We discuss the main findings of our preceding data analysis in terms of models for impulsive energy releases in the solar corona. For an overview of the processes involved in the radio and related hard X-ray emissions from those energy release events see, e.g., Benz (1993), Benz & Aschwanden (1992).
A general approach to energy release events in the solar corona is provided by the avalanche model (Lu & Hamilton 1991; Lu et al. 1993; Vlahos et al. 1995). This statistical model supposes that there is an elementary building block of the energy release processes below the limit of resolution, possessing a threshold which depends only on local conditions, and that the corona is in a state of so-called self-organized criticality, i.e., everywhere and at all times close to onset of an energy release event. This state results from the interplay of external driving (energy input) and more or less localized energy release, where the energy release may organize itself into avalanches of all sizes, since every occurrence of the elementary building block may trigger further occurrences in neighboring regions due to spatial redistribution of stresses and free energy. Interpreting flare events as such avalanches of the elementary process, the model succeeded to reproduce the observed power-law distribution of the occurrence rate of flares versus energy content (Crosby et al. 1993). In order to obtain the power-law, one has to assume that energy release processes in the corona have no characteristic length scale greater than that of the elementary building block (Lu & Hamilton 1991). Then the occurrence rate of avalanches is a monotone decreasing function of energy content. Although the very nature of the energy release process is not addressed by this approach, it basically supposes that coronal energy release processes of all sizes have the same physical nature. This supposition is supported by our finding of structural similarity of pre-burst, burst, and post-burst phases, respectively. We suppose that the analyzed radio bursts are also composed of elementary building blocks (flux enhancements) not resolved by the available instrumentation, as suggested by their time profiles. One important difference between the scalegrams (Figs. 11 (click here)-14 (click here)) and the flare occurrence rate distribution needs to be clarified, however. While the latter is derived from a time series (several years long) in which every event occurs separately (i.e., localized in time), each of the former is derived from a time series of one single flare event, which is understood as a superposition of elementary energy release contributions, overlapping in time. Small-scale radio flux enhancements are then underrepresented in the scalograms and scalegrams, since in general they do not stand out clearly in the time profile of overlapping small-scale and large-scale flux enhancements.
The wavelet analysis and the structure function analysis both go beyond the description in terms of event occurrence rates by investigating the structure of single events. This refers to the question whether the energy release in one single flare or burst is composed of a number of avalanches, i.e., is fragmentary. If this is the case, it would be of interest whether the fragments possess a certain size, which would show up as a dominant time scale in the scalegrams. No signs of preferred avalanche sizes (maxima in the scalegrams) were found within the considered range of time scales, which again supports one of the basic assumptions of the avalanche model. We note, however, that there are indications of enhanced occurrence of s spikes in hard X-ray time profiles of solar flares (Aschwanden et al. 1995). Work is currently underway to extend the time series analysis of radio bursts into this range of scales.
An understanding of the relative contribution of the various time scales to the total radio flux profile and possible evolutions of the structure of energy release events from the pre-burst to the main and post-burst phase can only be understood in terms of more detailed plasma physical models of the energy release process(es). We restrict ourselves to a brief reference to two currently discussed models, which have both been found consistent with observations of subsecond flux variations in solar bursts (Krüger et al. 1994), viz. the current sheet model and the electric circuit model. We remark that the current sheet model has recently found substantial observational support from X-ray observations of the Yohkoh satellite (Masuda et al. 1994; Tsuneta 1996).
The current sheet model involves the formation and dynamical evolution of current filaments at small spatial scales (Tajima et al. 1987; Kliem 1995). MHD simulations of current sheet instabilities at high magnetic Reynolds numbers show that the dynamical evolution leads to the pileup of gradients, i.e., the formation of smaller scales than initially present (Schumacher & Kliem 1996). Since the electric field, which accelerates the particles, peaks in those regions, an enhancement of the small-scale contributions in the flux of fast particles (which is finally reflected by the radio flux) can be expected. Of course, the release of a large amount of energy during the burst phase (a large avalanche) requires the contribution from a large volume (an extended current sheet), hence large temporal scales will significantly contribute to the overall time profile if the release process involves a roughly uniform characteristic velocity (which is the Alfvén velocity).
The electric circuit model involves a combination of MHD and kinetic instabilities (Zaitsev & Stepanov 1991, 1992) and, consequently, a broad range of time scales. Characteristically, scales are predicted by this model for subflares, which is roughly comparable to the maxima in the scalograms obtained here. A more thorough discussion of the two energy release models will become possible if the multiresolution analysis can be applied to burst data with a time resolution .
Finally we note that the reduced spectral index in the post-burst phases in Figs. 11 (click here) and 12 (click here) is consistent with an important role of magnetohydrodynamic turbulence in the energy release process. After the main energy release, the energy in the turbulent motions cascades down to smaller scales where it is dissipated into heat, which shows up as a reduced slope in the scalegram.