The rate of flux pile-up magnetic reconnection in reduced MHD

The rate of two-dimensional flux pile-up magnetic reconnection is known to be severely limited by gas pressure in a low-beta plasma of the solar corona. As earlier perturbational calculations indicated, however, the pressure limitation should be less restrictive for three-dimensional flux pile-up. In this paper the maximum rate of reconnection is calculated in the approximation of reduced magnetohydrodynamics (RMHD), which is valid in the solar coronal loops. The rate is calculated for finite-magnitude reconnecting fields in the case of a strong axial field in the loop. Gas pressure effects are ignored in RMHD but a similar limitation on the rate of magnetic merging exists. Nevertheless, the magnetic energy dissipation rate and the reconnection electric field can increase by several orders of magnitude as compared with strictly two-dimensional pile-up. Though this is still not enough to explain the most powerful solar flares, slow coronal transients with energy release rates of order 10²⁵– 10²⁶ erg s^–1and heating of quiet coronal loops are within the compass of the model.     

Magnetic Energy Release in Flux Pile-up Merging

The problem of pressure limitations on the rate of flux pile-up magnetic reconnection is studied. We first examine the recent suggestion of Jardine and Allen (1998) for moderating the build-up of magnetic pressure in the current sheet by considering inflows with nonzero vorticity. An analytic argument shows, however, that unbounded magnetic pressures in the limit of small resistivities can be avoided only at the cost of unphysical dynamic pressures in the plasma. Hence, the pressure limitation on the reconnection rate in a low-beta plasma cannot be avoided completely. Nevertheless, we demonstrate that reconnection can be more rapid in a new solution that balances the build-up in dynamic pressure against both the plasma and magnetic pressures. This exact MHD solution has the characteristics of merging driven by the coalescence instability. The maximum energy release rate of the model is capable of explaining a modest solar flare.     

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.     

Numerical simulation of reconnection in an emerging magnetic flux region

The resistive MHD equations are numerically solved in two dimensions for an initial-boundary-value problem which simulates reconnection between an emerging magnetic flux region and an overlying coronal magnetic field. The emerging region is modelled by a cylindrical flux tube with a poloidal magnetic field lying in the same plane as the external, coronal field. The plasma betas of the emerging and coronal regions are 1.0 and 0.1, respectively, and the magnetic Reynolds number for the system is 2 × 10³. At the beginning of the simulation the tube starts to emerge through the base of the rectangular computational domain, and, when the tube is halfway into the computational domain, its position is held fixed so that no more flux of plasma enters through the base. Because the time-scale of the emergence is slower than the Alfvén time-scale, but faster than the reconnection time-scale, a region of closed loops forms at the base. These loops are gradually opened and reconnected with the overlying, external magnetic field as time proceeds.The evolution of the plasma can be divided into four phases as follows: First, an initial, quasi-steady phase during which most of the emergence is completed. During this phase, reconnection initially occurs at the slow rate predicted by the Sweet model of diffusive reconnection, but increases steadily until the fast rate predicted by the Petschek model of slow-shock reconnection is approached. Second, an impulsive phase with large-scale, super-magnetosonic flows. This phase appears to be triggered when the internal mechanical equilibrium inside the emerging flux tube is upset by reconnection acting on the outer layers of the flux tube. During the impulsive phase most of the flux tube pinches off from the base to form a cylindrical magnetic island, and temporarily the reconnection rate exceeds the steady-state Petschek rate. (At the time of the peak reconnection rate, the diffusion region at the X-line is not fully resolved, and so this may be a numerical artifact.) Third, a second quasi-steady phase during which the magnetic island created in the impulsive phase is slowly dissipated by continuing, but low-level, reconnection. And fourth, a static, non-evolving phase containing a potential, current-free field and virtually no flow.During the short time in the impulsive phase when the reconnection rate exceeds the steady-state Petschek rate, a pile-up of magnetic flux at the neutral line occurs. At the same time the existing Petschek-slow-mode shocks are shed and replaced by new ones; and, for a while, both new and old sets of slow shocks coexist.     

Robust Scalings in Compressible Flux Pile-Up Reconnection

Flux pile-up magnetic reconnection is traditionally considered only for incompressible plasmas. The question addressed in this paper is whether the pile-up scalings with resistivity are robust when plasma compressibility is taken into account. A simple analytical argument makes it possible to understand why the transition from a highly compressible limit to the incompressible one is difficult to discern in typical simulations spanning a few decades in resistivity. From a practical standpoint, however, flux pile-up reconnection in a compressible plasma can lead to anomalous electric resistivity in the current sheet and flare-like energy release of magnetic energy in the solar corona.     

Optimized magnetic reconnection solutions in three dimensions

It has recently been shown that there is a well defined upper limit to the rate of magnetic merging for two-dimensional flux pile-up solutions. This rate, derived by equalizing the dynamic and magnetic pressures in the reconnection region and saturating the magnetic field in the current layer, leads to a significant enhancement of the classical Sweet–Parker merging limit. In this study we explore optimal merging rates in the case of three-dimensional fan and spine reconnection solutions. The ideas of optimization and saturation are first illustrated using an exact fan solution. We go on to show that while spine solutions seem ineffective as flare release mechanisms, optimized fan solutions have energy release characteristics typical of modest events.     

The role of the magnetic barrier in the Solar wind-magnetosphere interaction

The magnetized solar wind carries a large amount of energy but only a small fraction of it enters the magnetosphere and powers its dynamics. Numerous observations show that the interplanetary magnetic field (IMF) is a key parameter regulating the solar wind-magnetosphere interaction. The main factor determining the amount of energy extracted from the solar wind flow by the magnetosphere is the plasma flow structure in the region adjacent to the sunward side of the magnetopause. While compared to the energy of the solar wind flow the IMF magnetic energy is relatively weak, it is considerably enhanced in a thin layer next to the dayside magnetopause variously called the plasma depletion layer or magnetic barrier. Important features of this barrier/layer are (i) a pile-up of the magnetic field with (ii) a concurrent decrease of density, (iii) enhancement of proton temperature anisotropy, (iv) asymmetry of plasma flow caused by magnetic field tension, and (v) characteristic wave emissions (ion cyclotron waves). Importantly, the magnetic barrier can be considered as an energy source for magnetic reconnection. While the steady-state magnetic barrier has been extensively examined, non-steady processes therein have only been addressed by a few authors. We discuss here two non-steady aspects related to variations of the magnetic barrier caused by (i) a north-to-south rotation of the IMF, and (ii) by pulses of magnetic field reconnection at the magnetopause. When the IMF rotates smoothly from north-to-south, a transition layer is shown to appear in the magnetosheath which evolves into a thin layer bounded by sharp gradients in the magnetic field and plasma quantities. For a given reconnection rate and calculated parameters of the magnetic barrier, we estimate the duration and length scale of a reconnection pulse as a function of the solar wind parameters. Considering a sudden decrease of the magnetic field near the magnetopause caused by the reconnection pulse, we study the relaxation process of the magnetic barrier. We find that the relaxation time is longer than the duration of the reconnection pulse for large Alfvén-Mach numbers.     

The structure of the magnetopause

The solar wind is a magnetized flowing plasma that intersects the Earth's magnetosphere at a velocity much greater than that of the compressional fast mode wave that is required to deflect that flow. A bow shock forms that alters the properties of the plasma and slows the flow, enabling continued evolution of the properties of the flow on route to its intersection with the magnetopause. Thus the plasma conditions at the magnetopause can be quite unlike those in the solar wind. The boundary between this “magnetosheath” plasma and the magnetospheric plasma is many gyroradii thick and is surrounded by several boundary layers. A very important process occurring at the magnetopause is reconnection whereby there is a topological change in magnetic flux lines so that field lines can connect the solar wind plasma to the terrestrial plasma, enabling the two to mix. This connection has important consequences for momentum transfer from the solar wind to the magnetosphere. The initiation of reconnection appears to be at locations where the magnetic fields on either side of the magnetopause are antiparallel. This condition is equivalent to there being no guide field in the reconnection region, so at the reconnection point there is truly a magnetic neutral or null point. Lastly reconnection can be spatially and temporally varying, causing the region of the magnetopause to be quite dynamic.     

2 1/2-Dimensional Reconnection Model and Energy Release in Solar Flares

We employ a 2 1/2-dimensional reconnection model to analyse different aspects of the energy release in two-ribbon flares. In particular, we investigate in which way the systematic change of inflow region variables, associated with the vertical elongation of current sheet, affects the flare evolution. It is assumed that as the transversal magnetic field decreases, the ambient plasma-to-magnetic pressure ratio increases, and the reconnection rate diminishes. As the transversal field decreases due to the arcade stretching, the energy release enhances and the temperature rises. Furthermore, the magnetosonic Mach number of the reconnection outflow increases, providing the formation of fast mode standing shocks above the flare loops and below the erupting flux rope. Eventually, in the limit of a very small transversal field the reconnection becomes turbulent due to a highly non-linear response of the system to small fluctuations of the transversal field. The turbulence results in the energy release fragmentation which increases the release efficiency, and is likely to be responsible for the impulsive phase of the flare. On the other hand, as the current sheet stretches to larger heights, the ambient plasma-to-magnetic pressure ratio increases which causes a gradual decrease of the reconnection rate, energy release rate, and temperature in the late phase of flare. The described magnetohydrodynamical changes affect also the electron distribution function in space and time. At large reconnection rates (impulsive phase of the flare) the ratio of the inflow-to-outflow magnetic field strength is much smaller than at lower reconnection rates (late phase of the flare), i.e., the corresponding loss-cone angle becomes narrower. Consequently, in the impulsive phase a larger fraction of energized electrons can escape from the current sheet downwards to the chromosphere and upwards into the corona – the dominant flare features are the foot-point hard X-ray sources and type III radio bursts. On the other hand, at low reconnection rates, more particles stay trapped in the outflow region, and the thermal conduction flux becomes strongly reduced. As a result, a superhot loop-top, and above-the-loop plasma appears, as sometimes observed, to be a dominant feature of the gradual phase.     

Reconnection driven by an erupting filament in the May 14, 1981 flare

We present observations of the flare of May 14, 1981, which can be classified as a three-ribbon flare. After a detailed analysis in metric, decimetric, microwave, optical, and X-ray ranges we propose that the event was caused by a reconnection process driven by erupting filament. The energy was liberated in the current sheet above the filament in the region between the erupting flux and the overlying field. It is shown that plasma microinstabilities develop as the plasma enters the current sheet. The observations indicate that during the precursor phase a certain low-frequency turbulence, such as ion-accoustic turbulence had to be present.The reconnection rate was growing due to the increasing tension of the stretched overlying field. It is shown that the reconnection proceeded in the Sonnerup-Petschek regime during the precursor, and changed to the pile-up regime in the fast reconnection phase, when the maximal lateral expansion (50 km s^–1) of the H ribbons was observed. The proposed process of reconnection driven by an erupting filament can be applied to three- and four-ribbon flares.     

Generation of rotational discontinuities by magnetic reconnection associated with microflares

Magnetic reconnection can take place between two plasma regions with antiparallel magnetic field components. In a time-dependent reconnection event, the plasma outflow region consists of a leading bulge region and a trailing reconnection layer. Magnetohydrodynamic (MHD) discontinuities, including rotational discontinuities, can be formed in both the bulge region and the trailing layer. In this paper, we suggest that the rotational discontinuities observed in the solar wind may be generated by magnetic reconnection associated with microflares in coronal holes. The structure of the reconnection layer is studied by solving the one-dimensional Riemann problem for the evolution of an initial current sheet after the onset of magnetic reconnection as well as carrying out two-dimensional MHD simulations. As the emerging magnetic flux reconnects with ambient open magnetic fields in the coronal hole, rotational discontinuities are generated in the region with open field lines. It is also found that in the solar corona with a low plasma beta ( 0.01), the magnetic energy is converted through magnetic reconnection mostly into the plasma bulk-flow energy. Since more microflares will generate more rotational discontinuities and also supply more energy to the solar wind, it is expected that the number of rotational discontinuities observed in the solar wind would be an increasing function of solar wind speed. The observation rate of rotational discontinuities generated by microflares is estimated to be dN _RD/dt - f/63 000 s (f > 1) at 1 AU. The present mechanism favors the generation of rotational discontinuities with a large shock normal angle.     

Shocks produced by impulsively driven reconnection

Shock waves produced by impulsively driven reconnection may be important during flares or during the emergence of magnetic flux from the photosphere into the corona. Here we investigate such shock waves by carrying out numerical experiments using two-dimensional magneto-hydrodynamics. The results of the numerical experiments imply that there are three different categories of shocks associated with impulsively driven reconnection: (1) fast-mode, blast waves which rapidly propagate away from the reconnection site; (2) slow-mode, Petschek shocks which are attached to the reconnection site; and (3) fast-mode, termination shocks which terminate the plasma jets flowing out from the reconnection site. Fast-mode blast waves are a common feature of many flare models, but the Petschek shocks and jet termination shocks are specific to reconnection models. These two different types of reconnection shocks might contribute to chromospheric ablation and energetic particle acceleration in flares.     

Visco-Resistive Dissipation in Strongly Driven Transient Reconnection

Solar flare energy release mechanisms often neglect the role played by viscous effects. Here we perform incompressible planar reconnection simulations, driven by the Orszag–Tang vortex, for both classical and Braginskii forms of the viscosity. We show that strongly driven “saturated” flux pile-up current layers, which lead to weak reconnection rates at small resistivities, are accompanied by invariant global viscous losses. These results support the notion that viscous dissipation in flaring plasmas can account for a significant fraction of the flare energy release.     

Spontaneous magnetic reconnection

The present review concerns the relevance of collisionless reconnection in the astrophysical context. Emphasis is put on recent developments in theory obtained from collisionless numerical simulations in two and three dimensions. It is stressed that magnetic reconnection is a universal process of particular importance under collisionless conditions, when both collisional and anomalous dissipation are irrelevant. While collisional (resistive) reconnection is a slow, diffusive process, collisionless reconnection is spontaneous. On any astrophysical time scale, it is explosive. It sets on when electric current widths become comparable to the leptonic inertial length in the so-called lepton (electron/positron) “diffusion region”, where leptons de-magnetise. Here, the magnetic field contacts its oppositely directed partner and annihilates. Spontaneous reconnection breaks the original magnetic symmetry, violently releases the stored free energy of the electric current, and causes plasma heating and particle acceleration. Ultimately, the released energy is provided by mechanical motion of either the two colliding magnetised plasmas that generate the current sheet or the internal turbulence cascading down to lepton-scale current filaments. Spontaneous reconnection in such extended current sheets that separate two colliding plasmas results in the generation of many reconnection sites (tearing modes) distributed over the current surface, each consisting of lepton exhausts and jets which are separated by plasmoids. Volume-filling factors of reconnection sites are estimated to be as large as \({<}10^{-5}\) per current sheet. Lepton currents inside exhausts may be strong enough to excite Buneman and, for large thermal pressure anisotropy, also Weibel instabilities. They bifurcate and break off into many small-scale current filaments and magnetic flux ropes exhibiting turbulent magnetic power spectra of very flat power-law shape \(W_b\propto k^{-\alpha }\) in wavenumber k with power becoming as low as \(\alpha \approx 2\). Spontaneous reconnection generates small-scale turbulence. Imposed external turbulence tends to temporarily increase the reconnection rate. Reconnecting ultra-relativistic current sheets decay into large numbers of magnetic flux ropes composed of chains of plasmoids and lepton exhausts. They form highly structured current surfaces, “current carpets”. By including synchrotron radiation losses, one favours tearing-mode reconnection over the drift-kink deformation of the current sheet. Lepton acceleration occurs in the reconnection-electric field in multiple encounters with the exhausts and plasmoids. This is a Fermi-like process. It results in power-law tails on the lepton energy distribution. This effect becomes pronounced in ultra-relativistic reconnection where it yields extremely hard lepton power-law energy spectra approaching \(F(\gamma )\propto \gamma ^{-1}\), with \(\gamma \) the lepton energy. The synchrotron radiation limit becomes substantially exceeded. Relativistic reconnection is a probable generator of current and magnetic turbulence, and a mechanism that produces high-energy radiation. It is also identified as the ultimate dissipation mechanism of the mechanical energy in collisionless magnetohydrodynamic turbulent cascades via lepton-inertial-scale turbulent current filaments. In this case, the volume-filling factor is large. Magnetic turbulence causes strong plasma heating of the entire turbulent volume and violent acceleration via spontaneous lepton-scale reconnection. This may lead to high-energy particle populations filling the whole volume. In this case, it causes non-thermal radiation spectra that span the entire interval from radio waves to gamma rays.     

Solar coronal heating by magnetic cancellation – II. Disconnected and unequal bipoles

Two-dimensional numerical magnetohydrodynamic simulations of a cancelling magnetic feature (CMF) and the associated coronal X-ray bright point (XBP) are presented. Coronal magnetic reconnection is found to produce the Ohmic heating required for a coronal XBP. During the BP phase where reconnection occurs above the base, about 90–95 per cent of the magnetic flux of the converging magnetic bipole cancels at the base. The last ≈5 to 10 per cent of the base magnetic flux is cancelled when reconnection occurs at the base. Reconnection happens in a time-dependent way in response to the imposed converging footpoint motions. A potential field model gives a good first approximation to the qualitative behaviour of the system, but the magnetohydrodynamics (MHD) experiments reveal several quantitative differences: for example, the effects of plasma inertia and a pressure build-up in-between the converging bipole are to delay the onset of coronal reconnection above the base and to lower the maximum X -point height.     

Aspects of Three-Dimensional Magnetic Reconnection – (Invited Review)

In this review paper we discuss several aspects of magnetic reconnection theory, focusing on the field-line motions that are associated with reconnection. A new exact solution of the nonlinear MHD equations for reconnective annihilation is presented which represents a two-fold generalization of the previous solutions. Magnetic reconnection at null points by several mechanisms is summarized, including spine reconnection, fan reconnection and separator reconnection, where it is pointed out that two common features of separator reconnection are the rapid flipping of magnetic field lines and the collapse of the separator to a current sheet. In addition, a formula for the rate of reconnection between two flux tubes is derived. The magnetic field of the corona is highly complex, since the magnetic carpet consists of a multitude of sources in the photosphere. Progress in understanding this complexity may, however, be made by constructing the skeleton of the field and developing a theory for the local and global bifurcations between the different topologies. The eruption of flux from the Sun may even sometimes be due to a change of topology caused by emerging flux break-out. A CD-ROM attached to this paper presents the results of a toy model of vacuum reconnection, which suggests that rapid flipping of field lines in fan and separator reconnection is an essential ingredient also in real non-vacuum conditions. In addition, it gives an example of binary reconnection between a pair of unbalanced sources as they move around, which may contribute significantly to coronal heating. Finally, we present examples in TRACE movies of geometrical changes of the coronal magnetic field that are a likely result of large-scale magnetic reconnection. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1005248007615     

Magnetic Flux Cancellation Observed in the Sunspot Moat

In this paper we study the evolution of magnetic fields of a 1F/2.4C solar flare and following magnetic flux cancellation. The data are Big Bear Solar Observatory and SOHO/MDI observations of active region NOAA 8375. The active region produced a multitude of subflares, many of them being clustered along the moat boundary in the area with mixed polarity magnetic fields. The study indicates a possible connection between the flare and the flux cancellation. The cancellation rate, defined from the data, was found to be 3×10¹⁹ Mx h^–1. We observed strong upward directed plasma flows at the cancellation site. Suggesting that the cancellation is a result of reconnection process, we also found a reconnection rate of 0.5 km s^–1, which is a significant fraction of Alfvén speed. The reconnection rate indicates a regime of fast photospheric reconnection happening during the cancellation.     

The dynamics of driven magnetic reconnection in coronal arcades

The dynamics of interacting coronal loops and arcades have recently been highlighted by observations from theYohkoh satellite and may represent a viable mechanism for heating the solar corona. Here such an interaction is studied using two-dimensional resistive magnetohydrodynamic (MHD) simulations. Initial potential field structures evolve in response to imposed photospheric flows. In addition to the anticipated current sheet about theX-point separating the colliding flux systems, significant current layers are found to lie all the way along the separatrices that intersect at theX-point and divide the coronal magnetic field into topologically distinct regions. Shear flows across the separatrices are also observed. Both of these features are shown to be compatible with recent analytical studies of two-dimensional linear steady-state magnetic reconnection, even though the driven system that has been simulated is not strictly ‘open’ in the sense implied by steady-state calculations. The implications for future steady-state models are also discussed. The presence of the neutral point also brings into question any constant-density approximations that have previously been used for such quasi-steady coronal evolution models. This results from the intimate coupling between the neutral point and its separatrices communicated via the gas pressure. In terms of the detailed energetics during the arcade evolution, preliminary results reveal that on the order of 3% of the energy injected by the footpoint motions is lost purely through ohmic dissipation. We would therefore anticipate a local hot spot between the interacting flux systems, and a brightening distributed along the length of any separatrix field lines. Furthermore, as the resistivityη is reduced, the flux annihilation rate and the ohmic dissipation rate are found to scale independently ofη.     

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.     

Simulating Astrophysical Jets in Laboratory Experiments

Pulsed-power technology and appropriate boundary conditions have been used to create simulations of magnetically driven astrophysical jets in a laboratory experiment. The experiments are quite reproducible and involve a distinct sequence. Eight initial flux tubes, corresponding to eight gas injection locations, merge to form the jet, which lengthens, collimates, and eventually kinks. A model developed to explain the collimation process predicts that collimation is intimately related to convection and pile-up of frozen-in toroidal flux convected with the jet. The pile-up occurs when there is an axial non-uniformity in the jet velocity so that in the frame of the jet there appears to be a converging flow of plasma carrying frozen-in toroidal magnetic flux. The pile-up of convected flux at this “stagnation region” amplifies the toroidal magnetic field and increases the pinch force, thereby collimating the jet.