Chaotic streamlines in the flow of knotted and unknotted vortices

Abstract

This paper describes the motion and the flow induced by a thin tubular vortex coiled on a torus. The vortex is defined by the number of turns, p, that it makes round the torus symmetry axis and the number of turns, q, that it makes round the torus centerline. All toroidal filamentary vortices are found to progress along and to rotate round the torus symmetry axis in an almost steady manner while approximately preserving their shape. The flow, observed in a frame moving with the vortex, possesses two stagnation points. The stream tube emanating from the forward stagnation point and the stream tube ending at the backward stagnation point transversely intersect along a finite number of streamlines. This produces a three-dimensional chaotic tangle whose geometry depends primarily on the value of p. Inside this chaotic shell there are two major stability tubes: the first one envelopes the vortex whereas the second one runs parallel to it and possesses the same topology. When p > 2 there is an additional stability tube enveloping the torus centerline.