Wave Turbulence: Computational and Theoretical Investigations of a Story Far From Over

波浪湍流是关于理解描述随机海浪的弱非线性场方程解的长期统计行为。弱的非线性使得人们可以得到BBGKY系列傅立叶空间矩方程的自然闭合，其中包括一个描述波数密度如何通过共振在整个光谱中重新分布的动力学方程。这些方程有特别相关的有限通量解(Kolmogorov-Zakharov或KZ)，它捕捉到能量流等密度守恒是如何从源流向汇的。使波动湍流成为一个悬而未决的问题是，这些KZ解几乎不是在所有尺度上都一致有效的，因此人们面临的挑战是找出它们在波数空间中分解的区域发生了什么。这项提案中的工作继续着作者们填补这些缺失的空白并使理论完整的努力。这并不是一项容易的任务，因为一旦击穿发生，新的态可能包含强烈的非线性相干结构。特别是，我们正在研究凝结和白云形成的问题。波浪湍流是关于理解波浪系统的统计数据的。想象一下暴风雨过后的海面。有各种波长的波，向四面八方传播。绝对值得注意的是，一个人可以在很大范围内说出长期能谱是什么样子的。能量谱是一张图表，显示了不同长度和不同方向的波浪中有多少能量。尽管最初的曲线图可能反映了海洋最初是如何被激发的，但由于不同波长和方向的波之间的共振，海洋松弛到了一种统计上稳定的宇宙状态，称为Kolmogorov-Zakharov或KZ谱。但也不完全是。如果风暴足够强，在较小的尺度上，KZ谱不能描述海面的统计数据。事实上，观察者会发现，海上风暴越大，白浪(局部破碎的海浪将空气和水结合成泡沫乳状液)的密度就越高。挑战是找出在这些小范围内用什么来取代KZ频谱。我们正在开发一种关于这种行为的新理论。关于这项工作，我们还感兴趣的是，水手们几个世纪以来就知道了，但直到现在才被研究的一种现象，即在原本相对平静的海上突然出现反常的海浪。

Wave Turbulence is about understanding the long time statistical behavior of solutions of weakly nonlinear field equations describing a sea of random waves. The weak nonlinearity allows one to obtain a natural closure of the BBGKY hierarchy of the Fourier space moment equations which includes a kinetic equation which describes how the wavenumber density is redistributed throughout the spectrum by resonances. These equations have especially relevant finite flux solutions (Kolmogorov-Zakharov or KZ) which capture how conserved densities such as energy flow from sources to sinks. What makes Wave Turbulence an open problem is that these KZ solutions are almost never uniformly valid at all scales and so one is left with the challenge of finding out what happens in the regions of wavenumber space where they break down. The work in this proposal continues the authors' efforts to fill in these missing gaps and to make the theory complete. This is not an easy task because once breakdown occurs, the new states may contain strongly nonlinear coherent structures. In particular, we are working on the problem of condensation and whitecap formation. Wave turbulence is about understanding the statistics of systems of waves. Imagine the sea surface after a storm. There are waves of all wavelengths, travelling in all directions. It is absolutely remarkable that one can, over a large range of scales, say what the longtime energy spectrum looks like. The energy spectrum is a graph showing how much energy there is in waves of different lengths and different directions. Even though the initial graph may reflect how the sea is first excited, the sea, because of resonances between waves of different wavelengths and directions, relaxes to a statistically steady universal state called the Kolmogorov-Zakharov or KZ spectrum. But not quite. If the storm is strong enough, at the smaller scales, the statistics of the sea surface is not described by the KZ spectrum. In fact, an observer will see that the more stormy the sea is, the higher the density of whitecaps (locally breaking waves which combine air and water into a frothy emulsion). The challenge is to find out what replaces the KZ spectrum at these small scales. We are developing a new theory of this behavior. In connection with this work, we are also interested in a phenomenon which has been known by mariners for centuries but which only now is being studied, the sudden emergence of freak waves in an otherwise relatively placid sea