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

波浪湍流是关于理解弱非线性场方程的解的长期统计行为，描述了一个随机波浪的海洋。弱非线性允许人们获得自然封闭的BBGKY层次的傅立叶空间矩方程，其中包括一个动力学方程，它描述了波数密度是如何重新分布在整个频谱的共振。这些方程有特别相关的有限通量解（Kolmogorov-Zakharov或KZ），它捕捉了守恒密度如何从源流向汇。是什么使波湍流一个开放的问题是，这些KZ解决方案几乎从来没有在所有尺度上一致有效，所以一个是左与挑战，找出发生了什么事，在波数空间的区域，他们打破。本提案中的工作继续了作者的努力，以填补这些缺失的空白，并使理论完整。这不是一件容易的事，因为一旦发生击穿，新的状态可能包含强非线性相干结构。特别是，我们正在研究冷凝和白浪形成的问题。波浪湍流是关于理解波浪系统的统计。想象一下暴风雨后的海面。有各种波长的波，向各个方向传播。绝对值得注意的是，人们可以在很大的尺度范围内说出长时间的能谱是什么样子的。能量谱是一个曲线图，显示了不同长度和不同方向的波中有多少能量。尽管最初的图形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，海洋会松弛到一个统计上稳定的宇宙状态，称为Kolmogorov-Zakharov或KZ谱。但不完全是。如果风暴足够强，在较小的尺度下，海面的统计特性不能用KZ谱来描述。事实上，观察者会发现，大海的风暴越大，白浪的密度就越高（局部破碎的波浪，联合收割机将空气和水结合成泡沫状的乳状液）。挑战在于找出在这些小尺度下取代KZ谱的是什么。我们正在发展一种新的理论来解释这种行为。与这项工作有关的是，我们还对一种现象感兴趣，这种现象几个世纪以来一直为水手所知，但直到现在才被研究，即在相对平静的海洋中突然出现的反常波浪。

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