The degree of knottedness of tangled vortex lines

TLDR

In an unbounded fluid, the velocity field produced by a vorticity distribution that is nonzero only on two closed vortex filaments is examined. The authors demonstrate that the integral \(I=\int \mathbf{u}\!\cdot\!\boldsymbol{\omega}\,dV\) equals \(\alpha K_{1}K_{2}\), where \(\alpha\) is an integer giving the degree of linkage between the filaments (\(\alpha=0\) for unlinked, \(\pm1\) for singly linked). They prove that \(I\) is invariant for continuous, localized vorticity in barotropic, inviscid flow under conservative body forces, implying conservation of vortex‑line linkages, and illustrate this with steady flows such as spherical vortices with swirl whose vortex lines are knotted and linked. The paper also discusses two related magnetohydrodynamic invariants discovered by Woltjer (1958a, b).

Abstract

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 K 1 , K 2 . It is first shown that the integral \[ I=\int{\bf u}.{\boldmath \omega}\,dV \] is equal to α K 1 K 2 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 (1958 a, b ) are discussed in ±5.