Numerical simulations of nonlinear dynamical systems with large degree of freedom and turbulence

二维湍流的理论与数值研究在Charney-Hasegawa-Mima湍流中，在湍流发展过程中，能量流入波数空间中的小波数区域。结果表明，在反级联过程中形成了准晶体结构，涡度场的结构函数服从动态标度规律。在衰减湍流中，我们证明了总能量是渐近守恒的，涡度谱存在动力学标度规律，并建立了一个现象学理论。此外，利用涡旋系综的哈密顿动力学特性，从现象学上确定了表征涡旋系综衰减过程的指数。提出了高波数动力学中强耗散非线性介质的新模型，并对包括该模型在内的三个具有混沌元的模型进行了数值求解。当系统尺寸小于临界尺寸Lc时，空间均匀混沌振荡是稳定的。另一方面，当系统规模大于Lc时，它变得不稳定。在后一种情况下，我们观察到一个间歇，其统计数据与开关间歇的统计数据相同。此外，我们还发现了开关间歇的以下新特征：(1)我们发现了通过开关间歇产生的新的扩散。它被称为开-关扩散。(2)提出了开关变量傅里叶谱的一种新的动态标度律。(3)发现了开关断续发生前暂态过程的一个新的统计规律。

Theoretical and numerical study of two-dimensional turbulenceIn the Charney-Hasegawa-Mima turbulence, energy flows into a small wavenumber region in the wavenumber space in the process of the development of turbulence. We observed that a quasi-crystal structure is formed in this inverse cascade process, and that the structure function of the vorticity field obeys the dynamic scaling law. In the decaying turbulence we showed that the total energy is asymptotically conserved, and that there exists a dynamical scaling law for the vorticity spectrum and developed a phenomenological theory. Furthermore the exponent characterizing the decaying process of the vortix ensemble is phenomenologically determined by utilizing the Hamiltonian dynamics characteristics of the ensemble of vortices.Nonlinear dynamical systems with large degrees of freedomA new model for nonlinear media with a strong dissipation in high-wavenumber dynamics is proposed, and three models with chaotic elements, including the proposed one were numerically solved. When the systems size is less than the critical size Lc, the spatially uniform chaotic oscillation is stable. On the other hand, when the system size is larger than Lc, it becomes unstable. In the latter case, we observed an intermittency, whose statitics turned out to be identical to those of on-off intermittency. Furthemore, we found the following new characteristics of on-off intermittency : (1) We found a new diffusion generated via on-off intermittency. It was termed as on-off diffusion. (2) A new dynamical scaling law for the Fourier spectrum of on-off variable is proposed. (3) A new statistical law of transient process before the onset of on-off intermittency is found.