DEGREE OF KNOTTEDNESS OF TANGLED VORTEX LINES
DEGREE OF KNOTTEDNESS OF TANGLED VORTEX LINES
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DOI：
10.1017/s0022112069000991
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发表时间：
1969-01-01
影响因子：
3.7
通讯作者：
MOFFATT, HK
中科院分区：
文献类型：
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作者：
MOFFATT, HK
令u（x）为由于涡量分布w（x）而导致的无限大流体中的速度场，该涡量分布w（x）除了在强度为K1、K2的两个闭合涡丝中之外为零。它首先表明，积分 \[ I=\int{\bf u}. {\boldmath \omega}\，dV \] 等于α K1 K2，其中α是表示两个细丝连接程度的整数;如果它们未连接，则α = 0，如果它们是单链的，则α = ± 1。在守恒体积力作用下，建立了正压无粘流的连续局部涡度分布的I不变性。结果用随流体运动的涡线的联系守恒来解释，并简要地描述了I ± 0的定常流动的一些例子，特别注意一类有旋流的球形涡（这与静磁方程的已知解族非常相似）;在±5中讨论了Woltjer（1958 a，B）发现的两个相关的磁流体动力学不变量。
Let u(x) be the velocity field in a fluid of infinite extent due to a vorticity distribution w(x) which is zero except in two closed vortex filaments of strengths K1, K2. It is first shown that the integral \[ I=\int{\bf u}.{\boldmath \omega}\,dV \] is equal to αK1K2 where α is an integer representing the degree of linkage of the two filaments; α = 0 if they are unlinked, ± 1 if they are singly linked. The invariance of I for a continuous localized vorticity distribution is then established for barotropic inviscid flow under conservative body forces. The result is interpreted in terms of the conservation of linkages of vortex lines which move with the fluid.Some examples of steady flows for which I ± 0 are briefly described; in particular, attention is drawn to a family of spherical vortices with swirl (which is closely analogous to a known family of solutions of the equations of magnetostatics); the vortex lines of these flows are both knotted and linked.Two related magnetohydrodynamic invariants discovered by Woltjer (1958a, b) are discussed in ±5.