Manolis K. Georgoulis

Quantitative Forecasting of Major Solar Flares

We define the effective connected magnetic field, Beff, a single metric of the flaring potential in solar active regions. We calculated Beff for 298 active regions (93 X- and M-flaring, 205 nonflaring) as recorded by SOHO/MDI during a 10 yr period covering much of solar cycle 23. We find that Beff is a robust criterion for distinguishing flaring from nonflaring regions. A well-defined 12 hr conditional probability for major flares depends solely on Beff. This probability exceeds 0.95 for M-class and X-class flares if Beff > 1600 G and Beff > 2100 G, respectively, while the maximum calculated Beff-values are near 4000 G. Active regions do not give M-class and X-class flares if Beff < 200 G and Beff < 750 G, respectively. We conclude that Beff is an efficient flare-forecasting criterion that can be computed on a timely basis from readily available data.

Survey of Magnetic Helicity Injection in Regions Producing X-Class Flares

Virtually all X-class flares produce a coronal mass ejection (CME), and each CME carries magnetic helicity into the heliosphere. Using magnetograms from the Michelson Doppler Imager on the Solar and Heliospheric Observatory, we surveyed magnetic helicity injection into 48 X-flare-producing active regions recorded by the MDI between 1996 July and 2005 July. Magnetic helicity flux was calculated according to the method of Chae for the 48 X-flaring regions and for 345 non-X-flaring regions. Our survey revealed that a necessary condition for the occurrence of an X-flare is that the peak helicity flux has a magnitude >6 × 1036 Mx2 s-1. X-flaring regions also consistently had a higher net helicity change during the ~6 day measurement intervals than nonflaring regions. We find that the weak hemispherical preference of helicity injection, positive in the south and negative in the north, is caused by the solar differential rotation, but it tends to be obscured by the intrinsic helicity injection, which is more disorganized and tends to be of opposite sign. An empirical fit to the data shows that the injected helicity over the range 1039-10 43 Mx2 s-1 is proportional to magnetic flux squared. Similarly, over a range of 0.3-3000 days, the time required to generate the helicity in a CME is inversely proportional to the magnetic flux squared. Most of the X-flare regions generated the helicity needed for a CME in a few days to a few hours.

A New Technique for a Routine Azimuth Disambiguation of Solar Vector Magnetograms

We introduce a nonpotential magnetic field calculation (NPFC) technique to perform azimuth disambiguation in solar vector magnetograms. It is shown that resolving the 180° ambiguity would be a numerically fully determined problem if the vertical electric current density was known a priori. Since this is not the case, we enforce a minimum-magnitude current density solution. The NPFC disambiguation is otherwise assumption-free, with the quality of the results depending on the quality of the measurements. The NPFC method first infers the nonpotential magnetic field component responsible for the assumed vertical currents and then determines the vertical magnetic field whose potential extrapolation, added to the nonpotential field, best reproduces the observationally inferred horizontal magnetic field. The technique is fast, effective, and physically sound, so it may be instrumental in a routine, real-time, disambiguation of future space-borne solar vector magnetograms.

Statistical Properties of Magnetic Activity in the Solar Corona

The long-time statistical behavior of a two-dimensional section of a coronal loop subject to random magnetic forcing is presented. The highly intermittent nature of dissipation is revealed by means of magnetohydrodynamic (MHD) turbulence numerical simulations. Even with a moderate magnetic Reynolds number, intermittency is clearly present in both space and time. The response of the loop to the random forcing, as described either by the time series of the average and maximum energy dissipation or by its spatial distribution at a given time, displays a Gaussian noise component that may be subtracted to define discrete dissipative events. Distribution functions of both maximum and average current dissipation, for the total energy content, the peak activity, and the duration of such events are all shown to display robust scaling laws, with scaling indices δ that vary from δ -1.3 to δ -2.8 for the temporal distribution functions, while δ -2.6 for the overall spatial distribution of dissipative events.

25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper “Self-Organized Criticality: An explanation of 1/fnoise” by Bak et al. (1987), the idea has been applied to solar physics, in “Avalanches and the Distribution of Solar Flares” by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.

INTERPRETING ERUPTIVE BEHAVIOR IN NOAA AR 11158 VIA THE REGION'S MAGNETIC ENERGY AND RELATIVE-HELICITY BUDGETS

In previous works, we introduced a nonlinear force-free method that self-consistently calculates the instantaneous budgets of free magnetic energy and relative magnetic helicity in solar active regions (ARs). Calculation is expedient and practical, using only a single vector magnetogram per computation. We apply this method to a time series of 600 high-cadence vector magnetograms of the eruptive NOAA AR 11158 acquired by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory over a five-day observing interval. Besides testing our method extensively, we use it to interpret the dynamical evolution in the AR, including eruptions. We find that the AR builds large budgets of both free magnetic energy and relative magnetic helicity, sufficient to power many more eruptions than the ones it gave within the interval of interest. For each of these major eruptions, we find eruption-related decreases and subsequent free-energy and helicity budgets that are consistent with the observed eruption (flare and coronal mass ejection (CME)) sizes. In addition, we find that (1) evolution in the AR is consistent with the recently proposed (free) energy-(relative) helicity diagram of solar ARs, (2) eruption-related decreases occur before the flare and the projected CME-launch times, suggesting that CME progenitors precede flares, and (3) self terms of free energy and relative helicity most likely originate from respective mutual terms, following a progressive mutual-to-self conversion pattern that most likely stems from magnetic reconnection. This results in the non-ideal formation of increasingly helical pre-eruption structures and instigates further research on the triggering of solar eruptions with magnetic helicity firmly placed in the eruption cadre.

Derivation of Solar Flare Cellular Automata Models from a Subset of the Magnetohydrodynamic Equations

Cellular automata (CA) models account for the power-law distributions found for solar flare hard X-ray observations, but their physics has been unclear. We examine four of these models and show that their criteria and magnetic field distribution rules can be derived by discretizing the MHD diffusion equation as obtained from a simplified Ohms law. Identifying the discrete MHD with the CA models leads to an expression for the resistivity as a function of the current on the flux tube boundary, as may be expected from current-driven instabilities. Anisotropic CA models correspond to a nonlinear resistivity η(J), while isotropic ones are associated with hyperresistivity η(2J). The discrete equations satisfy the necessary conditions for self-organized criticality (Lu): there is local conservation of a field (magnetic flux), while the nonlinear resistivity provides a rapid dissipation and relaxation mechanism. The approach justifies many features of the CA models that were originally based on intuition.

Turbulence In The Solar Atmosphere: Manifestations And Diagnostics Via Solar Image Processing

Intermittent magnetohydrodynamical turbulence is most likely at work in the magnetized solar atmosphere. As a result, an array of scaling and multi-scaling image-processing techniques can be used to measure the expected self-organization of solar magnetic fields. While these techniques advance our understanding of the physical system at work, it is unclear whether they can be used to predict solar eruptions, thus obtaining a practical significance for space weather. We address part of this problem by focusing on solar active regions and by investigating the usefulness of scaling and multi-scaling image-processing techniques in solar flare prediction. Since solar flares exhibit spatial and temporal intermittency, we suggest that they are the products of instabilities subject to a critical threshold in a turbulent magnetic configuration. The identification of this threshold in scaling and multi-scaling spectra would then contribute meaningfully to the prediction of solar flares. We find that the fractal dimension of solar magnetic fields and their multi-fractal spectrum of generalized correlation dimensions do not have significant predictive ability. The respective multi-fractal structure functions and their inertial-range scaling exponents, however, probably provide some statistical distinguishing features between flaring and non-flaring active regions. More importantly, the temporal evolution of the above scaling exponents in flaring active regions probably shows a distinct behavior starting a few hours prior to a flare and therefore this temporal behavior may be practically useful in flare prediction. The results of this study need to be validated by more comprehensive works over a large number of solar active regions. Sufficient statistics may also establish critical thresholds in the values of the multi-fractal structure functions and/or their scaling exponents above which a flare may be predicted with a high level of confidence.

On the Self-Similarity of Unstable Magnetic Discontinuities in Solar Active Regions

We investigate the statistical properties of possible magnetic discontinuities in two solar active regions over the course of several hours. We use linear force-free extrapolations to calculate the three-dimensional magnetic structure in the active regions. Magnetic discontinuities are identified using various selection criteria. Independently of the selection criterion, we identify large numbers of magnetic discontinuities whose free magnetic energies and volumes obey well-formed power-law distribution functions. The power-law indices for the free energies are in the range [-1.6, -1.35], in remarkable agreement with the power-law indices found in the occurrence frequencies of solar flare energies. This agreement and the strong self-similarity of the volumes that are likely to host flares suggest that the observed statistics of flares may be the natural outcome of a preexisting spatial self-organization accompanying the energy fragmentation in solar active regions. We propose a dynamical picture of flare triggering consistent with recent observations by reconciling our results with the concepts of percolation theory and self-organized criticality. These concepts rely on self-organization, which is expected from the fully turbulent state of the magnetic fields in the solar atmosphere.

NON-NEUTRALIZED ELECTRIC CURRENT PATTERNS IN SOLAR ACTIVE REGIONS: ORIGIN OF THE SHEAR-GENERATING LORENTZ FORCE

Using solar vector magnetograms of the highest available spatial resolution and signal-to-noise ratio, we perform a detailed study of electric current patterns in two solar active regions (ARs): a flaring/eruptive and a flare-quiet one. We aim to determine whether ARs inject non-neutralized (net) electric currents in the solar atmosphere, responding to a debate initiated nearly two decades ago that remains inconclusive. We find that well-formed, intense magnetic polarity inversion lines (PILs) within ARs are the only photospheric magnetic structures that support significant net current. More intense PILs seem to imply stronger non-neutralized current patterns per polarity. This finding revises previous works that claim frequent injections of intense non-neutralized currents by most ARs appearing in the solar disk but also works that altogether rule out injection of non-neutralized currents. In agreement with previous studies, we also find that magnetically isolated ARs remain globally current-balanced. In addition, we confirm and quantify the preference of a given magnetic polarity to follow a given sense of electric currents, indicating a dominant sense of twist in ARs. This coherence effect is more pronounced in more compact ARs with stronger PILs and must be of sub-photospheric origin. Our results yield a natural explanation of the Lorentz force, invariably generating velocity and magnetic shear along strong PILs, thus setting a physical context for the observed pre-eruption evolution in solar ARs.