SELF-ORGANIZED CRITICALITY MODEL OF SOLAR PLASMA ERUPTION PROCESSES ANDREW V. PODLAZOV Keldysh Institute Applied Mathematics, Miusskaya 4, Moscow Russia E-mail: [email protected]

ALEXEI R. OSOKIN Sternberg Astronomical Institute, Universitetskiy 13, Moscow Russia E-mail: [email protected]

Abstract. We propose a new two-dimensional self-organized critical model of eruption process based on the concept of magnetic elements. Solar flares are considered as avalanches of annihilations of magnetic elements. This approach allows to describe eruptive processes in the solar atmosphere in the physically clear manner and easily simulate their basic properties. The model proposed yields a power law distribution of flare energy in a good agreement with observations. One can also expand the model to take into account new factors and ideas. Keywords: self-organized criticality; eruption process; solar flares; magnetic tubes

1. Introduction The distribution of flare energies is a power law, i.e. the probability of flare with energies E higher than a fixed one obeys the following relation U (E) ∼ E −(1+αE )

(1)

with αP = 0, 45 ± 0.15. Similar relations hold for distributions of fluences P and durations T with exponents of αP = 1, 75 ± 0.15 and αT = 0, 6 ± 0.4 (Kassinksy et al., 1997; Crosby et al., 1993; Georgoulis et al., 2001). Power law distributions like (1) describe many different nature systems. Presence of such distribution means that the system is in the critical state with no characteristic scales. Normally complex systems become critical via fine-tuning of control parameter to the critical point. However this mechanism does not work in the situation of general position and cannot explain ubiquity of power law distributions in nature. The origin of power law distributions was discovered by the theory of selforganized criticality (Back, 1996). Open nonlinear systems can spontaneously evolve into the critical state under quite simple conditions. In this state any perturbation of the system can cause an avalanche-like response of any magnitude. The first attempt to model the dynamics of solar flares was made by Lu and Hamilton (1991, 1993). They considered active region as a three-dimensional latAstrophysics and Space Science 282: 221–226, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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tice with cells containing the magnetic field. If the difference between the field in the cell and the average field over six neighbors exceeds some threshold the cell becomes unstable and the difference is redistributed among these seven cells with a release of energy. This redistribution can make neighbors unstable causing the avalanche of sequential redistributions. The LH-model self-organizes into a critical state yielding power law distributions of flares with the exponents quite close to the observed values. Nevertheless this model has several weak points: – the average filed in the system grows unlimitedly because the model is driven by adding of a random vector with nonzero mean to a randomly selected cell; – real physical meaning of the ‘magnetic field’ used in the model is unclear; – the chromosphere must be considered as two-dimensional system because magnetic field is radial there.

2. Solution So we propose a different approach to the self-organized critical description of solar flares. Our model is based upon new observation data on the structure of the magnetic fields in the solar atmosphere. While observed at a high resolution the solar surface resembles a carpet consisted of microtubes of magnetic field. Local energy releases are related with simplification of the carpet structure or with disappearance of microtubes. So we propose a different approach to the self-organized critical description of solar flares. Our model is based upon new observation data on the structure of the magnetic fields in the solar atmosphere. While observed at a high resolution the solar surface resembles a carpet consisted of microtubes of magnetic field. Local energy releases are related with simplification of the carpet structure or with disappearance of microtubes (Schrijver, 1997). We can characterize the microtubes only by theirs two-dimensional coordinates and theirs magnetic field by the amount of accumulated energy. Abramenko (2001) shows that it’s possible to consider these tubes of local maximums Bz (magnetic field vertical components) as a scalar. Characterized in such way microtubes will be referred to as magnetic elements. The absolute value and the charge of element describes accumulated energy and the direction of the field of the microtube respectively. In our model the release of energy corresponds to a partial annihilation of magnetic elements of opposite signs (oppositely directed microtubes). Our model is a simple cellular automaton representing the active region by means of a two-dimensional rectangular lattice with periodic boundary conditions. Each lattice cell can contain one or more magnetic elements or it can be empty. Model rules are as follows:

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Figure 1. Model simplification.

1. Two magnetic elements of opposite sign and equal absolute value appear simultaneously in two random cells. Their value is Poisson random deviate with mean Q. 2. Coming into a cell tests it for the presence of elements of opposite sign. If any opposite elements present then one of them chosen at random annihilates with the incoming element. The absolute values of annihilating elements are decremented by one and the unit of energy is released. If element’s value becomes zero the element disappears. 3. Any release of energy in a cell causes an outward disturbance wave. This wave carries out all elements from the cell to its neighbors picked out at random among eight adjacent cells. These elements can also cause annihilations there (step 2)) resulting in an ava-lanche of annihilations. Such avalanche is nothing else than flare. If the transfer of elements doesn’t give rise to new annihilations then the avalanche is over and the step 1 repeats. In this model we neglect any motion of plasma in an active region except from disturbance wave forced by the energy of annihilations. This assumption is reasonable since there is no significant convection in the chromosphere. We simulated the model for different Q values on the lattice with size 512X512. After establishing stationary mode we made 30 millions steps for statistics gathering. Figure 2 shows the result of simulations.

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Figure 2. Distribution of flare energy ρ(E). The results are for Q = 8, 16, 32, 64, 128, and 256. The larger Q the closer curves to the straight line corresponding to a power law in logarithmic coordinates.

3. Results and Conclusions The model self-organizes into the critical state regardless initial conditions. Strict power law distribution of form (1) corresponds to infinite Q limit. In case of finite Q the distribution law is given by finitesize scaling relation N(E) ∼ Q−β f (EQ−ν )

(2)

where scaling function f (x) diminishes like x −(1+α) while x is moderate and faster than any power of x otherwise. Distribution (2) must turn into relation (1) for moderate energies. Hence, α = β/ν − 1 (3) Relation (3) allows determining the exponent a accurately for finite values of Q. Figure 3 shows scaled probability U (E)Qβ vs. scaled energy EQ−ν . Obtained scaling exponents ν and β yields α = 0.37 ± 0.04. This result lies in the error bar of observed value but it is less than the majority of estimates. In our opinion the divergence of the exponents is due to the presence of a preferred direction in the flare region. Magnetic elements on the boundary of a flare tend to go outward. It is some kind of inertia. One could take it into account by means of introduction of additional rule: 4. At the step 3 each element makes A attempts to find a neighbor cell unburned during current flare. If any of the attempts is successful than the element moves to the cell found. Otherwise the usual routine is used.

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Figure 3. Finite-size scaling of distribution of flare energy. Same as Figure 2 but data are scaled with powers of Q. Scaling exponents ν = 1.50 ± 0.03 and β = 2.05 ± 0.03 provides the best overlap of curves for different Q.

Figure 4. Distribution of flare energy u(E) for the model with the rule (4) added. The results for Q = 128 and A = 0, 2and5. Larger A gives steeper line.

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Figure 4 shows the results of simulation of the modified model. Line slopes grow while attempts number increases. So we can presume that the presence of inertia can raise α and make it closer to observed values. Unfortunately due to the inertia effects the distribution gains hump in the range of high energies. This circumstance disallows to use finite-size scaling for accurate determination of α. Theoretical estimates made for the model yield α = 1/3 for A = 0 and α = 1/2 for A = ∞ (Podlazov and Osokin, 2000). Summarizing, we have introduced the new simple self-organized critical model of solar flares based on the concept of magnetic elements. We believe the introduction of additional conditions in our SOC model to be very promising. There will be more articles on this subject.

Acknowledgements We wish to thank Dr M.A. Livshits at IZMIRAN for kindly providing us with connection between the SOC-model evolution rules and the observations. This work was supported by Russian Foundation for Basic Research, grants 01-01-00628 and N99-02-18430.

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