Heinz Isliker

Particle acceleration in stochastic current sheets in stressed coronal active regions

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.

Stochastic acceleration in turbulent electric fields generated by 3D reconnection.

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 a 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 are unable to represent the full extent of particle acceleration.

PARTICLE ACCELERATION IN STRESSED CORONAL MAGNETIC FIELDS

This Letter presents an analysis of particle acceleration in a model of the complex magnetic field environment in the flaring solar corona. A slender flux tube, initially in hydrodynamic equilibrium, is stressed by random photospheric motions. A three-dimensional MHD code is used to follow the stochastic development of transient current sheets. These processes generate a highly fragmented electric field, through which particles are tracked using a relativistic test particle code. It is shown that both ions and electrons are accelerated readily to relativistic energies in times of order 10-2 s for electrons and 10-1 s for protons forming power-law distributions in energy.

MHD consistent cellular automata (CA) models - II. Applications to solar flares

In Isliker et al. (2000b), an extended cellular automaton (X-CA) model for solar flares was introduced. In this model, the interpretation of the models grid-variable is specied, and the magnetic eld, the current, and an approximation to the electric eld are yielded, all in a way that is consistent with Maxwells 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 eld exhibits a quasi-cylindrical symmetry. (2) The global magnetic topology forms either (i) closed magnetic eld lines around and along a more or less straight neutral line for the model in its standard form, or (ii) an arcade of eld lines above the bottom plane and centered along a neutral line, if the model is slightly modied. (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 diusive 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 diusivity usually has a value which is eectively zero, and it turns locally to an anomalous value if the mentioned threshold is exceeded, whereby diusion 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 elds. A typical example of the spatio-temporal evolution of the electric eld 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 modied: distributions of the released energy are still power-laws with slopes comparable to the ones of the non-modied 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 which are spread further around. They show thus a highly dynamic temporal evolution. It follows that the X-CA model represents an implementation of the flare scenario of Parker (1993) in a rather complete way, comprising aspects from small scale physics to the global physical set-up, making though some characteristic simplications which are unavoidable in the frame-work of a CA.

Electron-cyclotron wave scattering by edge density fluctuations in ITER

The effect of edge turbulence on the electron-cyclotron wave propagation in ITER is investigated with emphasis on wave scattering, beam broadening, and its influence on localized heating and current drive. A wave used for electron-cyclotron current drive (ECCD) must cross the edge of the plasma, where density fluctuations can be large enough to bring on wave scattering. The scattering angle due to the density fluctuations is small, but the beam propagates over a distance of several meters up to the resonance layer and even small angle scattering leads to a deviation of several centimeters at the deposition location. Since the localization of ECCD is crucial for the control of neoclassical tearing modes, this issue is of great importance to the ITER design. The wave scattering process is described on the basis of a Fokker–Planck equation, where the diffusion coefficient is calculated analytically as well as computed numerically using a ray tracing code.

Random walk through fractal environments

We analyze random walk through fractal environments, embedded in three-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e., of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D(F) of the fractal is less than 2, there is though, always a finite rate of unaffected escape. Random walks through fractal sets with D(F)< or =2 can thus be considered as defective Levy walks. The distribution of jump increments for D(F)>2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk increments. It is shown that the particles undergo anomalous, enhanced diffusion for D(F)<2, the diffusion is dominated by the finite escape rate. Diffusion for D(F)>2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality models give rise to enhanced diffusion. The analytical results are illustrated by Monte Carlo simulations.

The correlation of fractal structures in the photospheric and the coronal magnetic field

Context. This work examines the relation between the fractal properties of the photospheric magnetic patterns and those of the coronal magnetic fields in solar active regions. Aims. We investigate whether there is any correlation between the fractal dimensions of the photospheric structures and the magnetic discontinuities formed in the corona. Methods. To investigate the connection between the photospheric and coronal complexity, we used a nonlinear force-free extrapolation method that reconstructs the 3d magnetic fields using 2d observed vector magnetograms as boundary conditions. We then located the magnetic discontinuities, which are considered as spatial proxies of reconnection-related instabilities. These discontinuities form well-defined volumes, called here unstable volumes. We calculated the fractal dimensions of these unstable volumes and compared them to the fractal dimensions of the boundary vector magnetograms. Results. Our results show no correlation between the fractal dimensions of the observed 2d photospheric structures and the extrapolated unstable volumes in the corona, when nonlinear force-free extrapolation is used. This result is independent of efforts to (1) bring the photospheric magnetic fields closer to a nonlinear force-free equilibrium and (2) omit the lower part of the modeled magnetic field volume that is almost completely filled by unstable volumes. A significant correlation between the fractal dimensions of the photospheric and coronal magnetic features is only observed at the zero level (lower limit) of approximation of a current-free (potential) magnetic field extrapolation. Conclusions. We conclude that the complicated transition from photospheric non-force-free fields to coronal force-free ones hampers any direct correlation between the fractal dimensions of the 2d photospheric patterns and their 3d counterparts in the corona at the nonlinear force-free limit, which can be considered as a second level of approximation in this study. Correspondingly, in the zero and first levels of approximation, namely, the potential and linear force-free extrapolation, respectively, we reveal a significant correlation between the fractal dimensions of the photospheric and coronal structures, which can be attributed to the lack of electric currents or to their purely field-aligned orientation.

A scaling test for correlation dimensions

Abstract A quantitative test is presented to check scaling and convergence of the correlation integral and consistency of two different algorithms. An application to two known attractors demonstrates that it allows one to judge fast and reliably the quality of a conjectured scaling behaviour above all in the case of short or noisy data. Results concerning minimum data amount and maximum noise level confirm earlier work, the crucial parameter concerning data length turns out, however, to be not the number of points, but the number of cycles in phase space (peaks in the time series).

Simulating flaring events in complex active regions driven by observed magnetograms

Context. We interpret solar flares as events originating in active regions that have reached the self organized critical state, by using a refined cellular automaton model with initial conditions derived from observations. Aims. We investigate whether the system, with its imposed physical elements, reaches a self organized critical state and whether well-known statistical properties of flares, such as scaling laws observed in the distribution functions of characteristic parameters, are reproduced after this state has been reached. Methods. To investigate whether the distribution functions of total energy, peak energy and event duration follow the expected scaling laws, we first applied a nonlinear force-free extrapolation that reconstructs the three-dimensional magnetic fields from two-dimensional vector magnetograms. We then locate magnetic discontinuities exceeding a threshold in the Laplacian of the magnetic field. These discontinuities are relaxed in local diffusion events, implemented in the form of cellular automaton evolution rules. Subsequent loading and relaxation steps lead the system to self organized criticality, after which the statistical properties of the simulated events are examined. Physical requirements, such as the divergence-free condition for the magnetic field vector, are approximately imposed on all elements of the model. Results. Our results show that self organized criticality is indeed reached when applying specific loading and relaxation rules. Power-law indices obtained from the distribution functions of the modeled flaring events are in good agreement with observations. Single power laws (peak and total flare energy) are obtained, as are power laws with exponential cutoff and double power laws (flare duration). The results are also compared with observational X-ray data from the GOES satellite for our active-region sample. Conclusions. We conclude that well-known statistical properties of flares are reproduced after the system has reached self organized criticality. A significant enhancement of our refined cellular automaton model is that it commences the simulation from observed vector magnetograms, thus facilitating energy calculation in physical units. The model described in this study remains consistent with fundamental physical requirements, and imposes physically meaningful driving and redistribution rules.

Electron-cyclotron wave propagation, absorption and current drive in the presence of neoclassical tearing modes

We analyze the propagation of electron-cyclotron waves, their absorption and current drive when neoclassical tearing modes (NTMs), in the form of magnetic islands, are present in a tokamak plasma. So far, the analysis of the wave propagation and power deposition in the presence of NTMs has been performed mainly in the frame of an axisymmetric magnetic field, ignoring any effects from the island topology. Our analysis starts from an axisymmetric magnetic equilibrium, which is perturbed such as to exhibit magnetic islands. In this geometry, we compute the wave evolution with a ray-tracing code, focusing on the effect of the island topology on the efficiency of the absorption and current drive. To increase the precision in the calculation of the power deposition, the standard analytical flux-surface labeling for the island region has been adjusted from the usual cylindrical to toroidal geometry. The propagation up to the O-point is found to be little affected by the island topology, whereas the power absorbed and the driven current are significantly enhanced, because the resonant particles are bound to the small volumes in between the flux surfaces of the island. The consequences of these effects on the NTM evolution are investigated in terms of the modified Rutherford equation.