Logistic Avalanche Processes, Elementary Time Structures, and Frequency Distributions in Solar Flares

We analyze the elementary time structures (on timescales of ≈ 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) 10⁴ HXR pulses at ≥25 and ≥50 keV, (2) 4000 radio type III bursts, (3) 4000 pulses of decimetric quasi-periodic broadband pulsation events, and (4) 10⁴ 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 τ_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)∝F , where the power-law index α is directly related to the number of e-folding amplifications by the relation α = (1 + τ_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 flare 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 flares 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).