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Velocity, energy and helicity of vortex knots and unknots

Presented by: 
F Maggioni & SZ Alamri & CF Barenghi & RL Ricca [Univ. of Bergamo; Taibah Univ.; Newcastle Univ.; Univ. of Milano-Bicocca]
Date: 
Tuesday 24th July 2012 - 10:10 to 10:30
Venue: 
INI Seminar Room 1
Session Chair: 
Renzo Ricca
Abstract: 
In this talk we examine the effect of several geometric and topological aspects on the dynamics and energetics of vortex torus knots and unknots. The knots are given by small-amplitude torus knot solutions [1] to the Localized Induction Approximation (LIA) law. Vortex evolution is thus studied in the context of the Euler equations by direct numerical integration of the Biot-Savart law. Earlier stability results on vortex knots and unknots [2] are here extended [3]-[4], and the velocity, helicity and kinetic energy of different vortex knots and unknots are presented for comparison. Vortex complexity is parametrized by the winding number w given by the ratio of the number of meridian wraps to that of longitudinal wraps. We find that for w 1 knots and poloidal coils have approximately same speed and energy of the reference vortex ring. Kinetic helicity is dominated by writhe contributions and increases with knot complexity. The stabilizing effect of the Biot-Savart law for all knots and unknots tested is also confirmed. Our results provide information on relationships between geometry, topology and dynamics of complex vortex systems and apply to quantized vortices in superfluid 4He.

References

[1] Ricca, R.L. (1993) Torus knots and polynomial invariants for a class of soliton equations. Chaos 3, 83-91. [1995 Erratum. Chaos 5, 346.]

[2] Ricca, R.L., Samuels, D.C. & Barenghi, C.F. (1999) Evolution of vortex knots. J. Fluid. Mech. 391, 29-44.

[3] Maggioni, F., Alamri, S.Z., Barenghi, C.F. & Ricca, R.L. (2009) Kinetic energy of vortex knots and unknots. Il Nuovo Cimento C, 32(1), 133–142.

[4] Maggioni, F., Alamri, S., Barenghi, C.F. & Ricca R.L. (2010) Velocity, energy and helicity of vortex knots and unknots. Phys. Rev. E., 82(2), 026309–026317.

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons