Symmetry breaking and Hamiltonian bifurcation theory for three-dimensional instability of a vortex tube

1.从哈密顿谱理论的角度研究了椭圆涡旋和涡环的三维不稳定性。(1)纯剪切破坏了朗肯涡旋的SO(2)XO(2)对称性，使圆心形变为椭圆。我们成功地用贝塞尔函数显式地写下了无穷小扰动。阐明了与椭圆不稳定性的关系。(2)对于涡环，对称性破坏扰动是曲率效应。用封闭的形式写出了开尔文涡环上的微小扰动。我们发现了方位波数为二的开尔文波对之间存在参数共振的可能性。我们还进行了局部稳定性分析，用WKB方法阐明了高斯核上的不稳定模的结构与开尔文涡环上的不稳定模的结构有很大的不同。(3)我们推导了几个不稳定的MO…振幅的弱非线性演化方程利用与Z2xO(2)对称性有关的等变矢量场，在纯剪切的涡流管上产生了更多的DES。2.当几十纳秒的准分子激光脉冲照射在钴涂层衬底上时，涡旋丝同时被产生、猝灭和冻结。通过从定域感应近似下的稳定性分析估计时间尺度，我们从冻结图像的微观图像中提取了动态图像。(2)通过重复多次脉冲照射，得到了不稳定的涡环。3.对于平面波边带不稳定性，证明了，在线性临界点附近，基于包络方程、基于振幅方程和基于二次不稳定性分析的方法都是等价的。4.对于Rayleigh-Benard对流，利用中心流形约化方法导出了五阶振幅方程，并以密度对温度的二次依赖为展开参数，在简并点附近对其进行了分叉分析。较少

1.Three-dimensional instability of an elliptic vortex and a vortex ring was investigated from the viewpoint of the Hamiltonian spectral theory.(1)A pure shear breaks the SO(2)xO(2)-symmetry of the Rankine vortex and deforms the circular core into an ellipse.We have succeeded in explicitly writing down the infinitesimal disturbances in terms of the Bessel functions.The relation with the elliptical instability is clarified.(2)For a vortex ring, symmetry-bresking perturbation is the curvature effect. The infinitesimal disturbances on Kelvin's vortex ring are written out in a closed form.We have found a possibility of parametric resonance between pairs of Kelvin waves whose azimuthal wavenumbers are separated by two.A local stability analysis is also made, using the WKB method It is clarified that the structure of unstable modes on a Gaussian core is very different from that on Kelvin's vortex ring.(3)We have derived weakly nonlinear evolution equations of amplitudes of several unstable mo … More des on a vortex tube subjected to a pure shear, with the aid of equivariant vector fields associated with Z2xO(2) symmetry.2.When a pulse of excimer laser of a few ten nanoseconds is irradiated on a Co-coated substrate, vortex filaments are created, quenched and frozen at once.By estimating the time scales from a stability analysis in the localized induction approximation, we have extracted a dynamical picture from the micrographs of the frozen pictures.(2)By repeating the irradiation, of pulses several times, unstable vortex rings are obtained.3.For the side-band instability of a plane wave, it is proven that, near the linear critical point, the methods based on envelope equations, on amplitude equations and on a secondary-instability analysis are all equivalent.4.For the Rayleigh-Benard convection, the fifth-order amplitude equations are derived, using the center-manifold reduction, and their bifurcation analysis is made near the degeneracy point with quadratic dependence on temperature of the density being an unfolding parameter It is clarified that right-hexagonal, right-triangle and patch-work quilt patterns are unstable as a primary bifurcation solution of the generic normal form equations. Less