Statistical models of flares
Abstract
By 'statistical' models of flares we denote the global stochastic models of the dynamics of the energy-release process and its associated phenomena which consider flares to consist in a large number of constituent small-scale processes. The observations strongly support such a kind of models: a) Radio-and HXR-emission of flares are highly fragmented in space and time, suggesting that the flare process itself is spatially and temporarily fragmented (De Jager and De Jonge 1978, Benz 1985, Aschwanden et al. 1990). b) The temporal dynamics of flares has been shown to be 'complex' (relatively high-dimensional chaotic or stochastic) through time-series analysis of radio-emission (dimension-estimate and power-spectra: Isliker and Benz (1994), Isliker (1996), Ryabov et al. (1997); wavelet transform: Aschwanden et al. 1998, Schwarz et al. 1998). c) Spatially, there are only weak and local correlations between neighbouring burst-sites, reminiscent of a chain-reaction (analysis of nb-spikes spectrograms with symbolic dynamics: Schwarz et al. 1993). The most prominent global dynamical models of the energy-release process which comprise entire flares are Cellular Automata (CA) models (Lu and Hamilton 1991, Lu et al. 1993; extended to model nano-flares: Vlahos et al. 1995, Georgoulis and Vlahos 1996; including non-local communications: MacKinnon et al. 1996; an analytic approach: MacKinnon and MacPherson 1997). In these models, the local processes (reconnection) are modeled in a strongly simplified way, by simple evolution rules, so that inhomogeneous active regions can be modeled entirely. Alternatively, Isliker (1996) proposed a shot noise model for flares. This model is able to explain the temporal characteristics of the flare-process, however, it is formal, so-far, it has not been tied to physics, yet. A different class of stochastic models has been proposed to explain the dynamics of the corona as a whole, with randomly occurring flares (Rosner and Vaiana 1978, criticized in Lu 1995b; Litvinenko 1996; a new approach (a master equation for the flare occurrence probability): Wheatland and Glukhov 1998). In this approach, structures within a flare are not resolved, the aim is to explain the occurrence rate and total sizes of flares. The CA models are successful in explaining the distributions of the peak-fluxes, total fluxes, and durations of HXR-emission, which are all power-laws (see references in Aschwanden et al. 1998). In the radio range, peak-flux distributions of generalized power-law and exponential shape are observed, which generally are steeper than in the HXR (type I: Mercier and Trottet (1997); type III, decim. pulsations, nb-spikes: Aschwanden et al. 1998; type III: Isliker and Vlahos 1998; nb-spikes: Isliker and Benz 1998). Since radio-waves can be emitted in low energy events, the steep distributions might be a hint that small flares (micro-flares) have a steep distribution, too, and might therewith substantially contribute to coronal heating. It must be noted, however, that poor time- or frequency-resolution can lead to a steepening of the peak-flux distributions (Isliker and Benz 1998), an effect whose influence on the published events has to be discussed, still. Originally, the evolution rules of the CAs were only loosely motivated through physical considerations and basically taken from the 'sand-pile' paradigm, above all the connection between CA and MHD (the local theory of magnetic reconnection) was missing. Recently, Isliker et al. (1998) have shown that the evolution rules of the CAs correspond to localized, threshold dependent diffusion, implementing directly the solution of a diffusion equation, with unknown diffusivity and scales. Thus, CAs can be interpreted as an implementation of the (simplified) induction equation in a large, inhomogeneous medium. A complete flare model needs to incorporate not just the energy release process, but also the acceleration and transport of particles, as well as the generation of EM-emission. First steps towards this direction are done: Anastasiadis et al. 1997 studied acceleration of electrons in a CA environment. Vlahos and Raoult (1995), and Isliker and Vlahos (1998) constructed a model for type III events (electron beams): they assumed a population of accelerated electrons traveling in a fragmented coronal volume and emitting radio-waves due to plasma-instabilities. Radio-spectrograms were calculated, and it was shown that the resulting peak-flux distributions are compatible with observed ones. Still missing is a model for the emission of HXR, based on a CA, which would allow to compare more thoroughly the CA models to the observations. Future work should be towards a completion and extension of these first trials, aiming at a model where the evolution rules of the CAs are derived from MHD, particles are accelerated and propagate in a fragmented corona and subsequently emit in the radio- and HXR-range.
- Publication:
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CESRA Workshop on Coronal Explosive Events
- Pub Date:
- 1998
- Bibcode:
- 1998cee..workE...8I