Intensity Thresholds and the Statistics of the Temporal Occurrence of Solar Flares

Abstract

Introducing thresholds to analyze time series of emission from the Sun enables a new and simple definition of solar flare events, and their interoccurrence times. Rescaling time by the rate of events, the waiting and quiet time distributions both conform to scaling functions that are independent of the intensity threshold over a wide range. The scaling functions are well described by a two parameter function, with parameters that depend on the phase of the solar cycle. For flares identified according to the current, standard definition, similar behavior is found.PACS numbers: 96.60. Rd, 05.45.Tp, 05.65.+b The solar corona is a very high Reynolds number turbulent plasma producing intermittent bursts of radiation. Plasma forces twist the coronal magnetic fields until stresses are suddenly released, an avalanching process governed by magnetic reconnection [1]. The released magnetic energy induces radiative emission that can be detected as a flare. Flares exhibit scale invariant statistics. For instance, the probability distribution of flare energies is a power law spanning more than eight orders of magnitude [2,3], similar to the Gutenberg-Richter law for earthquakes. The distribution of magnetic concentration sizes on the photosphere is also scale invariant, and the coronal magnetic network embodies a scale-free network [4,5]. In fact, a model of self-organized criticality (SOC) with avalanches of reconnecting flux tubes reproduces the observed scale-free network structure [4,6].As part of the debate on the characterization of magnetohydrodynamic turbulence in this regime [1,4,5,6,7,8,9], interest has focused on comparing interoccurrence times between flares with those in models of SOC. Analyses of flare catalogs have indicated scale invariance of the waiting times, but the behavior was found to vary with the phase of the solar cycle [10] and with the methods used to analyze the catalogs. (See e.g. Ref. [10,11].) The prior belief that avalanches occur with Poissonian waiting times in the well-known BTW sandpile model [12] (giving an exponential distribution of waiting times) argued against the SOC hypothesis [8]. However, including a finite detection threshold leads to a power law distribution of quiet times even for the BTW model [13], when durations and quiet times are measured with the same clock. Since the turnover time scale for flux to be regenerated in the corona is of the order of ten hours [14], while the correlated waiting time intervals between flares can extend up to years, the physical mechanism(s) responsible for these correlations resides in the turbulent convective region beneath the photosphere that generates magnetic flux and drives it into the corona. Systematic studies of the temporal pattern of flares can give insight into the dynamics of magnetic flux in the convective region, or the solar dynamo, which is difficult to observe directly.Here we show that the interoccurrence times between flares has a hierarchical scaling structure when flares are defined as intervals during which the emissi...

Superstatistics, thermodynamics, and fluctuations

“…[1], a lot of efforts have been made for further theoretical elaboration [5][6][7][8][9][10][11][12]. At the same time, it has also been applied successfully to a variety of systems and phenomena, including hydrodynamic turbulence [9,13,14], pattern formation [15], cosmic rays [16], solar flares [17], mathematical finance [18][19][20], random matrices [21], complex networks [22], wind velocity fluctuations [23], and hydro-climatic fluctuations [24].…”

Superstatistics: Theoretical Concepts and Physical Applications

“…Swinney et al [2,38] extract lognormal superstatistics for turbulent flow between counterrotating disks. Paczuski et al [13] study data of solar flares on various time scales and embedd this into a superstatistical model based on χ 2 -superstatistics. Human behaviour when sending off print jobs might also stand in connection to such a superstatistics [39].…”

“…The two effects produce a superposition of two statistics, or in a short, a 'superstatistics' [1,2,3,4,5,6,7,8]. The concept of a superstatistics was introduced in [1], in the mean time many applications for a variety of complex systems have been pointed out [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. The stationary probability distributions of superstatistical systems typically exhibit non-Gaussian behaviour with fat tails, which can decay e.g.…”

“…Recent successful applications include hydrodynamic turbulence [9,10,2,11,12], pattern forming systems [14], cosmic rays [15], solar flares [13], mathematical finance [17,18,19], random matrix theory [20], networks [21], quantum systems at low temperatures [23], wind velocity fluctuations [16,25], hydro-climatic fluctuations [22] and delay statistics in traffic models [24]. The aim in the following is to explain the basic concepts and applications in an easy-going way.…”

Generalized statistical mechanics for superstatistical systems

“…The environment is represented by a suitable parameter entering the stochastic differential equation describing the mesoscopic system. The superstatistics concept can be applied in quite a general way, and a couple of interesting applications for a variety of complex systems have been pointed out recently [17][18][19][20][21][22][23][24][25][26][27][28]. Essential for this approach is the existence of sufficient time-scale separation so that the system has enough time to relax to a local equilibrium state and to stay within it for some time.…”