Topology of magnetic field lines: Chaos and bifurcations emerging from two-action systems

Miyaguchi, Tomoshige; Hosoda, M.; Imagawa, Katsuyuki; Nakamura, Katsuhiro

doi:10.1103/physreve.83.016205

Cited by 3 publications

(2 citation statements)

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“…In communication technology, for instance, backscattering techniques have been applied to probe topological differences in physical systems [17,30] and knotted antennas have been proposed to inject helicity into electro-magnetohydrodynamical plasma [28]. Magnetic fields are also used to investigate electric dispersion in conducting tissues present in the brain, heart and muscles [29] and are employed in dynamical systems theory, to analyze quasi-periodic or chaotic orbits associated with complex networks of field lines [1,2,19].…”

Section: Introductionmentioning

confidence: 99%

Induction effects of torus knots and unknots

Oberti¹,

Ricca²

2017

J. Phys. A: Math. Theor.

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The induction effects due to a steady source field in the shape of a torus knot or unknot filament are analysed in detail. Similar studies for rectilinear, circular or helical geometries have been done in the past, but very little is known for more complex geometries and topologies. Torus knots provide a rare example of closed, space curves of non-trivial topology, that admit a mathematically simple description; for this reason they represent an interesting case study to consider. Moreover, since torus knots are also a good mathematical model for studying braided field line structures, the present work provides useful information for a wide range of possible applications, from physical sciences (solar physics and astrophysics, vortex dynamics, fusion physics) to technology (telecommunication, new materials design, data analysis). The work is organized in 4 chapters.In chapter 1 we present a comprehensive study of geometric and topological properties of torus knots and unknots. By using a standard parametrization, we demonstrate the existence, and determine the location, of inflection points for a given critical configuration, and prescribe the condition for removing the singularity associated with torsion at the inflection point. We show that, to first approximation, total length grows linearly with the number of coils, and it is proportional to the minimum crossing number of the knot type. By taking the winding number, given by the ratio between meridian and longitudinal wraps, as measure of topological complexity of the knot, we analyse its influence on several global quantities, such as total length, curvature, torsion and writhe.In chapter 2 we analyse the influence of the winding number and other geometric properties on induction, energy and helicity. This is done by assuming the physical filament of infinitesimally small cross-section and by using the Biot-Savart law adapted for the particular parametrization chosen. Field line patterns of the induced field are obtained for a large family of knots/unknots on several crosssectional planes. The intensity of the induced field is shown to depend linearly on the number of toroidal coils. We provide bounds on energy, and an estimate of helicity in terms of writhe.In chapter 3 we compare local and global induction contributions in relation to the winding number, by providing asymptotic expansions of the integrand function. We show that in general local leading order terms are not sufficient to provide accurate global information; nevertheless, for some values of the winding number local and global behaviours are found to be in good agreement.In chapter 4 we investigate the influence of the winding number on the binormal component of the self-induction a point asymptotically near to the source field. Since in the limit the Biot-Savart integral becomes singular, we apply the analytical prescription of Moore and Saffman (1972) to regularize it. While to leading order the self-induction is proportional to local curvature, we derive an integral formula for nex...

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Section: Introductionmentioning

confidence: 99%

Induction effects of torus knots and unknots

Oberti¹,

Ricca²

2017

J. Phys. A: Math. Theor.

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“…Several authors (Moffat, 1990, Zugasti, 1994, Morrison, 2000, Aguirre, et al, 2007, Miyaguchi, 2011 have discussed the non-integrability of the field line eqns. (1).…”

Section: Introductionmentioning

confidence: 99%

Chaotic magnetic fields: Particle motion and energization

Dasgupta¹,

Ram²,

Li³

et al. 2014

AIP Conference Proceedings

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Magnetic field line equations correspond to a Hamiltonian dynamical system, so the features of a Hamiltonian systems can easily be adopted for discussing some essential features of magnetic field lines. The integrability of the magnetic field line equations are discussed by various authors and it can be shown that these equations are, in general, not integrable. We demonstrate several examples of realistic chaotic magnetic fields, produced by asymmetric current configurations. Particular examples of chaotic force-free field and non force-free fields are shown. We have studied, for the first time, the motion of a charged particle in chaotic magnetic fields. It is found that the motion of a charged particle in a chaotic magnetic field is not necessarily chaotic. We also showed that charged particles moving in a time-dependent chaotic magnetic field are energized. Such energization processes could play a dominant role in particle energization in several astrophysical environments including solar corona, solar flares and cosmic ray propagation in space.

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