SYMMETRIC CORONAL JETS: A RECONNECTION-CONTROLLED STUDY

The Astrophysical Journal, 715:1556 1565, 2010 June 1 C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi:10.1088/0004-637x/715/2/1556 SYMMETRIC CORONAL JETS: A RECONNECTION-CONTROLLED STUDY L. A. Rachmeler 1, E. Pariat 2, C. E. DeForest 3, S. Antiochos 4, and T. Török 2 1 University of Colorado at Boulder, Boulder, CO 80304-0391, USA; laurel.rachmeler@colorado.edu 2 LESIA, Observatoire de Paris, CNRS, UPMC, Université Paris Diderot, 5 Place Jules Janssen, 92190 Meudon, France 3 Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, USA 4 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA Received 2010 January 26; accepted 2010 April 15; published 2010 May 12 ABSTRACT Current models and observations imply that reconnection is a key mechanism for destabilization and initiation of coronal jets. We evolve a system described by the theoretical symmetric jet formation model using two different numerical codes with the goal of studying the role of reconnection in this system. One of the codes is the Eulerian adaptive mesh code ARMS, which simulates magnetic reconnection through numerical diffusion. The quasi-lagrangian FLUX code, on the other hand, is ideal and able to evolve the system without reconnection. The ideal nature of FLUX allows us to provide a control case of evolution without reconnection. We find that during the initial symmetric and ideal phase of evolution, both codes produce very similar morphologies and energy growth. The symmetry is then broken by a kink-like motion of the axis of rotation, after which the two systems diverge. In ARMS, current sheets formed and reconnection rapidly released the stored magnetic energy. In FLUX, the closed field remained approximately constant in height while expanding in width and did not release any magnetic energy. We find that the symmetry threshold is an ideal property of the system, but the lack of energy release implies that the observed kink is not an instability. Because of the confined nature of the FLUX system, we conclude that reconnection is indeed necessary for jet formation in symmetric jet models in a uniform coronal background field. Key words: magnetic reconnection Sun: activity Sun: corona 1. INTRODUCTION The solar surface is an extremely dynamic environment. Time-lapse movies of the corona show that the magnetic field is constantly changing and shifting: building up and releasing energy in the form of coronal mass ejections (CMEs), flares, and jets. Much of the energy required to drive this dynamic system is stored in the magnetic field (e.g., Klimchuk & Sturrock 1992; Forbes 2000; Longcope & Beveridge 2007; Schrijver 2009). This coronal magnetic energy likely comes from flux emergence and photospheric motions (e.g., Barnes & Sturrock 1972; Cui et al. 2007; Welsch et al. 2007; Archontis 2008). The speed of these motions is roughly 1 km s 1 over size scales of tens of megameters and so the energy injection is slow compared to the timescales of energy release (i.e., minutes for flares). Thus, there are two distinct phases of energy evolution in the corona. The first is the slow storage phase which lasts days, weeks, or months. The second is the energy release phase, which lasts seconds to hours. Magnetic reconnection is likely a key element of the second phase because it enables very quick release of magnetic free energy. During the energy storage phase of any coronal eruption, the plasma is well described by magnetohydrodynamics (MHD). The spatial scales are larger than any of the individual particle scales such as the Debye radius or the gyrofrequency. In the lower corona, up to a few hundred megameters, the plasma β parameter (ratio of plasma pressure to magnetic pressure) is much less than 1, so magnetic forces dominate (Gary 2001). The magnetic Reynolds number (ratio of advection to diffusion of a magnetic field) is much greater than 1 (R m 10 8 10 12 ) in the corona so the plasma is close to perfectly conducting, or diffusion free, and the plasma is frozen into the field. When the scale sizes in the system become sufficiently small, the ideal approximation breaks down. Reconnection then becomes a likely mechanism for releasing the stored energy. Magnetic reconnection is, by definition, a non-ideal process and requires a resistive MHD treatment of the plasma. More information about reconnection in the solar corona can be found in the following reviews: Biskamp (2000); Priest & Forbes (2000); Shay et al. (2001); Hood et al. (2002); Low (2003); Scholer et al. (2003). To understand the role of reconnection, we use a control case of reconnectionless simulations in one example of a two-phase system: a symmetric coronal jet. Dynamic processes in the corona are extremely difficult to model numerically due to the disparity between the two energy phases. Kinetic simulations capture the processes involved in the act of reconnection by including detailed particle physics. However, they are not yet useful for modeling global and active region events because the spatial scales are extremely small. To capture larger scales, Eulerian MHD codes are often used for solar processes. These codes model fluid dynamics under the MHD approximation with a grid that is fixed in space. MHD codes capture the essence of the energy release, but they do not handle the detailed physics of the diffusion region, where the MHD approximation breaks down. There are two methods of approach when dealing with reconnection in Eulerian MHD codes. The first is to model a resistive plasma with an explicit diffusion term in the induction equation (e.g., Archontis et al. 2004; Aulanier et al. 2005). This method allows modelers to quantify the effects of diffusion in their systems. Other coronal MHD codes do not include this diffusive term and instead attempt to model ideal MHD (Magara & Longcope 2003;Török & Kliem 2003;Fan&Gibson2004). Even in these quasi-ideal models, magnetic diffusion is not completely eliminated because of the intrinsic properties of the finite approximation to the differential operators, which leads to numerical diffusion. Simply stated, Eulerian codes cannot model a Lagrangian of B or any other parameter that has a 1556

No. 2, 2010 SYMMETRIC CORONAL JETS: A RECONNECTION-CONTROLLED STUDY 1557 scale smaller than the size of the grid and thus always contain some magnetic diffusion whether it is explicit in the induction equation or not. Some codes, e.g., Adaptively Refined Magnetohydrodynamic Solver (ARMS; DeVore 1991; DeVore & Antiochos 2008), use adaptive mesh refinement to minimize the grid size, and hence the diffusion, only in those areas where spatial gradients of the magnetic field are large. Thus, the diffusion in the system can be minimized without oversampling the entire computational volume. The diffusion in ARMS and other adaptive mesh codes is overall as small as it can be given practical constraints on step size and run time, and usually smaller than in a fixed grid code. Diffusion occurs primarily where the grid is the smallest and higher gradients of B have formed. Previous researchers treat the resulting diffusion as a phenomenological proxy to the real coronal reconnection behavior (DeVore & Antiochos 2008); the evolution proceeds in an approximately ideal manner everywhere except where 2 B is large. Even with these targeted refinements, it is impossible to completely remove the effect of magnetic reconnection due to numerical diffusion. Thus, even these ideal codes are still not ideal in the strictest sense and do unavoidably include numerical reconnection that may destabilize magnetic systems even if direct numerical diffusion is negligible in terms of flux transfer. Lagrangian codes, where the grid moves with the fluid, can model ideal MHD evolution, but their scope is limited to specific geometries because of grid distortion which, in the absence of re-gridding, degrades the locality of the differential operators. The Field Line Universal relaxer (FLUX) code was developed specifically to look at ideal MHD magnetic field configurations (DeForest & Kankelborg 2007). It uses a quasi-lagrangian grid that is not susceptible to either the grid problems of Lagrangian codes or the magnetic diffusion problems of Eulerian codes. The version of FLUX used for this work is limited in that it only calculates the magnetic forces and not the plasma forces, which is valid in very low-β plasma systems such as the lower corona (e.g., Gary 2001). Using an Eulerian code in conjunction with the FLUX code can constrain the system evolution and provide insights into the role of the reconnection. Without a diffusionless control case, it is not clear how reconnection affects the evolution of any given system and whether even a small amount of numerical diffusion leads to destabilization. FLUX can evolve the system strictly without reconnection, which, along with a reconnective code like ARMS, allows for a more complete understanding of active solar phenomena. To model one example of one dynamic coronal event that has two distinct energy phases, we look at coronal jets with ARMS and FLUX. Coronal jets have been observed in the EUV and in X-ray domains (Schmahl 1981; Shibata et al. 1992) as brightenings of collimated material that appear to move outward from the surface of the Sun. With the improved resolution of the Hinode X-Ray Telescope (Golub et al. 2007), X-ray jets have been observed at a rate of 10 per hour in the polar regions (Cirtain et al. 2007). A particularly energetic coronal jet has even been observed in hard X-ray (Bain & Fletcher 2009). Essentially all coronal jets are associated with a brightening at their point of origin (Shimojo et al. 1996; Shimojo & Shibata 2000). The motion of the jet from the bright point is in the direction of the surrounding field, which is radially outward if the jet originates in a coronal hole. The main energy source for coronal jets is magnetic (Shibata et al. 1997) as with other coronal eruptions, and the distribution of the magnetic flux in the source regions of the jets is usually multipolar (Shimojo et al. 1998). The energization is usually a complex combination of magnetic emergence, footpoint shear, and footpoint rotation. The energy buildup phase is thus a slow process compared to the lifetime of the jet, which is on the order of tens of minutes (Nisticòetal.2009). In the present work, we explore a coronal jet model where the system is energized by a slow symmetric rotation of a magnetic polarity embedded in a region of uniform, oppositely directed magnetic field. As with flares (e.g., Tian & Alexander 2006; Zhang et al. 2008), shearing motions are likely to be key elements of the energization of jets. Stereoscopic observations have revealed twisting motions before the onset of a jet (Patsourakos et al. 2008) and three-dimensional (3D) helical structure along the jet axis in about 40% of jets (Nisticò et al. 2009). Models of jet systems generally invoke reconnection as the energy release mechanism (Filippov et al. 2009). There is substantial observational indication that reconnection is involved in jet initiation (Shibata et al. 1997). Jets appear almost instantaneously in EUV or X-ray image sequences. This short timescale is likely linked to reconnection being the primary destabilizer. Also, jets are often seen with associated bright point flares, which are strongly indicative of reconnection sites (Shibata et al. 1992; Shimojo et al. 1996; Shimojo & Shibata 2000). The multipolar distribution of the magnetic field at the base of the jets (Shimojo et al. 1998) creates a favorable condition for magnetic reconnection in the corona. Magnetic extrapolation of observed jets effectively associated the location and propagation of UV and X-ray jets with the spine and fan-separatrices of 3D magnetic null points (Fletcher et al. 2001; Moreno-Insertis et al. 2008) as predicted by models of reconnection. The X-ray bright points themselves often appear topologically different before and after the jet (e.g., Shibata et al. 1992). The detection of important emission from non-thermal electrons from radio and hard X-ray data further confirms models involving magnetic reconnection (Bain & Fletcher 2009). Eulerian numerical models also offer arguments for reconnection in jet systems. MHD numerical simulations have confirmed that magnetic reconnection at a null point is indeed a viable mechanism to explain numerous observable properties of the jets (Yokoyama & Shibata 1995, 1996; Miyagoshi & Yokoyama 2004; Archontis et al. 2005, 2007; Moreno-Insertis et al. 2008; Gontikakis et al. 2009). These simulations have shown that the reconnection of the magnetic field lines (specifically the outflow acceleration of plasma due to the high magnetic tension force and the slingshot effect of the reconnected field) is only marginally responsible for the thermodynamic and kinematic properties of the jets. The X-ray jet emission is likely due to chromospheric evaporation in reconnected field lines (Shimojo & Shibata 2000; Shimojo et al. 2001; Miyagoshi & Yokoyama 2004). However, the evaporation cannot account for the very high velocities (of the order of the Alfvén speed) and the wavelike pattern, which is increasingly observed with jets (Kim et al. 2001; Cirtain et al. 2007; Nishizuka et al. 2008; Liu et al. 2009). Recently, Pariat et al. (2009, hereafter PAD09) presented a new set of 3D numerical simulations using ARMS that demonstrate that impulsive 3D reconnection at the null point leads to the generation of a large nonlinear Alfvénic wave. A significant fraction of the magnetic energy and helicity in the initially closed field is carried away by the wave after reconnection. This jet mechanism relies on the concept of the magnetic twist jet (Shibata et al. 1997) based on the early

1558 RACHMELER ET AL. Vol. 715 Figure 1. Initial potential configuration of the system as modeled by ARMS (left) and FLUX (right). The rendered field lines in ARMS were chosen to match the footpoints of the fluxons in FLUX. The background field has not been rendered. The height of the magnetic null in the ARMS system is 2.2 units, and the width of the closed field is 3.1 units. simulations by Shibata & Uchida (1985, 1986), even though the term untwisting jet would be more appropriate. The PAD09 model can explain the 3D helical structure recently observed with the twin STEREO satellites (Patsourakos et al. 2008; Nisticò et al. 2009; Kamio et al. 2010) and the Alfvénic velocities displayed by some jets (Cirtain et al. 2007; Nishizuka et al. 2008; Liu et al. 2009). The PAD09 model can also successfully reproduce the observed recurrent homologous jets and explain the close link between jets and plumes (Pariat et al. 2010). Using a different initial setup that simulates the emergence of a twisted flux tube instead of twisting boundary motions, Török et al. (2009) demonstrate that the interaction of the emerging flux tube with the large-scale coronal field leads to the generation of a similar untwisting nonlinear wave by nullpoint reconnection. This induces the subsequent formation of the classical anemone/3d null-point topology. In this work, we follow the model outlined in PAD09. The system begins with a circular magnetic parasitic polarity embedded in a uniform open field of opposite sign. The closed field is rotated such that azimuthal symmetry is preserved, limiting the development of current sheets while preserving a magnetic null point. Eventually, the system loses axisymmetry and kinks through a deformation of the rotational axis in the closed field. As soon as this symmetry is broken reconnection quickly produces an Alfvénic jet. As with other Eulerian simulations, this model intrinsically includes some numerical magnetic diffusion and therefore the question of whether reconnection is always necessary for jet formation has not been thoroughly investigated. We explore the possibility of creating coronal jets without reconnection. The topic we wish to address in this work involves the loss of symmetry. Can a jet be triggered solely by an ideal process such as a kink instability? What is the role of the kink that is observed at the onset of the jet generation in PAD09? Is this kink-like behavior similar to the classic kink instability? Does it release magnetic free energy, and can this energy then be used to form a jet without the help of reconnection? We begin by describing the physical system modeled in two numerical codes; ARMS is an Eulerian adaptive mesh code (Section 2.1), and FLUX is an ideal quasi-lagrangian code that preserves topology (Section 2.2). We then look at the sideby-side similarities and differences of the results obtained by the two different numerical approaches (Section 3) and their consequences on our understanding of the physics of coronal jets (Section 4). 2. SIMULATION MODELS The setup of this simulation is the same as that of PAD09. The initial magnetic field configuration is B z (x,y,z) = μ om o 4π 2(z z o ) 2 x 2 y 2 (x 2 + y 2 +(z z o ) 2 ) 5/2 B v, (1) with μ om o = 25, and a uniform background field of B 4π v = 1. The vertical dipole is submerged (z o = 1.5), making it appear unipolar on the surface. Figure 1 shows this initial configuration in both ARMS and FLUX where the uniform background field has not been rendered. The system is evolved with a symmetric rotation profile about the axis of the unipole with near solidbody rotation that drops rapidly to zero at the polarity inversion line (Figure 2). The velocity at the photosphere is given by v (t) = v o f (t) k B B r B l B z tanh ( k B B z B l B r B l ) ẑ B z (2) with f (t) = 1 [ ( 1 cos 2π t t )] l. (3) 2 t r t l This particular profile was chosen since it is derived from a simple potential that can easily be implemented in ARMS. The constants are as follows: v o = 3.7 10 5, k B = 15, B l = 0.1, B r = 13, t l = 100, and t r = 1100. The velocity function is set such that at t l, the velocity increases from zero, and it returns to

No. 2, 2010 SYMMETRIC CORONAL JETS: A RECONNECTION-CONTROLLED STUDY 1559 Figure 2. Rotation rate varies as a function of radius. The number of turns imparted to the closed field at t = 850 is shown as a function of radius. For this paper, when we refer to a given number of turns, N, it corresponds to N (r = 1). zero at t r. The velocity remains zero in ARMS runs after t r,but continues in a periodic manner in the FLUX runs (Figure 3) to enable analysis on the effect of rotation after the symmetry loss. In ARMS, this gives the system time to relax to equilibrium in the beginning, and stops the rotation after the symmetry is broken and the jet has formed. The magnetic parameters B l and B r insure that only the positive polarity is rotated. We will refer to both time, t, and the number of turns imparted to the closed field, N, in this paper. Since the rotation rate is a function of radius within the closed region, note that for N, we are referring specifically to the number of turns at r = 1 (Figure 2). 2.1. ARMS The simulation presented in this work is based on the previous simulation of PAD09. In brief, the simulation is performed with ARMS, which relies on the flux-corrected transport algorithms described in DeVore (1991). The ARMS simulations reported in this paper are performed in a Cartesian domain where the vertical axis is z. The simulation domain corresponds to [ 12; 12] [ 12; 12] [0; 24] in non-dimensional units and its volume is equal to 13, 824 square units. We impose a linetied condition at the bottom boundary. The side boundaries are closed and reflective with free slip for the tangential velocity. Finally, at the open top boundary, zero-gradient conditions are enforced. The time-dependent equations of ideal MHD, with the magnetic forces expressed in the Lorentz form, are solved on a dynamically solution-adaptive grid administered by the adaptive mesh toolkit PARAMESH (MacNeice et al. 2000). The nonuniform initial grid is the same as the one presented in PAD09. This initial grid refines and de-refines adaptively during the simulation; the adaptation occurs when the spatial gradient of J/B is either too large or too small. The grid adjusts to resolve thin current layers that drive and control the reconnection process. Two separate simulations were run with the ARMS code. Unless otherwise stated, the run that will be referred to has a maximum of six levels of refinement. The second run, ARMS2, has a maximum of five levels of refinement. Figure 3. Number of turns, N, imparted to the system as function of simulation time, t. The dashed line indicates N (r = 1). The dotted lines are N (r = 1.32) for the upper line and N (r = 0.82) for the lower line which, respectively, indicate the maximum and minimum number of turns at a given time. The solid line is the observed N (r = 1) in the ARMS code. The symbols indicate when each system broke symmetry: a square for FLUX, a circle for ARMS. Although no explicit resistive MHD terms are included in the ARMS simulations, numerical diffusion provides an effective resistivity in the model, in particular where the Laplacian of the magnetic field is high, which occurs at current sheets where the grid is the most refined. The domain is filled with a plasma with the highly conducting, low-pressure properties of the plasma of the inner corona. In order to stay as general as possible, we use non-dimensional units; however, a comparison with actual coronal units can be found in Section 5.2 of PAD09. The initial pressure is uniform (P = 10 2 ) as is the density (ρ = 1); an ideal equation of state is used and the temperature is therefore uniform, T = P/(ρR) = 1, where the gas constant is R = 10 2.TheAlfvén speed is c A = 0.28, and the plasma beta is β = P B 2 /2μ = 0.25 in the uniform field. 2.2. FLUX FLUX is a quasi-lagrangian code that solves for nonlinear force free equilibrium of a magnetic field with a given topology and connectivity. It is a static ideal MHD solver in the limiting case of a low-β plasma, which is an assumption that is commonly made for the lower solar corona (Gary 2001). Equilibrium configurations are calculated as a balance between the magnetic pressure force and the magnetic tension force. FLUX is different from Eulerian codes in that the grid is along fluxons, which are finite-magnetic-flux field lines. Each fluxon is broken in to linear segments called fluxels, and fluxels are joined at vertices. Field lines are discrete so the only way to produce reconnection is to separate and recombine two fluxons (a functionality which is not used in this work). The fieldaligned grid also means that FLUX is not susceptible to numeric diffusion across the magnetic field and therefore magnetic reconnection is impossible. More information about FLUX can be found in DeForest & Kankelborg (2007) or Rachmeler et al. (2009).

1560 RACHMELER ET AL. Vol. 715 The energy is calculated with the basic magnetic energy equation at each fluxel i: B 2 E = 8π d3 V = i Φ 2 i s i 8πA i (4) where s is the length of a fluxel and A is the cross sectional area that it occupies, also known as the Voronoi cell. The flux, Φ i,is the same for each fluxel because in these simulations all fluxons contain the same amount of magnetic flux. The distribution of that flux within a Voronoi cell is based on equi-angular partition such that dφ is constant. dθ The initial system is simulated with 250, 312, 470, 500, 640, and 4700 fluxons, and the system evolution with 312 and 640 fluxons. The plots and data are from the 312 fluxon system unless otherwise stated. The computational volume is a cylinder of radius 12 units (the initial closed field extends to a radius of approximately 3.4 and the polarity inversion line is at about 1.6), with a height of 50 units. The photosphere is at z = 0. The spatial units compare directly to those in PAD09, although the volume, which is expressed in those units that are consistent between the two simulations, is different in the FLUX simulation than in ARMS. The volume modeled in FLUX is about 22,620 square units, which is approximately 1.64 times larger than the volume in ARMS. The energy injected into the system is the same in both ARMS and FLUX, but the initial energy is dependent on the computational volume. Consequently in the rest of this work, we compare the magnetic free energy (energy above the potential magnetic energy of a system) as opposed to the total magnetic energy. The bottom and the side boundaries of the cylinder are linetied reflecting surfaces, and the upper boundary is a small portion of a hemisphere of radius 50 centered at the origin, which is open. Open boundaries in FLUX have fluxons whose last vertex is attached to the surface, but is able to move freely on it. Fluxon footpoints are placed with an error-accumulating dithering algorithm (see Riemersma 1998 for more on dithering algorithms). The routine walks along a plane-filling Hilbert curve (Hilbert 1891) and integrates the boundary field, B z,as it does so. Once a threshold of magnetic flux accumulation is reached, a fluxon is placed and its corresponding flux is subtracted from the running integral. This generates a pseudorandom fluxon placement that is consistent with field strength. It is important to note, however, that this routine places endpoints regardless of sign; each endpoint is then extended by shooting through the initial prescribed field. Areas of closed field are twice as dense as areas of open field because they receive both shot and placed endpoints. To compensate for that, the dithering algorithm is run once in the open field region and once in the closed field region and each algorithm places a pre-calculated number of fluxons such that the magnetic flux associated with the closed and open fluxons is the same to within 0.1%. We relaxed these systems to a stiffness coefficient of 0.05%. The stiffness is a parameter that is calculated as the average of the ratio of the net force to the sum of the magnitudes of the forces on each vertex. The stiffness coefficient goes to zero at the equilibrium configuration. When the boundary is moved incrementally, the system takes many small relaxation steps until the system is in equilibrium. The process is repeated to create a quasi-steady evolution of equilibrium states. In this work, the quasi-static FLUX systems are being compared to the dynamic ARMS systems. Using the plasma parameters in the ARMS systems, the velocity at the lower boundary reaches a maximum that is only about 1% of the local Alfvén speed, so the system is near equilibrium during the energy buildup phase. We are thus able to directly compare the quasi-static and dynamic evolution during this phase. 3. RESULTS On the whole, we found remarkable similarity between the systems in the two codes before the break in symmetry. The closed field twisted and expanded in the same way in each code until symmetry loss occurred, after which the evolution diverged. After the break in ARMS, the system rapidly reconnected, forming torsional Alfvén waves that caused a magnetic twist jet to travel upward out of the system; a more detailed description of the energy release can be found in PAD09. After the break in FLUX, the system showed marked kink-like behavior and the closed field remained at an approximately constant height without clear energy loss. 3.1. Initial Energy When calculating the magnetic energies, we found that the initial potential energy configuration yielded different results in the FLUX and the ARMS codes. The initial energy in the ARMS code is very close to the analytic potential energy for the simulation volume. In the FLUX system, the closed field was energized compared to the analytic solution. Theoretically, the closed region contains approximately 3 units of energy, but this same region contains 10 units in FLUX. We modeled several versions of the system, varying the number of fluxons each time (between 312 and 4700 fluxons), and we found that in each case the closed field contained between 8 and 11 units of energy. There was no obvious trend of initial energy with fluxon density so we do not believe that this is a grid resolution effect. The total initial energy is close to the analytic solution, so it appears that in FLUX the closed field contains more and the open field less energy than analytic values. One possibility for why the closed field in the FLUX system contains more energy is because the closed field is compressed compared to the ARMS system. This compression could lead to an increase in energy of this field. However, when we simulate the system with 4700 fluxons, the null point is located at a higher value of around 2.2 (the same as the theoretical value of the initial null height). Even though the null height in that simulation is higher, the energy in the closed field region is still comparable to the run with 312 fluxons. In any case, since the energy difference ( 7 units of magnetic energy) is an order of magnitude smaller than the free magnetic energy that is injected into the closed field ( 60 units) at the time of symmetry loss, this discrepancy should not strongly influence the evolution of the system. Another possibility is that the initial fluxons are not exactly poloidal, but contain some small azimuthal component due to the quasi-random nature of their placement. Even if the azimuthal component cancels on average, it can nonetheless induce and increase in energy. This may be present independent of the number of fluxons because even with a large number, the azimuthal component may not completely disappear. We looked at other analytic field configurations to qualify the initial energy discrepancy. In the case of uniform field in a cylinder, FLUX reproduced the analytic energy closely. In the case of the Gold Hoyle flux rope (Gold & Hoyle 1960), there was some energy discrepancy. The simulated system underestimated the analytical energy regardless of fluxon

No. 2, 2010 SYMMETRIC CORONAL JETS: A RECONNECTION-CONTROLLED STUDY 1561 Figure 4. Magnetic free energy as a function of time (left) and number of turns based on the analytical rotation rate (right). The run labeled ARMS2 has a coarser grid resolution than the run labeled ARMS. The ARMS lines and the black FLUX line show the total magnetic free energy, and the green line shows the magnetic energy in the closed field in the FLUX run. density. As with the uniform field, the higher the fluxon density, the closer the FLUX energy approximated the analytic energy. However, the FLUX energy approached a value less than the analytic one. As the twist of the flux rope increased, the energy gap decreased. 3.2. Free Energy Evolution Despite the discrepancy in the initial energy calculation, the energy evolution in the first phase of the simulation (t <900) proceeds in a similar fashion qualitatively between FLUX and ARMS (Figure 4). Indeed, in the FLUX calculation (black line) the free energy grows almost linearly after the photospheric velocity has been turned on (Section 2). The ARMS run behaves in the same way for both runs, which have different grid resolutions. The ARMS2 run (plotted in pink) has a grid that is coarser (5 levels of refinement) than the ARMS run (plotted in red, 6 levels of refinement). As expected, we observe that the slope of the linear increase is correlated with the numerical resistivity. The ideal FLUX run displays the steepest increase while the ARMS2 run, corresponding to the lowest grid resolution and effectively the highest resistivity, has the most gradual free energy increase. The higher the numerical resistivity, the more magnetic energy is dissipated. The slope discrepancy indicates that energy is dissipated in the ARMS runs even before the symmetry-breaking reconnection occurs. Another indication of this numerical diffusion is the discrepancy between the theoretical and actual value of the number of turns given to a field line anchored at r = 1intheARMS code. The theoretical rotation of a r = 1 field line (dashed line in Figure 3), integrated from the analytical rotation rate (Equation (2)), is higher than the number directly obtained from the output of the ARMS simulation (solid line). The latter was estimated by tracing a field line (from a stationary starting position at r = 2.3 outside the rotating area, whose other footpoint is at r = 1) at various time steps and determining the angular separation between the start and end footpoints. At a given time, the actual number of turns in the ARMS runs is slightly smaller than theoretical values because of field line slippage due to slight magnetic diffusion. The FLUX run preserves the analytic rotation rate. It is possible to quantitatively check and estimate the global numerical diffusion in the ARMS runs. We found that the free energy difference at t = 900 (just before the break in symmetry in ARMS) between FLUX and the better resolved of the two ARMS runs is on the order of 13 units. Since at that time 10 5 time steps had been calculated, we can estimate that ARMS induced a dissipation of 1.3 10 4 magnetic energy units per time step or 0.015 energy units per unit time. The total magnetic energy at t = 900 is on the order of 610 units, so the total dissipation is therefore 2.2% of the maximum total magnetic energy in the system, which corresponds to a relative dissipation rate of 2.4 10 3 % per unit time. It is interesting to note that in the FLUX runs, although all of the free magnetic energy is directly injected by the boundary motion into only the closed field, energy is still imparted to the open field. This can be seen in Figure 4 from the different growth rates of the free energy integrated over all of the fluxons (black continuous line), and integrated over only the closed fluxons (green line). The deviation between these two curves corresponds to the free energy transferred to the open field as a result of the growth of the closed field. Because of the increase of the magnetic pressure in the closed field, the domain expands and mechanically transfers some energy to the open field. The open fluxons cannot move out of the domain because the side boundaries are closed, and therefore the open field density increases causing an increase in energy. We see similar behavior in ARMS because of the rigid side boundaries. In addition to the energy evolution, the systems present very similar morphological evolution during the first phase (t <900, i.e., N turn < 1.3; Figure 5). To determine the similarity of the closed field in the two systems, we plotted a comparison of the positions of the null points as the systems evolve (Figure 6). The solid line corresponds to the height of the null in the ARMS code. It drops significantly starting at about t = 900 when the reconnection begins. The twisted field then escapes through the open upper boundary leaving a system that

1562 RACHMELER ET AL. Vol. 715 Figure 5. Side-by-side views of the system as modeled by ARMS (left) and FLUX (right) at t = 800. The rendered field lines in ARMS were chosen to match the footpoints of the fluxons in FLUX. For simplicity the uniform field is not rendered. The null height is 7 units in ARMS and 5.5 units in FLUX, the width of the closed field is 3.1 units. approximates the initial system by t = 1300 (v ARMS (t) = 0 after t = 1100). We cannot directly measure the height of a null in FLUX because there are no fluxons where there is no field. Thus, areas of null magnetic field are unresolved. As a proxy to the null height, we instead measured the height of the highest fluxel in the closed field region. This is plotted as a dashed line. It is significantly below the height of the null in ARMS. For a more direct comparison, we also plotted the equivalent field lines in ARMS; from the position of each stationary footpoint in the closed domain in FLUX, we traced a field line in the ARMS system. We then recorded the highest point of those field lines, which is shown as a dash-dotted line. Because it closely follows the null height, we conclude that this method is indeed an accurate way to assess the height of the null and that the closed field occupies a smaller volume in FLUX than in ARMS. This compressed closed field appears to be a grid effect, as the system with 4700 fluxons is not compressed, although the energy of that closed field is still higher than theoretical values. 3.3. System Kinking In the FLUX system, when the symmetry broke at N = 1.3 or t = 800, the closed field exhibited a kink-like behavior, with the top beginning to roll to one side, distorting the rotational axis (Figure 7). Once the kink began, the height of the closed field remained approximately constant. The 640 fluxon run was continually twisted until a time of t = 1800 or N = 3, and no jet was observed during this time. This kink is also seen briefly in the ARMS system at the onset of the jet generation at t 920. At this time, the axis of the closed magnetic structure, the inner spine, is seen to writhe. The moment of loss of symmetry appears to be a property of the system. In ARMS, at the time of the kink, a field line located at r = 1 effectively turns N = 1.4 times around the central axis and the free energy is equal to 60 units. The number of turns is similar in FLUX at N = 1.3, with a free energy of 63 units, as seen in Figure 3. With FLUX, we tested a linear instead of a sinusoidal time profile (Equation (3)), and the results were the Figure 6. Solid line shows the height of the center of the null in the ARMS run as a function of time. The dashed line is the height of the highest closed fluxel in the FLUX run. The dash-dotted line is the highest point of field lines traced in the ARMS code from the stationary fluxon footpoint locations, which compares directly with the dashed line. same. Thus N crit, where the system kinks, is not affected by the ramping up or down of the rotation rate. In ARMS, several runs have also been performed that find no jet is triggered as long as the number of turns is lower than N crit = 1.3 (PAD09). The initial setup in the FLUX system contains more seed noise in the symmetry than the ARMS setup does because of the Hilbert dither. This noise level difference might account for the slightly smaller N crit in FLUX. Another possibility is that the difference in N crit may be due to the dissipation in ARMS. If we compare the energy injected in both simulations at the time of the kink, we find 60 units for ARMS (at t = 920, N = 1.4) and

No. 2, 2010 SYMMETRIC CORONAL JETS: A RECONNECTION-CONTROLLED STUDY 1563 Figure 7. FLUX system after symmetry is broken at t = 1500, N = 2.16. Note that the top has rolled to one side in a kink-like manner. 63 units in FLUX (at t = 800, N = 1.3). The FLUX and the ARMS simulations are therefore consistent with regards to the kinking of the system. The energy and N crit similarity between the two systems indicate that the kinking of the system observed in ARMS is an ideal MHD process. In the ARMS simulation, since the writhe of the axis breaks the axisymmetric condition, the kinking behavior enables reconnection. It is unclear from the ARMS results alone what role the kink plays in terms of energy release and if it really an instability or not. 3.4. Kink Instability as a Jet Trigger? Due to its strict idealness, FLUX is able to directly study the ideal evolution of the system. Each step in FLUX corresponds to a quasi-equilibrium state (within the 0.05% stiffness; Section 2.2). Between each step, the small explicit displacements of the footpoints of a large number of fluxon induce a complex perturbation to the system. If a loss of equilibrium would result from this perturbation, FLUX would not be able to find an equilibrium and the simulation would not relax. However, in the presented system, at least for N<3, equilibrium configurations are always found. This indicates that no loss of equilibrium develops within the simulation even though the system does kink. In addition, we observe that the energy evolution between successive configurations is smooth. Within the precision of our experiment, no significant sudden energy or morphology variation is observed between two successive steps. The evolution of the energy seems continuous. The system does kink, exhibiting a clear break in the symmetry, but within the ideal methodology of FLUX, this evolution is neither unstable nor does it induce a catastrophic eruption. Contrary to speculation in PAD09, the kinking observed at the onset of the jet does not appear to be due to an ideal instability. We find with FLUX that the kinking does not significantly decrease the magnetic energy of the system. Indeed, there is practically no energetic evidence of an instability around the time of the kink, even in small-scale variations (Figure 4). The magnetic energy in the FLUX and the ARMS systems increases approximately linearly with number of turns before the kink at N crit. When symmetry breaks, the energy of the ARMS systems decreases rapidly because of reconnection. In the FLUX system, the total free energy continues to rise linearly albeit with a slightly less steep slope. This is in contradiction with expected results since kinking behavior usually reduces the global magnetic energy; a kinkunstable twisted flux tube will writhe and decrease the azimuthal magnetic energy relative to the axial energy. The energy difference is commonly converted into kinetic energy through the motion of the kinking flux tube (e.g., Török & Kliem 2005). However, most of the simulations of kink-unstable flux ropes also observe the formation of thin helical current sheets around the flux tube (Gerrard & Hood 2004; Török et al. 2004; Fan & Gibson 2004). In these types of simulations, the majority of the energy decrease observed after the onset of writhe is actually due to reconnection across current sheets, as in our ARMS simulation. Many simulations that show marked magnetic energy decreases during the kinking of a coronal flux rope of finite length do not start with a stable twisted system, but rather from a kinkunstable twisted configuration. In this case, the energy drop may be misleading because the flux rope already holds energy above its kinking threshold. The existence of a kinked but stable structure, like the one modeled with FLUX, has already been reported in other numerical simulations (e.g., Török & Kliem 2003; Fan 2005; Török et al. 2010). In Török& Kliem(2003), the twisted loop (N = 1.26 turns) presented in Figure 19(a) is writhed and can relax to a stable equilibrium (Figure 17 of Török & Kliem 2003). Fan (2005) also describes the formation, after emergence, of a mildly kinked equilibrium with some finite writhing of the flux rope axis. The author notes that the kinking is not directly responsible for the catastrophic evolution of the system that occurs later in the simulation. However, there is a lack of clear indicators of an instability occurring during writhe of the flux rope because the magnetic energy does not decrease and there is no obvious increase in velocity. It may be possible that the conversion from twist to writhe in that work, like in ours, is not due to a true instability. We nonetheless do not completely reject the existence of a real kink instability in our jet system. Our approach is limited by the zero-β approach of FLUX. Dynamically, when plasma material is included, the system is not able to reach the equilibrium found with FLUX because of current sheet formation. Since coronal plasma is not diffusionless, the kink would always be accompanied by reconnection and so the evolution is not purely ideal. Non-ideal evolution (reconnection) could thus drive an instability that is not accessible in the idea case. A kink instability may have occurred in our system but its energetic signature, which is not apparent in our simulation, must thus be smaller than the energy injected at each FLUX time step ( 0.8 energy units). The energy decrease induced by the kink instability may indeed be relatively small. For a kink-unstable Gold Hoyle flux rope with N = 1.5 and a significantly larger writhe than in our system, T. Török et al. (2010, in preparation) found an ideal magnetic energy release of only 10%. Ali & Sneyd (2001) studied kinking cylindrical magnetic flux tubes with a Lagrangian magneto-frictional approach and noted that less than 5% of the magnetic energy was released during kinking for all configurations they studied. For some configurations the

1564 RACHMELER ET AL. Vol. 715 energy loss could be as small as 0.073%. For resistive MHD codes, the kink instability and subsequent reconnection releases about 50% of the magnetic energy (e.g., Gerrard et al. 2001, and references therein). Similarly, in the process of the jet generation of PAD09, 80% of the free energy was eventually released. Unfortunately, little is known about how much energy the kink instability converts, since for most configurations energy release due to reconnection in current sheets cannot be separated from the purely ideal release due to the kinking, although previous research has shown that it could be as small as a few percent. Overall we find that the writhe of the system is not necessarily related to an instability. More importantly, we do not see any catastrophic behavior that could lead to a jet in FLUX. A jet can be defined as a very dynamical event involving a radial evolution away from the solar surface. With FLUX, two possible dynamic evolutions could have been interpreted as the signature of a jet: first, a sudden loss of equilibrium, i.e., the impossibility for FLUX to find an equilibrium after the twist as been incremented. Another possible evolution would have been a nonlinear increase of the height of the closed structure. Rather we find that in the FLUX simulations the closed field remains stable and confined until at least N = 3. After the system kinks, the closed field does not grow vertically, but rather horizontally (Figure 7). We indeed observe in Figure 6 that after t = 900, the apex of the highest field line in FLUX only rises marginally. The evolution of the system cannot be interpreted as the signature of an impulsive jet-like evolution. The comparison of the evolution with ARMS runs leads us to conclude that reconnection is the central and necessary mechanism of the generation of jets in geometries similar to this one. 4. CONCLUSIONS Despite the slight differences in energy and N(t), the two systems are still remarkably similar. The initial energy of the closed field in FLUX is higher than theoretical values, but the energy evolution proceeds as expected from ARMS. From the slopes of the energy increases in the FLUX, ARMS, and ARMS2 runs, we infer that some numerical diffusion occurs during the ideal phase in the ARMS runs and that this diffusion is dependent on the resolution of the grid. The magnetic energy dissipation rate due to this diffusion is calculated to be 0.015 magnetic energy units per unit time, which is negligible compared to the total amount of injected energy. This shows that ARMS models ideal systems effectively in the absence of current sheets. The FLUX system kinks before the ARMS systems (t FLUX = 800; t ARMS = 920), but at the time of symmetry break, the two systems have a similar amount of twist (N FLUX = 1.3; N ARMS = 1.4) and a similar amount of magnetic free energy units (E FLUX = 63; E ARMS = 60). Thus, the moment of symmetry break appears to be an ideal process that is not affected by reconnection or diffusion in the pre-jet system. The lack of energy release and the smooth change in magnetic energy and morphology during kinking in the ideal case also indicate that the kinking is not an instability. If this is true, it would be beneficial to re-evaluate other numerical models that assume the existence of a kink instability even though the magnetic energy does not decrease or the kinetic energy does not increase significantly during writhe. The ideal system was run until N = 3 without any signs of jet-like behavior. After kinking, the closed field remained at an approximately constant height while increasing in width. This leads to more energy being transferred from the closed field to the open field, but the total energy continued to rise steadily. Thus, without reconnection no jet was formed, even for high amounts of twist. When reconnection is allowed to occur in ARMS however, PAD09 showed that a jet is always seen for a relatively broad choice of parameters if sufficient twist (more than 1.3 turns) is injected. The reconnection is the key to the energy release in this symmetric model for jet formation. Magnetic reconnection is often cited as required for many coronal dynamic events. However, the results of Rachmeler et al. (2009), show, for example, that CME-like flux rope eruptions can occur without reconnection in a purely ideal simulation. The lack of constraint of numerical reconnection rates in Eulerian codes can lead to an over-reliance on reconnective destabilization. It is important to look at these events carefully and with a controlled numerical experiment to determine the role of reconnection in that specific model. The authors thank P. Démoulin for insightful comments and discussions. This work was supported in part by the NASA HTP, LWS-TR&T, and SHP-SR&T programs. Financial support by the European Commission through the SOLAIRE network (MTRM-CT-2006-035484) is gratefully acknowledged. The research leading to these results has received funding from the European Commission s Seventh Framework Program (FP7/ 2007-2013) under the grant agreement 218816 (SOTERIA project, www.soteria-space.eu). The ARMS numerical simulations were performed on DoD High Performance Computing Modernization Program resources at NRL-DC. FLUX is open source software available from http://flux.boulder.swri.edu. Thanks is also given to the PDL development team http://pdl.perl.org. REFERENCES Ali, F., & Sneyd, A. 2001, Geophys. Astrophys. Fluid Dyn., 94, 221 Archontis, V. 2008, J. Geophys. Res. (Space Phys.), 113, 3 Archontis, V., Hood, A. W., & Brady, C. 2007, A&A, 466, 367 Archontis, V., Moreno-Insertis, F., Galsgaard, K., & Hood, A. W. 2005, ApJ, 635, 1299 Archontis, V., Moreno-Insertis, F., Galsgaard, K., Hood, A., & O Shea, E. 2004, A&A, 426, 1047 Aulanier, G., Démoulin, P., & Grappin, R. 2005, A&A, 430, 1067 Bain, H. M., & Fletcher, L. 2009, A&A, 508, 1443 Barnes, C. W., & Sturrock, P. A. 1972, ApJ, 174, 659 Biskamp, D. 2000, Magnetic Reconnection in Plasmas (Cambridge: Cambridge Univ. Press) Cirtain, J. W., et al. 2007, Science, 318, 1580 Cui, Y., Li, R., Wang, H., & He, H. 2007, Sol. Phys., 242, 1 DeForest, C. E., & Kankelborg, C. C. 2007, J. Atmos. Sol. Terr. Phys., 69, 116 DeVore, C. R. 1991, J. Comput. Phys., 92, 142 DeVore, C. R., & Antiochos, S. K. 2008, ApJ, 680, 740 Fan, Y. 2005, ApJ, 630, 543 Fan, Y., & Gibson, S. E. 2004, ApJ, 609, 1123 Filippov, B., Golub, L., & Koutchmy, S. 2009, Sol. Phys., 254, 259 Fletcher, L., Metcalf, T. R., Alexander, D., Brown, D. S., & Ryder, L. A. 2001, ApJ, 554, 451 Forbes, T. G. 2000, J. Geophys. Res., 105, 23153 Gary, G. A. 2001, Sol. Phys., 203, 71 Gerrard, C. L., Arber, T. D., Hood, A. W., & Van der Linden, R. A. M. 2001, A&A, 373, 1089 Gerrard, C. L., & Hood, A. W. 2004, Sol. Phys., 223, 143 Gold, T., & Hoyle, F. 1960, MNRAS, 120, 89 Golub, L., et al. 2007, Sol. Phys., 243, 63 Gontikakis, C., Archontis, V., & Tsinganos, K. 2009, A&A, 506, L45 Hilbert, D. 1891, Math. Ann., 38, 459 Hood, A. W., Galsgaard, K., & Parnell, C. E. 2002, in SOLMAG 2002: Proc. of the Magnetic Coupling of the Solar Atmosphere Euroconference, ed. H. Sawaya-Lacoste (ESA SP-505; Noordwijk: ESA), 285