Studies of Turbulence as a Nonlinear Science

(1)旋转分层湍流中涡旋的形成和扩散：旋转和分层是大气和海洋中的两种基本物理机制。在这种流动中的涡旋形成和扩散的研究在提高我们对地球物理学和天体物理学湍流现象的理解方面起着至关重要的作用。我们研究的结果直接数值模拟的Navier-Stokes方程的旋转和稳定层结的影响。旋转和分层都倾向于使流动成为二维（或两个分量），但我们证实了它们在产生垂直结构时以不同的方式工作。至于扩散，我们观察到，分层抑制颗粒在垂直方向上的迁移，旋转也是如此，并且在强旋转和/或分层的情况下，单颗粒分散在水平方向上的线性增长持续存在。(2)常曲率曲面上的涡旋运动：二维涡旋运动 关于我们 给出了具有常曲率的标准黎曼曲面。利用赤平极射投影，可以清楚地分析出球平面和双曲平面上涡旋运动的联系和区别。本文给出了球面和双曲平面上点涡运动的哈密顿形式。作为解析解的一个例子，涡对（偶极子）的运动被认为是。结果表明，偶极子在两个表面上都画出一条测地线作为其轨迹。(3)衰减二维湍流的演化和自相似性：我们研究了二维衰减湍流的能谱的自相似性的后果，并得出结论，只有当空间区域对能量和涡度拟能转移有显著贡献时，传统的闭合才符合这一原则，因为随着时间的推移，从高斯混沌的初始时间开始，空间区域不断缩小。(4)非均匀椭圆涡旋的轴对称化过程：根据涡度梯度的平方，即回文型的增长，研究了二维非均匀椭圆涡旋的轴对称化过程。首先指出，如果用应变率张量表示，则回涡增长的方程与三维涡拟能增长的方程--涡拉伸方程具有相似的形式。然后分析了非均匀椭圆涡的回文涡旋产生。分析和数值验证了非均匀椭圆涡旋的四极结构，在正产生区，涡旋丝的喷射伴随着涡度的梯度增强过程. (5)随机高斯速度的压力分布：对随机高斯速度的压力分布进行了分析和数值研究。基于旋转对称的参数可以澄清压力特征函数的分析结构，并找到其所有奇点的幂，这反过来又可以获得PDF尾部的确切形式，包括指数和幂前指数因子。对于狭窄的速度谱（速度限制在一个壳在k-空间），特征函数被明确地发现，产生的所有订单的压力累积量。少

(1) Vortex Formation and Diffusion in Rotating Stratified Turbulence : Rotation and Stratification are two fundamental physical mechanisms in atmospheres and oceans. Studies of vortex formation and diffusion in such flows play a vital role in improving our understanding of geophysical and astrophysical turbulent phenomena. We examine results of direct numerical simulation of the Navier-Stokes equations with the effects of rotation and stably stratification. Both rotation and stratifications tend to make flows two dimensional (or two components), but we verified that they work in different ways in generating vertical structures. As for diffusion, we observed that stratification suppresses the particle migration in the vertical direction and so does rotation, and that the linear growth in time for single particle dispersion persists in the horizontal direction under strong rotation and/or stratification.(2) Vortex Motion on Surfaces with Constant Curvature : Vortex motion on two- dimensi … More onal Riemannian surfaces with constant curvature is formulated. By way of the stereographic projection, the relation and difference between the vortex motion on a sphere and on a hyperbolic plane can be clearly analyzed. The Hamiltonian formalism is presented for the motion of point vortices on a sphere and a hyperbolic plane. As an example of analytic solutions, the motion of a vortex pair (dipole) is considered. It is shown that a dipole draws a geodesic curve as its trajectory on both surfaces.(3) Evolution of decaying two-dimensional turbulence and self similarity : We examine the consequences of self-similarity of the energy spectrum of two-dimensional decaying turbulence, and conclude that traditional closures are consistent with this principle only if the regions of space contributing significantly to energy and enstrophy transfer comprise an ever diminishing region of space as time proceeds from the initial time of Gaussian chaos.(4) Axisymmetrization process for a non-uniform elliptic vortex : The axisymmetrization of a 2D non-uniform elliptic vortex is studied in terms of the growth of palinstrophy, the squared of the vorticity gradient. First, it is pointed out that the equation for the palinstrophy growth, if written in terms of the strain rate tensor, has a similar form to that of enstrophy growth in 3D - the vortex-stretching equation. Then palinstrophy production is analyzed particularly for non-uniform elliptic vortices. It is shown analytically and verified numerically that a non-uniform elliptic vortex in general has quadrupole structure for the palinstrophy production, and that in the positive production regions, vortex filaments are ejected following the gradient enhancement process for vorticity.(5) Pressure distribution for random Gaussian Velocities : Pressure distributions for random Gaussian velocities are studied both analytically and numerically. Arguments based on rotation symmetry allow to clarify the analytical structure of the characteristic function of pressure and to find the power of all its singularities, which in turn, allows to obtain the exact form of the PDF tail, including both exponent and power pre-exponent factors. For the narrow velocity spectrum (velocity restricted to a shell in k-space), the characteristic function is found explicitly, generating pressure cummulants of all orders. Less