**Magnetic** **reconnection** **and** **field** **l****in**e **topology** **in** **magnetic**relaxationSimon C**and**elaresi, ∗† 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 conf**in**ed away from the walls of thedevice us**in**g helical **magnetic** **field**s—i.e., those where theproduct of the **magnetic** **field** B with the vector potentialA is significant. Chosen for stability reasons [?], helical**field**s can be twisted, l**in**ked or knotted, **and** further analysishas proposed a lower limit for **magnetic** energy **in** thepresence of **magnetic** helicity [?]. Helical **magnetic** **field**scan also be observed on the Sun’s surface shortly beforecoronal mass ejections; **in**deed, the probability of an ejectionis shown to be higher if the **field** is helical [?].But total **magnetic** helicity does not uniquely determ**in**ethe **topology** of a **magnetic** **field**, **and** there have been recent**in**dications that additional topological **in**variants mayplay a dynamical role [?, ?].Problem **and** Approach. We study the dynamical evolutionof particular **magnetic** **field** configurations whichare topologically equivalent to certa**in** l**in**ks **and** knots,with emphasis on non-helical configurations. We aim todeterm**in**e if there is any other relevant quantity that describesthe **topology** of the **field** **and** can give additionalconstra**in**ts on its evolution. As a c**and**idate quantity weapply the topological flux function A [?], which is the**magnetic** vector potential **in**tegrated along **magnetic** **field****l****in**es. For a **field** with B z > 0 we can def**in**e a **field** **l****in**e∗ 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 K**in**gdommapp**in**g 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 po**in**ts of the mapp**in**g.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 parallelrout**in**es for stream**l****in**e trac**in**g **and** fixed po**in**t f**in**d**in**g.Initial Results As first milestone we report our successfullywork**in**g tool for stream**l****in**e trac**in**g **in** parallel. Thistool has been **in**corporated **in** the PencilCode **and** can nowbe freely used **and** further developed by its users.As an **in**itial test for the analysis rout**in**es, we **in**vestigatea braided **magnetic** **field** (Fig, 1), which is topologicallyequivalent to the trefoil knot: a helical **field** configuration.Most fixed po**in**ts are found by our rout**in**e(Fig. 2), although at present tolerances some are missed**in** weak **field** regions. We have also computed A at thefixed po**in**ts as a function of time **in** an **in**itial test run, verify**in**gthat resistive diffusion alone leads to the expectedslow exponential decay **in** the A values.Outlook. Although the speed of the stream**l****in**e trac**in**grout**in**e is not a bottleneck **in** our simulations we wish todo further speedups to save more computation time. Ongo**in**gwork is **in**creas**in**g the reliability of the fixed po**in**tf**in**d**in**g algorithm, such that all fixed po**in**ts are found **in**a reasonable computational time, no matter how weakthe **field** is. The work promises **in**terest**in**g results **in** thenear future when we use the code to study **magnetic** **field**1

configurations correspond**in**g to l**in**ks **and** knots with zero**magnetic** helicity. This should lead to new conclusionsabout how A, rather then the **magnetic** helicity, **in**fluencesthe 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 render**in**g of the **in**itial **magnetic** **field**strength of a braided **field** configuration.Figure 2: Color cod**in**g of the mapp**in**g F 1 (x, y). Thecolorsare green: F1 x >x,F y 1 >y, yellow: F 1 x >x,F y 1