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

波浪湍流：一个远未结束的故事的计算和理论研究

• 批准号: 0809189
• 负责人: Alan Newell
• 金额: $ 15万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0809189&HistoricalAwards=false
• 关键词: Wave; Turbulence; Computational; Theoretical; Investigations

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

波浪湍流是关于理解描述随机波浪海洋的弱非线性场方程解的长时间统计行为。弱非线性允许人们获得傅立叶空间矩方程的BBGKY层次的自然闭合，其中包括一个动力学方程，该方程描述了波数密度如何通过共振在整个频谱中重新分布。这些方程具有特别相关的有限通量解（Kolmogorov-Zakharov或KZ），它捕获了诸如能量从源流向汇的守恒密度。使波浪湍流成为一个开放问题的是，这些KZ解在所有尺度上几乎都不一致有效，因此人们面临的挑战是找出在波数空间的区域发生了什么，在这些区域它们被破坏了。本提案的工作继续作者的努力，以填补这些缺失的空白，使理论完整。这不是一项容易的任务，因为一旦发生击穿，新的状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。一个人能够在大范围的尺度上，说出长时间的能谱是什么样子，这绝对是了不起的。能谱是一个图表，显示了不同长度和不同方向的波中有多少能量。尽管最初的图表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浪之间的共振，海洋松弛到一个统计上稳定的普遍状态，称为Kolmogorov-Zakharov光谱或KZ光谱。但也不完全如此。如果风暴足够强，在较小的尺度上，海面的统计数据不能用KZ谱来描述。事实上，观察者会发现，海上风暴越大，白浪（局部破碎的波浪，将空气和水结合成泡沫状的乳液）的密度就越高。挑战在于找出在这些小尺度上取代KZ谱的是什么。我们正在发展一种关于这种行为的新理论。与这项工作有关的是，我们还对一种现象感兴趣，这种现象已经被水手们知道了几个世纪，但现在才开始研究，即在相对平静的海洋中突然出现反常的波浪