Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. 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. V C 2015 AIP Publishing LLC.

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.

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.

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 a nonzero 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.

Magnetohydrodynamic dynamos operating in stellar interiors produce the diverse range of magnetic activity observed in solar-like stars. Sophisticated dynamo models including realistic physics of convection zone flows and flux tube dynamics have been built for the Sun, for which appropriate observations exist to constrain such models. Nevertheless, significant differences exist in the physics that the models invoke, the most important being the nature and location of the dynamo-effect and whether it is spatially segregated from the location of the-effect. Spatial segregation of these source layers necessitates a physical mechanism for communication between them, involving unavoidable time delays. We construct a physically motivated reduced dynamo model in which, through the use of time delays, we mimic the generation of field components in spatially segregated layers and the communication between them. The model can be adapted to examine the underlying structures of more complicated and spatially extended numerical dynamo models with diverse-effect mechanisms. A variety of dynamic behaviors arise as a direct consequence of the introduction of time delays in the system. Various parameter regimes give rise to periodic and aperiodic oscillations. Amplitude modulation leads to episodes of reduced activity, such as that observed during the Maunder minima, the length and duration of which depend on the dynamo number. Regular activity is more easily excited in the flux transportYdominated regime (when the time delay is smaller than the dissipative timescale), whereas irregular activity characterizes solutions in the diffusion-dominated regime (when the time delay is larger than the dissipative timescale).

Aims. 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. Methods. 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. Results. 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.

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.Here we give a method for calculating the squashing factor in the braided fields E n . For this we must obtain the mapping of field lines from the lower boundary of the domain to the upper boundary. We make use of a property of the fields E n , specifically that E n is a concatenation of n times the 'basic' field E.

Aims. 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. Methods. Two magnetic loops, braided in different ways, are used as initial conditions in resistive MHD simulations and their subsequent evolution is studied. Results. 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.