Resonant interactions of nonlinear water waves in a finite basin

We study exact four-wave resonances among gravity water waves in a square box with periodic boundary conditions. We show that these resonant quartets are linked with each other by shared Fourier modes in such a way that they form independent clusters. These clusters can be formed by two types of quartets: (1) Angle resonances which cannot directly cascade energy but which can redistribute it among the initially excited modes and (2) scale resonances which are much more rare but which are the only ones that can transfer energy between different scales. We find such resonant quartets and their clusters numerically on the set of $1000\times 1000$ modes, classify and quantify them and discuss consequences of the obtained cluster structure for the wave-field evolution. Finite box effects and associated resonant interaction among discrete wave modes appear to be important in most numerical and laboratory experiments on the deep water gravity waves, and our work is aimed at aiding the interpretation of the experimental and numerical data.

Figure 1

(Color online) Wave numbers $m,n$ are shown in the rectangles as upper and lower numbers. The smallest tridents cluster formed by eight symmetry generated solutions, two presentation forms are given: With multiedges (left-hand panel) and without multiedges (right-hand panel).

Figure 2

(Color online) Structure of collinear resonances in the spectral domain $m,n$ such that $\left|\mathbf{k}\right|=\left|\sqrt{m^2+n^2}\right|\le 100$ (left-hand panel) and of noncollinear resonances in the same domain (right-hand panel). Wave vectors belonging to different resonance quartets are shown in different colors.

Figure 3

(Color online) Wave vectors are shown which form first nonaxial trident (nonaxial trident with the smallest possible wave numbers $m,n$).

Figure 4

(Color online) Wave vectors are shown which form first nontrident (nontrident with the smallest possible wave numbers $m,n$).

Figure 5

(Color online) Wave vectors belonging to one resonant quartet are depicted in the same color. Left-hand panel: A shortest cluster formed by both tridents and nontridents; cluster length is 8 (64 with eight symmetries), among them six (48 with eight symmetries) tridents and two (16 with symmetries) nontridents. Right-hand panel: A shortest cluster formed by both collinear and noncollinear quartets; cluster length is 8 (48 with eight and four symmetries), among them six (32 with eight and four symmetries) collinear and two (16 with eight symmetries) noncollinear quartets. No multiedges are shown.

Figure 6

(Color online) Example of a nontrident cluster of length 4. Wave numbers $m,n$ are shown in the circles as upper and lower numbers. Solid black, dashed blue, dotted-dashed red, and dotted green lines connect the wave vectors forming first, second, third, and fourth quartets of the cluster correspondingly.

Figure 7

(Color online) Schematic presentation of mixed resonance cascade.

Figure 8

(Color online) Wave numbers $m,n$ are shown in the circles as upper and lower numbers. Wave (64,0) takes part in two scale resonances, both nontridents. The upper number in the circle is $m$, the lower is $n$, and (red) thick lines drawn between vectors on the same side of Eqs. (3). Wave (119,120) takes part in one scale resonance and in 12 angle resonances.

Figure 9

(Color online) The multiplicities histogram for angle resonances.