Vortex knots

In this paper we present new results concerning the evolution and stability of vortex knots in the context of the Euler equations. For the first time, since Lord Kelvin’s original conjecture of 1875, we have direct numerical evidence of stability of vortex filaments in the shape of torus knots. The results are based on the analytical solutions of Ricca [1] for thin vortex filaments and numerical integration of the Biot-Savart induction law. Moreover, a comparative study of vortex knot evolution under the so-called Localized Induction Approximation (LIA), which is a low-order approximation to the Biot-Savart law, confirms the stability results predicted by the LIA analysis. In particular, we show that thin vortex knots which are unstable under LIA have a greatly extended lifetime when the Biot-Savart law is used, but thick vortex knots have the same stability behaviour for both equations of motion.