Abstract.In Isliker et al. (2000b), an extended cellular automaton (X-CA) model for solar flares was introduced. In this model, the interpretation of the model's grid-variable is specified, and the magnetic field, the current, and an approximation to the electric field are yielded, all in a way that is consistent with Maxwell's and the MHD equations. The model also reproduces the observed distributions of total energy, peak-flux, and durations. Here, we reveal which relevant plasma physical processes are implemented by the X-CA model and in what form, and what global physical set-up is assumed by this model when it is in its natural state (self-organized criticality, SOC). The basic results are: (1) On large-scales, all variables show characteristic quasi-symmetries: the current has everywhere a preferential direction, the magnetic field exhibits a quasi-cylindrical symmetry. (2) The global magnetic topology forms either (i) closed magnetic field lines around and along a more or less straight neutral line for the model in its standard form, or (ii) an arcade of field lines above the bottom plane and centered along a neutral line, if the model is slightly modified. (3) In case of the magnetic topology (ii), loading can be interpreted as if there were a plasma which flows predominantly upwards, whereas in case of the magnetic topology (i), as if there were a plasma flow expanding from the neutral line. (4) The small-scale physics in the bursting phase represent localized diffusive processes, which are triggered when a quantity which is an approximately linear function of the current exceeds a threshold. (5) The interplay of loading and bursting in the X-CA model can be interpreted as follows: the local diffusivity usually has a value which is effectively zero, and it turns locally to an anomalous value if the mentioned threshold is exceeded, whereby diffusion dominates the quiet evolution (loading), until the critical quantity falls below the threshold again. (6) Flares (avalanches) are accompanied by the appearance of localized, intense electric fields. A typical example of the spatio-temporal evolution of the electric field during a flare is presented. (7) In a variant on the X-CA model, the magnitude of the current is used directly in the instability criterion, instead of the approximately linear function of it. First results indicate that the SOC state persists and is only slightly modified: distributions of the released energy are still power-laws with slopes comparable to the ones of the non-modified X-CA model, and the large scale structures, a characteristic of the SOC state, remain unchanged. (8) The current-dissipation during flares is spatially fragmented into a large number of dissipative current-surfaces of varying sizes, which are spread over a considerably large volume, and which do not exhibit any kind of simple spatial organization as a whole. These current-surfaces do not grow in the course of time, they are very short-lived, but they multiply, giving rise to new dissipative current-surfaces whic...

We study the acceleration of electrons and protons interacting with localized, multiple, small-scale dissipation regions inside an evolving, turbulent active region. The dissipation regions are Unstable Current Sheets (UCS), and in their ensemble they form a complex, fractal, evolving network of acceleration centers. Acceleration and energy dissipation are thus assumed to be fragmented. A large-scale magnetic topology provides the connectivity between the UCS and determines in this way the degree of possible multiple acceleration. The particles travel along the magnetic field freely without loosing or gaining energy, till they reach a UCS. In a UCS, a variety of acceleration mechanisms are active, with the end-result that the particles depart with a new momentum. The stochastic acceleration process is represented in the form of Continuous Time Random Walk (CTRW), which allows to estimate the evolution of the energy distribution of the particles. It is found that under certain conditions electrons are heated and accelerated to energies above 1 MeV in much less than a second. Hard X-ray (HXR) and microwave spectra are calculated from the electrons' energy distributions, and they are found to be compatible with the observations. Ions (protons) are also heated and accelerated, reaching energies up to 10 MeV almost simultaneously with the electrons. The diffusion of the particles inside the active region is extremely fast (anomalous super-diffusion). Although our approach does not provide insight into the details of the specific acceleration mechanisms involved, its benefits are that it relates acceleration to the energy release, and it well describes the stochastic nature of the acceleration process.Comment: 37 pages, 10 figures, one of them in color; in press at ApJ (2004

Electron and proton acceleration in three-dimensional electric and magnetic fields is studied through test particle simulations. The fields are obtained by a three-dimensional magnetohydrodynamic simulation of magnetic reconnection in slab geometry. The nonlinear evolution of the system is characterized by the growth of many unstable modes and the initial current sheet is fragmented with formation of small scale structures. We inject at random points inside the evolving current sheet a Maxwellian distribution of particles. In relatively short time (less than a millisecond) the particles develop a power law tail. The acceleration is extremely efficient and the electrons absorb a large percentage of the available energy in a small fraction of the characteristic time of the MHD simulation, suggesting that resistive MHD codes, used extensively in the current literature, are unable to represent the full extent of particle acceleration in 3D reconnection.It is widely accepted that magnetic reconnection plays a significant role in converting magnetic energy to thermal energy and kinetic energy of electrons and protons in laboratory plasmas, the Earth's magnetosphere, the solar corona, and in extragalactic jets [1,2].In resistive magnetohydrodynamic models, resistivity breaks the frozen-in law in a boundary layer, allowing reconnection to occur. A current sheet can be spontaneously unstable to resistive instabilities, like the tearing modes, which lead to magnetic reconnection [3,4]. Many numerical codes have been developed to study the nonlinear evolution of tearing modes in two-dimensional approximations [5]. However, three-dimensional effects may become important in modifying the spatial structure of the current sheets and the reconnection rate [6,7]. Different numerical studies have been performed to investigate collisionless magnetic reconnection using fluid models, where magnetic reconnection is made possible by electron inertia, and kinetic simulations, in two and three dimensional configurations [8,9]. It has been shown that in the three-dimensional kinetic reconnection the characteristic time scale of the instability is much faster than that of the two-dimensional tearing mode instability.The change of the topology of the magnetic field due to magnetic reconnection allows the release of magnetic energy, which can be responsible for the acceleration of particles. In two-dimensional reconnection configurations particle acceleration has been extensively studied both analytically and numerically [10,11]. The acceleration is caused by the motion of particles along the electric field in the current sheet, but the magnetic field plays a significant role since it influences the trajectory and therefore the energy gain of the particles. Recently, it has become clear that it is essential to include in the model the longitudinal component of the magnetic field, which is parallel to the electric field in the current sheet [12]- [14]. Studies of particle acceleration with a longitudinal magnetic field component have also be...

We study sequences of periodic orbits and the associated phase space dynamics in a 4-D symplectic map of interest to the problem of beam stability in circular particle accelerators. The increasing period of these orbits is taken from a sequence of rational approximants to an incommensurate pair of irrational rotation numbers of an invariant torus. We find stable (ellipticelliptic) periodic orbits of very high period and show that smooth rotational tori exist in their neighborhood, on which the motion is regular and bounded at large distances away from the origin. Perturbing these tori in parameter and/or initial condition space, we find either chains of smaller rotational tori or certain twisted tube-like tori of remarkable morphology. These tube-tori and tori chains have small scale chaotic motions in their surrounding vicinity and are formed about invariant curves of the 4-D map, which are either single loops or are composed of several disconnected loops, respectively. These smaller chaotic regions as well as the nonsmoothness properties of large rotational tori under small perturbations, leading to eventual escape of orbits to infinity, are studied here by the computation of correlation dimension and Lyapunov exponents.

Aims. To perform numerical experiments of particle acceleration in the complex magnetic and electric field environment of the stressed solar corona. Methods. The magnetic and electric fields are obtained from a 3-D MHD experiment that resembles a coronal loop with photospheric regions at both footpoints. Photospheric footpoint motion leads to the formation of a hierarchy of stochastic current sheets. Particles (protons and electrons) are traced within these current sheets starting from a thermal distribution using a relativistic test particle code. Results. In the corona the particles are subject to acceleration as well as deceleration, and a considerable portion of them leave the domain having received a net energy gain. Particles are accelerated to high energies in a very short time (both species can reach energies up to 100 GeV within 5 × 10 −2 s for electrons and 5 × 10 −1 s for protons). The final energy distribution shows that while one quarter of the particles retain their thermal distribution, the rest have been accelerated, forming a two-part power law. Accelerated particles are either trapped within electric field regions of opposite polarities, or escape the domain mainly through the footpoints. The particle dynamics are followed in detail and it is shown how this dynamic affects the time evolution of the system and the energy distribution. The scaling of these results with time and length scale is examined and the Bremstrahlung signature of X-ray photons resulting from escaping particles hitting the chromosphere is calculated and found to have a main power law part with an index γ = −1.8, steeper than observed. Possible resolutions of this discrepency are discussed.