Power Law Flare Statistics Driven by Photospheric

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Power Law Flare Statistics Driven by Photospheric

Turbulence

Argyrios Koumtzis

Supervisor: Nikolaos Stergioulas, professor

Physics Department Faculty of Natural Sciences Aristotle University of Thessaloniki

Greece

July 2019

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Abstract

One of the outstanding open questions in the analysis of solar explosions is their statistical characteristics. The fact that the turbulent photospheric motions act as a

driver for the formation of fragmented currents has been suggested as a possible explanation. Another potential explanation was offered with the use of the concept of

the Self-Organized Criticality (SOC). In this thesis, we address three questions: (1) What are the statistical properties of the photospheric turbulent driver of the energy release in the solar active region? (2) What are the statistical properties of the solar explosions, driven by photospheric turbulent flows, when solar active regions are far from the SOC state? (3) how are the statistical properties of our system affected when

there is an interaction between the turbulent driver and the SOC state? We use a well known non linear force free extrapolation method to estimate the magnetic field topology of the active regions above the photosphere. The extrapolated magnetic

fields from a sequence of magnetograms are used to extract information for the turbulent photospheric driver. We use a Cellular Automaton model, developed twenty

seven years ago, to explore the applicability of the concept of SOC in the interpretation of the flare statistics, when the solar active regions are driven by turbulent photosphere and are far from or very close to the SOC state.According to our analysis the statistical properties of the driver of the solar active regions, extracted

for the extrapolated magnetic fields, follow a double power law probability distribution function. The peak luminosity and total released energy distributions of the solar explosions obey a power law even in the beginning of the simulation much before the active region reaches its dynamic equilibrium state. The indices of the estimated power laws from our analysis agree very well with the statistical analysis of

the explosions recorded in soft X-rays, hard X-rays and extreme ultraviolet radiation.

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*Acknowledgments First, I would like to thank my professor and mentor Loukas Vlahos for our excellent collaboration. His passion and enthusiasm for our work helped me to remain motivated along the way. Professor Nikos Stergoulas was also always more than willing to offer his assistance whenever I asked him to do so. Moreover, the completion of this work would have been impossible without the help of Heinz Isliker who could always find a bug in my code that I couldn’t discover myself. Finally, I cannot forget the unconditional support of my family and friends who were constantly by my side when I was disappointed by one of the many failures I encountered. Nikolas Lasithiotakis and Jason Mparmparesos are two of my friends who not only supported me psychologically, but also helped me with technical issues, devoting a lot of their free time.

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1 Introduction

With the term solar flare we describe the explosive events happening in the solar corona during which a significant amount of energy is released. Some of these phenomena are also combined with coronal mass ejections. Solar active regions are areas on the photosphere with intense magnetic fields where most of the flares are recorded. The reconfiguration of the complex magnetic field topology above these regions is thought to be responsible for the majority of the solar explosions. These events can be observed from hard x-Rays to radio waves as they differ significantly in the amount of released energy and duration. The solar flares and the coronal mass ejections are the major drivers of space weather. Thus, deepening our understanding of these phenomena is crucial in accurately forecasting them and lessening the severity of their consequences for our civilization.

The complexity of the physical processes taking place in the solar atmosphere and the lack of direct measurement of physical quantities hinder our efforts to fully compre- hend and predict solar flares. Another way to study this externally driven dynamical system is by the use of a statistical approach. More than forty years of solar observations reveal that the size distributions of flares obey robust power law distributions which remain unchanged during a solar cycle. More specifically, the frequency his- tograms of peak luminocity, total released energy and duration of an event are all described by well defined power law indices [Aschwanden and Parnell, 2002]. These power laws extend for several orders of magnitude covering the whole energy flare spectrum. In other words, solar explosions seem to follow the properties of other physical phenomena Such as avalanches, earthquakes etc. This generalization led several flare occurrence models to be inspired by the concept of Self-Organized Criticality (SOC) (see recent reviews by Aschwanden et al. [2016] and Vlahos and Isliker [2016]

and their cited references). Parker [1983, 1988] proposed that weak solar explosions (nanoflares and microflares) are driven by the random continuous motion of the foot- points of the magnetic field in the photospheric convection and larger flares are the result of the superposition of numerous nanoflares.

The formation of a 3D theoretical model that reproduces flare statistics requires the inclusion of a fully developed convection zone, a stratified atmosphere above it and the study of the interplay between magnetic field and plasma flows. At present such a multi scale study is far out of reach due to extensive requirements in computer facilities.

In the years following the Parker conjecture two different lines of research appeared which now seem to converge on similar overall conclusions.

Using the MHD equations, the formation of current sheets, assuming a random forcing or motion of the magnetic field lines at the photosphere [Mikic et al., 1989, Einaudi et al., 1996, Galsgaard and Nordlund, 1996, 1997a,b, Georgoulis et al., 1998, Rappazzo et al., 2010, Dahlburg et al., 2005, Kanella and Gudiksen, 2017, 2018, Knizh- nik et al., 2018] was studied in detail.

Einaudi et al. [1996] analyzed a 2D section of a coronal loop, subject to random forcing of the magnetic fields. Galsgaard and Nordlund [1996] solved the dissipative 3D MHD equations in order to investigate an initially homogeneous magnetic flux tube stressed by large scale sheared random motions at the two boundaries. They recorded spontaneous formation of current sheets at random places and at random times (turbu-

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lent reconnection) inside the structure. In a series of articles, Rappazzo et al. [2010, 2013], Rappazzo and Parker [2013], Rappazzo et al. [2017], following the steps of the work of Galsgaard and Nordlund [1996], analyzed the establishment of turbulent reconnection in the solar corona. The observational expectations from the intermit- tent heating in turbulent reconnection were also investigated in depth Dahlburg et al.

[2016].

Georgoulis et al. [1998] extended the simulations of Einaudi et al. [1996] for much longer periods. Their aim was to extract reliable statistical information about turbulent reconnection in the solar atmosphere. Their main result was that the distribution function of both, the maximum and average current dissipation, and of the total energy content, the peak activity and the duration of such events, all show a robust power law distributions. Using completely different settings Kanella and Gudiksen [2017, 2018], and Knizhnik et al. [2018] extended the search for the statistical properties of randomly driven by motions at the boundaries of simple potential magnetic topologies.

In summary, the analysis of the Parker conjecture using the resistive MHD equations followed the subsequent steps.

• The turbulent photospheric motion was simplified with the random sheared motions

• The magnetic topology was simplified with a loop with straight field lines or a magnetic dipole in the potential state

• The formation and dissipation of magnetic discontinuities was approximated with a current driven, anomalous electric resistivity.

• The statistical properties of the explosive events recorded by the MHD simulations show remarkably stable power law distributions in the energy released [Georgoulis et al., 1998, Kanella and Gudiksen, 2017, 2018, Knizhnik et al., 2018].

Twenty eight years ago Lu and Hamilton [1991] assumed that solar active regions are driven complex systems and used the then newly proposed concept of SOC [Bak et al., 1987] to model Parker conjecture. We will outline very briefly Lu and Hamilton’s (LH) model in the next section. An important part of the SOC model is that it assumes that solar active regions reach a dynamical equilibrium state called “SOC state” by self- organising or self tuning without an external control parameter, which is accomplished by a slow and continuous driver, which brings back the system after each explosion.

In other words the solar active regions when in SOC state, the energy of solar active regions is balanced between the slowly-driven input from the turbulent photosphere and the (spontaneously) impulsive energy release and thus energy is conserved in the system.

Numerous models appear in the literature, following the initial suggestion of LH [Aschwanden et al., 2016] but only few of them use a model for the driver which reproduces the turbulent flows in the photosphere [Georgoulis and Vlahos, 1996, Boffetta et al., 1999, Hughes et al., 2003, Fragos et al., 2004, Mendoza et al., 2014] .

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Mendoza et al. [2014] uses a lattice Boltzmann model on a square lattice of size L, with a forcing term reproduces the Kolmogorov energy spectrum regime. The foot- prints of the magnetic flux tubes are anchored in the photospheric flows and are twisted by the vorticity. The magnetic field is simple and the magnetic field lines are wrapped around a semi-circular flux tube. The energy is released based on the kink instability.

Although This model used some of the characteristics of the SOC models it is not clear if their system ever reached the SOC state or if the scaling free properties of the impulsive energy release are due partly to the properties of the driver and partly to the internal evolution of the loop system.

In this work we pose three questions: (a) Is the uniform random forcing a good rep- resentation of the turbulent photospheric motions for the driver for the energy released in the solar active regions or does the driver extracted by observations better describe the physical processes happening in the solar corona? (b) Can the solar active regions driven by the turbulent photosphere show a power law statistic in the energy release before reaching the SOC state (c) what happens when there is a synergy between a turbulant driver and the SOC state?

The structure of this thesis is as follows. In section 2 we outline the LH model and we stress in particular the characteristics of the solar active region before it reaches the SOC state. In section 3 we explore the driver we recover by analysing a series of non-linearly extrapolated magnetograms and using this result as the driver, especially in the pre-SOC state of the solar active region. In the final section we discuss our main results.

2 The Lu and Hamilton model

Our aim in this section is to explore the well known sand pile model introduced initially by Bak et al. [1987] almost thirty years ago. In order to achieve this, we first reproduce the results of the simple cellular automaton sand pile model which was proposed by Lu and Hamilton [1991]. A short presentation of the cellular automaton model rules and the procedure we followed is presented below. In these kinds of models the physical processes are simulated using a set of rules which approximate the real complex behaviour of a natural system, which in our case is the solar atmosphere. In general there are two types of rules, theloadingand therelaxation. The loading rules define how the main variable of the system changes at some grid sites, while the relaxation rules define how the variable is redistributed among the neighbouring grid sites. The rules should be local in both space and time; that is, they should involve only the local neighbourhood (no grid site at a distance greater than one), and just the configuration at the previous time step (no history or memory is included). In our case the main system variable is the magnetic field. In studying the respective sand pile model we use a 3D grid of size 64×64×64 pixels. The rules of the model are:

1. The loading rule: A random vectorδB with components in the rangeδB ∈ [−0.2,0.8] is added to a random position inside the grid, at the beginning of each timestep, so:

B⁺_i,j,k=B_i,_j,k+δB, for specifici,j,k; (1)

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The loading rule is often called the driving mechanism or simply the driver.

2. The relaxation rule: The relaxation rule depends on dB, the local magnetic gradient, which is the difference between the local magnetic field and the average of its nearest neighbours (denoted nn), namely

dBi,j,k=Bi,j,k−1 6

X

nn

Bnn, {i,j,k}=1, . . . ,N. (2) A grid site (i,j,k) is considered unstable if the magnitude of its local magnetic gradient exceeds a certaincriticalvalue, which was chosen to be equal to 7. It has been proven that the value of the threshold does not play any role on the final results.

|dBi,j,k|>Bc. (3)

When this occurs, the rules tend to reduce the magnetic gradient by redistributing magnetic field from the unstable grid site to its neighbours, according to the relations:

B⁺_i,j,k=Bi,j,k−6 7dBi,j,k, B⁺_nn=Bnn+1

7dB_i,j,k,

{i,j,k}=1, . . . ,N. (4)

where the values after redistribution are denoted with a plus sign (+) as super- script. The local field then equals the average of its neighbourhood.

After a significant number of time-steps the simulation box reaches the SOC state.

As already mentioned in the introduction the SOC state is a property of some dynamical systems which evolve towards an asymptotic state. These systems are self-organized into this critical state, with low dependence on their parameters. In our case, the total sum of the magnetic field and the total energy (sum of the magnitude of magnetic gradient) of the system remain constant when the system reaches SOC state. Finally, the last quantity we need to introduce to our model is theReleased energy:Themagnetic energyreleased during a redistribution event is

∆B²nn= 6

7|dBi,j,k|². (5)

2.1 Important quantities and definitions

When our system is in SOC state we are interested in studying the statistical properties of two different quantities by analyzing the time series data which have been produced by our simulation. These quantities are:

1. Peak energy/luminosity: During an avalanche, the numerical box should be scanned a number of times in order to become completely relaxed. In each time step an amount of energy is released. The peak energy is the largest released energy during an avalanche.

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2. Avalanche energy/total released energy/step energy: The avalanche energy is simply the sum of all the released energy during an avalanche.

The term time step is used to refer to one scanning of the system while the term perturbation step is used for time steps in which a perturbation is added to the grid.

2.2 Reaching SOC state

The initial field in our simulation does not play any role because as stated before this kind of system does not have memory. This is the reason why we use zero magnetic field as a starting condition in our simulation. The system needs several million perturbation steps so as to reach SOC state. The number of steps depends on the box size and the loading rules. In our case 1100 million perturbations were performed in order to reach the SOC state. We use two criteria to ensure that the system reached SOC state, (1) the stabilization of the total system energy and (2) the stabilisation of the total magnetic field (see Fig. 1a and 1b).

−10 0 10 20 30 40 50 60 70 Perturbation step (20 milions) 25

30 35 40 45 50 55

Total system energy average (sim.units) Total system energy evolution

−10 0 10 20 30 40 50 60 70 Perturbation step (20 milions) 0.25

0.30 0.35 0.40 0.45 0.50 0.55

Average of total field (simulation units) Total magnetic field evolution

Figure 1: (a) Evolution of total system energy, (b) Evolution of total system magnetic fieldv.

In Fig. 1a and 1b it is clear that the total system energy and total magnetic field rapidly increase at the beginning of the simulation but after some time the increasing rate decreases and it becomes zero when the system reaches the SOC state. At SOC state the system is in the dynamic equilibrium state which in practical terms means that some fluctuations are observed in the total energy and field but the mean total energy and the mean total field remain constant with a negligible standard deviation.

More specifically in our case the average of the total system energy in SOC state is 49.1±0.0186 The mean total magnetic field in SOC state is 55.3±0.001. In practice by using the term SOC state in our work we mean the state of the system after 1.1×10⁹ perturbation steps. In other words the assumption is that the solar active region driven by the turbulent photosphere reaches the SOC state very slowly.

2.3 Result comparison

Having proven that the system is in SOC state we perform another 80×10⁶perturbation steps to produce enough data for our statistical analysis of peak luminosity and total

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10² 10³ 10⁴ Peak luminocity (simulation units) 10^-6

10^-5 10^-4 10^-3 10^-2

Peak luminocity frequency

Peak luminocity distribution Slope = -1.36 Slope = -1.85

10³ 10⁴ 10⁵ 10⁶ 10⁷

Total released energy (simulation units) 10^-9

10^-8 10^-7 10^-6 10^-5 10^-4 10^-3

Energy frequency

Total released energy distribution Slope = -1.45

Figure 2: (a) Peak released energy distribution in SOC state, (b) Total released energy distribution in SOC state

released energy. According to the Figs. 2a, 2b, the statistical properties of peak luminosity and total energy are in total agreement with those presented by Lu and Hamilton [1991].

2.4 Properties of the system before the SOC state

It is interesting to study what happens in the simple LH model before reaching SOC state. Since the characteristic time of reaching the SOC state is very long the study of the statistical properties of the explosive energy releases (flares) is important since the behavior of the solar active regions before they reach SOC state, if ever, must be recorded by the observations. Statistical analysis of the first 7×10⁷ perturbation steps shows that only a small percentage of the avalanches result in peak luminosity above the noise level. In this 70 million steps only 0.5% resulted in avalanches with peak luminosity in the energy range studied in SOC state. In LH model where the magnitude of the loading is small compared to the threshold, the system should reach the SOC state in order to produce large avalanches. This is logical if one considers the nature of the phenomena we are dealing with. When the system is far from SOC state there are not many points within the system volume with a magnetic field gradient close to the critical threshold. Therefore, when a perturbation is added the avalanche probability is a lot lower than that in SOC state. Moreover, when some avalanches are triggered before SOC state they cannot so easily trigger other secondary magnetic field redistributions in neighbouring points. When we try to plot the distribution of the total released energy we face the same situation, there is not enough data in the energy range we are interested in.

3 Power law driven model

3.1 Identifying the nature and the properties of the loading rule

As already discussed in the introduction our goal in this article is to study the magnetic coupling of the turbulent photosphere with the solar coronal active regions. To examine

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how the evolution of the magnetic field at the photospheric level drives the activity in the solar corona, we used a nonlinear force-free (NLFF) extrapolation method following the lines of Dimitropoulou et al. [2011, 2013]. From the extrapolated magnetic field one can study the magnetic discontinuities [Moraitis et al., 2016] which can naturally provide indications for the statistical properties of the driver [Georgoulis and Vlahos, 1996].

For the determination of the properties of the loading rule we used NOAA active region (AR) 11261. This AR was very active during the first days of August 2011, producing many flares, up to M class, and also a few coronal mass ejections [CMEs;

Ye et al., 2018]. It thus constitutes a typical solar AR, appropriate for our study.

The magnetic field data we used was obtained from the Helioseismic and Magnetic Imager [HMI, Scherrer et al., 2012] instrument onboard the Solar Dynamic Observa- tory [SDO, Pesnell et al., 2012], and is freely accessible. We used five vector magnetograms of AR 11261 from August 4 2011, corresponding to the time interval 03:24 UT – 04:12 UT, and with the 12-minutes cadence of HMI. During this period an M9.3 flare occured that was accompanied by the release of a high-speed CME [Yang et al., 2013].

For each vector magnetogram we resolved the disambiguity in the perpendicular to the line of sight component of the magnetic field with the method of Georgoulis [2005] as revised in Metcalf et al. [2006]. We then selected a 256×256-pixels part of the AR with the original HMI resolution of 0.5” where the magnetic field is more important.

3.2 The basic idea of the nonlinear force free extrapolation method

There are multiple ways to approach the problem of reconstructing the coronal magnetic field above an active region (Force free linear and nonlinear methods, magneto hydrostatic equilibriuma methods and driven MHD simulations). In this work, we chose a force free method which is not demanding in terms of computational resources (It can easily run on a desktop) but which is accurate enough in describing the formation of discontinuities within the coronal volume. As a force free method it assumes that electric currents are running parallel to the magnetic field lines thus enabling the vanishing of the Lorence force. This condition known as the force free condition, combined with the divergence free nature of the magnetic field can be expressed as a system of differential equations.∇ ×~b=α~band∇~b=0. whereαis a constant along each field line but takes different values on different lines. The solution of this system can provide the coronal magnetic field. The boundary condition for this system is the magnetogram (the magnetic field on the level of the photosphere). The other five boundaries are un- known so they are treated differently using the waiting functions. The first step of this method is to calculate the potential field (α=0. This calculation uses only the component of the magnetogram which is perpendicular to the photosphere and thus parallel to the line of sight. After performing this calculation, the potential field is used as an initial condition for the next steps of the algorithm. The next move is to change the line of sight field on the base of our numerical box with the real vector field of the magnetogram. It is obvious then that at this point the field in the box is losing its divergence freeness and force freeness. Therefore, the final and the non trivial part of the algorithm is the use of an iteration process which minimises the divergence of the field and the angle between the current and the field. This optimization principal results in a

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magnetic field the structure of which shows satisfying agreement with the stereoscopic observations of the corona. These observations although they do not provide information for the magnitude of the coronal field they reveal its 3DD configuration enabling a qualitative comparison between observations and different modeling techniques.

Figure 3: Original magnetogram for the eruptive NOAA AR 11261 on August 4, 2011 at 03:48 UT together with the NLFF field lines.

The three-dimensional (3D) coronal magnetic field is extrapolated to the height of 128 pixels (64”), by applying the NLFF field extrapolation method of Wiegel- mann [2004]. We used the multigrid-like option in the NLFF code after preproccess- ing the magnetograms [Wiegelmann et al., 2006] with the standard set of parameters µ = [1,1,10⁻³,10⁻²,0], and without smoothing the original magnetograms. In the NLFF code we also used the weighting function in the lateral and the top boundaries with a thickness of 12 pixels in each boundary. An example of the 3D morphology of the NLFF field is shown in Fig. 3 for the snapshot at 03:48 UT.

The resulting magnetic field datacubes for the five snapshots were further examined with respect to their quality in terms of solenoidality and force-freeness. In more detail, we considered two parameters introduced by Wheatland et al. [2000], the average frac- tional flux increase, fi, as a solenoidality indicator, and the current-weighted average angleθJas a force-free one. The mean values obtained for the different snapshots are fi=1.2 10⁻³andθJ=19^o, typical for this extrapolation method. These values indicate that the used magnetic fields are of sufficient quality for the purposes of this work.

From the five snapshots of the magnetic field we can construct four differences by subtracting every two consecutive NLFF fields. These differences are considered as the loading,δB, which we also split into its Cartesian components. As an example, we show in Fig. 4 the form of theδB_xloading component that is computed from the snapshots at 03:36 UT and 03:48 UT. We notice that the magnitude ofδB_xhas a similar dependence on height with the respective magnetic field component, with values decreasing with height.

For each loading component and for all four available loadings we calculate their

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Figure 4: Morphology of the loadingδBx(measured in Gauss) at three different heights in the volume, as obtained from the difference of the snapshots at 03:36 UT and 03:48 UT.

frequency histogram. We show in Fig. 5 a typical histogram created by such a process for the three loading components that are produced from the snapshots at 03:36 UT and 03:48 UT. We notice that the loading components follow an almost symmetrical distribution around 0, and extend for more than two orders of magnitude in both positive and negativeδB’s. We approximate all these distributions with two different broken power laws, one for the positive and one for the negativeδB’s. In all the broken power law fittings, we consider only the loadings that are within the magnitude rangeδB ∈ [10¹,10^2.8] G so that the distributions are smooth enough for the fittings to be reasonable. In the example shown in Fig. 5, the power law indices are∼ −0.8 in the low-loading regime and∼ −2.2 in the high-loading regime, while the break of the power law is at loadings of∼ ±100 G. The fittings are very successful in all cases, and similar values are obtained from the fittings to the other loadings as well, as is discussed in the next section.

3.3 Rules of the model

The real difference of our model with the LH model is the nature of the loading rule.

In the LH model small vectors are added to the system. The magnitude of these is some times smaller than the gradient threshold, and their components follow a uniform distribution. The loading rule in our case is still a random vector, whose components

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Figure 5: Distribution of thex- (left),y- (middle), andzcomponents of the loading (right), as this is obtained from the difference of the snapshots at 03:36 UT and 03:48 UT. The labels provide the values of the slopes and of the break for the fitted broken power laws. Loading units are in Gauss.

however, follow a more complicated distribution that stems from the observationally- derived distributions discussed in Section 3.1. We derive the mean values of the fitting parameters from a total of 24 sets, 4δB’s with 3 components each and 2 possible di- rections each. The slopes of the driver are defined then as these average slopes, namely α1 =−0.8 andα2 =−2.2. Furthermore, we assign the probabilities of each slope to be proportional to the number of the respective loadings. Since on average the number corresponding to the high-loading regime is 15 times smaller than the respective number in the low-loadings, the probability of each slope follows the same pattern¹⁵₁₆ for the low loading and ₁₆¹ for the high loading in the simulation code. The reason for choos- ing this loading rule is that, as shown in Section 3.1, it describes more realistically the way in which the magnetic field changes according to the NLFF extrapolations. There- fore, by using this probability function we have a better approximation of the natural processes happening in the solar corona as this is driven by the processes occurring in the photospheric boundary. The other rules and procedures of our model are exactly the same with the LH model so there is no reason to restate them here.

In the rest of the article the analysis is focused on the statistics of the peak luminosity and total released energy, the same quantities we analysed in the LH model case. These are the most trustworthy observable quantities that enable us to compare our results with relevant observations. In practice, we use two different settings in our work: a) A symmetrical broken power law distribution, and b) an asymmetrical one, following the probability function of the driver. First we investigate the effect of the symmetrical driver and after understanding its impact to our results we continue by introducing an asymmetry to it.

3.4 Symmetrical power-law driven model

3.4.1 System evolution

After establishing the rules of our model the first step of our simulation is to see if the system reaches the SOC state by monitoring the evolution of the total magnetic field and the total system energy. This is shown in Fig. 6 and it was done as described in Section 2.2.

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−20 0 20 40 60 80 100 120 140 160 Perturbation step (20 milions) 0.840

0.845 0.850 0.855

Total system energy average (LH SOC) Total system energy evolution

−10 0 10 20 30 40 50 60 70 80 Perturbation step (20 milions) 0.000

0.005 0.010 0.015 0.020 0.025 0.030

Average of total field (LH SOC) Total magnetic field evolution

Figure 6: Evolution of the system’s total energy (left) and total magnetic field (right).

In this case the behaviour of the system is somewhat different. At first, we have to take into consideration the fact that the driver is completely symmetrical in contrast to the LH model where there is always a specific loading direction. This symmetry does not allow the formation of permanent structures in the system like the well known sand pile in SOC state. This is why even after 3×10⁹ perturbation steps the system energy is 85% of the system energy in the respective SOC state for the LH, and the total field is only 2.5% of the total field of the SOC state. For these reasons we will not call the state of the system after 3×10⁹ perturbation steps as SOC state, but rather as final state. It will be useful for our understanding to visualize how the magnitude of the magnetic field inside the simulation volume changes, so as to have a general picture of the small-scale structures forming in the system.

0 10 20 30 40 50 60 0

10 20 30 40 50 60

Isostathmic magnetic surfac

10 20 30 40 50 60 70 80

0 10 20 30 40 50 60 0

10 20 30 40 50 60

15 30 45 60 75 90 105

0 10 20 30 40 50 60 0

10 20 30 40 50 60

1530 4560 7590 105120 135

0 10 20 30 40 50 60 0

10 20 30 40 50 60

bz Isostathmic surface

−75

−50

−25 0 25 50 75 100

Figure 7: Top: Magnetic field magnitude distribution at the heightz=31 after 4×10⁸ (left), 16×10⁸(middle), and 30×10⁸perturbation steps (right). Bottom:Bzcomponent distribution at the heightz = 31 after 30×10⁸ perturbation steps. The units of the colorbars are G.

In Fig. 7 we show the form of the magnetic field on the plane z = 31, near the

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center of the volume. Similar images can be obtained for the other components, and at different planes in the volume of the system as well. In the bottom plot of Fig. 7 we depict thezcomponent of the field in the same plane. In both cases (magnitude orzcomponent of the magnetic field) we can see that instead of a big structure like the sand pile, smaller coherent areas are formed. We can describe the situation like an anomalous landscape with hills and valleys. During the simulation the morphology changes radically as existing hills are destroyed and new ones are formed. This is expected if we take into consideration the fact that in our model, in contrast to the classic LH model, there is 50% probability for positive and negative increments. This means that if, for instance, there is a coherent area with a positive magnetic field and a negative vector is added there, the height of the hill will be reduced. This mechanism explains why our system cannot become organised on a global scale and only small scale structures are formed.

3.4.2 Peak luminosity and total released energy distributions

After studying the evolution of the system driven by a power-law driver, we discover that the peak luminosity and total released energy distributions again obey well fitted power laws. Therefore, we proceed to find their indices. We repeat this process two times, one for the first 2×10⁵perturbation steps (initial state) and one for the last×10⁵ perturbation steps (final state).

In Figs. 8a, b the slopes obtained from our simulation are estimated. The first slopes are a lot steeper compared to the LH model, although their extent is very small, while the second ones are a lot flatter and extend for almost 2 orders of magnitude. The rest of the data is at the noise level. We study these two slopes here because in this region 82%

of the total energy is released and it is reasonable to say that it contains phenomena of different scales. The first slope is equal to −2.81±0.03. We could relate this first region and its statistics to the hypothetical nano-flares, which may also explain why such steep distributions are not observed. The second slope obtained is−1.68±0.02, very close to the second slope obtained by LH model and by most of the observations.

In this case however, the breakthrough of our model is that the system exhibits this power laws statistics from the beginning of the simulation without the need of the SOC state concept. Moreover, these slopes in the initial state are changed very little when the system reaches the final state, to−2.82±0.04 and−1.68±0.04. This is an interesting result if we consider the fact that major changes and redistributions happen in the system during this 3×10⁹ perturbation steps, as discussed in Section 3.4.1.

We can also deduce that this stable behaviour depends on the driver and not on the state of the system. This conclusion is based on the fact that the system state changes radically and it seems to have little effect on the obtained frequency distributions, while the properties of the driver are defined before the simulation and remain unchanged throughout its duration. Similarly, we perform the same analysis with the total released energy analysing the two different data sets (initial and final state).

In Figs. 8c, d, we can observe that the behaviour of the total released energy is very similar to that of the peak luminosity. There are again two different distinctive energy regions which correspond to two different power law distributions. There is again the steeper part in the beginning of the distribution with a slope of−3.1±0.03,

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10³ 10⁴ 10⁵ 10⁶ Peak luminosity (simulation units) 10^-8

10^-7 10^-6 10^-5 10^-4 10^-3

Peak luminosity frequency

Peak luminosity distribution Slope = -2.81 Slope = -1.68

10³ 10⁴ 10⁵ 10⁶

Peak luminosity (simulation units) 10^-8

10^-7 10^-6 10^-5 10^-4 10^-3

10³ 10⁴ 10⁵ 10⁶

10^-7 10^-6 10^-5 10^-4 10^-3

Energy frequency

Total released energy distribution Slope = -3.11 Slope = -1.64

10³ 10⁴ 10⁵ 10⁶

10^-7 10^-6 10^-5 10^-4 10^-3

Energy frequency

Total released energy distribution Slope = -3.14 Slope = -1.65

Figure 8: (a) Peak luminosity distribution in initial state. (b) Peak luminocity distribution in the final state. (c) Total released energy distribution in initial state. (d) Total released energy distribution in the final state.

and the flatter one, with the slope−1.64±0.007 in the larger energies. In this case the second slope obtained is somewhat different than the LH one, and more significantly, the combination of the peak luminosity slope and of the total released energy is not common in the observations. Usually, the peak luminosity slope is steeper than the total released energy slope but this is not true when the system is far from the SOC state. Furthermore, the distribution of total released energy extends for 3.5 orders of magnitude in the LH model while here we do not detect avalanches with such big energies. Finally, the mean duration time is computed in initial and in final state. Its value (dt =2.2) is an order of magnitude smaller than the mean duration time in LH model.

In contrast to what happens in the LH model before the SOC state, a powerful driver is able to cause large avalanches without depending on the state of the system. Notice also that in this comparison in LH case we analysed 7×10⁷perturbation steps while in the other case 2×10⁵perturbation steps. This happens because after this number of steps both systems are in the same state in terms of system energy, with energy values of 38. In this way we ensure that they have the same “distance” from the SOC state.

After presenting the basic results of our model, we summarize and compare them to real observations. The basic characteristics of the avalanches occurring in our model are:

1. The power law slopes of the peak luminosity are equal to -2.81 and -1.68 for the initial state, and -2.82 and -1.68 in the final state.

2. The power law slopes of the total released energy are equal to -3.11,-1.64 in the

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initial state, and -3.13, -1.65 in the final state.

3. The duration time is smaller than the LH model duration time.

As stated in the Introduction, the power law slopes obtained by the data analysis of the observations exhibit a significant variability. Moreover, in general there is a larger difference between the slopes of peak luminosity and total released energy. However this is the case for all the observations in hard X-rays, which are related to the high energy particles and their generation are not directly related with the magnetic energy released by the current dissipation in the magnetic discontinuities. In soft X-rays and in ultraviolet there are some cases where the slope of total released energy is steeper than the one of peak luminosity and very few cases where they have very similar values. A dataset analysed by Aschwanden and Parnell [2002] resulted in fittings with parameters very close to these obtained by the analysis of our simulation results. The peak luminosity slope was found to be−1.75±0.07 and the total released energy slope−1.7±0.17.

There are no recorded observations for the nano-flares so we cannot use this energy region to validate our model. However, such steep distributions are estimated that they could provide sufficient energy to heat the solar corona.

3.5 Asymmetrical power-law driven model

Having studied the behaviour of the system when it is driven by a symmetrical driver we now continue our analysis by driving the system with an asymmetrical broken power law driver, as the loading distributions indicate. This asymmetry is small compared to the width of the entire distribution as stated in Section 3.1, but it defines a specific loading direction and allows the system to form global scale structures. The asymmetry may also relate to the quality and method of NLFF extrapolation. The rules are the same as the rules of the model described in Section 3.3, we only added an asymmetry to all the components of the driver which is equal to 5 G the¹₂ of the lower limit of our power law driver. We again follow the steps delineated in Section 2.2. The evolution of total system energy and total field in this case are very similar to this of the LH model. After 60×10⁶perturbation steps (almost 20 times fewer than in LH case) the system reaches a final state, really close to the classic SOC state. The system energy is 99.4% of the LH system energy and the total field is 99.5% of the total field in the LH case. Moreover, in Fig. 3.5 we can see the image of the well-known sand pile. The magnetic surface in view is the perpendicular plane to the z axis with a height equal to 31. As stated before, the system succeeds in organising itself on a global scale because of the asymmetry. However, the addition of large loadings with opposite signs does not allow the system to reach exactly the typical SOC state as one would expect. Although the evolution of the system and its final state present significant similarities with those of the LH model, the power law driver not only forces the system to reach its final state a lot faster, but it also produces power law size distributions even in the beginning of the simulation. More specifically, analysis of the first 2×10⁵perturbation steps results in identical total energy and peak luminosity distributions with those in Figs. 8a, c. It seems that in the beginning of the simulation where the system is far from self organ- isation the asymmetry does not have any impact on the resulting distributions. The effect of the asymmetry is that it forces the system to become self organised and the

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final state is so close to SOC state that we can see significant differences compared to the distributions produced by the symmetrical driver. As we can see in Figs. 3.5c, d the results in SOC state are closer to those of the LH model. The peak luminosity distribution is fitted again with two distinctive slopes−2.9±0.01 and−1.8±0.02. The total released energy in SOC state exhibits more complicated statistics. We can detect three different regions. We would expect to find a steeper slope in the smaller energies as is the case in the initial state but here the first slope is very flat and contains a very small amount of energy so we can consider it to be of minor importance. The second region contains 50% of the total released energy. Its slope is -1.45 in total agreement with the LH model. The last region contains 42% of the total released energy. Its slope is -2.21 which is also the value of the second slope of the driver. In the LH case the total energy distribution is dominated by our second slope while here there is an interaction of the state of the system with the driver which affects the total released energy distribution.

Such a break in the total released energy slope has been reported in several articles.

0 10 20 30 40 50 60 0

10 20 30 40 50 60

Isostathmic magnetic surface

6001200 18002400 30003600 42004800 5400

0 10 20 30 40 50 60 0

10 20 30 40 50 60

bz Isostathmic surface

400 800 1200 1600 2000 2400 2800 3200

10³ 10⁴ 10⁵ 10⁶

Peak luminosity (simulation units) 10^-8

10^-7 10^-6 10^-5 10^-4 10^-3

10³ 10⁴ 10⁵ 10⁶ 10⁷

10^-8 10^-7 10^-6 10^-5 10^-4

Energy frequency

Total released energy distribution Slope = -0.59 Slope = -1.45 Slope = -2.21

Figure 9: (a),(b) Isostathmic magnetic surfaces. (c) Peak luminosity distribution in SOC state asymmetrical model. (d) Total released energy distribution in SOC state asymmetrical model.

4 Summary and conclusions

We can summarise our work by the following points:

A: Symmetrical power-law driven model

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1. We used observational data so as to explore the properties of a realistic loading rule for the first time. We show that according to this analysis the loading rule follows a broken power law probability distribution function.

2. The peak luminosity and total released energy distributions obey a power law even in the beginning of the simulation a lot before the system reaches its final state. The exponents of these power laws are close to the soft X-rays and the extreme ultraviolet statistical analysis of observed explosive events.

3. Despite the many changes happening in the system in terms of total energy, total magnetic field and magnetic field topology, the frequency distributions remain remarkably constant throughout the simulation.

4. Our results (the combination of peak luminosity and total released energy slopes) are consistent with some observations which are not yet explained by other cellular automata or MHD models.

5. We also find flares of smaller sizes, the statistics of which are described by steeper power law distributions. We propose that this energy region corresponds to the hypothetical nano-flares and this is the reason why we do not detect such steep distributions when we observe the solar explosions.

B: Asymmetrical power-law driven model

1. In the beginning of the simulation far from final and SOC state, the behaviour of the system is identical to that of the symmetrically driven model.

2. The system exhibits global scale self organization forming the well known sand pile structure.

3. The events in final state release more energy and their statistics are in agreement with the LH model and with relevant observations.

4. The distribution of the total released energy in the final state shows the signatures of the power law driver and of the SOC state. Moreover, we do not detect steep power laws in the total released energy as most of the energy is released in other energy regions.

We propose that the reason the frequency size distributions in the ultraviolet spectrum show a larger variability than the distributions in higher energies is that the active regions where flares in the ultraviolet spectrum are observed are driven by symmetrical drivers. In this case, the slopes of the driver have a significant impact on the resulting frequency distributions causing this large variability. Moreover, if we continue this line of thought we can argue that these systems remain close to the nano- or micro-flare activity mainly since the driver is symmetric and never reaches the SOC state. When we drive the system with a preferred direction and come closer to SOC state we detect larger avalanches, the statistics of which are closer to the statistics of the majority of the observations. We can say that the asymmetry plays the role of deciding which scenario will be followed. There are still many questions open along the lines of our work, such

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as: How do the characteristics of the driver change with respect to the time evolution of the magnetograms of an active region? Are the results reported here universal? Can we use the setting reported in this article to drive a 3D MHD code? We plan to return to these issues.

5 Appendix

The fortran code developed can be found in the link below: https://www.dropbox.com/s/05byfhtl4ospy7e/Powerlawdriven.txt?dl= 0

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Power Law Flare Statistics Driven by Photospheric