Proof of Six-Wave Resonance Conditions of Ocean Surface Gravity Waves in Deep Water

A necessary big step up in the modern water wave theories and their widespread application in ocean engineering is how to obtain 6-wave resonance conditions and to prove it. In the light of the existing forms and characteristics of 3-wave, 4-wave and 5-wave resonance conditions, the 6-wave resonance conditions are proposed and proved for currently a maximum wave-wave resonance interactions of the ocean surface gravity waves in deep water, which will be indispensable to both the Kolmogorov spectrum of the corresponding universal wave turbulence and a synthetic 4-5-6-wave resonant model for the ocean surface gravity waves.


Introduction
Resonance occurs widely in nature, and is observed in almost every branch of physics and many interdisciplines as well as in every field of engineering technology. In ocean engineering, the study of resonance phenomena is also a hot topic. There have been many resonance forms, such as wave−wave resonance interaction which can help us understand the formation of open sea freak wave (Janssen, 2003), parametric resonance which is very important for the safety of floating structures (Yang and Xu, 2018;Li et al., 2017;Liu et al., 2019), harbor resonance which is crucial in the locally coastal hydrodynamics and ship navigation (Gao et al., 2016) and so on. For the motion of ocean surface waves, the most basic mechanism is wave−wave resonance. Based on the wave−wave resonance mechanism of ocean surface gravity waves in deep water founded by Phillips (1960), many famous results were obtained which became the cornerstones of the modern water wave theories (Hasselman, 1962;Zakharov, 1968;Hammack and Henderson, 1993;Krasitskii, 1994;Dyachenko et al., 1995;Janssen, 2004;Newell and Rumpf, 2011;Huang, 2013;Aubourg and Mordant, 2016). Therefore, a more complete water wave theory may be obtained by extending the existing wave−wave resonance conditions. Specifically, the 4-wave resonance conditions of ocean surface gravity waves in deep water are derived from the cu-bically nonlinear dynamic equation of the theory of surface gravity waves (Phillips, 1960) and the experimental verification can refer to Longuet-Higgins and Smith (1966) and McGoldrick et al. (1966). Subsequently, 5-wave interaction process was discussed by Longuet-Higgins and Cokelet (1978) and McLean (1982aMcLean ( , 1982b using numerical computations of the exact water wave equations. It leads to a new type of instabilities (McLean, 1982a(McLean, , 1982bMcLean et al., 1981), and the experiments concerning five-wave interactions can refer to Su (1982) , Su et al. (1982) and Su and Green (1984). With four-order nonlinear dynamic equation, the proposal and proof of 5-wave resonance conditions were given by Krasitskii and Kozhelupova (1995). So far, no reports concerning 6-wave resonance interactions of water waves have been found. Naturally, this paper is devoted to the study of establishing and proving the 6-wave resonance conditions for an extension of the modern water wave theories and, more importantly, the Kolmogorov spectral analytic solution for universal wave turbulence theory (Zakharov et al., 1992;Nazarenko, 2011;Newell and Rump, 2011;Nazarenko and Lukaschuk, 2016). It is worth pointing out that one-dimensional 6-wave interactions have been dealt with by, for example, Bortolozzo et al. (2009) for optical wave turbulence and Laurie (2010) for quantum turbulence. No attempt, however, has been made here to develop a water wave theory for three-dimensional 6-wave reson-ance conditions, which will be the aim of this paper following the method given by Krasitskii and Kozhelupova (1995).

Resonance conditions and its proof
For simplicity of study, this paper deals with ocean surface gravity waves in deep water. When the dynamic equation contains nonlinearity of five-order there occurs the possibility for 6-wave interactions. It produces six possible linear combinations of frequencies and wave vectors, and only three of them may result in resonant interactions as follows: The other three have only trivial solutions as follows: Here, is the horizontal wave vector, is the dispersion relation, and is the acceleration because of gravity. In this paper, we mainly deal with the resonance conditions (1) and (2) (resonance conditions (2) and (3) are equivalent in the sense that one of them is derived from the other by re-indexing of wave vectors).
For Eqs. (4) and (5), set . Obviously, the curve passes through the origin and , , that is for , the function is monotonic and convex upward. The curve of is shown in Fig. 1. From Fig. 1, one can obtain the following inequality k 1 = 0 k 2 = 0 where equality is the case only when or . With sequential use of Eq. (7), one can obtain which implies that Eqs. (4) and (5) have only trivial solutions.
2.1 Solutions of resonance condition (1) Firstly, let us consider a special case, a totality of collinear vector where is an arbitrary vector and an evident particular solution of resonance condition (1). Let us consider the general case of six non-collinear vectors as shown in Fig. 2.
The following will proceed from the same line of reasoning as in Hasselmann (1962) and Krasitshii and Kozhelupova (1995), and will consider interactions between various triples of vector and and a fixed triple and . Set { ω(k 1 )+ω(k 2 )+ω(k 3 ) =ω(k 4 )+ω(k 5 )+ω(k 6 ) = γω(k) , and the vector is fixed. Norming all the vectors by and leaving the previous notations for dimensionless vectors, one can rewrite system (9) in the form e k where is the unit vector aligned with vector .
First, we consider the equations then we can obtain α s x s = |s| where is the angle between vectors and , and .
In Ω(k) Fig. 1. The curve of . x ′ s the new coordinate system with the axis aligned with the vector and with the origin at its midpoint (Fig. 2), we can obtain β k 1 s where is the angle between vectors and . From Eq.
Isolines of this function corresponding to fixed and determine the geometric locus of endpoints of vector in the coordinates . The example is shown in Fig. 3. Thus, we can determine wave vectors and Moreover, wave vector is determined by Eq. (10). and can be determined by the same way, we will not go into detail. s = 0.8, α = π/6, δ = 1.1, 1.2, 1, 265, 1.3, √ 2, 1, 5, Fig. 3, and respectively. All vectors are normed by k = .
2.2 Solutions of resonance condition (2) Let us consider a special case, a totality of collinear vectors , where is an arbitrary vector and an evident particular solution of resonance condition (2).
Let us consider the general case of six non-collinear vectors as shown in Fig. 4.
Firstly, set then we have , y = k 1 sin α, k 1 x where α is the angle between vectors and . Using Eq. (17), we can obtain Isolines of this function corresponding to fixed determine the geometric locus of endpoints of vector in the coordinates . The example is displayed in Fig. 5. Thus, we can determine wave vectors and . γ = 1.3, 1.35, √ 2, 1.5, 1.6, In Fig. 5, respectively. All vectors are normed by . Secondly, we consider the following equations: (20) Then we have

Here, is the angle between vectors and . Again let
Thus, we obtain where is the angle between vectors and . In the new coordinate system with axis aligned with vector and with the origin at its midpoint (Fig. 4), we can obtain which implies that where is the angle between vectors and . Isolines of this function corresponding to fixed and determine the geometric locus of endpoints of vector in the coordinates . The example is shown in Fig. 6 In Fig. 6, and respectively. All vectors are normed by Finally, it is worth stressing that we focus on the resonance conditions (1) and (2), and the solutions we obtained not only have particularity, but also have typicality, and that it can represent general characteristics. Therefore, the results can be extended to n-wave resonance as follows: m ⩾ 2, n ⩾ 4 where . Like the processes of Eqs. (19)−(25), we can also obtain the solutions of Eq. (26).

Conclusion
On the basis of research progress of modern ocean surface gravity waves in deep water, we put forward and prove the 6-wave resonance conditions which have generalized the previous studies. Then we can both establish a model of ocean surface gravity waves in deep water which contains 4-5-6-wave resonance interactions, and obtain a Kolmogorov spectral analytic solution of wave turbulence of ocean surface gravity waves.