiv:
- arXiv:1011.2476
- Bibcode:
- 2011PhRvL.107r4501W
- Keywords:
-
- 47.35.Bb;
- 05.45.-a;
- Gravity waves;
- Nonlinear dynamics and chaos;
- Physics - Fluid Dynamics;
- 76B15;
- 76B07;
- 65N35
- E-Print:
- 4 pages, 5 figures. Final version accepted for publication