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

无粘二维准地转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发。在这项研究中，我们用谱法进行了数值模拟，比以前的方法有更多的傅里叶模式，并重新检查了数值数据，这些数据被解释为表明奇点的出现。我们的数值结果表明，以前的数值模拟没有足够的模态数来解析解的奇异行为，变量的变化甚至表明解在有限时间内不会爆炸。我们还对粘性QG系统进行了数值模拟，结果表明不存在与无粘溶液爆炸有关的串级现象。这些结果不支持有限时间奇点的出现，但表明了不可见解的规律性。研究了湍流壳模型中混沌的现象学理论，得到了Lyapunov谱的渐近公式。该理论的基础是该系统中李雅普诺夫向量的支持在傅里叶空间中具有明显的局域性，这使我们能够将李雅普诺夫谱与Kolmogorov相似律联系起来。当吸引子维数较大时，该公式与数值结果吻合较好。结果表明，壳模型是高维混沌中少有的能得到李雅普诺夫谱渐近公式的例子。我们也将此方法应用于Navier-Stokes湍流，得到了无粘极限下Lyapunov谱的渐近公式。

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.