Avalanche models for solar flares (Invited Review)

This paper is a pedagogical introduction to avalanche models of solar flares, including a comprehensive review of recent modeling efforts and directions. This class of flare model is built on a recent paradigm in statistical physics, known as self-organized criticality. The basic idea is that flares are the result of an ‘avalanche’ of small-scale magnetic reconnection events cascading through a highly stressed coronal magnetic structure, driven to a critical state by random photospheric motions of its magnetic footpoints. Such models thus provide a natural and convenient computational framework to examine Parker's hypothesis of coronal heating by nanoflares. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1013301521745     

Solar and stellar flares

Short-lived increases in the brightness of many red dwarfs have been observed for the last 30 yr, and a variety of more or less exotic models have been proposed to account for such flares. Information about flares in the Sun has progressed greatly in recent years as a result of spacecraft experiments, and properties of coronal flare plasma are becoming increasingly better known. In this paper, after briefly reviewing optical, radio and X-ray observations of stellar flares, we show how a simplified model which describes conductive plus radiative cooling of the coronal flare plasma in solar flares has been modified to apply to optical and X-ray stellar flare phenomena. This model reproduces many characteristic features of stellar flares, including the mean UBV colors of flare light, the direction of flare decay in the two-color diagram, precursors, Stillstands, secondary maxima, lack of sensitivity of flare color to flare amplitude, low flux of flare X-rays, distinction between so-called spike flares and slow flares, Balmer jumps of as much as 6–8, and emission line redshifts up to 3000 km s^–1. In all probability, therefore, stellar flares involve physical processes which are no more exotic (and no less!) than those in solar flares. Advantages of observing stellar flares include the possibilities of (i) applying optical diagnostics to coronal flare plasma, whereas this is almost impossible in the Sun, and (ii) testing solar flare models in environments which are not generally accessible in the solar atmosphere.     

Predicting Solar Flares by Data Assimilation in Avalanche Models

Data assimilation techniques, developed in the past two decades mainly for weather prediction, produce better forecasts by taking advantage of both theoretical/numerical models and real-time observations. In this paper, we explore the possibility of applying the four-dimensional variational data assimilation (4D-VAR) technique to the prediction of solar flares. We do so in the context of a continuous version of the classical cellular-automaton-based self-organized critical avalanche models of solar flares introduced by Lu and Hamilton (Astrophys. J. 380, L89, 1991). Such models, although a priori far removed from the physics of magnetic reconnection and magnetohydrodynamical evolution of coronal structures, nonetheless reproduce quite well the observed statistical distribution of flare characteristics. We report here on a large set of data assimilation runs on synthetic energy release time series. Our results indicate that, despite the unpredictable (and unobservable) stochastic nature of the driving/triggering mechanism within the avalanche model, 4D-VAR succeeds in producing optimal initial conditions that reproduce adequately the time series of energy released by avalanches and flares. This is an essential first step toward forecasting real flares.     

Center-to-Limb Variability of Hot Coronal EUV Emissions During Solar Flares

It is generally accepted that densities of quiet-Sun and active region plasma are sufficiently low to justify the optically thin approximation, and this is commonly used in the analysis of line emissions from plasma in the solar corona. However, the densities of solar flare loops are substantially higher, compromising the optically thin approximation. This study begins with a radiative transfer model that uses typical solar flare densities and geometries to show that hot coronal emission lines are not generally optically thin. Furthermore, the model demonstrates that the observed line intensity should exhibit center-to-limb variability (CTLV), with flares observed near the limb being dimmer than those occurring near disk center. The model predictions are validated with an analysis of over 200 flares observed by the EUV Variability Experiment (EVE) on the Solar Dynamics Observatory (SDO), which uses six lines, with peak formation temperatures between 8.9 and 15.8 MK, to show that limb flares are systematically dimmer than disk-center flares. The data are then used to show that the electron column density along the line of sight typically increases by \(1.76 \times 10^{19}~\mbox{cm}^{-2}\) for limb flares over the disk-center flare value. It is shown that the CTLV of hot coronal emissions reduces the amount of ionizing radiation propagating into the solar system, and it changes the relative intensities of lines and bands commonly used for spectral analysis.     

High-energy observations of solar flares

Extensive observations of solar flares made in high energy bands during the maximum of the present solar cycle are discussed with a special reference to the results from HINOTORI, and with attention to the relevant flare models. The hard X-ray (HXR) images from HINOTORI showed mostly coronal emission at 20–25 keV suggesting that the HXR is emitted from multiple coronal loops, consistent with the non-thermal electron beam model in a high density corona. The thermal HXR model seems to be inconsistent with some observations. Three types of flares which have been classified from the Hinotori results are described, along with newly discovered hot thermal component of 30–40 million K which contributes thermal HXR emission. A summary is given for the characteristics of the energy release in an impulsive burst; and an empirical model is described, which explains simultaneous energy releases in multiple loops and successive movements of the release site as suggested from the HXR morphology. The discovery of large blue-shifted hot plasma from the soft X-ray line spectrum leads to some quantitative arguments for the evaporating flare model. An electron-heated flare atmosphere appears to explain various observations consistently.Invited paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Frequency distributions and correlations of solar X-ray flare parameters

We have determined frequency distributions of flare parameters from over 12000 solar flares recorded with the Hard X-Ray Burst Spectrometer (HXRBS) on the Solar Maximum Mission (SMM) satellite. These parameters include the flare duration, the peak counting rate, the peak hard X-ray flux, the total energy in electrons, and the peak energy flux in electrons (the latter two computed assuming a thick-target flare model). The energies were computed above a threshold energy between 25 and 50 keV. All of the distributions can be represented by power laws above the HXRBS sensitivity threshold. Correlations among these parameters are determined from linear regression fits as well as from the slopes of the frequency distributions. Variations of the frequency distributions were investigated with respect to the solar activity cycle.Theoretical models for the frequency distribution of flare parameters depend on the probability of flaring and the temporal evolution of the flare energy build-up. Our results are consistent with stochastic flaring and exponential energy build-up, with an average build-up time constant that is 0.5 times the mean time between flares. The measured distributions of flares are also consistent with predicted distributions of flares from computer simulations of avalanche models that are governed by the principle of self-organized criticality.     

The Radio Supernebula in NGC 5253

It was recently pointed out that the distribution of times between solar flares (the flare waiting-time distribution) follows a power law for long waiting times. Based on 25 years of soft X-ray flares observed by Geostationary Operational Environmental Satellite instruments, it is shown that (1) the waiting-time distribution of flares is consistent with a time-dependent Poisson process and (2) the fraction of time the Sun spends with different flaring rates approximately follows an exponential distribution. The second result is a new phenomenological law for flares. It is shown analytically how the observed power-law behavior of the waiting times originates in the exponential distribution of flaring rates. These results are argued to be consistent with a nonstationary avalanche model for flares.     

A device for precise orientation of solar images

The reconnecting current sheet model for energy accumulation and release during solar flares results in the flare frequency distribution that is a power-law function of total flare energy, with the index 7/4 for sufficiently large energies. The distribution is predicted to be much steeper in the low-energy region, implying the significance of microand nanoflares for coronal heating.     

Non-thermal processes in large solar flares

We analyze particle acceleration processes in large solar flares, using observations of the August, 1972, series of large events. The energetic particle populations are estimated from the hard X-ray and γ-ray emission, and from direct interplanetary particle observations. The collisional energy losses of these particles are computed as a function of height, assuming that the particles are accelerated high in the solar atmosphere and then precipitate down into denser layers. We compare the computed energy input with the flare energy output in radiation, heating, and mass ejection, and find for large proton event flares that:

1. The ～10–10² keV electrons accelerated during the flash phase constitute the bulk of the total flare energy.
2. The flare can be divided into two regions depending on whether the electron energy input goes into radiation or explosive heating. The computed energy input to the radiative quasi-equilibrium region agrees with the observed flare energy output in optical, UV, and EUV radiation.
3. The electron energy input to the explosive heating region can produce evaporation of the upper chromosphere needed to form the soft X-ray flare plasma.
4. Very intense energetic electron fluxes can provide the energy and mass for interplanetary shock wave by heating the atmospheric gas to energies sufficient to escape the solar gravitational and magnetic fields. The threshold for shock formation appears to be ～10³¹ ergs total energy in >20 keV electrons, and all of the shock energy can be supplied by electrons if their spectrum extends down to 5–10 keV.
5. High energy protons are accelerated later than the 10–10² keV electrons and most of them escape to the interplanetary medium. The energetic protons are not a significant contributor to the energization of flare phenomena. The observations are consistent with shock-wave acceleration of the protons and other nuclei, and also of electrons to relativistic energies.
6. The flare white-light continuum emission is consistent with a model of free-bound transitions in a plasma with strong non-thermal ionization produced in the lower solar chromosphere by energetic electrons. The white-light continuum is inconsistent with models of photospheric heating by the energetic particles. A threshold energy of ～5×10³⁰ ergs in >20 keV electrons is required for detectable white-light emission.

The highly efficient electron energization required in these flares suggests that the flare mechanism consists of rapid dissipation of chromospheric and coronal field-aligned or sheet currents, due to the onset of current-driven Buneman anomalous resistivity. Large proton flares then result when the energy input from accelerated electrons is sufficient to form a shock wave.     

Lasco and eit Observations of the Bastille day 2000 Solar Storm

The period of 10–14 July 2000 saw a large number of energetic solar events ending with a very energetic flare that was associated with a large solar energetic particle event and a fast halo coronal mass ejection (CME) that produced the largest geomagnetic disturbance since 1989. This paper tries to summarize the complex coronal activity observed during this period, in order to establish a background for a number of papers in this topical issue. The GOES X-ray data are presented. Data animations of observations from EIT and LASCO C2 and C3 are presented on the accompanying CD-ROM. The observations around the time of the three X-class flares are considered. EIT observations of the Bastille Day flare show coronal brightening followed by dimming. LASCO had good data coverage for all three events. For one of the flares, no coronal response was seen. The other two flares are associated with halo CMEs. The timing suggests that the start of the flares and CMEs are simultaneous to approximately 30 min. Analysis of the LASCO and EIT images following the Bastille Day flare show the arrival of energetic particles at SOHO at approximately 10:41 UT on 14 July. Individual features of these CMEs have been tracked and the height–time plots used to estimate the dynamics of the CMEs. The initial speed and deceleration of the halo CMEs estimated from the fitting of height–time plots are compared with the in-situ observations at L1. The three flares are identified as the solar sources of three shocks observed at 1 AU. Finally, it is stressed that global heliospheric effects during periods of exceptional activity should consider a cumulative scenario rather than events in isolation.     

A magnetic bald-patch flare in solar active region 11117

With SDO observations and a data-constrained magnetohydrodynamics(MHD)model,we identify a confined multi-ribbon flare that occurred on 2010 October 25 in solar active region 11117 as a magnetic bald patch(BP)flare with strong evidence.From the photospheric magnetic field observed by SDO/HMI,we find there are indeed magnetic BPs on the polarity inversion lines(PILs)which match parts of the flare ribbons.From the 3D coronal magnetic field derived from an MHD relaxation model constrained by the vector magnetograms,we find strikingly good agreement of the BP separatrix surface(BPSS)footpoints with the flare ribbons,and the BPSS itself with the hot flaring loop system.Moreover,the triggering of the BP flare can be attributed to a small flux emergence under the lobe of the BPSS,and the relevant change of coronal magnetic field through the flare is reproduced well by the pre-flare and post-flare MHD solutions,which match the corresponding pre-and post-flare AIA observations,respectively.Our work contributes to the study of non-typical flares that constitute the majority of solar flares but which cannot be explained by the standard flare model.     

Evidence of Magnetic Reconnection in Solar Flares and a Unified Model of Flares

The solar X-ray observing satellite Yohkoh has discovered various new dynamic features in solar flares and corona, e.g., cusp-shaped flare loops, above-the-loop-top hard X-ray sources, X-ray plasmoid ejections from impulsive flares, transient brightenings (spatially resolved microflares), X-ray jets, large scale arcade formation associated with filament eruption or coronal mass ejections, and so on. It has soon become clear that many of these features are closely related to magnetic reconnection. We can now say that Yohkoh established (at least phenomenologically) the magnetic reconnection model of flares. In this paper, we review various evidence of magnetic reconnection in solar flares and corona, and present unified model of flares on the basis of these new Yohkoh observations. This revised version was published online in July 2006 with corrections to the Cover Date.     

The curious temporal behavior of the frequency of different class flares

We investigate the frequency of all (X-ray flare events higher than class B1.0), B, C, M and X-class flares, respectively, derived from the National Geophysical Data Center (NGDC) list of solar flares between May 1983 and September 2014, which corresponds to the two complete solar cycles (SCs) 22 and 23 as well as the rise and maximum phases of SC 24. Analysis shows that the temporal behavior for these various class flares is quite different. The main findings of this study, confirmed by using the Hinode flare catalog where possible, are as follows. (1) The B-class flares are in complete antiphase with all, C, M and X-class flares. (2) While, there is a small decreasing trend in the peak values of the smoothed monthly C-class flare numbers from SC 22 to 24, the occurrence rate of M and X-class flares dropped by almost half and two-thirds, respectively, during SC 23 and remained almost the same during SC 24. This class-dependent temporal behavior provides support for dynamo models that involve the coexistence of a deep global and a superficial local dynamo.     

Predictive Capabilities of Avalanche Models for Solar Flares

We assess the predictive capabilities of various classes of avalanche models for solar flares. We demonstrate that avalanche models cannot generally be used to predict specific events because of their high sensitivity to the embedded stochastic process. We show that deterministically driven models can nevertheless alleviate this caveat and be efficiently used for predictions of large events. Our results suggest a new approach for predictions of large (typically X-class) solar flares based on simple and computationally inexpensive avalanche models.     

Interpretation of the observed motions of hard X-ray sources in solar flares

In connection with the RHESSI satellite observations of solar flares, which have revealed new properties of hard X-ray sources during flares, we offer an interpretation of these properties. The observed motions of coronal and chromospheric sources are shown to be the consequences of three-dimensional magnetic reconnection at the separator in the corona. During the first (initial) flare phase, the reconnection process releases an excess of magnetic energy related predominantly to themagnetic tensions produced before the flare by shear plasma flows in the photosphere. The relaxation of a magnetic shear in the corona also explains the downward motion of the coronal source and the decrease in the separation between chromospheric sources. During the second (main) flare phase, ordinary reconnection dominates; it describes the energy release in the terms of the “standard model” of large eruptive flares accompanied by the rise of the coronal source and an increase in the separation between chromospheric sources.     

Magnetic reconnection and coronal transients

Every two-ribbon flare observed during the Skylab period produced an observable coronal transient, provided the flare occurred close enough to the limb. The model presented here treats these two events as a combined process. Transients that occur without flares are believed to involve magnetic fields that are too weak to produce significant chromospheric emission. Adopting the hypothesis that the rising flare loop systems observed during two-ribbon flares are exhibiting magnetic reconnection, a model of a coronal transient is proposed which incorporates this reconnection process as the driving force. When two oppositely directed field lines reconnect a lower loop is created rooted to the solar surface (the flare loop) and an upper disconnected loop is produced which is free to rise. The magnetic flux of these upper loops is proposed as the driver for the transient. The force is produced by the increase in magnetic pressure under the filament and transient.A quantitative model is developed which treats the transient configuration in terms of four distinct parts- the transient itself with its magnetic field and material, the region just below the transient but above the filament, the filament with its magnetic field, and the reconnected flux beneath the filament. Two cases are considered - one in which all the prominence material rises with the transient and one in which the material is allowed to fall out of the transient. The rate of rise of the neutral line during the reconnection process is taken from the observations of the rising X-ray flare loop system during the 29 July, 1973 flare. The MHD equations for the system are reduced to four non-linear ordinary coupled differential equations which are solved using parameters believed to be realistic for solar conditions. The calculated velocity profiles, widths, etc., agree quite well with the observed properties of coronal transients as seen in white light. Since major flares are usually associated with a filament eruption about 10–15 min before the flare and since this model associates the transient with the filament eruption, we suspect that the transient is actually initiated some time before the actual flare itself.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Multi-Wavelength Signatures of Magnetic Reconnection of a Flare-Associated Coronal Mass Ejection

The evolution of an X2.7 solar flare, that occurred in a complex β γ δ magnetic configuration region on 3 November 2003 is discussed by utilizing a multi-wavelength data set. The very first signature of pre-flare coronal activity is observed in radio wavelengths as a type III burst that occurred several minutes prior to the flare signature in Hα. This type III burst is followed by the appearance of a loop-top source in hard X-ray (HXR) images obtained from RHESSI. During the main phase of the event, Hα images observed from ARIES solar tower telescope, Nainital, reveal well-defined footpoint (FP) and loop-top (LT) sources. As the flare evolves, the LT source moves upward and the separation between the two FP sources increases. The co-alignment of Hα with HXR images shows spatial correlation between Hα and HXR footpoints, whereas the rising LT source in HXR is always located above the LT source seen in Hα. The evolution of LT and FP sources is consistent with the reconnection models of solar flares. The EUV images at 195 Å taken by SOHO/EIT reveal intense emission on the disk at the flaring region during the impulsive phase. Further, slow-drifting type IV bursts, observed at low coronal heights at two time intervals along the flare period, indicate rising plasmoids or loop systems. The intense type II radio burst at a time in between these type IV bursts, but at a relatively greater height, indicates the onset of CME and its associated coronal shock wave. The study supports the standard CSHKP model of flares, which is consistent with nearly all eruptive flare models. More importantly, the results also contain evidence for breakout reconnection before the flare phase.     

Statistical Models for the Solar Flare Interval Distribution in Individual Active Regions

This article discusses statistical models for the solar flare interval distribution in individual active regions. We analyzed solar flare data in 55 active regions that are listed in the Geosynchronous Operational Environmental Satellite (GOES) soft X-ray flare catalog for the years from 1981 to 2005. We discuss some problems with a conventional procedure to derive probability density functions from any data set and propose a new procedure, which uses the maximum likelihood method and Akaike Information Criterion (AIC) to objectively compare some competing probability density functions. Previous studies of the solar flare interval distribution in individual active regions only dealt with constant or time-dependent Poisson process models, and no other models were discussed. We examine three models – exponential, lognormal, and inverse Gaussian – as competing models for probability density functions in this study. We found that lognormal and inverse Gaussian models are more likely models than the exponential model for the solar flare interval distribution in individual active regions. The possible solar flare mechanisms for the distribution models are briefly mentioned. We also briefly investigated the time dependence of probability density functions of the solar flare interval distribution and found that some active regions show time dependence for lognormal and inverse Gaussian distribution functions. The results suggest that solar flares do not occur randomly in time; rather, solar flare intervals appear to be regulated by solar flare mechanisms. Determining a solar flare interval distribution is an essential step in probabilistic solar flare forecasting methods in space weather research. We briefly mention a probabilistic solar flare forecasting method as an application of a solar flare interval distribution analysis. The application of our distribution analysis to a probabilistic solar flare forecasting method is one of the main objectives of this study.     

Characteristics of flares producing metric type II bursts and coronal mass ejections

We attempt to study the origin of coronal shocks by comparing several flare characteristics for two groups of flares: those with associated metric type II bursts and coronal mass ejections (CMEs) and those with associated metric type II bursts but no CMEs. CMEs accompany about 60% of all flares with type II bursts for solar longitudes greater than 30°, where CMEs are well observed with the NRL Solwind coronagraph. H flare areas, 1–8 Å X-ray fluxes, and impulsive 3 cm fluxes are all statistically smaller for events with no CMEs than for events with CMEs. It appears that both compact and large mass ejection flares are associated with type II bursts. The events with no CMEs imply that at least many type II shocks are not piston-driven, but the large number of events of both groups with small 3 cm bursts does not support the usual assumption that type II shocks are produced by large energy releases in flare impulsive phases. The poor correlation between 3 cm burst fluxes and the occurrence of type II bursts may be due to large variations in the coronal Alfvén velocity.Sachs/Freeman Associates, Inc., Bowie, MD 20715, U.S.A.