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Extreme geomagnetic activities: a statistical study
Earth, Planets and Space volume 72, Article number: 124 (2020)
Abstract
Statistical distributions are investigated for magnetic storms, sudden commencements (SCs), and substorms to identify the possible amplitude of the one in 100year and 1000year events from a limited data set of less than 100 years. The lists of magnetic storms and SCs are provided from Kakioka Magnetic Observatory, while the lists of substorms are obtained from SuperMAG. It is found that majorities of events essentially follow the lognormal distribution, as expected from the random output from a complex system. However, it is uncertain that largeamplitude events follow the same lognormal distributions, and rather follow the powerlaw distributions. Based on the statistical distributions, the probable amplitudes of the 100year (1000year) events can be estimated for magnetic storms, SCs, and substorms as approximately 750 nT (1100 nT), 230 nT (450 nT), and 5000 nT (6200 nT), respectively. The possible origin to cause the statistical distributions is also discussed, consulting the other space weather phenomena such as solar flares, coronal mass ejections, and solar energetic particles.
Introduction
It is important to understand the characteristics and possible amplitudes of extreme events of substorms, sudden commencements (SCs), and magnetic storms to mitigate the space weather hazard, especially from geomagnetically induced currents (Kataoka and Ngwira 2016; Pulkkinen et al. 2017). However, it is still hard to predict the amplitude of unprecedented extreme events by physicsbased simulations, and the statistical analysis is necessary to estimate the quantitative amplitude of possible extreme events.
One of the extreme geomagnetic activity events was observed associated with an episodic solar flare on 1 September 1859 (Carrington 1859), which has been considered as a measure of extreme events. A lowlatitude magnetometer measured ~ 1600 nT spike during the magnetic storm on 2 September 1859 (Tsurutani et al. 2003). Siscoe et al. (2006) estimated the 1h averaged value as − Dst = 850 nT, which is well below the theoretical upper limit of − Dst = 2500 nT (Vasyliunas 2011). From the statistical comparison among several space weather phenomena of magnetic storms, solar flares, and coronal mass ejections, Riley (2012) estimated a probability of 12% to have another Carrington event in coming 10 years. Kataoka (2013) applied the same analysis to the 90year list of magnetic storms and estimated the probability of another Carrington storm in 10 years as 4 ~ 6%.
It is possible that we will have extreme magnetic storms even larger than the Carrington storm in future. For example, from the detailed analysis of an auroral painting from Kyoto, Japan, Kataoka and Iwahashi (2017) estimated that the amplitude of a historical magnetic storm occurred on 17 September 1770 can be comparable to or even larger than the Carrington storm. As the latest example, the amplitude of a magnetic storm in May 1921 was estimated to be comparable to the Carrington event (Hapgood 2019; Love et al. 2019), suggesting that the Carringtonclass events may be more frequent than previously expected.
Another useful concept to define the extreme events is socalled “one in 100year event”. Tsubouchi and Omura (2007) applied an extreme value theory to estimate the amplitude of the 100year event as − Dst = 550 ~ 750 nT. More recently, Love et al. (2015) estimated the 100year event as − Dst = 880 nT, i.e., the Carrington event corresponds to the 100year event. The best efforts have been repeatedly conducted to estimate 100year event and even 1000year event by carefully extrapolating the tail distributions, although such results are highly uncertain especially from the limited data set of the Dst index for only a half century (Love 2020).
In this study, acknowledging an advantage of Japan’s longterm monitoring effort of geomagnetic activities at Kakioka Magnetic Observatory (KAK), their complied event lists of magnetic storms and SCs are used for the statistics to estimate the possible amplitudes of the 100year and 1000year events. Using the KAK lists, the possible powerlaw distributions of the amplitude of magnetic storms as well as SCs were discussed by Minamoto et al. (2015).
The similar statistical analysis can also be applied for substorms as well. It has been discussed for a long time that the amplitude of substorms basically follows lognormal distributions (e.g., Liou et al. 2018). From the statistical analysis, Nakamura et al. (2015) estimated the possible maximum amplitude of substorms as AE = ~ 4100 nT.
The purpose of this paper is to estimate the possible amplitude of extreme events of magnetic storms, SCs, and substorms from those statistical distributions. The event lists used in this study are explained in detail in “Data set” section. The method of analysis how to fit the lognormal and powerlaw distributions to the data set is described in “Method of analysis” section. Obtained results are provided in “Results” section. The possible origins to cause the lognormal and powerlaw distributions are discussed in “Discussion” section, by consulting the statistical distributions of the other space weather phenomena such as solar flares, coronal mass ejections (CMEs), and solar proton events (SEPs). Finally, concluding remarks are described in “Conclusions” section.
Data set
The event lists of magnetic storms and SCs are available at the KAK website (https://www.kakiokajma.go.jp/obsdata/metadata/ja/products/list/event/kak). The lists are manually accumulated everyday by the professional operators at KAK. The occurrence properties of the identified magnetic storms and SCs are displayed in Figs. 1, 2. Although the lists essentially include the local variation, the longterm 96year data with the unchanged identification criteria is very unique to investigate the 100year and 1000year extreme events. Further, in the viewpoint of space weather countermeasure against the extreme events, the local enhancement itself is also of great interest to mitigate the possible maximum hazard.
The standard data for measuring the amplitude of substorms are the AE index. However, the AE index becomes unreliable when large substorms occurred at lower magnetic latitude than that of the AE stations located at high latitude of 65–70 deg. Recently, the SME index was developed from globally distributed magnetometers ranging from 40 deg to 80 deg magnetic latitude (Gjerloev 2012) to provide a better replacement to evaluate the amplitude of such a large substorm events. The substorm list was also provided from SuperMAG website (http://supermag.jhuapl.edu/). The data set used in this study is 34year data from January 1986 when the number of SME stations was large enough (> 30 stations) to better identify extreme events. A total of ~ 6×10^{4} substorm events were identified in the list for the 34year time interval. The substorm amplitude of each event was calculated as the 15min mean value of the SME index starting 10 min after the substorm onset (Newell and Gjerloev 2011). The basic occurrence property and the amplitude distribution of the SME index were documented by Newell and Gjerloev (2011).
In this paper, some more space weather event lists are consulted to discuss the possible origins of the statistical distributions. The event list of solar flares is available at NOAA’s website (https://www.ngdc.noaa.gov/stp/solar/solarflares.html). The event list of CMEs is available at NASA’s website (https://cdaw.gsfc.nasa.gov/CME_list/). The list of SEP events is available at NOAA and NASA (https://umbra.nascom.nasa.gov/SEP/).
Method of analysis
In general, interactions among many elements or their nonlinearity bring out a new characteristic from the complex network. In the complex system, largeamplitude events tend to follow a powerlaw distribution (e.g., Riley 2012; Gopalswamy 2018). The powerlaw distribution of the event amplitude x is defined as
where α denotes the spectral index and A is a constant. A useful way to investigate the largeamplitude rare events is a cumulative distribution function (CDF) which is defined as
We use the maximum likelihood estimation (Riley 2012; Kataoka 2013) to obtain the slope as
where the x_{min} is the minimum value to be used for the fitting.
Rare events or largeamplitude events always arise from the majority. The majority usually follows a lognormal distribution in a complex system because it is characterized as a random walk of multiplications rather than that of additions. The lognormal distribution is defined as
where μ is the geometric mean value and σ is the standard deviation. The CDF of the lognormal distribution can be written as
where N_{T} is the total number of events and the error function is
A standard method of minimum variance fitting (scipy.optimize.curve_fit) is used in this study to find the bestfit CDF.
In order to estimate the possible amplitudes of the 100year or 1000year events from a limited data set of less than 100 years, both the lognormal CDF (Eq. 2) and powerlaw CDF (Eq. 5) are fitted to the data set. In this study, the timestationarity of the distributions is then assumed to extend the limited time interval of the data set to 100 years or 1000 years. The lognormal distribution gives relatively conservative estimates, while the powerlaw distribution generally gives upperlimit estimates (Riley and Love 2017).
Results
Magnetic storms
Figure 3 shows that the majority of magnetic storms roughly follows the lognormal distribution, and the largeamplitude population of − dH > 200 nT may also follow the powerlaw. Note that the lognormal fit for above 200 nT level can be meaningful to give an estimate of the extreme amplitude because those large storms were caused by only CMEs, while relatively weak magnetic storms are driven by both CMEs and corotating interaction regions (Richardson et al. 2006; Kataoka and Miyoshi 2006). The excess from the lognormal distribution at relatively weak level can therefore be interpreted to be the mixed population of magnetic storms.
The lognormal fit for > 200 nT storms gives the 100year and 1000year events as − dH = 750 nT and 1100 nT, respectively, while the powerlaw fit gives the 100year and 1000year events as − dH = 1100 nT and 2200 nT, respectively. The largest amplitude in the list is − dH = 661 nT that occurred on 24 March 1940. Note also that the record largest event of − dH > 700 nT on 4 July 1941 at KAK was not used in this study because of its ambiguity. The largest events of − dH > 400 nT are listed in Table 1. The 13 March 1989 storm is the largest Dst event since 1957 with the peak of final Dst index of − 589 nT. Nagatsuma (2015) estimated the intensity of the southward Bz = ~ 50 nT for the March 1989 storm, which is about the largest intensity of interplanetary magnetic field at 1 AU.
Sudden commencements
Figure 4 shows that SC events follow a lognormal distribution within the amplitude range from 5 nT to 70 nT, while the largeamplitude SC events deviate from the lognormal distribution and follow a powerlaw distribution. The lognormal fit gives the 100year and 1000year events as dH = 160 nT and 240 nT, respectively. The powerlaw fit gives the 100year and 1000year events as dH = 230 nT and 450 nT, respectively. The largest amplitude is dH = 220 nT that occurred on 13 November 1960. Note also that the record largest event of dH = 310 nT on 24 March 1940 was not in the KAK list because it does not include the second SC according to the rule of KAK (Araki 2014). The largest events of dH > 100 nT are listed in Table 2.
Substorms
Figure 5 shows that the substorm events essentially follows the lognormal distribution. The lognormal fit gives the 100year and 1000year events as 5000 nT and 6000 nT, respectively. The powerlaw fit gives the 100year and 1000year events as 6200 nT and 8500 nT, respectively. The record largest amplitude is SME = 3929 nT that occurred on 30 October 2003 during the socalled Halloween event. The largest events of SME > 3000 nT are listed in Table 3.
Solar flares, CMEs, and SEP
Consulting other space weather phenomena, the same statistical analysis is applied to solar flares, CMEs, and SEP events. The results shown in Figs. 6, 7, 8 basically follow the analysis of Gopalswamy (2018), and the only difference is the types of distributions fitted to the cumulative distribution function. Gopalswamy (2018) used the Weibull distribution (exponential curve) and powerlaw distribution, while this study uses the lognormal and powerlaw distributions to give somewhat larger estimates as follows.
The lognormal fit shown in Fig. 6 gives the 100year and 1000year event sizes as X70 and X200, respectively. These values are larger than the estimates of Gopalswamy (2018) in which the 100year and 1000year event sizes are ~ X40 and ~ X100, respectively. The lognormal fit shown in Fig. 7 gives the 100year and 1000year speed as 4500 km/s and 6000 km/s, while Gopalswamy (2018) gives the 100year and 1000year event as 3800 km/s and 4700 km/s, respectively. The lognormal fit shown in Fig. 8 gives the 100year and 1000year events as ~ 2.5 × 10^{5} pfu and ~ 1.5 × 10^{6} pfu, while the extrapolated curve of Gopalswamy (2018) gives the 100year and 1000year events as ~ 2×10^{5} pfu and ~ 1×10^{6} pfu, respectively.
Discussion
In summary, conservative amplitudes can be estimated from the lognormal distribution, while the possible excess (likely upper limit to the behavior of the tail) can be estimated from the powerlaw distribution. The possible amplitudes of 100year and 1000year events based on the lognormal and powerlaw distributions are summarized in Table 4.
The possible origins of the statistical distributions are briefly discussed as follows. The powerlaw fitting can be meaningful for rare events or largeamplitude events, and there are possible reasons to cause the excess from lognormal distribution at the tail. For SCs, the main cause of the excess from lognormal distribution are spikes (Araki 1997, 2014), which can be interpreted as the amplification of the preliminary impulse phase due to the velocityjump effect over the densityjump effect, based on the parameter survey of a global magnetohydrodynamic (MHD) simulation (Kubota et al. 2015). Although it has been well known that the amplitudes of SC are proportional to the change of dynamic pressure, densityjump effect dominates the change of dynamic pressure for weak SCs, while velocityjump effect dominates for large SCs. The rapidly changing largeamplitude spike appears when the shock downstream speed becomes high (Kubota et al. 2015). In addition, we must admit that there are a few more missing extreme events from the statistics. For example, an extreme SC event on 24 March 1940 is not included in the KAK list, because it was the second SC event (Araki 2014). This particular example reminds us of the complex interaction among multiple CMEs and ambient solar wind to enhance the geoeffectivity (e.g., Kataoka et al. 2015; Shiota and Kataoka 2016), which may also additionally contribute to the further excess from the standard lognormal distribution.
For magnetic storms, unusual spikes may also cause the excess, although the physics and timescale are totally different from SCs of course. There is an example that a huge spike of > 1600 nT appeared in the Carrington storm, in which additional ionospheric current or fieldaligned current may locally contribute (Akasofu and Kamide 2005). Even if the spikes are not originated from the ring current, it does matter to prepare against the possible hazards. Those spikes may also contribute to make an excess from the lognormal extrapolation at the tail.
It was clearly demonstrated that substorms of the Earth’s magnetosphere essentially follow the lognormal distribution (Fig. 5). On the other end of the solar–terrestrial system, solar flares and CMEs may resemble the plasma explosions against substorms. Here, we discuss whether there are similarities among their statistical distributions.
It is interesting to note here that the majority of solar flares ranging from Cclass to X10class follows a powerlaw distribution rather than a lognormal distribution, although C1 class and the tail beyond X10 flares show the deviations (Fig. 6). The fewer samples of C1 class flares may be the missing counts when the background level is comparable to C1 flares during highly active conditions. Note also that we have only a few samples for > X10 flares in the last 40 years. The powerlaw distribution over a wide range may be interpreted, considering a difference between the Earth’s magnetosphere and sunspots’ magnetosphere, i.e., the active region. Active regions have a large variation of the spatial scales ranging multi orders of magnitude, and the fractal reconnection patterns naturally arise in the scalefree MHD system, in contrast to the onlyone magnetospheric system of the Earth. In other words, if significant limitations in the scalefree system exist, lognormal distribution may clearly appear.
Figures 7, 8 show that CMEs and SEP events essentially follow the lognormal distributions. These distributions are different from that of solar flares, which can be originated from the fact that only selective active regions can launch CMEs against the strong solar gravitation, and only selective CMEs can launch SEPs. For SEP events, however, an additional factor of a complex interplanetary propagation of energetic particles may broaden the distribution from the standard lognormal distribution.
In a simplified view, geomagnetic activities are the product of the interaction between the solar wind and magnetosphere. The solar wind parameters follow lognormal distributions (Burlaga and Lazarus 2000; Burlaga 2001), while the Earth’s magnetic moment does not essentially change in short time. This is one of the reasons that majority of the geomagnetic activities follows the lognormal distribution rather than the powerlaw distribution. In other words, if the magnetic moment changes rapidly and follows the lognormal distribution like sunspots, the occurrence of substorms may essentially follow a powerlaw distribution. This idea can be tested by a global MHD simulation of the Earth’s magnetosphere by changing the magnetic moment.
Recently, is became possible to continuously run the global MHD simulation of the magnetosphere for more than several months, using the observed solar wind data as the input to reproduce a number of substorms. For example, the occurrence properties of the simulated substorms were statistically compared against the observed one for a whole month in January 2015 (Haiducek et al. 2017, 2020). Future works should also include the similar direction with different simulation codes to examine the difference of the statistical distributions.
Conclusions
It was found that the amplitudes of magnetic storms, SCs, and substorms essentially follow the lognormal distributions, with the largeamplitude events showing a possible excess from the lognormal distributions, which follow the powerlaw distributions. This is interpreted as a natural consequence as a random output from a complex system. Based on both the lognormal and powerlaw distributions, the amplitudes of the 100year (1000year) events can be approximately estimated for magnetic storms, SCs, substorms as 750 nT (1100 nT), 230 nT (450 nT), and 5000 nT (6200 nT), respectively.
Availability of data and materials
The event lists of magnetic storms and SCs are available at the website of Kakioka Magnetic Observatory (https://www.kakiokajma.go.jp/obsdata/metadata/en/products). Substorm list and SME index data is available at SuperMAG website (http://supermag.jhuapl.edu/). The event list of solar flares is provided from NOAA/NGDC (https://www.ngdc.noaa.gov/stp/solar/solarflares.html). The event list of CMEs is provided from NASA/GSFC (https://cdaw.gsfc.nasa.gov/CME_list/). The event list of SEP is provided from NOAA/SWPC (https://umbra.nascom.nasa.gov/SEP/).
Abbreviations
 CDF:

Cumulative distribution function
 CME:

Coronal mass ejection
 KAK:

Kakioka Magnetic Observatory
 MHD:

Magnetohydrodynamics
 SC:

Sudden commencement
 SEP:

Solar energetic particle
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Acknowledgements
RK thanks Daikou Shiota for fruitful discussions. RK thanks Kakioka Magnetic Observatory for providing the event lists of magnetic storms and SCs. RK thanks SuperMAG development team and collaborators for providing the SME index. The event list of solar flares is provided from NOAA/NGDC. The event list of CMEs is provided from NASA/GSFC. The event list of SEP is provided from NOAA/SWPC.
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Kataoka, R. Extreme geomagnetic activities: a statistical study. Earth Planets Space 72, 124 (2020). https://doi.org/10.1186/s40623020012618
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Keywords
 Magnetic storms
 Sudden commencements
 Substorms
 Solar flares
 Coronal mass ejections
 Solar energetic particles
 Lognormal distributions
 Powerlaw distributions