Singular behaviors of two-dimensional numerical turbulence and chaos in a shell model.

壳模型中二维数值湍流和混沌的奇异行为。

基本信息

批准号：
07832004
负责人：
YAMADA Michio
金额：
$ 1.34万
依托单位：
University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07832004/
关键词：
Ouasi-geostrophic approximations Inviscid equation Solution blow up Shell model Chaos Lypapunov spectrum Kolmogorov similarity law High-dimensional attractor 非常粘性方程式 2次元乱流特異性非粘性流

项目摘要

Inviscid two-dimensional quasi-geostroph (QG) flow has been considered to have a finite-time singularity. Some mathematical result and numerical simulations on the quasi-geostrophic system suggested that temperature gradient of a solution to QG equation blows up in a finite time. In this research we performed a numerical simulation by spectral method with more Fourier modes than in previous ones, and re-examined the numerical data which was interpreted to indicate the appearance of the singularity. Our numerical result shows that the previous numerical simulations did not have a sufficient number of modes to resolve the singular behavior of the solution, and a change of variable even suggests that the solution does not blow up in a finite time. We also performed a numerical simulation on viscous QG system, which indicates that there is no cascade phenomenon connected to the blow up of the inviscid solution. These results does not support the appearance of a finite-time singularity, but suggests the regularity of the invisid solution. We also investitated a phenomenological theory of chaos in shell model of turbulence, and obtained an asymptotic formula for Lyapunov spectrum. This theory is based on the fact that the support of Lyapunov vectors in this system in sharply localized in Fourier space, which permits us to relate the Lyapunovspectrum to Kolmogorov similarity law. This formula agrees with numerical result better in the case of larger attractor dimension. This result shows that the shell-model is a rare example of high-dimensional chaos in which the asymptotic formula for Lyapunov spectrum can be obtained. We applied this method also to Navier-Stokes turbulence and obtanied an asymptotic formula for Lyapunov spectrum in the inviscid limit.

无粘二维准地转体（QG）流被认为具有有限时间奇点。准地转系统的一些数学结果和数值模拟表明，QG方程解的温度梯度在有限时间内爆炸。在这项研究中，我们利用比以前更多的傅立叶模式进行谱法进行数值模拟，并重新检查了被解释为表明奇点出现的数值数据。我们的数值结果表明，之前的数值模拟没有足够数量的模态来解决解的奇异行为，并且变量的变化甚至表明解不会在有限时间内爆炸。我们还对粘性 QG 系统进行了数值模拟，这表明不存在与无粘溶液爆炸相关的级联现象。这些结果并不支持有限时间奇点的出现，但表明了隐形解的规律性。我们还研究了湍流壳模型中的混沌唯象理论，得到了李雅普诺夫谱的渐近公式。该理论基于以下事实：该系统中李雅普诺夫向量的支持明显集中在傅里叶空间中，这使我们能够将李雅普诺夫谱与柯尔莫哥洛夫相似律联系起来。在吸引子尺寸较大的情况下，该公式与数值结果吻合较好。这一结果表明，壳模型是一个罕见的高维混沌例子，其中可以获得李亚普诺夫谱的渐近公式。我们也将此方法应用于纳维-斯托克斯湍流，并获得了无粘极限下李雅普诺夫谱的渐近公式。