Logistic avalanche processes, elementary time structures, and frequency distributions in solar flares

We analyze the elementary time structures (on timescales of approximate to 0.1-3.0 s) and their frequency distributions in solar flares using hard X-ray (HXR) data from the Compton Gamma Ray Observatory (CGRO) and radio data from the radio spectrometers of Eidgenoessische Technische Hochschule (ETH) Zurich. The four analyzed data sets are gathered from over 600 different solar flares and include about (1) 10(4) HXR pulses at greater than or equal to 25 and greater than or equal to 50 keV, (2) 4000 radio type III bursts, (3) 4000 pulses of decimetric quasiperiodic broadband pulsation events, and (4) 10(4) elements of decimetric millisecond spike events. The time profiles of resolved elementary time structures have a near-Gaussian shape and can be modeled with the logistic equation, which provides a quantitative measurement of the exponential growth time tau(G) and the nonlinear saturation energy level W-S of the underlying instability. Assuming a random distribution (Poisson statistics) of saturation times t(S) (with an e-folding constant t(Se)), the resulting frequency distribution of saturation energies W-S or peak energy dissipation rates F-S = (dW/dt)(t=tS) has the form of a power-law function, i.e., N(F-S) proportional to F-S(-alpha), where the power-law index alpha is directly related to the number of e-folding amplifications by the relation alpha = (1 + tau(G)/t(Se)). The same mathematical formalism is used to generate power-law distributions, as in Rosner & Vaiana, but the distribution of energies released in elementary hare instabilities is not related to the global energy storage process. We assume Poissonian noise for the unamplified energy levels in unstable flare cells, implying an exponential frequency distribution of avalanche energies W-S or fluxes F-S in the absence of coherent amplifications. Also, in the case of coherent amplification, the Poissonian noise introduces exponential rollovers of the power law at the low and high ends of the frequency distributions. We fit both power-law and exponential functions to the observed frequency distributions of elementary pulse fluxes N(F) in each flare separately. For HXR pulses, one-half of the flares are better fitted with power-law frequency distributions, demanding coherent amplification of the underlying energy dissipation mechanism, e.g., current exponentiation occurring in the magnetic tearing instability. The majority of type III burst pares are best fitted with power-law distributions, consistent with the interpretation in terms of beam-driven coherent emission. The frequency distributions of decimetric pulsations and decimetric millisecond spikes are found to fit exponential functions, in contrast to the expected power laws for coherent emission mechanisms generally proposed for these radio burst types. A coherent emission mechanism can be reconciled with the observed exponential frequency distributions only if nonlinear saturation occurs at a fixed amplification factor for all elementary pulses or spikes, for example, if it is produced by an oscillatory compact source (in the case of decimetric pulsations) or by a fragmented source with similar spatial cell sizes (in the case of decimetric millisecond spikes).