Statistical physical Study of Anomalous Diffusion Phenomena in Inviscid Fluid.

由于计算机技术的进步，非线性非平衡系统的一些性质得到了澄清。从那时起，基于耗散动力系统的非线性动力学研究开始了，并受到了日本科学家的鼓励，特别是物理学家、数学家、生物学家、化学家、工程师、地质学家，甚至社会主义者。我们的研究目的是建立一个统计物理理论的异常扩散现象在一个无粘流体。利用充足的计算机资源，我们成功地进行了二维衰减湍流的理论推导和直接计算机模拟。在湍流中，已知涡旋的时间发展有三个阶段。在第一阶段，流体自组织成一组相干涡。一旦相干涡出现，第二阶段就开始了。在第二阶段，相干涡控制系统的动力学。当它们靠近时，在一个临界距离内，同符号漩涡合并并成为一个更大的漩涡。一对相反符号的涡旋最终构成了一个偶极子结构。在最后阶段，偶极子结构扩散衰减。在本研究中，我们明确了第二阶段的标度规律。我们还成功地建立了尺度指数xi的理论，并确定了xi的数值，该数值与数值模拟结果一致。

Due to the progress of a computer technology, several properties of nonlinear-nonequilibrium systems are clarified. Since then the studies of nonlinear dynamics on the bases of dissipative dynamical system were started and encouraged for Japanese scientists, particularly, physicists, mathematicians, biologists, chemists, engineers, geologists and even for socialists.The goal of our study is to construct a statistical physical theory of anomalous diffusion phenomena in an inviscid fluid. By making use of sufficient computer resources, we have performed successfully theoretical formulation and direct computer simulation of the two-dimensional decaying turbulence. In the turbulence, it is known there are three stages in the temporal developement of the vortices. In the first stage, fluid self-organizes into a set of coherent vortices. Once the coherent vortices emerge, the second stage begins. In the second stage, the coherent vortices governs dynamics of the system. When they become close, within a critical distance, same sign vortices merge and become a larger vortex. A pair of opposite-sign vortices ultimately construct a dipole structure. In the final stage, the dipole structure decays diffusively. In the present study, we clarified the scaling law hold in the second stage. We also succeeded to formulate a theory to the scaling exponent xi and decided the numerical value of xi that is consistent with the result of numerical simulations.