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The Parker hypothesis (Parker (1972)) assumes that heating of coronal loops occurs due to reconnection, induced when photospheric motions braid field lines to the point of current sheet formation. In this contribution we address the question of how the nature of photospheric motions affects heating of braided coronal loops. We design a series of boundary drivers and quantify their properties in terms of complexity and helicity injection. We examine a series of long-duration full resistive MHD simulations in which a simulated coronal loop, consisting of initially uniform field lines, is subject to these photospheric flows. Braiding of the loop is continually driven until differences in behaviour induced by the drivers can be characterised. It is shown that heating is crucially dependent on the nature of the photospheric driver - coherent motions typically lead to fewer large energy release events, while more complex motions result in more frequent but less energetic heating events.

We investigate the evolution of field line helicity for magnetic fields that connect two boundaries without null points, with emphasis on localized finite-B magnetic reconnection. Total (relative) magnetic helicity is already recognized as an important topological constraint on magnetohydrodynamic processes. Field line helicity offers further advantages because it preserves all topological information and can distinguish between different magnetic fields with the same total helicity. Magnetic reconnection changes field connectivity and field line helicity reflects these changes; the goal of this paper is to characterize that evolution. We start by deriving the evolution equation for field line helicity and examining its terms, also obtaining a simplified form for cases where dynamics are localized within the domain. The main result, which we support using kinematic examples, is that during localized reconnection in a complex magnetic field, the evolution of field line helicity is dominated by a work-like term that is evaluated at the field line endpoints, namely the scalar product of the generalized field line velocity and the vector potential. Furthermore, the flux integral of this term over certain areas is very small compared to the integral of the unsigned quantity, which indicates that changes of field line helicity happen in a well-organized pairwise manner. It follows that reconnection is very efficient at redistributing helicity in complex magnetic fields despite having little effect on the total helicity., Comment: 11 pages, 8 figures. Accepted for publication in Physics of Plasmas

Parker has hypothesised that, in a perfectly ideal environment, complex photospheric motions acting on a continuous magnetic field will result in the formation of tangential discontinuities corresponding to singular currents. We review direct numerical simulations of the problem and find the evidence points to a tendency for thin but finite thickness current layers to form, with thickness exponentially decreasing in time. Given a finite resistivity these layers will eventually become important and cause the dynamical process of energy release. Accordingly, a body of work focusses on evolution under continual boundary driving. The coronal volume evolves into a highly dynamic but statistically steady state where quantities have a temporally and spatially intermittent nature and where the Poynting flux and dissipation are decoupled on short timescales. Although magnetic braiding is found to be a promising coronal heating mechanism much work remains to determine its true viability. Some suggestions for future study are offered., Comment: 11 figures, 23 pages. To be published in Philosophical Transactions A (2015)

We investigate the relaxation of braided magnetic loops in order to find out how the type of braiding via footpoint motions affects resultant heating of the loop. Two magnetic loops, braided in different ways, are used as initial conditions in resistive MHD simulations and their subsequent evolution is studied. The fields both undergo a resistive relaxation in which current sheets form and fragment and the system evolves towards a state of lower energy. In one case this relaxation is very efficient with current sheets filling the volume and homogeneous heating of the loop occurring. In the other case fewer current sheets develop, less magnetic energy is released in the process and a patchy heating of the loop results. The two cases, although very similar in their setup, can be distinguished by the mixing properties of the photospheric driver. The mixing can be measured by the topological entropy of the plasma flow, an observable quantity., Comment: 16 pages, 13 Figures. To be published in Astronomy and Astrophysics

We present a model for separator reconnection due to an isolated reconnection process. Separator reconnection is a process which occurs in the neighbourhood of a distinguished field line (the separator) connecting two null points of a magnetic field. It is, for example, important for the dynamics of magnetic flux at the dayside magnetopause and in the solar corona. We find that, above a certain threshold, such a reconnection process generates new separators which leads to a complex system of magnetic flux tubes connecting regions of previously separated flux. Our findings are consistent with the findings of large numbers of separators in numerical simulations. We discuss how to measure and interpret the reconnection rate in a configuration with multiple separators., Comment: 37 pages, 11 figures

The final state of turbulent magnetic relaxation in a reversed field pinch is well explained by Taylor's hypothesis. However, recent resistive-magnetohydrodynamic simulations of the relaxation of braided solar coronal loops have led to relaxed fields far from the Taylor state, despite the conservation of helicity. We point out the existence of an additional topological invariant in any flux tube with non-zero field: the topological degree of the field line mapping. We conjecture that this constrains the relaxation, explaining why only one of three example simulations reaches the Taylor state., Comment: 8 pages, 4 figures, accepted for publication in Physical Review Letters

Aims: Our aim is to investigate the resistive relaxation of a magnetic loop that contains braided magnetic flux but no net current or helicity. The loop is subject to line-tied boundary conditions. We investigate the dynamical processes that occur during this relaxation, in particular the magnetic reconnection that occurs, and discuss the nature of the final equilibrium. Methods: The three-dimensional evolution of a braided magnetic field is followed in a series of resistive MHD simulations. Results: It is found that, following an instability within the loop, a myriad of thin current layers forms, via a cascade-like process. This cascade becomes more developed and continues for a longer period of time for higher magnetic Reynolds number. During the cascade, magnetic flux is reconnected multiple times, with the level of this `multiple reconnection' positively correlated with the magnetic Reynolds number. Eventually the system evolves into a state with no more small-scale current layers. This final state is found to approximate a non-linear force-free field consisting of two flux tubes of oppositely-signed twist embedded in a uniform background field., Comment: To be published in A&A

The response of the solar coronal magnetic field to small-scale photospheric boundary motions including the possible formation of current sheets via the Parker scenario is one of open questions of solar physics. Here we address the problem via a numerical simulation. The three-dimensional evolution of a braided magnetic field which is initially close to a force-free state is followed using a resistive MHD code. A long-wavelength instability takes place and leads to the formation of two thin current layers. Magnetic reconnection occurs across the current sheets with three-dimensional features shown, including an elliptic magnetic field structure about the reconnection site, and results in an untwisting of the global field structure., Comment: 18 pages, 9 figures

The squashing factor Q (Titov et al. 2002), a property of the magnetic field line mapping, has been suggested as an indicator for the formation of current sheets, and subsequently magnetic reconnection, in astrophysical plasmas. Here we test this hypothesis for a particular class of braided magnetic fields which serve as a model for solar coronal loops. We explore the relationship between Quasi-Separatrix Layers (QSLs), that is, layer-like structures with high Q value, electric currents and integrated parallel currents, the latter being a quantity closely related to the reconnection rate. It is found that as the degree of braiding of the magnetic field is increased the maximum values of Q increase exponentially. At the same time, the distribution of Q becomes increasingly filamentary, with the width of the high-Q layers exponentially decreasing. This is accompanied by an increase in the number of layers so that as the field is increasingly braided the volume becomes occupied by a myriad of thin QSLs. QSLs are not found to be good predictors of current features in this class of braided fields. Indeed, despite the presence of multiple QSLs, the current associated with the field remains smooth and large-scale under ideal relaxation; the field dynamically adjusts to a smooth equilibrium. Regions of high Q are found to be better related to regions of high integrated parallel current than to actual current sheets., Comment: 17 pages, 8 figures

Force-free magnetic fields are important in many astrophysical settings. Determining the properties of such force-free fields -- especially smoothness and stability properties -- is crucial to understanding many key phenomena in astrophysical plasmas, for example energy release processes that heat the plasma and lead to dynamic or explosive events. Here we report on a serious limitation on the computation of force-free fields that has the potential to invalidate the results produced by numerical force-free field solvers even for cases in which they appear to converge (at fixed grid resolution) to an equilibrium magnetic field. In the present work we discuss this problem within the context of a Lagrangian relaxation scheme that conserves magnetic flux and div(B) identically. Error estimates are introduced to assess the quality of the calculated equilibrium. We go on to present an algorithm, based on re-writing the curl operation via Stokes' theorem, for calculating the current which holds great promise for improving dramatically the accuracy of the Lagrangian relaxation procedure., Comment: Submitted to ApJ

In this series of papers we examine magnetic reconnection in a domain where the magnetic field does not vanish and the non-ideal region is localised in space. In a previous paper we presented a technique for obtaining analytical solutions to the stationary resistive MHD equations in such a situation and examined specific examples of non-ideal reconnective solutions. Here we further develop the model, noting that certain ideal solutions may be superimposed onto the fundamental non-ideal solutions and examining the effect of imposing various such flows. Significant implications are found for the evolution of magnetic flux in the reconnection process. It is shown that, in contrast to the two-dimensional case, in three-dimensions there is a very wide variety of physically different steady reconnection solutions., Comment: 20 pages, 8 figures

The braiding of the solar coronal magnetic field via photospheric motions - with subsequent relaxation and magnetic reconnection -- is one of the most widely debated ideas of solar physics. We readdress the theory in the light of developments in three-dimensional magnetic reconnection theory. It is known that the integrated parallel electric field along field lines is the key quantity determining the rate of reconnection, in contrast with the two-dimensional case where the electric field itself is the important quantity. We demonstrate that this difference becomes crucial for sufficiently complex magnetic field structures. A numerical method is used to relax a braided magnetic field to an ideal force-free equilibrium; that equilibrium is found to be smooth, with only large- scale current structures. However, the equilibrium is shown to have a highly filamentary integrated parallel current structure with extremely short length- scales. An analytical model is developed to show that, in a coronal situation, the length scales associated with the integrated parallel current structures will rapidly decrease with increasing complexity, or degree of braiding, of the magnetic field. Analysis shows the decrease in these length scales will, for any finite resistivity, eventually become inconsistent with the stability of a force- free field. Thus the inevitable consequence of the magnetic braiding process is shown to be a loss of equilibrium of the coronal field, probably via magnetic reconnection events., Comment: 19 pages with 7 figures