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几个性质被阐明。从那时起，日本科学家，特别是物理学家、数学家、生物学家、化学家、工程师、地质学家，甚至社会主义者开始了基于耗散动力系统的非线性动力学研究。我们的研究目的是建立一个关于无粘流体中反常扩散现象的统计物理理论。在充分利用计算机资源的情况下，成功地对二维衰变湍流进行了理论推导和直接计算机模拟。在湍流中，涡旋的时间发展有三个阶段。在第一阶段，流体自组织形成一系列连贯的旋涡。一旦相干涡旋出现，第二阶段就开始了。在第二阶段，相干涡旋控制着系统的动力学。当它们靠近时，在一个关键距离内，相同的符号涡会合并并成为一个更大的涡旋。一对反号旋涡最终构成偶极结构。在最后阶段，偶极子结构扩散衰变。在本研究中，我们阐明了标度律在第二阶段的成立。我们还成功地建立了标度指数xi的理论，并确定了与数值模拟结果一致的xi的数值。

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.