Correlation of the sunspot number and the waiting time distribution of solar flares, coronal mass ejections, and solar wind switchback events observed with the Parker Solar Probe
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Correlation of the sunspot number and the waiting time distribution of solar flares, coronal massejections, and solar wind switchback events observed with the Parker Solar Probe
Markus J. Aschwanden and Thierry Dudok de Wit Solar and Stellar Astrophysics Laboratory (LMSAL), Palo Alto, CA 94304, USA Laboratory de Physique et Chimie de l’Environnement et de l’Espace, LPC2E, CNRS/CNES/University of Orl´eans, 3A Av. de laRecherche Scientifique, 45071 Orl´eans cedex 2, France
ABSTRACTWaiting time distributions of solar flares and coronal mass ejections (CMEs) exhibit power law-likedistribution functions with slopes in the range of α τ ≈ . − .
2, as observed in annual data setsduring 4 solar cycles (1974-2012). We find a close correlation between the waiting time power lawslope α τ and the sunspot number (SN) , i.e., α τ = 1.38 + 0.01 × SN. The waiting time distributioncan be fitted with a Pareto-type function of the form N ( τ ) = N ( τ + τ ) − α τ , where the offset τ depends on the instrumental sensitivity, the detection threshold of events, and pulse pile-up effects.The time-dependent power law slope α τ ( t ) of waiting time distributions depends only on the globalsolar magnetic flux (quantified by the sunspot number) or flaring rate, independent of other physicalparameters of self-organized criticality (SOC) or magneto-hydrodynamic (MHD) turbulence models.Power law slopes of α τ ≈ . − . Parker Solar Probe (PSP) . We conclude that the annual variability of switchback events in theheliospheric solar wind is modulated by flare and CME rates originating in the photosphere and lowercorona.
Keywords:
Solar flares — Solar wind — Statistics INTRODUCTIONWaiting times, also called elapsed times , inter-occurrence times , inter-burst times , or laminar times , are defined bythe time interval between two subsequent events, i.e., τ i = ( t i +1 − t i ), measured from a time series t i = 1 , ..., n t ofevents. Simple examples are: (i) periodic processes (where the waiting time is constant and is equal to the time period);(ii) random, Poissonian, or Markov point processes (where the distribution of waiting times follows an exponentialfunction); (iii) exponentially growing avalanche processes (where the waiting time distribution matches a scale-freepower law-like function, as it is common in self-organized criticality (SOC) processes), (iv) magneto-hydrodynamic(MHD) turbulence processes (where power spectra can be represented by piece-wise power law functions of waitingtime distributions, or (v) sympathetic flaring, which is an effect that is not consistent with independent flaring events(Wheatland et al. 1998). Hence, the study of waiting time distributions, applied to solar flares here, can be a powerfultool to identify and disentangle the relevant physical processes, in particular in connection with physical scaling laws(Aschwanden 2020).In solar physics, the waiting distribution functions of solar flares has been found to be dominantly power law-like(Boffeta et al. 1999; Lepreti et al. 2001; Grigolini et al. 2002; Wheatland 2000a, 2003; Aschwanden and McTiernan2010), unless the sample of waiting times covers a too small range or is incompletely sampled otherwise, due to selectioneffects (e.g., in Pearce et al. 1993; Crosby 1996), as demonstrated in comparison with larger and more complete datasets (Aschwanden and McTiernan 2010). The observationally established result of power law functions in the waiting Corresponding author: Markus J. [email protected]@cnrs-orleans.fr time distribution of solar flares rules out a stationary Poisson process and requires an alternative explanation in terms ofnon-stationary Poisson processes (Wheatland 2000a; 2003), self-organized criticality (SOC) models (Aschwanden andMcTiernan 2010; Aschwanden and Freeland 2012; Aschwanden et al. 2014), or MHD turbulence (Boffetta et al. 1999;Lepreti et al. 2001; Grigolini et al. 2002). It was argued that SOC avalanches occur statistically independently (Baket al. 1987, 1988), and thus would predict an exponential waiting time distribution (Boffetta et al. 1999). However,inclusion of a driver with time-dependent variations, such as the solar cycle variability (Aschwanden 2011b), leads toa non-stationary Poisson process with power law behavior (Wheatland 2000a, 2003). In this Paper we study the timevariability of the power law slope of waiting times in more detail and find a strong correlation between the level ofsolar activity in terms of the sunspot number SN(t), the annual flaring rate λ ( t ), and the power law slope α τ ( t ) ofthe waiting time distribution, which can be explained by their common magnetic drivers. Obviously, the variabilityof solar activity represents the most dominant physical process that modulates the value of the power law slope α τ ( t )in solar flare waiting time distributions, an argument that has not received much attention (except in Wheatland andLitvinenko 2002; Wheatland 2003), but it puts previous modeling of waiting time distributions into a new light.New results emerge also from the variability of the solar wind, especially from the renewed interest in switchbackevents, as observed by the Parker Solar Probe (PSP) mission. Switchbacks are sudden deflections of the magneticfield that have been found to be ubiquitous in the inner heliosphere. These events are likely to play an important rolein structuring the young solar wind (Mozer et al. 2002; Tenerani et al. 2020; Horbury et al. 2020; Dudok de Wit etal. 2020). Their origin, however, remains elusive.The structure of the paper entails observations and data analysis in Section 2, discussions in Section 3, and conclusionsin Section 4. OBSERVATIONS AND DATA ANALYSIS2.1.
Previous Studies of Solar Flare Hard X-rays
In Table 1 we compile power law slopes α τ of published waiting time distributions of solar flares, which we brieflydescribe in turn.The study of Pearce et al. (1993) determines a total of 8319 waiting times (within a time range of τ = 1 −
60 minduring the observation period of 1980-1985), using data from the
Hard X-ray Burst Spectrometer (HXRBS) onboardthe
Solar Maximum Mission (SMM) . Similarly, Biesecker (1994) detects a total of 6596 waiting times, using
Burst AndTransient Source Experiment (BATSE) onboard the
Compton Gamma Ray Observatory (CGRO) . A much smallerdata set with 182 waiting times was obtained from the
Wide Angle Telescope for Cosmic Hard X-rays (WATCH) onboard the GRANAT satellite (Crosby 1996). More extended statistics in hard X-ray wavelengths was gathered fromthe
International Cometary Explorer (ICE) onboard the
International Sun/Earth Explorer (ISEE-3) (Wheatland etal. 1998), HXRBS/SMM (Aschwanden and McTiernan 2010), BATSE/CGRO (Grigolini et al. 2002; Aschwanden andMcTiernan 2010), and the
Ramaty High Energy Solar Spectroscopic Imager (RHESSI) (Aschwanden and McTiernan2010).These studies are all based on hard X-ray solar flare catalogs, where waiting times are found within a range of τ ≈ . − F HXR > ∼
100 cts s − at hardX-ray energies of E HXR > ∼
25 keV. Most waiting time distributions are self-similar (or scale-free) and consequentlyexhibit a power law scaling. We find only two exceptions, one is an exponential case, and another is a double-exponential case, see column 5 in Table 1. Thus, most of the waiting time distributions of hard X-ray data can befitted with a power law function, with a typical slope of α τ ≈ . α τ < ∼ . Previous Studies of Solar Flare Soft X-rays
More waiting time distributions of solar flares were sampled in soft X-ray wavelengths, making use of the 1-8 ˚A fluxdata observed with the
Geostationary Orbiting Earth Satellite (GOES) , which yields an uninterrupted time series ofup to 47 years (from 1974 to present). Subsets of GOES time series in different year ranges were analyzed by Boffettaet al. (1999), Wheatland (2000a, 2003), and Lepreti et al. (2001). These data are ideal for waiting time statistics,since the duty cycle of GOES data is very high (94%) thanks to the geostationary orbit. The number of solar flaresabove a threshold of the C-class level (10 − W m − ) amounts to 35,221 events, while a deep survey with automatedflare detection and with ≈
10 times higher sensitivity revealed a total of 338,661 events (Aschwanden and Freeland2012). This data set provides the largest statistics of waiting times and is investigated in Section 2.5. All waitingtime distributions could be fitted with a “thresholded” power law distribution (Aschwanden 2015), also called Paretofunction (Eq. 6), or Lomax function (Lomax 1954; Hosking and Wallis 1987), which yields here power law slopes inthe range of α τ ≈ . − . σ τ ≈ . − .
2, which is much smaller than the spread of the slope values α τ ≈ . − .
2, andthus cannot be explained with a single theoretical constant.2.3.
Previous Studies of Coronal Mass Ejections
There is a general consensus now that solar flare events and coronal mass ejections (CMEs) have very high mutualassociation rates, especially for eruptive flare events, so that the event statistics of one event group can be used asa proxy of the other event group. Hence we expect that the waiting time distributions of solar flares and CMEs aresimilar. Wheatland (2003) sampled the waiting time distributions of CMEs based on
Large Angle Solar SpectrometricCoronagraph (LASCO) data from the
Solar and Heliospheric Observatory (SOHO) spacecraft. These waiting timedistributions could be fitted with thresholded power law (or Pareto) functions also, with power law slopes in the rangeof α τ ≈ . − .
0, sampled in different time phases of the solar cycle (see column 6 in Table 1). It turns out that theflare waiting times produce similar power law slopes α τ as the CME waiting times, if one compares data from thesame phase of the solar cycle. 2.4. Solar Cycle Dependence of Waiting Times
Wheatland (2003) examined the distribution of waiting times between subsequent CMEs in the LASCO CMEcatalog for the years 1996-2001 and found a power-law slope of α τ ≈ . ± .
11 for large waiting times (at ≥
10 hours).Wheatland (2003) noted that the power law index of the waiting-time distribution varies with the solar cycle: forthe years 1996-1998 (a period of low activity), the power-law slope is α τ ≈ . ± .
14, and for the years 1999-2001(a period of higher activity), the slope is α τ ≈ . ± .
20. Wheatland (2003) concluded that the observed CMEwaiting time distribution, and its variation with the solar cycle, may be understood in terms of CMEs occurring as atime-dependent Poisson process. The same can be said for solar flares, since flares and CMEs are highly correlated.In order to prove this hypothesis, we investigate the functional relationship between the waiting time power lawslope α τ α τ as a function of the annually averagedsunspot numbers SN in Fig. 1 and find a linear relationship of α τ = 1 .
38 + 0 . × SN , (1)as confirmed by the high value of the Pearson’s cross-correlation coefficient (CCC=0.987). The sunspot number variesfrom SN ≈
40 during the solar cycle minimum to SN ≈
160 during the solar cycle maximum. Interestingly, this resultis robust in the sense that it predicts the same linear relationship even when different event detection thresholds,different phenomena (flares, CMEs, solar wind switchbacks), and different wavelengths (hard X-rays, soft X-rays) areused. This empirical linear relationship explains a variety of observed power law slopes for waiting times, covering arange of α τ ≈ . − . Annual GOES Waiting Time Statistics
The published power law slopes mentioned in the previous Section mostly cover multi-year ranges. In the followingtask we examine the degree of correlation between the annual power law slopes of waiting times and the annual sunspotnumbers from the entire 47-year data set of GOES flares. We break the data sets down into 39 annual groups (from1974 to 2012), for the same time epoch as we performed automated flare detection in a previous study (Aschwandenand Freeland 2012). For each of the 39 data sets we fit a thresholded power law (or Pareto-type) distribution) (Eq. 6),parameterized with ( N , τ , α τm ).The published power law slopes α τ are usually fitted in the inertial range ( x , x ) and have a mean value thatcorresponds to the (logarithmic) midpoint x ≈ ( x + x ) / α τm at the upper end x , where the slope is steepest. In order to make the two methods compatible, we can define an empirical correctionfactor q m , for which we find an average value of q m ≈ . α τ = q m α τm ≈ . α τm . (2)Applying this correction to the upper limits α τm (crosses in Fig. 3a), the corrected power law slopes α τ (crosses inFig. 3b) become fully compatible with the published power law values α τ (diamonds in Figs. 1, 3a, and 3b) based onstandard power law fitting methods in the observed inertial ranges.We present the fits of the Pareto distribution functions (Eq. 6) in Fig. 2 and list the years, the number of events n ev ,the power law slope α τ of the waiting times, the goodness-of-fit χ , the lower x and upper bounds x of the fittedrange, the decades of the inertial range log ( x /x ), and the annual sunspot number SN in Table 2.Plotting the time evolution of the power law slope α τ ( t ) in annual steps (Fig. 4a), we see a solar cycle variation thatis similar to the time variation of the annual sunspot number SN(t) (Fig. 4b), or the flaring rate per year (Fig. 4c).Fitting a linear regression between the two parameters, we find a linear relationship of α τ = 1 .
442 + 0 . × SN(Fig. 5a), with a correlation coefficient of CCC=0.961. This correlation corroborates the result of a linear relationshipbetween the power law slope and the sunspot number from previous work (Fig. 1).The value of the power law slope α τ , after the inertial range correction, appears not to depend on event detectionthresholds, efficiency of event detection, and pulse pile-up effects. For long-duration flare events, short flares couldpile-up upon on long tails, violating the separation of time scales, and possibly steepening the power law slope ofwaiting times. Our automated flare detection is about 10 times more sensitive than the standard NOAA flare catalog,while the GOES C-class level is often used as a threshold in statistical studies.2.6. Switchback Events detected with the Parker Solar Probe
More recently, the Parker Solar Probe (PSP) (Fox et al. 2015) has revealed the omnipresence of so-called switchbackevents in Alfv´enic solar wind streams within a distance of 50 solar radii from the Sun. These events show up as suddendeflections of the (otherwise mostly radial) magnetic field of the young solar wind. Because switchbacks can easily beidentified as discrete events, they offer yet another opportunity for investigating how waiting time distributions areinfluenced by solar activity.Dudok de Wit et al. (2020) analyzed a time interval that is centered on the first perihelion pass of November 6, 2018and runs from November 1st to November 10, 2018. The vector magnetic field onboard PSP is measured by the MAGmagnetometer from the FIELDS instrument suite (Bale et al. 2016). The waiting time and residence time scales ofswitchbacks occur predominantly in the inertial range of the solar wind: the sampled waiting time distribution coversa range of τ ≈ − − hours. The power law slope varies in a range of α τ ≈ . − . n ev =19,543 individual switchback events that havebeen automatically selected by feature recognition using magnetic field data sampled at 0.1 s. Switchback events aredefined here as a step function in the deflection of the magnetic field with respect to the orientation of the Parkerspiral. These events are required to be aligned for some duration before and after the event. Waiting times with τ <
10 s are believed to be biased by smaller events, and waiting times with τ < α τ = 1 . ± .
01 and a goodness-of-fit χ = 2 .
21, which indicates that a power law with an exponential cutoff is a morerealistic description of the waiting time distribution. DISCUSSION3.1.
Pareto-Type Waiting Time Distribution Function
The statistics of waiting times bears information that enables us to discriminate between two statistical distribu-tion functions: (i) random processes with Poissonian noise, and (ii) clustering of events within individual intervals(such as Omori’s law for earthquakes, which exhibits precursors and aftershock events during a major complex earth-quake event). A Poissonian process in the time domain is a sequence of randomly distributed and thus statisticallyindependent events, producing a waiting time distribution of time intervals τ that follows an exponential function, N ( τ ) dτ = λ exp ( − λ τ ) dτ , (3)where λ = 1 /τ is the mean flaring rate or mean reciprocal waiting time, and N ( τ ) is the probability density function(or differential occurrence frequency). Therefore, if individual events are produced by a physical random process, theoccurrence frequency of waiting times should follow such an exponential-like function (Eq. 3).In reality, however, almost all observed waiting time distributions of solar flare events exhibit a power law-likedistribution for the probability function N ( τ ), rather than an exponential function. In an attempt to match thisobservational constraint, a non-stationary flare occurrence rate λ ( t ) was defined that varies as a function of time inpiece-wise time intervals or Bayesian blocks (Wheatland et al. 1998; Wheatland 2000a, 2001, 2003). Some examplesof such non-stationary Poisson processes are given in Aschwanden and McTiernan (2010) and Aschwanden (2011a,Section 5.2). For instance, it includes an exponentially growing (or decaying) flare rate λ ( t ) that produces a Pareto-typedistribution for the waiting time distribution with a power law slope of 3, N ( τ ) dτ = 2 λ (1 + λ τ ) dτ . (4)It includes also a flare rate that varies highly intermittently in form of δ -functions, which produces a Pareto distributionalso, but with a different power law slope of 2, N ( τ ) dτ = λ (1 + λ τ ) dτ , (5)In order to generalize these two solutions (Eqs. 4 and 5) into a single function, we can define a Pareto-type distributionwith a variable power law slope α τ , as parameterized in Eqs. (1) and (2), N ( τ ) dτ = N ( τ + τ ) − α τ dτ . (6)A more general expression of the waiting time distribution that includes finite system size effects, in terms of anexponential-like cutoff near the longest waiting time intervals τ < ∼ τ e , N ( τ ) dτ = N ( τ + τ ) − α τm exp (cid:18) − ττ e (cid:19) dτ . (7)which yields a superior fit, as shown in the case of solar wind switchbacks (Fig. 6b). This equation fully describes theobserved waiting time distributions (as shown in Figs. 2 and 6), expressed by five parameters ( N , τ , τ e , α τm , SN).The mean waiting time τ , approximately represents the lower bound of the inertial range and separates the rangeof incompletely sampled events (see Fig. 6b) from the scale-free power law range of completely sampled events. Thewaiting time τ e demarcates the e-folding cutoff due to finite system size effects. τ is an offset that depends on theinstrumental sensitivity, the flux threshold in the automated detection of waiting times, and pulse-pileup effects. Forlong-duration flare events, short flares may break up long waiting times into smaller waiting times, which steepensthe power law slope. Since the sunspot number SN is an observable that is known from earlier centuries up to today(Table 2), there are only four free variables left ( N , τ , τ e , α τm ), which can be derived empirically by fitting Eqs. (7)to an observed data set. Note that the maximum power law slope α τm relates to the mean power law slope α τ by acorrection factor given in Eq. (2).The fact that most waiting time distributions exhibit a power law-like function, rather than an exponential-likefunction, clearly requires a non-stationary Poisson process (Wheatland et al;. 1998; Wheatland 2000a, 2001), whichimplies that the flaring rate λ ( t ) has a substantial time variability. The time variability of the mean flaring rate hasbeen found to be highly correlated with the annual sunspot number (Fig. 4), and thus is modulated by Hale’s magneticcycle of solar activity. Apparently, the mean waiting time between solar flares depends on the global magnetic flux(quantified by the sunspot number), but not on the flare size (or peak count rate) according to observations, in contrastto theoretical expectations of energy storage models (Rosner and Vaiana 1978; Wheatland 2000b). 3.2. Flare Model of Waiting Times
All fitted waiting time distributions shown in Fig. 2 are modulated by annual variations of the solar cycle. Onshorter (than annual) time scales, the flaring rate varies also, as statistics based on Bayesian-block decomposition reveal(Wheatland et al. 1998; Wheatland 2000b, 2001, 2003; Wheatland and Litvinenko 2002). A flare model that possiblycould explain the waiting time distribution function was proposed (Wheatland and Litvinenko 2002; Wheatland andCraig 2006), based on the assumptions of: (i) Alfv´enic time scale for crossing the magnetic reconnection region, (ii)2-D geometry of reconnetion region (or separatrix); and (iii) correlation of flare energy build-up (or storage) and flarewaiting time. This model, however, is not consistent with observations, which show no correlation between flare sizesand flare waiting times (Crosby 1996; Wheatland 2000b; Georgoulis et al. 2001), not even between subsequent flares ofthe same active region (Crosby 1996; Wheatland 2000b). Consequently, such theoretical flare energy storage models(Rosner and Vaiana 1978; Lu 1995) have been abandoned. A more likely model involves interchange reconnectionbetween coronal loops and open magnetic fields (Zank et al. 2020).3.3.
Self-Organized Criticality Models
The fractal-diffusive avalanche model of a slowly-driven self-organized criticality (FD-SOC) system (Aschwanden2012, 2014; Aschwanden and Freeland 2012; Aschwanden et al. 2016), expanded from the original version of Bak etal. (1987, 1988), is based on a scale-free (power law) size distribution function of avalanche (or flare) length scales L , N ( L ) dL ∝ L − d dL , (8)with d the Euclidean spatial dimension (which can have values of d=1, 2, or 3). This reciprocal relationship betweenthe spatial size L of a switchback structure and the occurrence frequency N ( L ) is visualized in Fig. 8, for the case ofa space-filling avalanche mechanism, but it holds for rare events in terms of relative probabilities also.The transport process of an avalanche is described by classical diffusion according to the FD-SOC model, whichobeys the scaling law, L ∝ T β/ , (9)with β = 1 for classical diffusion. Substituting the length scale L ∝ T β/ with the duration T of an avalanche event,using Eq. (8-9) and the derivative dL/dT = T β/ − , predicts a power law distribution function for the size distributionof time durations T , N ( T ) dT = N ( T [ L ]) (cid:18) dLdT (cid:19) dT = T − [1+( d − β/ dT = T − α τ dT ≈ T − dT , (10)for d = 3 and β = 1, defining the waiting time power law slope α τ , α τ = 1 + ( d − β/ . (11)We can now estimate the size distribution of waiting times by assuming that the avalanche durations represent upperlimits to the waiting times τ during flaring time intervals, while the waiting times become much larger during quiescenttime periods. Such a bimodal size distribution with a power law slope of α τ < ∼ τ ≤ τ e ), andan exponential-like cutoff function at long waiting times ( τ ≥ τ e ), is depicted in Fig. 7, N ( τ ) dτ = ( τ − for τ ≪ τ e τ − exp( − τ /τ e ) for τ > ∼ τ e . (12)Thus, this FD-SOC model predicts a power law with a slope of α τ ≈ . α τ ≈ . − .
2, varying as a function of the solaractivity, but the solar cycle modulation is not quantified in any SOC model (Aschwanden 2019b).3.4.
MHD Turbulence Processes
Boffetta et al. (1999) argue that the statistics of solar flare (laminar or quiescent) waiting times indicate a physicalprocess with complex dynamics with long correlation times, such as in chaotic models, in contradiction to stationarySOC models that predict Poisson-like statistics. They consider chaotic models that include the destabilization of thelaminar phases and subsequent restabilization due to nonlinear dynamics, as invoked in their shell model of MHDturbulence.Similarly, Lepreti et al. (2001) attribute the origin of the observed waiting time distribution to the fact that thephysical process underlying solar flares is statistically self-similar in time and is characterized by a certain amount of“memory”. They find that the power law distribution can be modeled by a L´evy function which can explain a powerlaw exponent of α τ = 3 (Eq. 4) in the waiting time distribution.Grigolini et al. (2002) develop a technique called diffusion entropy method to reproduce the observed waiting timedistribution function, which evaluates the entropy of the diffusion process generated by the time series. Note thatclassical diffusion (Eq. 9) has been employed in SOC models (Aschwanden 2012), which may be related to the diffusionentropy method, since both models produce a similar scaling of short waiting times, with power law slopes in the rangeof α τ ≈ . − . α τ > α τ < Switchback events observed with Parker Solar Probe
The novel phenomenon of the so-called switchback events were sampled in situ with the PSP in the solar wind at adistance of ≈ R ⊙ = 0 .
166 AU. Switchback events have durations of less than 1 s to more than an hour. Hallmarksof switchback events are reversals in the radial field component B r (with respect to the Parker spiral geometry), whichcan produce deflection angles from a few degrees to nearly 180 ◦ in the fully anti-sunward direction (Dudok de Wit etal. 2020). A switchback event can be quantified either by the magnetic potential energy, E p = − B · < B > , (13)or by the normalized deflection angle µ ,, z = 12 (1 − cos µ ) for 0 ≤ z ≤ . (14)Moreover, we can sample the waiting times τ i = ( t i +1 − t i ) of subsequent events, which resulted into a power law-likeinertial range of τ ≈ −
500 s, an exponential cutoff at τ ≈ − τ ≈ . −
10s (Fig. 6b). The power law slope of the waiting time distribution, α τ = 1 . ± .
01 (Fig. 6), is measured in the year2018, close to the minimum of the solar cycle, and follows the same trend as solar flares and CMEs (Fig. 1).This close similarity of the slopes obtained with solar flares and with switchbacks is intriguing and could be thesignature of common drivers. However, as of today the origin of switchbacks is unclear. They may be generatedeither locally in the upper solar corona or by instabilities such as plasma jets occurring much deeper in the corona.In addition, there are also several differences in the way flares and switchbacks are registered: the flaring rate, forexample, includes independent and sympathetic flares occurring everywhere on the solar disc and at the limb, whileswitchback events are recorded at one single point in space only, along the satellite orbit. In addition, the impact oflocal solar wind conditions, and the relative speed of PSP on the rate of switchback events still has to be properlyinvestigated. Therefore, while the similarity of the waiting time distributions is likely to be deeply rooted in theunderlying physical processes, it is premature to conclude about the connection between the two types of events.Long-term memories, expressed by the residence time of switchback events are then expected to scale with the flareor CME duration T , which is predicted from SOC models to follow a power law distribution function of N ( T ) ∝ T − (Eq. 10), and a proportional distribution of N ( τ ) ∝ τ − for short waiting times (Eq. 12). Both the waiting time τ and the residence time T distribution of these switchback deflections tend to follow a power law and are remarkablysimilar (Dudok de Wit et al. 2020). The long memory we observe is most likely associated with the strong spatialconnection between adjacent magnetic flux tubes and their common photospheric footpoints (Dudok de Wit et al. 2020).Consequently, it has been proposed that switchback events are modulated by impulsive flare (or CME) events in thelower corona (Roberts et al. 2018; Tenerani et al. 2020; Zank et al. 2020). CONCLUSIONSIn this study we investigate the statistics of waiting time distributions of solar flares, CMEs, and solar wind switch-back events. The motivation for this type of analysis method is the diagnostics of stationary and non-stationaryPoissonian random processes, SOC systems, and MHD turbulence systems. The observational analysis is very simple,since only an event catalog with the starting times t i of the events is necessary to sample waiting times τ = ( t i +1 − t i ).We obtain the following results: 1. Using the statistics of hard X-ray solar flares (using flare catalogs from HXRBS/SMM, BATSE/CGRO, WATCH,ICE/ISEE-3, RHESSI) we find power law distribution functions with slopes in the range of α τ ≈ . − . α τm = 1 . ± . × SN and a cross-correlation coefficient of CCC=0.987(Fig. 1). This trend clearly indicates that the waiting time power law slope α τ is foremost correlated with thesunspot number (or the flaring rate), which is fully consistent with previous findings (Wheatland and Litvinenko2002; Wheatland 2003).2. Using the statistics of soft X-ray flares, sampled by GOES over 47 years in annual intervals, but with a 10 timeshigher sensitivity, we perform fits with a Pareto-type distribution function (Fig. 2), which consists of an inertial(power law) range, an exponential cutoff range, and a range of under-sampling. The fits clearly show power lawslopes that are modulated by the four solar cycles, strongly correlated with the annual sunspot number and theannual flaring rate (Figs. 4, 5), consistent with the hard X-ray flare results.3. We sample 19,452 magnetic field switchback events from data observed with the Parker Solar Probe and finda power law slope of α τ = 1 . ± .
01. A theoretical value of α τ = 2 . τ < ∼
500 s) by a self-organized criticality model during contigous flaring time episodes, while an exponentiallydroping cutoff is expected for long waiting times ( τ ≈ − Acknowledgements:
Part of the work was supported by NASA contract NNG04EA00C of the SDO/AIA instrumentand NNG09FA40C of the IRIS instrument.REFERENCESAschwanden, M.J. and McTiernan, J.M. 2010,
Reconciliation of waiting time statistics of solar flares observed in hardX-rays , ApJ 717, 683Aschwanden, M.J. 2011a
Self-Organized Criticality in Astrophysics. The Statistics of Nonlinear Processes in theUniverse , ISBN 978-3-642-15000-5, Springer-Praxis: New York, 416p.Aschwanden, M.J. 2011b,
The state of self-organized criticality of the Sun during the last 3 solar cycles. I. Observations ,SoPh 274, 99Aschwanden, M.J. 2012,
A statistical fractal-diffusive avalanche model of a slowly-driven self-organized criticalitysystem
A&A 539:A2.Aschwanden, M.J. and Freeland, S.M. 2012,
Automated solar flare statistics in soft X-rays over 37 years of GOESobservations: The invariance of SOC during 3 solar cycles , ApJ 754:112.Aschwanden, M.J. 2014,
A macroscopic description of self-organized systems and astrophysical applications , ApJ 782,54Aschwanden, M.J. 2015,
Thresholded power law size distributions of instabilities in astrophysics , ApJ 814:19.Aschwanden, M.J. 2016,
25 Years of SOC: Solar and astrophysics
SSRv 198:47.Aschwanden, M.J. 2019a,
Self-organized criticality in solar and stellar flares: Are extreme events scale-free ?
ApJ 880,105.Aschwanden, M.J. 2019b,
Nonstationary fast-diven, self-organized criticality in solar flares , ApJ 887:57Aschwanden, M.J. 2020,
Global energetics of solar flares. XII. Energy scaling laws , ApJ (in press).Aschwanden M.J. and G¨udel, M. 2021,
Self-organized criticality in stellar flares , (subm.)Bak, P., Tang, C., and Wiesenfeld, K. 1987,
Self-organized criticality - An explanation of 1/f noise , Physical ReviewLett. , 381-384.Bak, P., Tang, C., and Wiesenfeld, K. 1988,
Self-organized criticality , Physical Rev. A , 364-374.Bale, S.D., Goetz, K., Wygant, J.R. et al. 2016,
The FIELDS instrument suite for Solar Probe Plus , SSRv 204, 49Biesecker, D.A. 1994
On the occurrence of solar flares observed with the burst and transient source experiment
PhDThesis, University of New HampshireBoffetta, G., Carbone, V., Giuliani, P., Veltri, P., and Vulpiani, A. 1999,
Power Laws in Solar Flares: Self-OrganizedCriticality or Turbulence?
Phys.Rev.Lett. 83, 4662Crosby, N.B., Georgoulis, M., and Villmer, N. 1996,
A comparison between the WATCH flare data statistical properitiesand predictions of the statistical flare model , in Proc. 8th SOHO Workshop (eds. J.C. Vial and B. Kaldeich-Schuermann), ESA 446, Estec Nordwijk, p.247Dudok de Wit, P., Krasnoselskikh, V.V., Bale, S.D., et al. 2020,
Switchbacks in the near-Sun magnetic field. Longmemory and impact on the turbulence cascade , ApJSS 246:39Fox, N.J., Velli, M.C., Bale, S.D., Decker, R., Driesman, A., Howard et al. 2016,
The Solar Probe Plus Mission:Humanity’s First Visit to Our Star , Space Science Reviews, 204, 7.Georgoulis, M.K., Vilmer, N., Croby, N.B. 2001,2
A Comparison Between Statistical Properties of Solar X-Ray Flaresand Avalanche Predictions in Cellular Automata Statistical Flare Models , A&A 367, 326Grigolini, P., Leddon,D., and Scafetta,N. 2002,
Diffusion entropy and waiting time statistics of hard X-ray solar flares ,Phys.Rev.Lett E, 65/4. id. 046203Horbury, S., Woolley, T., Laker R., Matteini, L. et al. 2020
Sharp Alfvenic impulses in the Near-Sun solar wind ,ApJSS 246:45Hosking, J.M.R. and Wallis, J.R. 1987, Technometrics 29, 339.Krasnoselskikh, V., Larosa, A., Agapitov, O., Dudok de Wit, T., 2020,
Localized magnetic Field structures and theirboundaries in the near-Sun solar wind from Parker Solar Probe measurements , ApJ 893, 93.Lepreti, F., Carbone, V., and Veltri,P. 2001,
Solar flare waiting time distribution: varying-rate Poisson or Levyfunction?
ApJ 555, L133Lomax, K.S. 1954, , J. Am. Stat. Assoc. 49, 847Lu, E.T. 1995,
Constraints on energy storage and release models for astrophysical transients and solar flares , ApJ 447,416.Mozer, F.S., Agapitov, O.V., Bale, S.D., Bonnell, J.W. et al. 2020,
Switchbacks in the solar magnetic field: Theirevolution, their contentk, and their effects on the plasma , ApJSS 246:680Pearce, G., Rowe, A.K., and Yeung, J. 1993,
A statistical analysis of hard X-ray solar flares . Astrophys. Space Science208, 99.Roberts M.A., Uritsky, V.M., DeVore, C.R., and Karpen,J.T. 2018,
Simulated encounters of the Parker Solar Probewith a Coronal-hole Jet , ApJ 866, 14Rosner, R., and Vaiana, G.S. 1978,
Cosmic flare transients: constraints upon models for energy storage and releasederived from the event frequency distribution , ApJ 222, 1104Tenerani, A., Velli, M., Matteini, L., et al. 2020,
Magnetic Field Kinks and Folds in the Solar Wind , ApJS 246, 32Wheatland, M.S., Sturrock,P.A., and McTiernan,J.M. 1998,
The waiting-time distribution of solar flare hard X-rays
ApJ 509, 448-455.Wheatland,M.S. 2000a,
The origin of the solar flare waiting-time distribution , ApJ 536, L109.Wheatland,M.S. 2000b,
Do solar flares exhibit an internal-size relationship , SoPh 191, 381-389.Wheatland, M.S. 2001,
Rates of flaring in individual active regions , SoPh 203, 87-106.Wheatland, M.S. and Litvinenko, Y.E. 2002,
Understanding Solar Flare Waiting-Time Distributions , SoPh 211, 255-274.Wheatland, M.S. 2003,
The Coronal Mass Ejection Waiting-Time Distribution
SoPh 214, 361-373.Wheatland, M.S., and Craig, I.J.D. 2006,
Including Flare Sympathy in a Model for Solar Flare Statistics , SoPh 238,73-86.Zank, G.P., Nakanotani, M., Zhao, L.L., Adhikari, L., and Kasper, J., 2020,
The origin of switchbacks in the solarcorona , ApJ 903:11
Table 1.
Waiting time distributions measured from solar flares hard X-ray events, soft X-ray events, coronal mass ejections, andradio bursts. The waiting time distribution (WTD) functions are abbreviated as: PL=powerlaw, E=exponential, PE=powerlawwith exponential cutoff, DE=double exponential.
Observations Observations Number Waiting WTD Powerlaw Referencesyear of events spacecraft or range timeinstrument τ α τ . − . ± . . − . − . ± .
13 Crosby (1996)1978-1986 ICE/ISEE-3 6916 0 . −
20 hrs DE Wheatland et al. (1998)1980-1989 HXRBS/SMM 12,772 0 . −
500 hrs PL 2 . . −
200 hrs PL 2 . − . ± .
01 Grigolini et al. (2002)2002-2008 RHESSI 11,594 2 − . − . ± . − . ± .
05 Wheatland (2000a), Lepreti et al. (2001)1996-2001 GOES 1-8 A 4645 1 − . ± .
11 Wheatland (2003)1996-1998 GOES 1-8 A ... 1 − . ± .
08 Wheatland (2003)1999-2001 GOES 1-8 A ... 1 − . ± .
19 Wheatland (2003)1975-2001 GOES 1-8 A ... 1 − . ± . − . ± . − . ± . − . ± .
11 Wheatland (2003)1996-1998 SOHO/LASCO ... 1 − . ± .
14 Wheatland (2003)1999-2001 SOHO/LASCO ... 1 − . ± .
20 Wheatland (2003)2018 PSP ... 10 − − hrs PL 1 . − . − − hrs PE 1 . ± .
01 This work2
Table 2.
Annual waiting time distribution slopes and sunspot numbers measured from solar flares
Year Number power law best-fit lower upper decades Sunspotof events slope chi-square bound bound log( x /x ) number N ev α χ x [hrs] x [hrs] SN1974 158 1.53 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± P o w e r l a w s l ope w a i t i ng t i m e a ∆ t WW WCMECME CMEAM G BLSW a ∆ t = 1.381+ 0.010 * SNCCC= 0.987 S o l a r m i n i m u m S o l a r m a x i m u m BL: Flares (Boffetta et al. 1999; Lepreti 2001)G: Flares (Grigolini et al. 2002)CME: LASCO CME (Wheatland 2003)W: GOES flares (Wheatland 2003)AM: Flares (Aschwanden & McTiernan 2010)SW: Solar wind (Dudoc de Wit 2018)
Figure 1.
Scatterplot of power law slope α τ of waiting time distributions versus the sunspot number SN, with linear regressionfit (solid line) and cross-correlation coefficient CCC. F r equen cy Year=1974 158a = 1.53_a = 1.53+0.12 0.0010.0100.1001.00010.000100.00010 -2 Year=1975 2864a = 1.32_a = 1.32+0.02 0.0010.0100.1001.00010.000100.0001000.00010 -4 -2 Year=1976 1829a = 1.32_a = 1.32+0.03 0.01 0.10 1.00 10.0010 -2 Year=1977 5541a = 1.72_a = 1.72+0.02 0.001 0.010 0.100 1.000 10.00010 -2 Year=1978 16896a = 2.65_a = 2.65+0.020.001 0.010 0.100 1.000 10.00010 -2 F r equen cy Year=1979 18795a = 3.17_a = 3.17+0.02 0.001 0.010 0.100 1.000 10.00010 -2 Year=1980 15341a = 3.14_a = 3.14+0.03 0.001 0.010 0.100 1.000 10.00010 -2 Year=1981 15528a = 3.16_a = 3.16+0.03 0.001 0.010 0.100 1.000 10.00010 -2 Year=1982 15963a = 2.82_a = 2.82+0.02 0.01 0.10 1.00 10.0010 -2 Year=1983 9566a = 2.49_a = 2.49+0.030.01 0.10 1.00 10.0010 -2 F r equen cy Year=1984 6891a = 1.93_a = 1.93+0.02 0.01 0.10 1.00 10.00 100.0010 -2 Year=1985 2776a = 1.65_a = 1.65+0.03 0.01 0.10 1.00 10.00 100.0010 -2 Year=1986 2337a = 1.53_a = 1.53+0.03 0.01 0.10 1.00 10.00 100.0010 -4 -2 Year=1987 4867a = 1.71_a = 1.71+0.02 0.001 0.010 0.100 1.000 10.00010 -2 Year=1988 13028a = 2.67_a = 2.67+0.020.01 0.10 1.00 10.0010 -2 F r equen cy Year=1989 17533a = 3.63_a = 3.63+0.03 0.01 0.10 1.00 10.0010 -2 Year=1990 15281a = 3.37_a = 3.37+0.03 0.001 0.010 0.100 1.00010 -2 Year=1991 16857a = 3.13_a = 3.13+0.02 0.01 0.10 1.00 10.0010 -2 Year=1992 12487a = 2.77_a = 2.77+0.02 0.01 0.10 1.00 10.0010 -2 Year=1993 9386a = 2.21_a = 2.21+0.020.01 0.10 1.00 10.0010 -2 F r equen cy Year=1994 4844a = 1.67_a = 1.67+0.02 0.01 0.10 1.00 10.00 100.0010 -2 Year=1995 2651a = 1.59_a = 1.59+0.03 0.01 0.10 1.00 10.00 100.0010 -2 Year=1996 1287a = 1.44_a = 1.44+0.04 0.01 0.10 1.00 10.00 100.0010 -2 Year=1997 3775a = 1.88_a = 1.88+0.03 0.01 0.10 1.00 10.0010 -2 Year=1998 11170a = 2.61_a = 2.61+0.020.001 0.010 0.100 1.00010 -2 F r equen cy Year=1999 14683a = 2.67_a = 2.67+0.02 0.001 0.010 0.100 1.000 10.00010 -2 Year=2000 16160a = 3.04_a = 3.04+0.02 0.001 0.010 0.100 1.000 10.00010 -2 Year=2001 16023a = 2.92_a = 2.92+0.02 0.01 0.10 1.00 10.0010 -2 Year=2002 16174a = 3.29_a = 3.29+0.03 0.01 0.10 1.00 10.0010 -2 Year=2003 12283a = 2.60_a = 2.60+0.020.01 0.10 1.00 10.0010 -2 F r equen cy Year=2004 8956a = 2.15_a = 2.15+0.02 0.01 0.10 1.00 10.00 100.0010 -2 Year=2005 6705a = 1.89_a = 1.89+0.02 0.01 0.10 1.00 10.00 100.0010 -2 Year=2006 3100a = 1.60_a = 1.60+0.03 0.01 0.10 1.00 10.00 100.0010 -4 -2 Year=2007 1433a = 1.54_a = 1.54+0.04 0.01 0.10 1.00 10.00100.001000.0010 -4 -2 Year=2008 190a = 1.51_a = 1.51+0.110.01 0.10 1.00 10.00 100.00Waiting time10 -4 -2 F r equen cy Year=2009 531a = 1.60_a = 1.60+0.07 0.01 0.10 1.00 10.00Waiting time10 -2 Year=2010 3660a = 1.79_a = 1.79+0.03 0.01 0.10 1.00 10.00Waiting time10 -2 Year=2011 11270a = 2.56_a = 2.56+0.02 0.001 0.010 0.100 1.000 10.000Waiting time10 -2 Year=2012 13106a = 2.44_a = 2.44+0.02
Figure 2.
Annual waiting time distributions of GOES 1-8 ˚A soft X-ray fluxes for the years 1974 to 2012 (histograms), withleast-square fits of Pareto distributions (thick solid curve) and corresponding power law slopes a (dotted lines) P o w e r l a w s l ope w a i t i ng t i m e a ∆ t α τ m α τ (a) P o w e r l a w s l ope w a i t i ng t i m e a ∆ t α τ /α τ m =0.62 α τ (b) Figure 3.
The power law slope α τ of the waiting time distribution is obtained with standard power law fitting methods(diamond symbols), as well as with a Pareto-fitting method, α τm (cross symbols). The Pareto bias correction amounts to anempirical factor of α τ /α τm = 0 . P o w e r l a w s l ope w a i t i ng t i m e a ∆ t (a) S un s po t nu m be r (b) × × × × F l a r e r a t e pe r y ea r (c) Figure 4.
Time evolution of power law slope α τ ( t ) of waiting time distributions as a function of the time (a), time evolutionof annual sunspot number during the last four solar cycles (b), and annual flaring rate N ev . P o w e r l a w s l ope w a i t i ng t i m e a ∆ t CCC= 0.961y = 1.442+ 0.009 SN (a) × × × × Flare rate per year N ev P o w e r l a w s l ope w a i t i ng t i m e a ∆ t CCC= 0.960y = 1.307+0.00011 N ev (b) Figure 5.
Linear regression fits of the waiting time power law slope α τ versus the sunspot number SN (a) and versus theannual flare rate N ev (b) is shown (solid line), measured from automated flare detections of GOES flares in annual time intervals(diamonds). Solar wind switchback events (PSP) sw [s]10 -4 -2 O cc u rr en c e f r equen cy Model = PMN ev = 19452a = 1.30_a = 1.30+0.02 χ dif = 3.64 (a) Undersampling Inertial range Exponential cutoff
Solar wind switchback events (PSP) sw [s]10 -4 -2 O cc u rr en c e f r equen cy Model = FMN ev = 19452a = 1.21_a = 1.21+0.01 χ dif = 2.21 (b) Undersampling Inertial range Exponential cutoff
Figure 6.
The waiting time distributions of 19,452 magnetic field switchback events (histograms) observed with the ParkerSolar Probe. The observed distributions are fitted with two theoretical models: (a) the Pareto distribution model (PM), and(b) the Pareto distribution with an exponential cutoff (PF model). Note that the best fit favors the PF model (b) with a powerlaw slope of α τ = 1 . ± .
01 and a goodness-of-fit χ = 2 . Quiet time interval ∆ t q Active time interval ∆ t F l u x Time N u m be r o f e v en t s l og ( N ) Waiting time log( ∆ t) T T Figure 7.
The concept of a dual waiting time distribution is illustrated, consisting of active time intervals ∆ t < T thatcontribute to a powerlaw distribution, which is equal to that of time duration distributions, N ( T ). Random-like quiescent timeintervals ∆ t contribute to an exponential cutoff function. Verticular lines in the upper panel indicate the start times of events,between which the waiting times ∆ t are measured (Aschwanden 2014). N(L=1/8)=L -2 =1/8 -2 =64N(L=1/4)=L -2 =1/4 -2 =16N(L=1/2)=L -2 =1/2 -2 =4 Figure 8.
The reciprocal relationship between the geometric length scales L in two-dimensional Euclidean space and occurrencefrequency N ( L ) ∝ L − is depicted for three different length scales L = 1 / , / , /
2, leading to occurrence frequencies of N ( L ) ∝∝
2, leading to occurrence frequencies of N ( L ) ∝∝ L − ∝∝