Studies of Turbulence as a Nonlinear Science

(1)旋转分层湍流中的涡旋形成和扩散：旋转和分层是大气和海洋中的两种基本物理机制。研究涡旋的形成和扩散对提高我们对地球物理和天体物理湍流现象的认识起着至关重要的作用。我们检验了考虑旋转和稳定分层影响的Navier-Stokes方程的直接数值模拟结果。旋转和分层都倾向于使流动二维（或两个组成部分），但我们证实它们以不同的方式产生垂直结构。在扩散方面，我们观察到分层抑制了颗粒在垂直方向上的迁移，旋转也抑制了颗粒的迁移，在强旋转和/或分层的情况下，单个颗粒在水平方向上的分散时间保持线性增长。(2)常曲率曲面上的涡旋运动：推导了二维更多向常曲率黎曼曲面上的涡旋运动。通过立体投影，可以清楚地分析涡旋在球面和双曲平面上运动的联系和区别。给出了点涡在球面和双曲平面上运动的哈密顿形式。作为解析解的一个例子，考虑了涡旋对（偶极子）的运动。结果表明，偶极子在两个表面上都画了一条测地线曲线作为其轨迹。(3)二维衰减湍流和自相似性的演化：我们研究了二维衰减湍流能谱的自相似性的结果，并得出结论，传统闭包只有在对能量和熵传递有重要贡献的空间区域包含一个从高斯混沌初始时间开始随着时间的推移而不断减小的空间区域时才符合这一原理。(4)非均匀椭圆涡的轴对称化过程：从涡度梯度平方的增量角度研究了二维非均匀椭圆涡的轴对称化问题。首先，本文指出，如果用应变速率张量来表示回力生长的方程，其形式与三维中的熵生长方程——涡-拉伸方程相似。然后着重分析了非均匀椭圆涡旋的回旋产生。分析和数值验证表明，非均匀椭圆涡一般具有四极结构，在正涡产生区，涡丝沿涡度梯度增强过程喷射出来。(5)随机高斯速度下的压力分布：对随机高斯速度下的压力分布进行了解析和数值研究。基于旋转对称的论证允许澄清压力特征函数的分析结构，并找到其所有奇点的幂，从而允许获得PDF尾的精确形式，包括指数和幂前指数因子。对于窄速度谱（速度限制在k空间中的壳层），特征函数被显式地找到，产生所有阶的压力累积量。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