Constructing a class of topological solitons in magnetohydrodynamics

Thompson, Amy; Swearngin, Joe; Wickes, Alexander; Bouwmeester, Dirk

doi:10.1103/physreve.89.043104

Cited by 13 publications

(28 citation statements)

References 34 publications

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“…An ideal MHD soliton, as defined in [13,24], is a static configuration of magnetic field B, fluid velocity u, and pressure p that satisfies the ideal, incompressible MHD equations. The fluid field and pressure that solve this can be inferred from the momentum equation, which can be written as:…”

Section: Plasma Torus Knotsmentioning

confidence: 99%

“…, is well suited to generate new solutions to Maxwells equations with torus knotted field lines [22]. The class of torus knot solitons, of which the Kamchatnov-Hopf soliton is an element, are the solutions to the ideal MHD equations where the velocity and fluid field are identical to the magnetic field of these EM solutions [24].…”

Section: Plasma Torus Knotsmentioning

confidence: 99%

“…Another way of obtaining such solutions, for massless fields of various spins, is to use twistor theory [23]. The magnetic fields of the t=0 solutions in [22], have been used to construct novel plasma torus knots [24], solutions to the ideal incompressible MHD equations.…”

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confidence: 99%

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Magnetic surface topology in decaying plasma knots

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Torus-knot solitons have recently been formulated as solutions to the ideal incompressible magnetohydrodynamics (MHD) equations. We investigate numerically how these fields evolve in resistive, compressible, and viscous MHD. We find that certain decaying plasma torus knots exhibit magnetic surfaces that are topologically distinct from a torus. The evolution is predominantly determined by a persistent zero line in the field present when the poloidal winding number ¹ n 1 p . Dependence on the toroidal winding number n t is less pronounced as the zero line induced is contractible and disappears. The persistent zero line intersects the new magnetic surfaces such that, through the Hopf-Poincaré index theorem, the sum of zeroes on the new surfaces equals their (in general non-zero) Euler characteristic. Furthermore we observe the formation of magnetic islands between the surfaces. These novel persistent magnetic structures are of interest for plasma confinement, soliton dynamics and the study of dynamical systems in general.It is remarkable how abstract topological concepts are directly relevant to many branches of science. A prime example is the Hopf map [1], a non-trivial topological structure that has found applications in liquid crystals [2], molecular biology [3], superconductors [4], superfluids [5], Bose-Einstein condensates [6,7], ferromagnets [8], optics [9][10][11], and plasma physics [12,13]. This article deals with topological aspects of novel persistent plasma configurations that emerge from decaying plasma torus knots.Due to the generally high electrical conductivity of plasma described by magnetohydrodynamics (MHD), large electrical currents can flow and plasmas are heavily influenced by the resulting magnetic forces. The zerodivergence magnetic fields can lead to closed magnetic field lines, field lines that ergodically fill a magnetic surface, and field lines that chaotically fill a region of space. In ideal (zero-resistance) MHD the magnetic flux through a perfect conducting fluid element cannot change, leading to frozen in magnetic fields in the plasma [14]. This implies that in ideal MHD magnetic topology and magnetic helicity is conserved [15][16][17].In 1982 Kamchatnov described an intrinsically stable plasma configuration [13] with a magnetic topology based on fibers of the Hopf map [18]. This type of MHD equilibrium, where the fluid velocity is parallel to the field and equal to the local Alfvén speed, was shown by Chandrasekhar to be stable [19], even in specific cases in the presence of dissipative forces [20]. Quasi stable self-organizing magnetic fields with similar magnetic topology to Kamchatnov's field (but different flow) have recently been demonstrated to occur in full-MHD simulations [12]. Here the final configuration is not a Taylor state, which is consistent with recent findings in [21].Recently the class of topologically non-trivial solutions to Maxwell's equations has been extended by including torus knotted fields [22]. Another way of obtaining such solutions, for massles...

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Section: Plasma Torus Knotsmentioning

confidence: 99%

Section: Plasma Torus Knotsmentioning

confidence: 99%

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Magnetic surface topology in decaying plasma knots

et al. 2017

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“…That duality, termed "electromagnetic democracy" [10], has been central in the work of knotted field configurations [5,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Related field configurations have also appeared in plasma physics [27][28][29][30], optics [31][32][33][34][35], classical field theory [36], quantum physics [37,38], various states of matter [39][40][41][42][43] and twistors [44,45].…”

Section: Introductionmentioning

confidence: 99%

Spin-Orbital Momentum Decomposition and Helicity Exchange in a Set of Non-Null Knotted Electromagnetic Fields

Arrayás

Trueba

2018

Symmetry

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Abstract:We calculate analytically the spin-orbital decomposition of the angular momentum using completely nonparaxial fields that have a certain degree of linkage of electric and magnetic lines. The split of the angular momentum into spin-orbital components is worked out for non-null knotted electromagnetic fields. The relation between magnetic and electric helicities and spin-orbital decomposition of the angular momentum is considered. We demonstrate that even if the total angular momentum and the values of the spin and orbital momentum are the same, the behavior of the local angular momentum density is rather different. By taking cases with constant and non-constant electric and magnetic helicities, we show that the total angular momentum density presents different characteristics during time evolution.

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“…Since our solutions are based on the Bateman construction, it follows from a recent result by Goulart [29] that they are also exact solutions of nonlinear electrodynamics. We expect that the implementation of knot theory to generate topologically nontrivial structures in electromagnetism introduced here will lead to generalizations in linearized gravity [12,30], Bose-Einstein condensate configurations [31], and plasma configurations [32,33].…”

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confidence: 99%

Knotted optical vortices in exact solutions to Maxwell's equations

et al. 2017

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We construct a family of exact solutions to Maxwell's equations in which the points of zero intensity form knotted lines topologically equivalent to a given but arbitrary algebraic link. These lines of zero intensity, more commonly referred to as optical vortices, and their topology are preserved as time evolves and the fields have finite energy. To derive explicit expressions for these new electromagnetic fields that satisfy the nullness property, we make use of the Bateman variables for the Hopf field as well as complex polynomials in two variables whose zero sets give rise to algebraic links. The class of algebraic links includes not only all torus knots and links thereof, but also more intricate cable knots. While the unknot has been considered before, the solutions presented here show that more general knotted structures can also arise as optical vortices in exact solutions to Maxwell's equations.

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Constructing a class of topological solitons in magnetohydrodynamics

Cited by 13 publications

References 34 publications

Magnetic surface topology in decaying plasma knots

Magnetic surface topology in decaying plasma knots

Spin-Orbital Momentum Decomposition and Helicity Exchange in a Set of Non-Null Knotted Electromagnetic Fields

Knotted optical vortices in exact solutions to Maxwell's equations

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