Frequency distributions: from the sun to the earth

Abstract

Abstract. The space environment is forever changing on all spatial and temporal scales. Energy releases are observed in numerous dynamic phenomena (e.g. solar flares, coronal mass ejections, solar energetic particle events) where measurements provide signatures of the dynamics. Parameters (e.g. peak count rate, total energy released, etc.) describing these phenomena are found to have frequency size distributions that follow power-law behavior. Natural phenomena on Earth, such as earthquakes and landslides, display similar power-law behavior. This suggests an underlying universality in nature and poses the question of whether the distribution of energy is the same for all these phenomena. Frequency distributions provide constraints for models that aim to simulate the physics and statistics observed in the individual phenomenon. The concept of self-organized criticality (SOC), also known as the "avalanche concept", was introduced by Bak et al. (1987Bak et al. ( , 1988, to characterize the behavior of dissipative systems that contain a large number of elements interacting over a short range. The systems evolve to a critical state in which a minor event starts a chain reaction that can affect any number of elements in the system. It is found that frequency distributions of the output parameters from the chain reaction taken over a period of time can be represented by power-laws. During the last decades SOC has been debated from all angles. New SOC models, as well as non-SOC models have been proposed to explain the powerlaw behavior that is observed. Furthermore, since Bak's pioneering work in 1987, people have searched for signatures of SOC everywhere. This paper will review how SOC behavior has become one way of interpreting the power-law behavior observed in natural occurring phenomenon in the Sun down to the Earth.

Stochastic Coupling of Solar Photosphere and Corona

“…The matching intervals of scales computed based on Equations (15) and (16) are shown by horizontal dash-dotted vertical lines in Figure 5 and vertical lines in Figure 7. It can be seen that the distributions decay algebraically, consistent with earlier observations of power-law statistics in solar photosphere and corona (Crosby et al 1993;Abramenko et al 2002;Aschwanden & Parnell 2002;Aschwanden 2011;Crosby 2011). The broadband power-law behavior of MDI distributions suggests that the dynamics of the LOS photospheric magnetic field is an inherently multiscale process (Uritsky & Davila 2012).…”

“…SOC seeks to explain power-law distributions of flare parameters via cooperative interactions of a large number of nonlinearly coupled degrees of freedom representing unstable coronal loops and/or loop strands (Crosby et al 1993;Crosby 2011;Charbonneau et al 2001;Aschwanden et al 2000b;Aschwanden & Parnell 2002;Morales & Charbonneau 2008;Uritsky et al 2007). It has been suggested that the nonpotential magnetic field configurations existing in the corona release their free energy through a chain interaction of multiple spatially localized instabilities such as those associated with nanoflare heating (Parker 1988;Klimchuk 2006;Viall & Klimchuk 2011).…”

Occurrence rate of extreme magnetic storms

“…In accordance with numerous papers (see, for example, recent review by Crosby []), various events on the Sun and the Earth have similar shape of distributions: a top at small values, then a gentle slope and power law tail at large values. For instance, Tsubouchi and Omura [] showed that the distribution function of minimal D s t index for strong magnetic storms from 1957 to 2001 F ( D s t min ) has power law index γ ( F )∼−4.9.…”

“…Therefore, we also make a square approximation in log‐log space (see Table and thick lines in Figure ). The square law dependencies are too steeply decreasing, providing lower limits, and they do not agree with the conclusion about power law tails at large values [ Tsubouchi and Omura , ; Crosby , ; Riley , ]. We make two additional approximations for MC: (1) using the two last points ( γ ( P MC )∼−2.5, dash‐dotted line in Figure ) and (2) with fixed γ ( P MC )=−2.9 which gives us a waiting time T MC (−1700)∼1500 years, as calculated on the basis of F ∗ data (dashed line in Figure ).…”

“…One of the main problems of solar‐terrestrial physics and space weather is the study and prediction of magnetic storms including extreme ones (see, e.g., recent papers and reviews by Echer et al [], Alves et al [], Podladchikova and Petrukovich [], Yakovchouk et al [], Yermolaev et al [], and references therein). A useful technique for solving these types of problems is the calculation of frequency (occurrence rate) distributions of events based on observations in the form d N = F ( x )d x , where d N is the number of events recorded with the parameter x of interest between x and x +d x , and F ( x ) is a frequency distributions (see, e.g., recent reviews and papers by Koons [], Tsubouchi and Omura [], Crosby [], Gorobets and Messerotti [], Riley [], Love [], and references therein). The D s t index (or its proxies) calculated from the observations of the horizontal magnetic field at four low‐latitude to midlatitude ground stations is used to indicate the value of magnetic storms [ Sugiura , ; Sugiura and Kamei , ].…”