Wave turbulence is by nature a multiple timescale problem for which there is a natural asymptotic closure. The main result of this analytical theory is the kinetic equation that describes the longtime statistical behavior of such turbulence composed of a set of weakly nonlinear interacting waves. In the case of gravitational waves, it involves four-wave interactions and two invariants, energy, and wave action. Although the kinetic equation of gravitational wave turbulence has been published with the Hadad-Zakharov metric, along with their physical properties, the detailed derivation has not been shown. Following the seminal work of Newell [Rev. Geophys. 6, 1 (1968).] for gravity/surface waves, we present the multiple timescale method, rarely used to derive the kinetic equations, and clarify the underlying assumptions and methodology. This formalism is applied to a wave amplitude equation obtained using an Eulerian approach. It leads to a kinetic equation slightly different from the one originally published, with a wave equation obtained using a Hamiltonian approach; we verify, however, that the two formulations are fully compatible when the number of symmetries used is the same. We also show that the exact solutions (Kolmogorov-Zakharov spectra) exhibit the same power laws and cascade directions. Furthermore, the use of the multiple timescale method reveals that the system retains the memory of the initially condition up to a certain level (second order) of development in time.

中文翻译：

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引力波湍流：四次波相互作用的多时间尺度方法

波湍流本质上是一个多时间尺度问题，存在自然渐近闭合。该分析理论的主要结果是描述由一组弱非线性相互作用波组成的湍流的长期统计行为的动力学方程。就引力波而言，它涉及四波相互作用和两个不变量：能量和波作用。尽管引力波湍流的动力学方程及其物理性质已经与 Hadad-Zakharov 度量一起发表，但详细的推导尚未显示。继纽厄尔 [ Rev. Geophys] 的开创性工作之后。 6 , 1 (1968).] 对于重力/表面波，我们提出了很少用于推导动力学方程的多时间尺度方法，并阐明了基本假设和方法。这种形式应用于使用欧拉方法获得的波幅方程。它得出的动力学方程与最初发表的方程略有不同，波动方程是使用哈密顿方法获得的；然而，我们验证，当使用的对称数相同时，这两种公式是完全兼容的。我们还表明，精确解（Kolmogorov-Zakharov 谱）表现出相同的幂律和级联方向。此外，多时间尺度方法的使用表明，系统保留了初始条件的记忆，直到一定的发展水平（二阶）。