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The Astrophysical Journal, 729:101 (6pp), 2011 March 10 C 2011.

doi:10.1088/0004-637X/729/2/101

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

PARTICLE ACCELERATION BY MAGNETIC RECONNECTION IN A TWISTED CORONAL LOOP Mykola Gordovskyy and Philippa K. Browning Jodrell Bank Centre for Astrophysics, University of Manchester, Manchester M13 9PL, UK Received 2010 September 23; accepted 2011 January 6; published 2011 February 14

ABSTRACT Photospheric motions may lead to twisted coronal magnetic fields which contain free energy that can be released by reconnection. Browning & Van der Linden suggested that such a relaxation event may be triggered by the onset of ideal kink instability. In the present work, we study the evolution of a twisted magnetic flux tube with zero net axial current following Hood et al. Based on the obtained magnetic and electric fields, proton and electron trajectories are calculated using the test-particle approach. We discuss resulting particle distributions and possible observational implications, for example, for small solar flares. Key words: acceleration of particles – magnetic reconnection – Sun: flares Online-only material: color figures an unstable twisted magnetic flux tube; in their model the reconnection occurs in the nonlinear phase of the ideal kink instability, as current sheets develop. The energy release in this model is rather uniform along the reconnecting flux tube. This model gives an opportunity to develop a comprehensive model for the single-loop events. Recently, Gordovskyy et al. (2010a, 2010b) developed a two-dimensional time-dependent model of proton and electron acceleration using a combination of MHD and test-particle approaches. In the present study, we extend this method to three dimensions (3D) in order to study magnetic reconnection in an unstable twisted coronal loop and the resulting particle acceleration.

1. INTRODUCTION High-energy particles are one of the main features of solar flares. In major events electrons and ions can be accelerated to the energies of 100 MeV and 10 GeV, respectively (Mandzhavidze & Ramaty 1993; Krucker et al. 2008). The explanation of the acceleration process is one of the key problems of solar physics. It is widely accepted that the main mechanism behind the energy release in solar flares is magnetic reconnection in a current sheet occurring due to the interaction of opposite magnetic fluxes in the upper corona. The standard model of solar flares (see, e.g., Hudson 1978; Forbes 2000; Vlahos 2007) implies that particles are primarily accelerated in the upper corona in a volume small compared with the volume of the flaring atmosphere. Some of these high-energy particles are ejected into the interplanetary space and others precipitate toward the dense chromosphere. However, this standard model faces several significant difficulties. For example, the number of accelerated particles derived from observations of non-thermal emission is comparable to, or even higher than, the total number of particles in the reconnection region (e.g., Brown et al. 2009). Furthermore, if the particle beam is not completely neutral (i.e., the flux of negative charges is not equal to the flux of positive charges) it will lose most of its energy due to the self-induced electric field before reaching the chromosphere to produce the observed hard X-ray emission (Emslie 1980; Diakonov & Somov 1988; Zharkova & Gordovskyy 2005). These difficulties may be avoided if the particles are re-accelerated during their transport (Brown et al. 2009) or the acceleration process is distributed within the flaring loop or arcade (see, e.g., Turkmani et al. 2006). Uniform acceleration would also help to overcome the problem of charge neutrality, even if the ion and electron velocities are different. Observations show that many small impulsive flares do not demonstrate the complicated structure of large arcade-type events, instead these flares are confined to a single flaring loop (e.g., Uchida et al. 2001). The mechanism behind the energy release and particle acceleration in such events is not exactly clear. The majority of models addressing the problem focus on current dissipation or magnetic reconnection occurring due to some instability developing in the coronal loop. Browning & Van der Linden (2003), Browning et al. (2008), and Hood et al. (2009) considered magnetic reconnection in

2. MAGNETIC RECONNECTION IN A TWISTED FLUX TUBE In this section, we study magnetic reconnection in a twisted magnetic flux tube following Browning et al. (2008) and Hood et al. (2009). Similarly to Hood et al. (2009), we consider the initial configuration with force-free magnetic field to be defined by a set of Bessel functions: ⎧ ⎨B1 J0 (α1 r ) Bz = B2 J0 (α2 r ) + C2 Y0 (α2 r ) ⎩ B2 J0 (α2 L0 ) + C2 Y0 (α2 L0 ) ⎧ ⎨B1 J1 (α1 r ) Bθ = B2 J1 (α2 r ) + C2 Y1 (α2 r ) ⎩ 0

⎫ (r < Rc ) ⎬ (Rc < r < L0 ) (1) ⎭ ( r L0 ) , ⎫ (r < Rc ) ⎬ (Rc < r < L0 ) . (2) ⎭ ( r L0 ) .

This model (see Figure 1) has a strong azimuthal component within the shell r = L0 , while outside this cylinder the magnetic field is purely vertical (i.e., potential field). In the present paper, we use the same model parameters as in Case 1 in (Hood et al. 2009): α1 = 2.5, α2 = −0.9418, and Rc = 0.5 L0 . With this set of parameters the azimuthal component of magnetic field Bθ vanishes at r = L0 and the initial magnetic field configuration is unstable in respect of the line-tied ideal kink instability (Browning & Van der Linden 2003; Browning et al. 2008). The coefficients B1 , B2 , and C2 are determined so that the magnetic field components are 1

The Astrophysical Journal, 729:101 (6pp), 2011 March 10

Gordovskyy & Browning

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Figure 1. Initial magnetic field (panel (a)) and current density (panel (b)) components for the considered model. (A color version of this figure is available in the online journal.)

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Figure 2. Azimuthally averaged z- (panel (a)) and θ - (panel (b)) components of magnetic field at z = 0 cross-section of the domain. Solid lines correspond to the initial configuration; dashed and dot-dashed lines are for t = 200t0 and t = 320t0 , respectively.

continuous, though the current density jumps at r = Rc and r = L0 (see Browning et al. 2008). The evolution of the system is followed using the numerical simulations with the Lare3D code (Arber et al. 2001). The model with the dimensions x = −2.5 L0 , . . ., 2.5 L0 , y = −2.5 L0 , . . ., 2.5 L0 , andz = −10.0 L0 , . . ., 10.0 L0 is represented by the numerical grid of 256 × 256 × 512 elements. The standard set of resistive MHD equations (see, e.g., Equations (4)–(8) in Gordovskyy et al. 2010b) is solved with the resistivity defined as a step-like function: it is η = 10−3 η0 if the local current density is greater than the threshold value jcr and is equal to zero otherwise. The threshold current is set to be slightly higher than the maximum current density in the initial configuration: jcr = 3.6. The plasma velocity is zero at all the six boundaries of the domain. The magnetic field is line-tied (i.e., B = const.) at the upper (z = 10.0 L0 ) and lower (z = −10.0 L0 ) boundaries 2 representing the photosphere, while it is “free” ( ∂∂ nB2 = 0) at the four side boundaries. However, provided the model domain is big enough, the perturbations associated with magnetic relaxation do not reach the four side boundaries and the corresponding boundary conditions are not very important. In order to speed up the development of the kink instability, we introduce small non-zero initial velocities:

1 2π z 2π z vr (t = 0 ) = sin + sin 4 L0 4 2.7 L0

2π r ψ (r, z ), × sin 2.2 L0

2π z 4.1 L0 × ψ (r, z ),

vz (t = 0 ) = sin

where ψ (r, z ) = 2 × 10−3 v0 exp(− (2.5zL0 )2 − (0.6rL0 )2 ). It should be noted, however, that exact profile of initial velocity field does not play a significant role: it only determines the time needed for the kink instability to develop but does not affect properties of accelerated particles. The following scaling parameters were adopted in the present model: length L0 = 106 m, magnetic field B0 = 4 × 10−3 T, and density ρ0 = 3.3 × 10−12 kg m−3 . The corresponding characteristic Alfv´en velocity and time are v0 = 1.95 × 106 m s−1 and t0 = 0.53 s, and the characteristic current −1 is j0 = μ−1 = 3.2 × 10−3 A m−2 . The resistivity is 0 B 0 L0 normalized by η0 = μ0 L0 v0 . Evolution of the magnetic field and current is shown in Figures 2 and 3. Initially, it takes some time (50t0 –150t0 depending on the initial velocity distribution) for kink instability to develop. During the next ∼100t0 period the reconnection occurs as a helical current sheet forms in the nonlinear phase of the kink instability (Figure 3(c)). It is worth mentioning that the magnetic reconnection in this model consists of two features. First, during the fast reconnection stage the magnetic twist is substantially reduced. Thus, the maximum twist angle decreases from ≈ 9π to ≈ 1.5π (see Figure 3(b)). This is mainly due to reconnection within the twisted flux tube, occurring initially at the helical current sheet. Second, the field lines of the inner, twisted flux tube reconnect with the outer lines. Due to the second effect, the reconnecting flux tube becomes wider with time. This is apparent from Figure 2(b): strong θ component expands to r ∼ 1.6 L0 . As the reconnection proceeds, the helical current sheet breaks up, becoming rather chaotic and filamentary toward the end. Thus, in the later stages of reconnection, there are many localized small-scale current sheets distributed 2

2π z 1 2π z vθ (t = 0 ) = cos + cos 3.5 L0 4 2.2 L0

2π r ψ (r, z ), × sin 2.2 L0

2π r 2π z 1 cos + cos 2.5 L0 3 1.5 L0

2

2



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Figure 3. Evolution of the magnetic field and current density in the model. Panel (a) shows the magnetic field lines. Only two field lines are selected on panel (b) in order to demonstrate change of connectivity. Panel (c) shows current density isosurfaces (yellow color corresponds to |j | = 1.0 and blue color corresponds to |j | = 4.0). Panel (d) shows the current density at one of the model x–y cross-sections (z = 5 L0 ). (A color version of this figure is available in the online journal.)

Initially, test particles are uniformly distributed within the domain in respect of x, y, and z. Particle absolute velocities |v| are√uniformly distributed in the range 0.2vth –3.2vth . Here, vth = 2kB Tini /m with Tini = 0.8 MK. Each particle is assigned with a statistical weight which is used to calculate energy and pitch-angle spectra. This weight w depends on the initial test2 2 particle velocity as w = |v|2 /vth exp(−|v|2 /vth ), so that the initial distribution is Maxwellian. Particle magnetic moments μ are calculated based on their initial pitch angles sin θ = v⊥ /|v|: sin θ )2 , with the initial pitch-angle distribution of test μ = (|v| 2B particles corresponding to isotropic distribution in 3D velocity sin θ space: d dN ∼ cos . In the present model up to ∼5 × 105 test sin θ θ particles were used in simulations.

throughout the model volume (Figures 3(c) and (d); as noted in Hood et al. 2009). At the end of the simulations, the reconnection has largely ceased and the field is approaching a relaxed equilibrium state. 3. PROTON AND ELECTRON ACCELERATION The time-dependent electric and magnetic fields from the MHD simulations are used as input to calculate proton and B, and their electron trajectories. All the components of E, derivatives can be determined at any given location {x, y, z} for any time moment t using the simple linear interpolation between adjacent grid points and time snapshots (see Equation (14) in Gordovskyy et al. 2010b). 3



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Figure 4. Typical proton (panel (a)) and electron (panel (b)) energies along with the parallel electric field experienced by these particles vs. time. The characteristic electric field is E0 = v0 B0 = 7.9 × 103 V m−1 . (A color version of this figure is available in the online journal.)

Figure 5. Distribution of protons and electrons with the energy >10 keV near the “footpoint” boundaries (|z| > 9.5 L0 ).

two “typical” protons and two electrons. It is interesting to note that the energy gain in this model is not gradual process. Instead, the energies jump up when the test particles experience “collisions” with localized electric field concentrations. This is more evident during the later stages of the reconnecting current sheet evolution when the electric currents have filamentary structure. Generally, both protons and electrons are accelerated in this way. However, since electrons travel on average much faster (by a factor of ∼40) than protons, they experience more “collisions” with strong electric field and, as the result, their energies jump much more frequently. Figure 5 shows the distribution of particles with energies >10 keV in x–y plane close to the “footpoint boundaries” (i.e., at |z| > 9.5 L0 ). Despite the fact that the acceleration

Since the particle gyroradii are much smaller than the model scale length, it is justified to use the guiding center approximation. The trajectories were calculated using the relativistic set of guiding center motion equations (Northrop 1963; see also Equations (9)–(13) in Gordovskyy et al. 2010b). Mirroring boundary conditions are applied at the upper and lower boundaries (corresponding to the loop footpoints). In reality this would represent a loop with a strong magnetic convergence at the chromospheric level, although this convergence is not directly included in our magnetic field. The side boundaries are transparent for test particles, though in fact virtually no particles reach these boundaries. Variation of particle energies with time, along with the parallel electric field experienced by them, is shown in Figure 4 for 4



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Figure 6. Energy spectra for protons (panel (a)) and electrons (panel (b)) at the end of fast reconnection (t = 300t0 ). Thin dashed lines denote the initial Maxwellian spectra.

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Figure 7. Pitch-angle distributions for protons (panel (a)) and electrons (panel (b)) at the end of the fast reconnection. Green line is for particles in the energy range 10–100 keV, blue line is for the energy range 100 keV–1 MeV, and red line is for particles with energies >1 MeV. Black dashed line shows the pitch-angular distribution for isotropic case. All the distributions are normalized to N (0 ) = 1 for the sake of convenience. (A color version of this figure is available in the online journal.)

does not occur close to the footpoints (i.e., near the upper and lower boundaries), high-energy particles quickly become uniformly distributed within the flux tube. At the beginning of the reconnection, the electric current is concentrated at the cylindrical surface near the flux tube outer radius r ≈ L0 . This results in the high-energy particles forming an annular structure near the footpoints. However, this effect is transient and during a relatively short period (∼20t0 ), high-energy particles uniformly fill the whole flux tube column. However, because the crosssection of the flux tube increases with time, the same happens to the volume filled by high-energy particles. It can be seen that high-energy particles trying to escape the domain are nearly uniformly distributed within a circle centered on the axis of the domain. The radius of this circle grows as the reconnection proceeds. Thus, during approximately 150t0 , it increases from ∼0.4 L0 to 1.5–1.7 L0 . Now let us consider the resulting energy and pitch-angle distributions. Figure 6 shows the energy spectra at the end of the fast reconnection stage t = 300t0 . The bulk of protons and electrons remains at thermal Maxwellian distributions at lower energies while a small fraction (approximately 6%) goes into the high-energy tail. The high-energy parts of particle spectra are power laws between ∼10 keV and 1 MeV and decay quickly at higher energies. The power-law parts are hard both for protons and electrons—the spectral indices are between 1.5 and 2.0. As expected in the case of field-aligned acceleration, the accelerated particles are strongly collimated to the field lines (Figure 7). The width of the pitch-angle distribution depends on the particle energy. Thus, electrons with energies between 10 keV and 100 keV have width of the f (sin θ ) distribution of ∼0.1; for the energy range 100 keV–1 MeV it is ∼0.04, and for electrons with the energies >1 MeV it is ∼0.01. (Here we

define the width of distribution as the value of sin θ where f is 10 times smaller than f for an isotropic distribution.) Proton pitch-angular distributions are slightly wider. For the same energy ranges 10–100 keV, 100 keV–1 MeV, and >1 MeV the widths are 0.12, 0.05, and 0.02, respectively. 4. DISCUSSION The above simulations show that the magnetic reconnection in the unstable twisted loop with parameters typical for the corona (see Section 2) can accelerate ∼5%–10% of protons and electrons into non-thermal distribution with energies from around 1 keV up to 10 MeV. The obtained energy and pitch-angle distributions are typical for field-aligned acceleration of adiabatic particles (e.g., Wood & Neukirch 2005; Gordovskyy et al. 2010a, 2010b). The energy spectra are very hard power-law shapes with the indices ∼1.5 both for protons and electrons. The accelerated particles are strongly collimated along the field lines with the width of pitch-angle distribution decreasing from ∼0.1 for the particles with E ∼ 50 keV to ∼0.01 for the particles with MeV energies. Such a correlation between energy and pitch angles along with magnetic convergence near the flux-tube footpoints would affect the depth-energy structure of the hard X-ray sources in flares. This can be used as one of the observational tests for the considered model. The volume of the magnetic flux tube in the present model is of the order ∼1020 cm3 . Taking into account the plasma density (see Section 2), one can conclude that the total number of highenergy particles produced in the current model is ∼1028 –1029 . Therefore, the model considered in the present study can be appropriate for small flares. 5



Another interesting feature of the present model is the acceleration that leads to nearly uniform distribution of highenergy protons and electrons in the reconnecting flux tube. It is quite an important outcome, since the uniform particle distribution means that there is no charge separation and, therefore, no energy losses due to the return current effect. This result is rather similar to one obtained by Turkmani et al. (2006) who used a stationary electromagnetic field configuration taken from MHD simulations of a stressed magnetic flux tube. Furthermore, distributed particle acceleration means that energy losses due to other transport effects (such as collisions) would be reduced as well. Therefore, reconnection in twisted coronal loops can be an effective source of high-energy particles and the considered scenario can be used as one of the possible model for small singleloop flares. The simple model presented here is a significant “proof of principle” to this effect. More simulations with more realistic configurations (e.g., with magnetic field convergence) are needed and this is the subject of our future work.

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