Geometric and topological aspects associated with induction effects of field lines in the shape of torus knots/unknots are examined and discussed in detail. Knots are assumed to lie on a mathematical torus of circular cross-section and are parametrized by standard equations. The induced field is computed by direct integration of the Biot-Savart law. Field line patterns of the induced field are obtained and several properties are examined for a large family of knots/unknots up to 51 crossings. The intensity of the induced field at the origin of the reference system (center of the torus) is found to depend linearly on the number of toroidal coils and reaches maximum values near the boundary of the mathematical torus. New analytical estimates and bounds on energy and helicity are established in terms of winding number and minimum crossing number. These results find useful applications in several contexts when the source field is either vorticity, electric current or magnetic field, from vortex dynamics to astrophysics and plasma physics, where highly braided magnetic fields and currents are present.

Oberti, C., Ricca, R. (2017). Induction effects of torus knots and unknots. JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL, 50(36), 1-12 [10.1088/1751-8121/aa80b0].

Induction effects of torus knots and unknots

RICCA, RENZO
2017

Abstract

Geometric and topological aspects associated with induction effects of field lines in the shape of torus knots/unknots are examined and discussed in detail. Knots are assumed to lie on a mathematical torus of circular cross-section and are parametrized by standard equations. The induced field is computed by direct integration of the Biot-Savart law. Field line patterns of the induced field are obtained and several properties are examined for a large family of knots/unknots up to 51 crossings. The intensity of the induced field at the origin of the reference system (center of the torus) is found to depend linearly on the number of toroidal coils and reaches maximum values near the boundary of the mathematical torus. New analytical estimates and bounds on energy and helicity are established in terms of winding number and minimum crossing number. These results find useful applications in several contexts when the source field is either vorticity, electric current or magnetic field, from vortex dynamics to astrophysics and plasma physics, where highly braided magnetic fields and currents are present.
Articolo in rivista - Articolo scientifico
Biot-Savart law; electric currents; magnetic braids; topological fluid mechanics; torus knots; vortex filaments; winding number;
Biot–Savart law, torus knots, winding number, magnetic braids,topological fluid mechanics, vortex filaments, electric currents
English
2017
50
36
1
12
365501
open
Oberti, C., Ricca, R. (2017). Induction effects of torus knots and unknots. JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL, 50(36), 1-12 [10.1088/1751-8121/aa80b0].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/164855
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