The Dynamical and Kinetic Equations of Four-Five-Six-Wave Resonance for Ocean Surface Gravity Waves in Water with a Finite Depth

The development of ocean surface wave theories has focused on the mechanism of wave–wave resonance [1,2,3,4,5]. Two integro-differential equations [6], based on this mechanism, are central to wave turbulence theory. One of them is the dynamical equation (also known as the “Zakharov equation”) [7,8], which is based on the Hamiltonian function of wave energy and the linear dispersion relation, and the other one is the kinetic equation (also known as the “Hasselmann equation”) [9,10], which is based on the wave numbers (or wave action) [7,9,10,11,12,13,14,15,16,17]. The former is deterministic, and the latter is stochastic. The broad-banded wave turbulence theory is contrasted with the classical narrow-banded high-order Stokes waves with a nonlinear dispersion relation [18] and its extended theories; however, it is also implicated in [19,20]. At present, the wave turbulence theories of ocean surface waves are limited to the three-wave resonance of surface capillary waves in deep water [21] and the four-wave [9] and four-five-wave resonance [7] of surface gravity waves in a finite water depth. Therefore, the next step is to establish a wave turbulence theory of four-five-six-wave resonance for surface gravity waves in a finite water depth.

The existing spatial scale of the ocean can satisfy the six-wave resonance interactions. If a wave length, a wave steepness [19], and a horizontal scale of the ocean are taken as typical values, respectively, 100 m, 0.2, and $2\times {10}^6$ m, then the number of resonance waves n is $\frac{100}{{\epsilon }^n}=2\times {10}^6$, $n=\frac{lg0.5-4}{lg\epsilon }\approx 6.15.$ It can be expected that the steeper the resonance waves are, the higher the order of the resonance waves are, and the higher the number of the resonance waves are until the resonance waves are broken [22]. If the effect of the surface capillary waves is considered separately [21,23], the resonant number of ocean surface capillary waves can reach infinity in a wide ocean area, which makes it possible to construct a high-order ocean surface capillary–gravity wave system with six-wave resonance. Therefore, a six-wave resonance system for ocean surface gravity waves is necessary. The fifth-order Korteweg–de Vries equation of solitary wave phenomena originating from water waves may admit an infinite number of locally propagating waves under some singular conditions [24].

It has been shown that the higher-order effect is not only possible but also observed [22]. A quite common phenomenon seen by people on the sea or river surface under the action of fresh wind comprises the so-called ’horseshoe’ or ’crescent-shaped’ patterns, which appear, owing to the class II instability for the five-wave and higher-order interactions [25,26,27]. Low-order theories sometimes distort the real physical picture and produce apparent chaos, which disappears at a higher order. Therefore, the high-order approximations of the water–wave problem are necessary [28] for reliable studies of the transition from order to chaos. In the one-dimensional case, the amplitudes of four-wave interactions in the effective Hamiltonian are exactly equal to zero, and the five-wave interaction is the first nonvanishing term [29,30], i.e., an effective five-wave Hamiltonian [31]. It is also found that the six-wave amplitude of the compact 1D Zakharov equation is not zero; that is, the equation is not integrable [32]. The high-order spectral (HOS) of numerical simulation [19] can be effective, and the theory of high-order wave resonance should be more so.

If we look at other waves besides water waves, one-dimensional six-wave resonance has already appeared in light wave turbulence [33] and quantum wave turbulence [34]. To this end, the classical dynamical equation and its kinetic equation [7] of the classical four-five-wave resonance of the surface gravity waves in a finite water depth are extended to the four-five-six-wave resonance. At the same time, the classical fifth-order expansion of the Hamiltonian in integral-power series in a normal variable for ocean surface capillary–gravity waves in a finite water depth is extended to the sixth order, thus laying a necessary foundation for the dynamical equation and its kinetic equation for surface capillary–gravity waves in a finite water depth in the future.

The nonlinear Schrödinger equations play important roles in constructing the mathematical model of ocean surface gravity waves, and they can be derived from the dynamical equations [8,19,35]. Based on the results, we can establish a fifth-order nonlinear Schrödinger equation of ocean surface gravity waves. Afterward, the corresponding abundant solutions can be obtained through various methods [36,37,38,39,40].

In this paper, we follow Krasitskii’s method to ensure that the coefficients of the two equations obtained meet the desired symmetries. The paper contains six sections. In Section 2, the sixth-order integral-power series of Hamiltonian and canonical transformation are introduced. In Section 3, the expansion of the Hamiltonian with accuracy up to the sixth-order terms for ocean surface capillary–gravity waves with a finite water depth is presented. In Section 4, the dynamical equations of ocean surface gravity wave that contain four-five-six-wave interactions and the forms of the coefficients concerning the canonical transformation are obtained. In Section 5, taking the method of quasi-Gaussian approximation, the kinetic equations of four-five-six-wave resonance for ocean surface gravity waves are established. Finally, the potential applications are discussed in Section 6.

The Hamiltonian description of ocean surface waves was introduced by Zakharov [8] as follows: where $\mathit{x}=(x,y)$ is the horizontal coordinate, $\zeta (\mathit{x},t)$ is the vertical displacement, $\varphi (\mathit{x},z,t)$ is the velocity potential, $\psi (\mathit{x},t)=\varphi (\mathit{x},\zeta ,t)$ is the velocity potential evaluated on the surface, z is the vertical coordinate directed upwards with its origin on the undisturbed surface $z=0$, $\frac{\delta H}{\delta \psi }$ is the functional derivative of H with respect to $\psi$, and H is the Hamiltonian (the total energy) expressed in $\zeta (\mathit{x},t)$ and $\psi (\mathit{x},t)$. Here, the Hamiltonian $(H=K+\Pi )$ is the sum of the kinetic $\left(K\right)$ and potential $\left(\Pi \right)$ energies divided by the fluid density. These are given by where $\nabla =(\frac{\partial }{\partial x},\frac{\partial }{\partial y})$ is the horizontal gradient operator, h is the constant depth of water, g is the acceleration due to gravity, and $\gamma$ is the ratio of the surface tension coefficient to the fluid density. The velocity potential $\varphi (\mathit{x},z,t)$ must satisfy the following equations:

The Fourier transforms of $\zeta (\mathit{x},t)$ and $\psi (\mathit{x},t)$ are introduced by where $\zeta (\mathit{k},t)$ and $\psi (\mathit{k},t)$ are, respectively, the Fourier transforms of $\zeta (\mathit{x},t)$ and $\psi (\mathit{x},t)$, $\mathit{k}=(k_x,k_y)$ is the horizontal wave vector, and the asterisk denotes the complex conjugate, $\mathrm{i}=\sqrt{-1}$. In the following, the time variable t is omitted for simplicity. Please note that the Fourier transformation is the canonical one; then, the canonical Equations $\left(1\right)$ can be transformed into the canonical ones with the pair of canonically conjugate variables $\zeta \left(\mathit{k}\right)$ and $\psi \left(\mathit{k}\right)$ as follows:

To normalize Equation (5), the canonical transformation is introduced by where $\mathcal{U}\left(\mathit{k}\right)={\left[\frac{q\left(\mathit{k}\right)}{2\omega \left(\mathit{k}\right)}\right]}^{\frac{1}{2}},\mathcal{N}\left(\mathit{k}\right)={\left[\frac{\omega \left(\mathit{k}\right)}{2q\left(\mathit{k}\right)}\right]}^{\frac{1}{2}}$, and $\omega \left(\mathit{k}\right)$ is the linear dispersion relation defined by

Equations (5) can be reduced to a single equation using (6). Now, the Hamiltonian H is a functional of $a\left(\mathit{k}\right)$ and $a^{*}\left(\mathit{k}\right)$. In this work, attention is paid to waves of small but finite amplitudes (weakly nonlinear waves). Assuming small wave slopes, we can formally expand the Hamiltonian $H=H(a,a^{*})$ into integral power series in powers of a and $a^{*}$ (see in Section 3). Here, we consider this expansion with accuracy up to and including the sixth-order terms as follows: where $H_j(j=3,4,5,6)$ stands for the Hamiltonian of j-wave interactions, and where $U^{\left(n\right)},V^{\left(n\right)}$, and $W^{\left(n\right)}$ were given by Krasitskii [7], and $X^{\left(n\right)}$ will be given in Section 3. In the compact notation, the arguments ${\mathit{k}}_j$ in $a,$$U^{\left(n\right)}$,$V^{\left(n\right)}$, $W^{\left(n\right)}$, $X^{\left(n\right)}$, and $\delta$-functions are replaced by the subscript j, with the subscript zero assigned to $\mathit{k}$. For example, $a_j=a({\mathit{k}}_j,t),{\omega }_j=\omega \left({\mathit{k}}_j\right),{\delta }_{0-1-2}=\delta (\mathit{k}-{\mathit{k}}_1-{\mathit{k}}_2),U_{0,1,2}^{\left(n\right)}=U^{\left(n\right)}(\mathit{k},{\mathit{k}}_1,{\mathit{k}}_2),d{k}_0=d\mathit{k},d{k}_{012}=d{\mathit{k}}_{0}d{\mathit{k}}_{1}d{\mathit{k}}_2$, etc., and the integral signs denote corresponding multiple integrals with the limits from $-\infty$ to $+\infty$.

It is important to assume that the coefficients $U^{\left(n\right)},V^{\left(n\right)},W^{\left(n\right)}$, and $X^{\left(n\right)}$ satisfy the natural symmetry conditions, which ensure that the Hamiltonian coefficients do not change as the integral variable subscripts change. For example, the coefficient $U_{0,1,2}^{\left(1\right)}$ is symmetric under the transposition of the arguments 1 and 2, $U_{0,1,2}^{\left(3\right)}$ under all transpositions of $0,1$, and 2, $V_{0,1,2,3}^{\left(1\right)}$ under the transpositions of $1,2$, and 3, $V_{0,1,2,3}^{\left(2\right)}$ under the transpositions of the arguments inside the groups $(0,1)$ and $(3,4)$, $V_{0,1,2,3}^{\left(4\right)}$ under the transpositions of all its indices, $W_{0,1,2,3,4}^{\left(1\right)}$ under all transpositions of $1,2,3$, and 4, $W_{0,1,2,3,4}^{\left(2\right)}$ under the transpositions of the arguments inside the groups $(0,1)$ and $(2,3,4)$, $X_{0,1,2,3,4,5}^{\left(1\right)}$ under all transpositions of $1,2,3,4$, and 5, $X_{0,1,2,3,4,5}^{\left(2\right)}$ under the transpositions of the arguments inside the groups $(0,1)$ and $(2,3,4,5)$, and so on. The coefficients should ensure the Hamiltonian is a real quantity. In view of this, there are additional symmetry conditions

It should be mentioned here that the coefficients calculated directly are asymmetric, so it is necessary to pay attention to skills in the calculation process (the definite integral has nothing to do with the form of the integral variable) to construct the symmetric coefficients (see Section 3).

By virtue of $\left(8\right)$ and $\left(9\right)$, the following evolution equation is obtained: where

To simplify the expression of the Hamiltonian, the canonical transformation $a\left(\mathit{k}\right)=a[b\left(\mathit{k}\right),b^{*}\left(\mathit{k}\right)]$ was introduced [7]. Thus, we can obtain a new Hamiltonian $\tilde{H}$ that satisfies where the canonical transformation satisfies Poisson’s brackets,

Here, $\delta (\mathit{k}-{\mathit{k}}^{\prime })$ on the right side of $\left(15\mathrm{b}\right)$ stands for the Dirac function.

According to previously established canonical transformation principles, the form of canonical transformation is postulated by

Supposing that $A^{\left(n\right)},B^{\left(n\right)},C^{\left(n\right)},$ and $D^{\left(n\right)}$ satisfy the natural symmetry conditions, their detailed expressions will be given in Section 4. Then, the new Hamiltonian $\tilde{H}$ can be obtained by substituting (16) into (7). In the process, the coefficients $U^{\left(n\right)},V^{\left(n\right)},W^{\left(n\right)},$ and $X^{\left(n\right)}$ are replaced, respectively, by ${\tilde{U}}^{\left(n\right)},{\tilde{V}}^{\left(n\right)}$, ${\tilde{W}}^{\left(n\right)}$, and ${\tilde{X}}^{\left(n\right)}$ and satisfy symmetry conditions like the previous ones.

On the basis of the transformation, as mentioned above, and the resonance conditions, and choosing appropriate $A^{\left(n\right)},B^{\left(n\right)},C^{\left(n\right)},$ and $D^{\left(n\right)}$, one can deduce the dynamical equation of ocean surface waves concerning three-four-five-six-wave resonance by eliminating the nonresonant terms at the right-hand side of Equation (12). The dynamical equation of ocean surface gravity waves that contain four-five-wave resonance interactions was obtained by Krasitskii [7]. In Section 4, the corresponding dynamical equation, which contains four-five-six-wave resonance interactions, will be obtained.

In this section, the precise form of the expansion coefficients $U^{\left(n\right)},V^{\left(n\right)},W^{\left(n\right)}$, and $X^{\left(n\right)}$ are obtained by complex calculations. The first three were given by Krasitskii [7]. Here, the expansion of H with accuracy up to the sixth-order terms for ocean surface capillary–gravity waves with a finite water depth is presented. The general solution of the Laplace equation satisfying the bottom boundary condition can be presented by the following Fourier integral: