G. Hornig

Evolution of magnetic flux in an isolated reconnection process

A realistic notion of magnetic reconnection is essential to understand the dynamics of magnetic fields in plasmas. Therefore a three-dimensional reconnection process is modeled in a region of nonvanishing magnetic field and is analyzed with respect to the way in which the connection of magnetic flux is changed. The process is localized in space in the sense that the diffusion region is limited to a region of finite radius in an otherwise ideal plasma. A kinematic, stationary model is presented, which allows for analytical solutions. Aside from the well-known flipping of magnetic flux in the reconnection process, the localization requires additional features which were not present in previous two- and 2.5-dimensional models. In particular, rotational plasma flows above and below the diffusion region are found, which substantially modify the process.

Kinematic reconnection at a magnetic null point: spine-aligned current

Magnetic reconnection at a three-dimensional null point is the natural extension of the familiar two-dimensional X-point reconnection. A model is set up here for reconnection at a spiral null point, by solving the kinematic, steady, resistive magnetohydrodynamic equations in its vicinity. A steady magnetic field is assumed, as well as the existence of a localised diffusion region surrounding the null point. Outside the diffusion region the plasma and magnetic field move ideally. Particular attention is focussed on the way that the magnetic flux changes its connections as a result of the reconnection. The resultant plasma flows are found to be rotational in nature, as is the change in connections of the magnetic field lines.

Magnetic topology and the problem of its invariant definition

The evolution of an ideal plasma conserves magnetic lines of force and hence magnetic topology. However, magnetic topology, i.e. the structure and linkage of magnetic flux, is a property of the magnetic field alone. Therefore, the conservation of topology can also be a property of non‐ideal plasmas for which the plasma flow is not line conserving. A general definition of magnetic topology is given and it is shown that it yields a large set of non‐ideal topology‐conserving systems. In the application of the notion of magnetic topology to real plasmas problems arise concerning the stability of topology. Instability may inhibit one from defining the topology of a given real, i.e. not exactly prescribed, magnetic field configuration and makes it difficult to detect changes of magnetic topology, such as reconnection processes. This problem of structural instability of magnetic topology also appears in connection with changes of the frame of reference. A change of the frame of reference may lead to a transition...

Dynamics of braided coronal loops - I. Onset of magnetic reconnection

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.

Magnetic Braiding and Parallel Electric Fields

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 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 toward an ideal force-free equilibrium; the field is found to remain smooth throughout the relaxation, with only large-scale current structures. However, a highly filamentary integrated parallel current structure with extremely short length-scales is found in the field, with the associated gradients intensifying during the relaxation process. 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 the coronal field. Thus the inevitable consequence of the magnetic braiding process is a loss of equilibrium of the magnetic field, probably via magnetic reconnection events.

Topological constraints on magnetic relaxation.

The final state of turbulent magnetic relaxation in a reversed field pinch is well explained by Taylors 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.

MAGNETIC BRAIDING AND QUASI-SEPARATRIX LAYERS

The squashing factor Q, 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.

A Time Delay Model for Solar and Stellar Dynamos

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 α-effect and whether it is spatially segregated from the location of the Ω-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 α-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 transport-dominated 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).

Heating of braided coronal loops

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.

The magnetic structure of B???0-reconnection

Magnetic reconnection can be interpreted as a process in which the electromagnetic field is frozen into a four-velocity field in Minkowski space. For reconnection to occur the four-velocity field has to be a special type of stagnation flow. Prescribing this type of flow in a finite spatial domain allows the modelling of localized reconnection events and the investigation of examples of reconnection in regions without magnetic nulls. In the present contribution, we start with a simple twisted magnetic flux tube. Reconnection occuring along a part of the axis of the tube results in a structure of the magnetic field which is a superposition of a two-dimensional X-type magnetic field well-known from stationary 2D reconnection models, and a component resulting from the magnetic field parallel to the axis. For localized reconnection, the latter component of the magnetic field evolves in a non-trivial way. This evolution is important for the spatial variation of the parallel electric field integrated along the magnetic field lines. The integrated electric field gives an upper limit for the energy to which particles can be accelerated in a reconnection event and its distribution shows to be localized in very thin structures.