Linking Numbers in Vortex and Magnetic Knots
speaker: Francesca Maggioni (Università di Bergamo)abstract:
In this talk we review some recent results on the role of linking number in the contexts of perfect fluids and ideal magnetohydrodynamics. Vortex torus knots in Euler fluids or magnetic tight knots in magnetohydrodynamics provide examples of complex structures where geometry and topology affect the dynamics, energetics and helicity 3 of the system. In the context of Euler fluids we shall be concerned with vortex filaments in the shape of torus knots/unknots obtained as solution of the Localized Induction Approximation LIA and their evolution is analysed by numerically integrating the Biot-Savart law 2. Conversely, magnetic knots in magnotohydrodynamics are identified with a tubular knot filled by a magnetic field, decomposed in poloidal and toroidal components, in which framing is given by the linking of the field lines with the knot axis 1.
Generic behaviors of the energy (kinetic or magnetic) and writhe and twist helicity in relation to the winding number w and linking number of the system are determined in both the contexts and compared for several knots with increasing complexity. For vortex knots we found that helicity is dominated by writhe contribution and that for w < 1 vortex knots move faster and carry more energy than a reference vortex ring of the same size and circulation, whereas for w > 1 knots have approximately same speed and energy of the reference vortex ring. For magnetic knots the helicity of the field is determined as a function of toroidal and poloidal fluxes and the constrained minimum magnetic energy is calculated by an exact analytical expression, function of ropelength and framing and compared for the first prime knots up to 10 crossings.
These results add new information on the interplay of geometric and topological aspects on the dynamics of complex systems.
REFERENCES:
1Maggioni, F., & Ricca, R.L., (2009) On the groundstate energy of tight knots. Proc. Roy. Soc. A, 465(2109), 27612783.
2Maggioni, 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.
3Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Calugareanu invariant, Proc. R. Soc. London A, 439, 11 429.
timetable:
Thu 19 May, 15:50 - 16:20, Aula Dini
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