Collapse of three vortices on a sphere
Collapse of three vortices on a sphere
复制标题
球体上三个涡旋的塌缩
DOI：
10.1017/s0022112085002324
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发表时间：
1999
影响因子：
3.7
通讯作者：
P. Newton
中科院分区：
文献类型：
--
作者：
R. Kidambi;P. Newton
本文分析了在半径为R的球面上运动的三点涡的自相似坍缩，并与文献（AREF H.，三个涡的运动，物理流体，22（1979）393-400; NOVIKOV E.一、涡系的动力学和统计学。JETP，41（1975）937-943; NOVIKOV E. A.和塞多夫·Y，漩涡崩溃了，索夫。JETP，50（1979）297-301; SYNGE J.L.，在三个涡的运动中，可以。J. Math.，1（1949）257-270）。一个重要的守恒量是涡度矢量c =（i = 1 3 Γ i x i）/i = 1 3 Γ i的中心，它必须有长度R才能发生坍缩。坍缩轨道成对出现，称为伙伴状态，它们有两个不同的坍缩时间τ - < τ +。对于给定的配置，所实现的崩溃时间取决于由涡旋位置矢量形成的平行六面体体积的符号，因此取决于涡旋（Γ 1，Γ 2，Γ 3）是以右手方向还是左手方向布置。从一个给定的坍缩构型，我们可以通过反转Γ i的符号，或者通过使用与初始构型相关的离散对称性，使所有相对距离不变，但反转平行六面体体积的符号，来获得伙伴态。在平面中，只有一个与给定构型相关联的塌缩时间-伙伴状态是自相似地扩展的状态（AREF H.，三个涡旋的运动，物理流体，22（1979）393-400）。导出了塌缩轨迹的计算公式，并与平面公式进行了比较。然后将坍缩轨迹投影到球极平面上，在球极平面上导出一个新的控制涡运动的哈密顿系统。在这个投影平面上，解不是自相似的。最后，利用三线坐标研究了系统的坍缩过程，将系统简化为平面系统。
The self-similar collapse of three point vortices moving on the surface of a sphere of radius R is analysed and compared with known results from the corresponding planar problem described in (AREF H., Motion of three vortices, Phys. Fluids, 22 (1979) 393-400; NOVIKOV E. A., Dynamics and statistics of a system of vortices, Sov. Phys. JETP, 41 (1975) 937-943; NOVIKOV E. A. and SEDOV Y., Vortex collapse, Sov. Phys. JETP, 50 (1979) 297-301; SYNGE J. L., On the motion of three vortices, Can. J. Math., 1 (1949) 257-270). An important conserved quantity is the center of vorticity vector c = (Σ i = 1 3 Γ i x i )/Σ i = 1 3 Γ i , which must have length R for collapse to occur. Collapse trajectories occur in pairs, called partner states, which have two distinct collapse times τ - < τ + . The collapse time that is achieved for a given configuration depends on the sign of the parallelpiped volume formed by the vortex position vectors, hence depends on whether the vortices (Γ 1 , Γ 2 , Γ 3 ) are arranged in a right-handed or left-handed sense. From a given collapsing configuration, one can obtain the partner state by reversing the signs of the Γ i 's, or, alternatively, by using a discrete symmetry associated with the initial configuration that leaves all relative distances unchanged, but reverses the sign of the parallelepiped volume. In the plane, there is only one collapse time associated with a given configuration-the partner state is one that expands self-similarly (AREF H., Motion of three vortices, Phys. Fluids, 22 (1979) 393-400). Formulas for the collapsing trajectories are derived and compared with the planar formulas. The collapse trajectories are then projected onto the stereographic plane where a new Hamiltonian system is derived governing the vortex motion. In this projected plane, the solutions are not self-similar. In the last section, the collapse process is studied using tri-linear coordinates, which reduces the system to a planar one.