Physical role of topological constraints in localized magnetic relaxation

Abstract

Predicting the final state of turbulent plasma relaxation is an important challenge, both in astrophysical plasmas such as the Sun's corona and in controlled thermonuclear fusion. Recent numerical simulations of plasma relaxation with braided magnetic fields identified the possibility of a novel constraint, arising from the topological degree of the magnetic field-line mapping. This constraint implies that the final relaxed state is drastically different for an initial configuration with topological degree 1 (which allows a Taylor relaxation) and one with degree 2 (which does not reach a Taylor state). Here, we test this transition in numerical resistivemagnetohydrodynamic simulations, by embedding a braided magnetic field in a linear force-free background. Varying the background force-free field parameter generates a sequence of initial conditions with a transition between topological degree 1 and 2. For degree 1, the relaxation produces a single twisted flux tube, whereas for degree 2 we obtain two flux tubes. For predicting the exact point of transition, it is not the topological degree of the whole domain that is relevant, but only that of the turbulent region.

Twisted versus braided magnetic flux ropes in coronal geometry

“…In (b) and (c) there are several observable regions of a particular sign of L f , in particular there are two large regions with a dominant sign L f , one positive one negative, though neither are homogeneous. Whilst it is not the clear splitting into two tubes found in Wilmot-Smith et al (2011), Yeates et al (2015, it is indicative that the tube has not relaxed to a linear force-free state. Unlike the twisted field it is not clear what the effect of reconnection with the background field has had on the distribution.…”

“…For a linear force-free field j · B = αB · B and L f is just the linear force-free parameter α, which would be constant throughout the domain. This quantity was calculated for relaxing braided and twisted cylindrical fields in Wilmot-Smith et al (2011), Yeates et al (2015. These authors found that a twisted field relaxed to a force-free state with a single sign of α within the tube, although the surrounding background field prevented it from reaching the spatially constant α predicted by the Taylor relaxation hypothesis.…”

“…Of most relevance to this study are a series of numerical experiments which compare the effect of pre-existing braided structures in a cylindrical domain on the field's eventual evolution, by comparison to twisted cylindrical fields (e.g. Yeates et al 2010Yeates et al , 2015Wilmot-Smith et al 2011;Wyper & Pontin 2014). One notable finding was that the braided fields typically released more energy than their twisted counterparts (for the same given starting energy).…”

“…The simulations presented here use a uniform resistivity of 5 × 10 −4 , a value found from experience to be just above the numerical resistivity at this resolution. This system and parameter set is the same as that used in Yeates et al (2015). The initial velocity is assumed to vanish everywhere so that any motion will result from the field attempting to relax to equilibrium.…”

The emergence of braided magnetic fields

“…A second approach to increasing the complexity of the field is to define fields which partly overlap, creating a composite field with more complex internal topology. This is used in what follows to develop a version of the braided field used in Yeates et al (2010), Wilmot-Smith et al (2011), Yeates et al (2015). The technical details of the process by which this field is interpolated onto a Cartesian grid are discussed in Prior and Yeates (2016a).…”