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 highresolution 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.

Computation of three-dimensional standing water waves

“…However, there appear to be significant differences, most notably in the relation between crest acceleration and wave height. While our computations currently lack the resolution to examine the crest formation in more detail, this result suggests that the behavior as the crest continues to sharpen may be significantly different from the behavior in the two-dimensional case [20].…”

“…In previous two-dimensional studies [20], a similar numerical continuation approach has been used, although using only linear extrapolation from two previous solutions. Here, given the high cost of each minimization, and the fact that a good initial guess may reduce the number of Jacobian computations, high-order extrapolation was deemed advantageous.…”

“…As discussed in Ref. [20], for crest accelerations beyond 0.99, the deep water bifurcation curve fragments into a number of disjoint branches corresponding to different resonant mode patterns that emerge near the crest tip due to complex dynamics. However, these features are too small to be seen in Fig.…”

“…11(b) is based on twodimensional data from Ref. [20], and shows that wave steepness initially grows linearly with crest acceleration before flattening out and passing through a turning point at a crest acceleration of 0.93. As discussed in Ref.…”

“…11(a), this is in close agreement with the computed results although some deviations become visible at higher amplitudes. In studies of two-dimensional standing waves [14,20], it has been useful to examine the relationship between the initial downward acceleration of the crest, and the wave steepness, defined as half of the crest-to-trough height difference. The dashed line in Fig.…”