Magnetic surface topology in decaying plasma knots

Abstract

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...