Magnetic reconnection and field line topology in magnetic ... - EPCC

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Magnetic reconnection and field line topology in magnetic ... - EPCC

Magnetic reconnection and field line topology in magneticrelaxationSimon Candelaresi, ∗† Anthony Yeates ‡June 22, 2012Background. The dynamics of magnetised plasmas arestrongly affected by the magnetic field’s topological structure,rather than its geometry alone. In Tokamak fusiondevices, the plasma is confined away from the walls of thedevice using helical magnetic fields—i.e., those where theproduct of the magnetic field B with the vector potentialA is significant. Chosen for stability reasons [?], helicalfields can be twisted, linked or knotted, and further analysishas proposed a lower limit for magnetic energy in thepresence of magnetic helicity [?]. Helical magnetic fieldscan also be observed on the Sun’s surface shortly beforecoronal mass ejections; indeed, the probability of an ejectionis shown to be higher if the field is helical [?].But total magnetic helicity does not uniquely determinethe topology of a magnetic field, and there have been recentindications that additional topological invariants mayplay a dynamical role [?, ?].Problem and Approach. We study the dynamical evolutionof particular magnetic field configurations whichare topologically equivalent to certain links and knots,with emphasis on non-helical configurations. We aim todetermine if there is any other relevant quantity that describesthe topology of the field and can give additionalconstraints on its evolution. As a candidate quantity weapply the topological flux function A [?], which is themagnetic vector potential integrated along magnetic fieldlines. For a field with B z > 0 we can define a field line∗ NORDITA, AlbaNova University Center, Roslagstullsbacken 23,SE-10691 Stockholm, Sweden† Department of Astronomy, Stockholm University, SE 10691 Stockholm,Sweden‡ Department of Mathematical Sciences, Durham University,Durham, DH1 3LE, United Kingdommapping F z (x, y) from the lower boundary (z = z 0 )tosome z. TherelevantvaluesforthetopologicalfluxfunctionA are those for which F 1 (x, y) =F 0 (x, y), i.e.thefixed points of the mapping.The evolution of the system is governed by the usualequations of resistive magnetohydrodynamics for compressibleisothermal media. We apply the thoroughlytested numerical PencilCode and enhance it with new parallelroutines for streamline tracing and fixed point finding.Initial Results As first milestone we report our successfullyworking tool for streamline tracing in parallel. Thistool has been incorporated in the PencilCode and can nowbe freely used and further developed by its users.As an initial test for the analysis routines, we investigatea braided magnetic field (Fig, 1), which is topologicallyequivalent to the trefoil knot: a helical field configuration.Most fixed points are found by our routine(Fig. 2), although at present tolerances some are missedin weak field regions. We have also computed A at thefixed points as a function of time in an initial test run, verifyingthat resistive diffusion alone leads to the expectedslow exponential decay in the A values.Outlook. Although the speed of the streamline tracingroutine is not a bottleneck in our simulations we wish todo further speedups to save more computation time. Ongoingwork is increasing the reliability of the fixed pointfinding algorithm, such that all fixed points are found ina reasonable computational time, no matter how weakthe field is. The work promises interesting results in thenear future when we use the code to study magnetic field1


configurations corresponding to links and knots with zeromagnetic helicity. This should lead to new conclusionsabout how A, rather then the magnetic helicity, influencesthe dynamics of the field.Acknowledgements. The work was carried out underthe HPC-EUROPA2 project (project number: 228398)with the support of the European Commission - CapacitiesArea - Research Infrastructures.Figure 1: Volume rendering of the initial magnetic fieldstrength of a braided field configuration.Figure 2: Color coding of the mapping F 1 (x, y). Thecolorsare green: F1 x >x,F y 1 >y, yellow: F 1 x >x,F y 1

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