Studies of Turbulence as a Nonlinear Science

(1)旋转分层湍流中的旋涡形成和扩散：旋转和层化是大气和海洋中的两种基本物理机制。研究这种流动中涡旋的形成和扩散对于提高我们对地球物理和天体物理湍流现象的理解具有重要的作用。我们检验了具有旋转和稳定层结影响的N-S方程的直接数值模拟结果。旋转和层化都倾向于使流动成为二维(或两个分量)，但我们证实，它们在形成垂直结构时以不同的方式起作用。对于扩散，我们观察到层结抑制了垂直方向上的粒子迁移，旋转也抑制了粒子的迁移，在强旋转和/或层结作用下，单个粒子弥散的时间线性增长在水平方向上保持不变。(2)常曲率表面上的涡旋运动：二维…上的涡旋运动给出了更多具有常曲率的正方形黎曼曲面的公式。通过赤平投影，可以清楚地分析球面上的涡旋运动与双曲面上的涡旋运动的联系和区别。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，我们考虑了涡偶的运动。结果表明，偶极子在两面绘制一条测地线作为其运动轨迹。(3)衰减二维湍流的演化和自相似：我们考察了二维衰减湍流能量谱自相似的结果，并得出结论：传统的闭合仅当对能量和拟能传输有重要贡献的空间区域包括从高斯混沌的初始时间开始随着时间的推移而不断减小的空间区域。(4)非均匀椭圆涡的轴对称化过程：从涡度梯度的平方变速增长的角度研究了二维非均匀椭圆形涡旋的轴对称化。首先，本文指出，如果用应变率张量表示，涡旋张量增长方程与三维涡旋张量增长方程具有相似的形式。然后，对非均匀椭圆涡的回旋效应进行了分析。分析和数值验证表明，非均匀椭圆涡旋一般具有四极结构，在正涡度产生区，涡丝遵循涡度梯度增强过程被抛出。(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解析和数值研究。基于旋转对称性的论证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，这反过来又允许获得PDF尾部的准确形式，包括指数和幂指数前因子。对于较窄的速度谱(k-空间中仅限于壳体的速度)，显式地找到了特征函数，产生了所有阶次的压力累积量。较少

(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