Current build-up as a result of the kink instability in a loop

The kink instability may be responsible for compact loop flares since the instability is triggered once the twist in a coronal loop exceeds a critical value. During the non-linear evolution of the instability a large current builds up, reconnection can occur and the magnetic energy released due to reconnection may explain the rapid heating of the flare. However, there has been some debate over the nature of the current concentration and, in particular, whether the current saturates or whether it is a current sheet, and what influences these possible states. In this paper we consider two similar equilibria having a twist function which rises to a peak and then falls off. One is steeper than the other allowing us to investigate whether the steepness of the peak has any effect on the nature of the current. For each profile, we run the code on five different grid resolutions and see how the maximum of the current scales with grid resolution. We also look for behavior in the x-component of the velocity which might be similar to the step-function behavior associated with singularities in the linear kink instability. For both profiles we find that the current scales almost linearly with resolution and that v _x drops steeply at the position of the current concentration. This suggests that, for these particular profiles, there are indications of current sheet formation and that the steepness in the peak of the twist does not affect the nature of the current.     

The Flare-Energy Distributions Generated by Kink-Unstable Ensembles of Zero-Net-Current Coronal Loops

It has been proposed that the million-degree temperature of the corona is due to the combined effect of barely detectable energy releases, called nanoflares, that occur throughout the solar atmosphere. Unfortunately, the nanoflare density and brightness implied by this hypothesis means that conclusive verification is beyond present observational abilities. Nevertheless, we investigate the plausibility of the nanoflare hypothesis by constructing a magnetohydrodynamic (MHD) model that can derive the energy of a nanoflare from the nature of an ideal kink instability. The set of energy-releasing instabilities is captured by an instability threshold for linear kink modes. Each point on the threshold is associated with a unique energy release; thus we can predict a distribution of nanoflare energies. When the linear instability threshold is crossed, the instability enters a nonlinear phase as it is driven by current sheet reconnection. As the ensuing flare erupts and declines, the field transitions to a lower energy state, which is modelled by relaxation theory; i.e., helicity is conserved and the ratio of current to field becomes invariant within the loop. We apply the model so that all the loops within an ensemble achieve instability followed by energy-releasing relaxation. The result is a nanoflare energy distribution. Furthermore, we produce different distributions by varying the loop aspect ratio, the nature of the path to instability taken by each loop and also the level of radial expansion that may accompany loop relaxation. The heating rate obtained is just sufficient for coronal heating. In addition, we also show that kink instability cannot be associated with a critical magnetic twist value for every point along the instability threshold.     

Kink unstable coronal loops: current sheets,current saturation and magnetic reconnection

The kink instability in a coronal loop is a possible explanation of a compact loop flare as it may cause a current sheet to form allowing reconnection to take place and release the free magnetic energy stored in the loop. However, current sheets do not form in all cases. Ali and Sneyd (2001) investigated three different classes of equilibrium (determined by the form of the twist) using a magneto-frictional code. They searched for the equilibria to which the loop might evolve once it had become unstable to the kink instability. They found indications of current-sheet formation for only one class of equilibrium studied. However, as they pointed out, since their code searched for equilibria they were unable to say for certain that the loop would evolve in this way. In this paper we have considered the same three classes of equilibria but have used a code which follows the non-linear 3D MHD (magnetohydrodynamic) evolution of the loop. We have investigated whether or not there are indications of current-sheet formation. In the cases where there is evidence of this we have found that reconnection does occur and releases sufficient magnetic energy to explain a compact loop flare.     

Effect of photospheric twisting motions on a confined toroidal loop

In this paper we investigate the effect of twisting motions at the photosphere on a curved loop confined by overlying field lines. We find that the twisting motions form a twisted loop along which a current builds up. However, the toroidal curvature of the loop appears to have a stabilising effect as there is no sign of the kink instability for a twist of 3.4  whereas in a straight cylinder for the same twist profile the critical twist is approximately 2.5 . When we include resistivity in the simulation there are indications that reconnection occurs and releases a substantial proportion of the free magnetic energy.     

Reconnection of a Kinking Flux Rope Triggering the Ejection of a Microwave and Hard X-Ray Source II. Numerical Modeling

Numerical simulations of the helical (m=1) kink instability of an arched, line-tied flux rope demonstrate that the helical deformation enforces reconnection between the legs of the rope if modes with two helical turns are dominant as a result of high initial twist in the range Φ≳6π. Such a reconnection is complex, involving also the ambient field. In addition to breaking up the original rope, it can form a new, low-lying, less twisted flux rope. The new flux rope is pushed downward by the reconnection outflow, which typically forces it to break as well by reconnecting with the ambient field. The top part of the original rope, largely rooted in the sources of the ambient flux after the break-up, can fully erupt or be halted at low heights, producing a “failed eruption.” The helical current sheet associated with the instability is squeezed between the approaching legs, temporarily forming a double current sheet. The leg – leg reconnection proceeds at a high rate, producing sufficiently strong electric fields that it would be able to accelerate particles. It may also form plasmoids, or plasmoid-like structures, which trap energetic particles and propagate out of the reconnection region up to the top of the erupting flux rope along the helical current sheet. The kinking of a highly twisted flux rope involving leg – leg reconnection can explain key features of an eruptive but partially occulted solar flare on 18 April 2001, which ejected a relatively compact hard X-ray and microwave source and was associated with a fast coronal mass ejection.     

The evolution of line-tied coronal arcades including a converging footpoint motion

It has been demonstrated in the past that single, two-dimensional coronal arcades are very unlikely driven unstable by a simple shear of the photospheric footpoints of the magnetic field lines. By means of two-dimensional, time-dependent MHD simulations, we present evidence that a resistive instability can result if in addition to the footpoint shear a slow motion of the footpoints towards the photospheric neutral line is included. Unlike the model recently proposed by van Ballegooijen and Martens (1989), the photospheric footpoint velocity in our model is nonsingular and the shear dominates everywhere. Starting from a planar potential field geometry for the arcade, we find that after some time a current sheet is formed which is unstable with respect to the tearing instability. The time of its onset scales with the logarithm of the magnetic diffusivity assumed in our calculation. In its nonlinear phase, a quasi-stationary situation arises in the vicinity of the x-line with an almost constant reconnection rate. The height of the x-line above the photosphere and the distance of the separatrix footpoints remain almost constant in this phase, while the helical flux tube, formed above the neutral line, continuously grows in size.     

IDEAL CURRENT LAYERS AND KINK INSTABILITY IN LINE-TIED CORONAL LOOPS

The development of the kink instability in line-tied coronal loops is studied using a cylindrical MHD code. When the twist of magnetic field lines of the initial configuration exceeds a critical value, an ideal kink mode develops and drives this unstable equilibrium towards a secondary bifurcated equilibrium containing an electric current concentration. Contrary to a periodic untied configuration where a current sheet is ideally generated, the current layer is non-singular with a non-zero thickness and a finite amplitude. This current concentration extends along all the loop length and takes the form of an helical ribbon of intense current. The numerical results give an algebraic linear-like scaling of the characteristics of the current layer (amplitude and thickness) as a function of the aspect ratio of the loop. An interpretation in terms of axial field-line bending of the three-dimensional kinked equilibrium is proposed.     

The study of magnetic reconnection in solar spicules

This work is devoted to study the magnetic reconnection instability under solar spicule conditions. Numerical study of the resistive tearing instability in a current sheet is presented by considering the magnetohydrodynamic (MHD) framework. To investigate the effect of this instability in a stratified atmosphere of solar spicules, we solve linear and non-ideal MHD equations in the x?z plane. In the linear analysis it is assumed that resistivity is only important within the current sheet, and the exponential growth of energies takes place faster as plasma resistivity increases. We are interested to see the occurrence of magnetic reconnection during the lifetime of a typical solar spicule.     

Numerical simulations of driven MHD waves in coronal loops

The solar corona, modeled by a low-, resistive plasma slab, sustains MHD wave propagations due to footpoint motions in the photosphere. Simple test cases are undertaken to verify the code. Uniform, smooth and steep density, magnetic profile and driver are considered. The numerical simulations presented here focus on the evolution and properties of the Alfvén, fast and slow waves in coronal loops. The plasma responds to the footpoint motion by kink or sausage waves depending on the amount of shear in the magnetic field. The larger twist in the magnetic field of the loop introduces more fast-wave trapping and destroys initially developed sausage-like wave modes. The transition from sausage to kink waves does not depend much on the steep or smooth profile. The slow waves develop more complex fine structures, thus accounting for several local extrema in the perturbed velocity profiles in the loop. Appearance of the remnants of the ideal singularities characteristic of ideal plasma is the prominent feature of this study. The Alfvén wave which produces remnants of the ideal x ^–1 singularity, reminiscent of Alfvén resonance at the loop edges, becomes less pronounced for larger twist. Larger shear in the magnetic field makes the development of pseudo-singularity less prominent in case of a steep profile than that in case of a smooth profile. The twist also causes heating at the edges, associated with the resonance and the phase mixing of the Alfvén and slow waves, to slowly shift to layers inside the slab corresponding to peaks in the magnetic field strength. In addition, increasing the twist leads to a higher heating rate of the loop. Remnants of the ideal log ¦x¦ singularity are observed for fast waves for larger twist. For slow waves they are absent when the plasma experiences large twist in a short time. The steep profiles do not favour the creation of pseudo-singularities as easily as in the smooth case.     

KINK MODES AND CURRENT SHEETS IN CORONAL LOOPS

We present simulations of the non-linear evolution of the m=1 kink mode in line-tied coronal loops. We focus on the structure of the current concentrations which develop as a consequence of the instability in two different types of magnetic field configuration, one containing a net axial current and the other with a vanishing total axial current. In the first case, current sheets develop one third of the way from footpoint to loop apex (where the non-linear kink mode folds on itself) within the body of the current channel, while in the second case the current sheet develops at the loop apex at the interface between the current containing channel and the outer axial potential field. In both cases line-tying, while playing a stabilizing role in the linear theory, acts as a destabilizing agent for the non-linear resistive evolution. The unwrapping of magnetic field lines in the vanishing axial current model appears to be consistent with the geometry of compact recurrent loop flares.     

Magnetic reconnection model for X-ray flare loop interaction

The subtle interactions between the magnetohydrodynamics (MHD) and transverse plasmons are investigated. It is shown that there is a resistive instability by the plasmon's soliton in a current sheet, which eventually turns into an eruptive instability at the magnetic field reconnection. In the case of ion-acoustic turbulence, the high temperature current sheet model must adopt the aromalous conductivity instead of the Coulomb conductivity. The numerical results are consistent with the observations obtained by Hanaoka (1994). Thus the flare caused by X-ray loop coalescence can be basically interpreted by this model of magnetic field reconnection driven by ponderomotive force.     

On Formation of the Sigmoidal Structure in Solar Active Region NOAA 8100

Using soft X-ray images taken by the Soft X-ray Telescope on board Yohkoh, line-of-sight magnetograms taken by SOHO/MDI and vector magnetograms taken at Beijing Astronomical Observatory, we have studied the formation of the sigmoidal structure in active region NOAA 8100 on 3–4 November 1997. The sigmoidal structure appeared after the occurrences of a series of flares accompanied by new magnetic flux emergence. This implies that reconnection may play a role in formation of this sigmoid structure. We calculated the self-helicity (twist) and mutual helicity of the active region before and after the formation of the sigmoidal structure and found that the mutual helicity decreased. The twist of the sigmoidal structure was higher than the twist of the emerging magnetic flux and exceeded the critical twist for kink in stability. This result suggests that the reconnection increased the twist of magnetic flux tubes by converting mutual helicity to self-helicity, supporting the previous studies by Berger (1998, 1999).     

Kink instability of solar coronal loops as the cause of solar flares

Solar coronal loops are observed to be remarkably stable structures. A magnetohydrodynamic stability analysis of a model loop by the energy method suggests that the main reason for stability is the fact that the ends of the loop are anchored in the dense photosphere. In addition to such line-tying, the effect of a radial pressure gradient is incorporated in the analysis.Two-ribbon flares follow the eruption of an active region filament, which may lie along a magnetic flux tube. It is suggested that the eruption is caused by the kink instability, which sets in when the amount of magnetic twist in the flux tube exceeds a critical value. This value depends on the aspect ratio of the loop, the ratio of the plasma to magnetic pressure and the detailed transverse magnetic structure. For a force-free field of uniform twist the critical twist is 3.3, and for other fields it is typically between 2 and 6. Occasionally active region loops may become unstable and give rise to small loop flares, which may also be a result of the kink instability.     

An unstable arch model of a solar flare

The theoretical consequences of assuming that a current flows along flaring arches consistent with a twist in the field lines of these arches are examined. It is found that a sequence of magneto-hydrodynamic (MHD) and resistive MHD instabilities driven by the assumed current (which we refer to as the toroidal current) can naturally explain most manifestations of a solar flare.The principal flare instability in the proposed model is the resistive kink (or tearing mode in arch geometry) which plays the role of thermalizing some of the field energy in the arch and generating X-configured neutral points needed for particle acceleration. The difference between thermal and nonthermal flares is elucidated and explained, in part, by amplitude-dependent instabilities, generally referred to as overlapping resonances. We show that the criteria for the generation of flare shocks strongly depend on the magnitude and gradient steepness of the toroidal current, which also are found to determine the volume and rate of energy release. The resulting model is in excellent agreement with present observations and has successfully predicted several flare phenomena.     

The Magnetic Structure of a Coronal X-Ray Bright Point

X-ray bright points are small dynamic loop structures that are observed all over the solar corona. The high spatial and temporal resolution of the TRACE instrument allows bright points to be studied in much greater detail than previously possible. This paper focuses on a specific bright point which occurred for about 20 hours on 13–14 June 1998 and examines its dynamic structure in detail. This example suggests that the mechanisms that cause bright points to form and evolve are more complex than previously thought. In this case, reconnection probably plays a major part during the formation and brightening of the loop structure. However, later on the foot points rotate injecting twist into the bright point which may cause an instability to occur with dynamic results. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1017907406350     

Breakout and Tether-Cutting Eruption Models Are Both Catastrophic (Sometimes)

We present a simplified analytic model of a quadrupolar magnetic field and flux rope to model coronal mass ejections. The model magnetic field is two-dimensional, force-free and has current only on the axis of the flux rope and within two current sheets. It is a generalization of previous models containing a single current sheet anchored to a bipolar flux distribution. Our new model can undergo quasi-static evolution either due to changes at the boundary or due to magnetic reconnection at either current sheet. We find that all three kinds of evolution can lead to a catastrophe, known as loss of equilibrium. Some equilibria can be driven to catastrophic instability either through reconnection at the lower current sheet, known as tether cutting, or through reconnection at the upper current sheet, known as breakout. Other equilibria can be destabilized through only one and not the other. Still others undergo no instability, but they evolve increasingly rapidly in response to slow steady driving (ideal or reconnective). One key feature of every case is a response to reconnection different from that found in simpler systems. In our two-current-sheet model a reconnection electric field in one current sheet causes the current in that sheet to increase rather than decrease. This suggests the possibility for the microscopic reconnection mechanism to run away.     

Flare-like energy release by flux pile-up reconnection

Flux pile-up magnetic merging solutions are discussed using the simple robust arguments of traditional steady-state reconnection theory. These arguments determine a unique scaling for the field strength and thickness of the current layer, namely B _s^–1/3, l^2/3, which are consistent with a variety of plasma inflow conditions. Next we demonstrate that flux pile-up merging can also be understood in terms of exact magnetic annihilation solutions. Although simple annihilation models cannot provide unique reconnection scalings, we show that the previous current sheet scalings derive from an optimized solution in which the peak dynamic and magnetic pressures balance in the reconnection region. The build-up of magnetic field in the current sheet implicit in flux pile-up solutions naturally leads to the idea of saturation. Hydromagnetic pressure effects limit the magnetic field in the sheet, yielding an upper limit on the reconnection rate for such solutions. This rate is still far superior to the Sweet–Parker merging rate, which can be derived by seeking solutions that avoid all forms of saturation. Finally we compare time dependent numerical simulations of the coalescence instability with the optimized flux pile-up models. This comparison suggests that merging driven by the relatively slow approach of large flux systems may be favored in practice.     

Magnetohydrodynamic study of shock waves in the current sheet of a solar coronal magnetic loop

A Lagrangian Remap (LareXd) Code is employed to investigate the shock wave formation in the current sheet of a solar coronal magnetic loop and its effect on the magnetic reconnection. We constructed the slow shock structure in the presence of viscosity and heat conduction parallel and perpendicular to the magnetic field and pairs of slow shocks propagate away from the central current sheet, the so-called diffusion region. Significant jumps in plasma density, pressure, velocity and magnetic field occur across the main shock while the temperature appears in the foreshock. In the presence of dissipative effects, the distinct jumps disappear and the shock profiles show smooth transition between the downstream and the upstream regions while the plasma density and the pressure show very narrow and a sharp decrease with time. These results can be applied to the heating of the solar corona, the structure of the magnetic reconnection and the solar wind.     

Coronal Mass Ejection of 15 May 2001: II. Coupling of the Cme Acceleration and the Flare Energy Release

We analyze the relationship between the dynamics of the coronal mass ejection (CME) of 15 May 2001 and the energy release in the associated flare. The flare took place behind the east limb and was disclosed by a growing system of hot soft X-ray (SXR) loops that appeared from behind the limb around the onset of the rapid acceleration of the CME. The highly correlated behavior of the SXR light-curve derivative and the time profile of the CME acceleration reveals an intrinsic relationship between the CME dynamics and the flare energy release. Furthermore, we found that the CME acceleration peak occurs simultaneously with the fastest growth (100 km s^-1) of X-ray loops, indicating that the reconnection plays an essential role in the eruption. Inspecting the CME/flare morphology we recognized in the Yohkoh-SXT images an oval feature that formed within the rising structure at the onset of the rapid acceleration phase, simultaneously with the appearance of the X-ray loops. The eruptive prominence was imbedded within the lower half of the oval, suggestive of a flux-rope/prominence magnetic configuration. We interpret the observed morphological evolution in terms of a reconnection process in the current sheet that presumably formed below the erupting flux-rope at the onset of the CME acceleration. Measurements of the tip-height of the cusped X-ray loop system and the height of the lower edge of the oval, enable us to trace the stretching of the current sheet. The initial distance between the oval and the loops amounted to 35 – 40 Mm. In about 1 h the inferred length of the current sheet increased to 150 – 200 Mm, which corresponds to a mean elongation speed of 35 – 45 km s^-1. The results are discussed in the framework of CME models that include the magnetic reconnection below the erupting flux-rope.     

Motions of Flare Ribbons and Loops in Various Magnetic Configurations

Kopp–Pneuman-type magnetic configurations, which include a vertical current sheet, with various background fields are investigated. Dissipation of the current sheet as a result of magnetic reconnection produces bright flare ribbons on the solar disk and a growing flare loop system in the corona. In principle, the growth of flare loop system is governed by a reconnection process only, and the behavior of flare ribbons is also controlled by the background field. The flare ribbons may appear either separate or attached to one another at the onset of the flare depending on the background field distribution on the boundary surface. We calculate the decrease in height that magnetic field lines undergo after they have reconnected to form closed loops. Following previous practice, we refer to this decrease as field line shrinkage. Unlike the motions of flare ribbons, the shrinkage of flare loops depends weakly on the background field. Individual loops always shrink fastest at the moment it is produced by reconnection and just starts to leave the current sheet. The earlier the loop forms, the more and faster it shrinks. The relevant observations are explained on the basis of our calculations, and the aspects of the explanation that need improvement are also discussed.