THE IMPACT OF MICROSCOPIC MAGNETIC RECONNECTION ON PRE-FLARE ENERGY STORAGE

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1 The Astrophsical Journal, 77:L158 L16, 9 December C 9. The American Astronomical Societ. All rights reserved. Printed in the U.S.A. doi:1.188/4-637x/77//l158 THE IMPACT OF MICROSCOPIC MAGNETIC RECONNECTION ON PRE-FLARE ENERGY STORAGE P. A. Cassak 1 and J. F. Drake 1 Department of Phsics, West Virginia Universit, Morgantown, WV 656, USA; Paul.Cassak@mail.wvu.edu Department of Phsics and Institute for Research in Electronics and Applied Phsics, Universit of Marland, College Park, MD 74, USA; drake@plasma.umd.edu Received 9 September 19; accepted 9 November 3; published 9 December 4 ABSTRACT It is widel accepted that magnetic reconnection releases a large amount of energ during solar flares. Studies of reconnection usuall assume that the length scale over which the global macroscopic) magnetic field reverses is identical to the thickness of the reconnection site. However, in spatiall extended high-lundquist number plasmas such as the solar corona, this scenario is untenable; the reconnection site is microscopic and embedded inside the macroscopic current set up b global fields. We use numerical simulations and scaling arguments to show that embedded effects on reconnection could have a profound influence on energ storage before a flare. From largescale high-lundquist number resistive magnetohdrodnamics simulations of reconnection with a diffusion region on a much smaller scale than the macroscopic current sheet, we find that the generation of secondar islands is governed b the local magnetic field immediatel upstream of the diffusion region rather than the potentiall much larger) global field. This diminishes the production of secondar islands and leads to a thicker diffusion region than those predicted using the global field strength. Such considerations are crucial for understanding the onset of solar eruptions and how energ accumulates before such eruptions. We argue that if reconnection with secondar islands is fast, the energ storage times before an eruption are too small to explain observations. If reconnection with secondar islands remains slow, embedded effects cause the diffusion region to begin far wider than kinetic scales, so energ storage before a flare can occur while collisional Sweet Parker) reconnection with secondar islands proceeds. Ke words: Sun: activit Sun: chromosphere Sun: corona Sun: flares magnetic fields MHD 1. INTRODUCTION Energ release during solar flares is thought to be allowed b magnetic reconnection. A long-standing question was how reconnection releases energ at the observed rate, which is much faster than the classical Sweet Parker model Sweet 1958; Parker 1957, 1963). A leading candidate is collisionless Hall) reconnection Birn et al. 1). An equall important question is how energ accumulates beforehand, a conundrum in light of reconnection. If reconnection is alwas fast and begins easil, reconnection would release an free energ immediatel, precluding major energ storage. Energ accumulation, therefore, requires that reconnection can be slow or completel inactive. It was recentl proposed that Sweet Parker reconnection, a valid model for some parameters as confirmed b simulations Biskamp 1986) and experiments Ji et al. 1998; Trintchouk et al. 3; Furno et al. 5), occurs continuousl before a flare; its slowness allows energ to accumulate Cassak et al. 5, 6, 7b, 8; Uzdensk 7). The transition from Sweet Parker to Hall reconnection is catastrophic, impling sudden onset of energ release. Consistent with this scenario, flux emergence Longcope et al. 5) and coronal eruption Liu et al. 9) observations displa slow energ release before fast. Transitions at kinetic scales are observed in experiments Ren et al. 5; Egedal et al. 7). Stellar flare data impl kinetic effects are relevant Cassak et al. 8). This discussion tacitl assumes Sweet Parker scaling holds for coronal parameters with Lundquist number S = 4πc A /ηc 1 14, where η is the resistivit, and L and c A are the characteristic length scale and Alfvén speed. However, this is false. Thin current sheets are unstable to secondar island formation when S > S 1 4 Biskamp 1986). Secondar islands, called plasmoids upon ejection from the current laer, are observed in man contexts Reeves et al. 8; Lin et al. 8; Linton & Moldwin 9). No quantitative model of their impact on Sweet Parker reconnection exists, but evidence suggests it gets faster Matthaeus & Lamkin 1986; Lazarian & Vishniac 1999; Lapenta 8; Loureiro et al. 9; Bhattacharjee et al. 9). Secondar islands make sheets thinner Shibata & Tanuma 1), as confirmed b simulations Daughton et al. 9a, 9b; Cassak et al. 9). These effects are ical; if secondar islands make reconnection fast or hasten the transition to collisionless reconnection, it is unclear how pre-flare energ accumulates. In this Letter, we address how energ storage during collisional reconnection is affected b secondar islands, taking into account the large separation between the thicknesses of the equilibrium current laer and the reconnection dissipation laer. Coronal current sheets can be 1 3 km thick Lin et al. 9), but the diffusion region is microscopic and embedded deepl within the macroscopic current sheet. This has been studied in magnetospheric contexts Pritchett & Coroniti 1995; Sitnov et al. ; Yin & Winske ). In embedded sstems, the magnetic field immediatel upstream of the diffusion region B up is weaker than the asmptotic large-scale) field B asmp Sha et al. 4). The diffusion region sees a much smaller effective Lundquist number S up than the asmptotic value S asmp. The S in the Biskamp 1986) erion S > S should be S up rather than S asmp. We confirm this prediction using high-s magnetohdrodnamic MHD) simulations of reconnection embedded in a wide current sheet. Embedded effects suppress secondar island generation and broaden the diffusion region relative to expectations based on S asmp. To address the impact on flares, we need quantitative models incorporating secondar islands. No accepted theor exists, but recent work b Daughton et al. 9a) and Loureiro et al. 7) provides a starting point. We argue that secondar islands are ubiquitous in coronal reconnection. If Sweet Parker L158

2 No., 9 MICROSCOPIC RECONNECTION AND ENERGY STORAGE L x Figure 1. Out-of-plane current densit J z for the w = 1 simulation with η =.3 at t = 4 and t = 55. Onl a small fraction of the domain is plotted and its aspect ratio is altered for clarit. reconnection with secondar islands is fast, issues of onset and energ storage remain unsolved; if slow, embedded effects ensure sufficient time to store energ before a flare.. SIMULATIONS AND RESULTS Two-dimensional resistive MHD simulations are performed using F3D Sha et al. 4) in a periodic domain of size L x L = with a grid scale of.1. Lengths, magnetic fields, and densities are normalized to arbitrar values L, B, and n ; velocities, times, electric fields, and resistivities are normalized to the Alfvén speed c A = B /4πm i n ) 1/, the Alfvén time L /c A, E = c A B /c, and η = 4πc A L /c, where m i is the ion mass. The equilibrium is a double Harris sheet B x ) = tanh[ L /4)/W ] tanh[ 3L /4)/W ] 1 with current sheet thickness W. The densit asmptotes at 1 and is non-uniform to balance total pressure. The temperature T = 1 is uniform and is not evolved. Resistivit is constant and uniform throughout all simulations. Ohmic heating is ignored. Viscosit is omitted, but fourth-order diffusion with coefficient is used in all equations to damp noise at the grid scale. A single X-line is seeded using a coherent magnetic perturbation. Initial random perturbations on the magnetic field of amplitude break smmetr so secondar islands are ejected. There is no out-ofplane guide) magnetic field. In collisional reconnection, L, the half-length of the Sweet Parker diffusion region. In periodic sstems, L x / We find closer to 75. To test the Biskamp 1986) erion, we use a base simulation with η =.3, corresponding to S asmp.5 1 4, which is unstable to secondar islands when B up B asmp. The classical Sweet Parker thickness is δ SP.47, so reconnection is initiall embedded if W >.47. As a first test, we run a simulation with W =.4. The diffusion region is immediatel unstable to secondar tearing as expected, splintering into 15 pieces at earl times. Next, we use W = 5, 1, and 15. For these runs, S asmp > S, but S up < S because B up is substantiall decreased. The simulations reveal that secondar islands do not form initiall, suggesting that S up is the control parameter. Figure 1 shows the out-of-plane current densit J z for the W = 1 simulation at t = 4 and t = 55, revealing secondar island formation onl later in the evolution. δ L/δ c) d) B up e) E N X-lines w = 1, η = t Figure. Quantities as a function of time t for the w = 1, η =.3 simulation. Diffusion region half) thickness δ, aspect ratio of the diffusion region L/δ, c) number of X-lines, d) magnetic field strength upstream of the main X-line, and e) reconnection electric field. The vertical dashed line denotes the time at which secondar islands appear. In embedded reconnection, the initial half-thickness is much larger than δ SP based on B asmp. To estimate, classical Sweet Parker theor predicts Parker 1957) S 1/ ηc up, 1) 4πc A,up where c A,up is the Alfvén speed based on B up. If the diffusion region is deepl embedded W ), the field profile is roughl linear, B up B asmp /W. Eliminating B up gives S 1/3 asmp W ) 1/3, ) so δ SP based on B asmp for large S. The dnamical evolution is completel self-generated. In embedded reconnection, the field upstream of the diffusion region is stronger than B up. This stronger field is frozen-in to the reconnection inflow. Therefore, the reconnection inflow itself carries in the stronger field. From Equation 1), increasing B up causes thinner diffusion regions. When B up is large enough to cross the Biskamp erion, secondar island production begins. This dnamics can be seen in Figure. Panel shows the diffusion region thickness δ, measured as the e-folding distance of J z in the inflow direction across the main X-line, as a function of time for the W = 1 simulation, which is representative. Following transients which last until t 3, δ decreases steadil in time. Extrapolating the linear part in back in time to t =, we find 1., in good agreement with 1.3 predicted from Equation ). Since the diffusion region half-length L, measured as the e-folding distance of J z in the outflow direction, is relativel

3 L16 CASSAK & DRAKE Vol. 77 δ L /δ c) d) B up e) E N X-lines w = 1, η = t Figure 3. Same quantities as plotted in Figure for the simulation with w = 1 but with a smaller resistivit of η =.75. constant, the aspect ratio L/δ steadil increases panel ). Onl when L/δ 13 does secondar island formation ensue. This is shown in c), with the number N of X-lines as a function of time. To find N, X-line candidates are identified as locations where the reconnected field B changes sign along the neutral line. The number of active X-lines is hand-counted to determine N. After initial transients, the sstem has a single X-line for time units. When L/δ exceeds 13, N increases. Panel d) shows B up, measured as the reconnecting field strength δ upstream of the main X-line, which steadil increases as reconnection proceeds. The onset of secondar islands at L/δ 13 corresponding to S up ) agrees with the Biskamp 1986) erion. To confirm B up is the relevant field, we perform a simulation with η reduced b a factor of 4 corresponding to S asmp 1 5 ), which should require a smaller B up to induce secondar tearing. Figure 3 shows plots like Figure. As before, transients last until t, then the sheet thins as B up increases. The extrapolated.8 agrees well with the prediction of.8 from Equation ). There is onl a single X-line until t 3, when secondar islands form. At this time, L/δ 13 as in the previous simulation. However, B up.6, rather than.9, consistent with expectations. We conclude that the field that controls the onset of secondar islands is B up, not B asmp. The reconnection electric field E is plotted in panel e) of Figures and 3, with the classical Sweet Parker prediction E SP based on B asmp plotted as the dashed line. Clearl, E exceeds E SP upon onset of secondar islands. However, we cannot ascertain whether E reaches a stead value. Larger simulations are required to determine whether E is slower than or comparable to the fast reconnection rate of E.1. It is crucial to ensure that secondar islands do not introduce excessive grid scale dissipation, which could adversel affect the results. Cuts of J z in the inflow direction for the data shown in Figure 1 are shown in Figure 4 before and after secondar J z before J z after Figure 4. Cut across J z in the inflow direction of the two current sheets pictured in Figure 1 confirming sufficient numerical resolution. islands arise. There are over 16 and 1 grid cells across the diffusion region, respectivel, confirming that grid scale effects are negligible. 3. APPLICATIONS TO THE CORONA The present work has important implications for pre-flare energ storage in the solar corona. The most important issues are whether secondar islands 1) make Sweet Parker reconnection fast, ) hasten the transition to collisionless reconnection, and 3) make the reconnection electric field exceed the Dreicer field, which would invalidate the use of a Spitzer resistivit. We address each in turn. At present, no accepted quantitative model for the impact of secondar islands exists. Daughton et al. 9a) provide a possible starting point b assuming that fragments of the diffusion region obe the Sweet Parker model, allowing one to predict the scaling with the number of islands N. With N islands, the length L of each segment scales like L /N. From Equation 1), the thickness of the fragmented laer scales like δ δ SP, 3) N 1/ where δ SP is the classical Sweet Parker prediction. The laer becomes thinner, as noted previousl Shibata & Tanuma 1). Since E δ/l, E E SP N 1/, 4) where E SP is the classical Sweet Parker prediction. As previousl noted, E increases due to secondar islands. Evidence that this scaling holds in a time averaged sense is presented in a companion stud Cassak et al. 9). There is evidence that the number of islands N scales with a power of S. The analtical model b Loureiro et al. 7) predicted N S 3/8. Amending their result to include embedded effects, we write ) α Sup N, 5) S where S 1 4 and α generalizes the model b characterizing potential models of secondar island generation. Recent numerical results b Samtane et al. 9) agree with the prediction, while Daughton et al. 9a) report α.6 in particle-in-cell simulations, and Cassak et al. 9) report α.7.93 in MHD simulations.

4 No., 9 MICROSCOPIC RECONNECTION AND ENERGY STORAGE L161 While the appropriateness of these models for coronal parameters has not been shown, it is instructive to investigate the predictions to see the effect of embedding and secondar islands. We consider properties of the reconnection for various α. To compare with coronal data, we use representative pre-flare parameters of B asmp 1 G, n cm 3, 1 9 cm, and T 1 6 K Cassak et al. 6). Using a parallel Spitzer resistivit, this gives S asmp For the macroscopic current sheet thickness, we use W s 1 3 km Lin et al. 9). These values do not appl to a particular event; the focus is on orders of magnitude, not specific values of predictions. The reconnection rate. Using E SP Sup 1/ with Equations 4) and 5) implies E Sup 1 α)/, which is quite sensitive to α. If α 1, then E weakl depends on S and reconnection is fast. If α < 1, then E scales with a negative power of S, so reconnection is slow. To make this quantitative, ignore embedded effects for simplicit which make reconnection slower). The classical normalized) prediction is E SP Sasmp 1/ 1 7, six orders of magnitude slower than observed energ release rates Parker 1963). To put this in perspective, if α = 3/8, then E Sup 5/16 giving E 63E SP for coronal parameters, which is four orders of magnitude slower than observed rates; see Cassak et al. 9) for further discussion. The transition to collisionless reconnection. Collisionless reconnection begins when δ reaches a kinetic length scale Ademir 1991; Mandt et al. 1994; Ma & Bhattacharjee 1996), which we call δ k. The transition is abrupt; for anti-parallel reconnection, δ k = d i = c/ω pi, the ion inertial scale Cassak et al. 5), where ω pi is the ion plasma frequenc. With a strong guide field δ k = ρ s = c s /Ω ci, the ion Larmor radius Cassak et al. 7a), where c s is the ion sound speed and Ω ci is the ion cclotron frequenc. Reconnection in the corona has a strong guide field Uzdensk 7), so the δ k = ρ s is appropriate. If reconnection with secondar islands remains slow α < 1), one must ask how secondar islands impact onset of collisionless reconnection. We estimate the length scales for the corona. The kinetic scale is ρ s 1 cm. From Equation ), the initial thickness of the diffusion region is δ SP,i W s L SP /S asmp) 1/3 1 4 cm. The initial upstream field is B up,i B asmp δ SP,i /W s.1 G, giving an initial S up,i Since S up,i S 1 4, secondar islands occur during coronal Sweet Parker reconnection from the outset despite embedded effects. For definiteness, if α = 3/8, Equation 5) predicts initiall N i 14 islands. From Equation 3), δ SP,i /N 1/ i 1 cm. This is 1 times wider than ρ s, so the fragmented diffusion regions are initiall wider than kinetic scales. Reconnection at initiation is collisional. To estimate the ical upstream field B up,c needed to make δ δ k, set Equation 3) equal to δ k and eliminate N using Equation 5). We make contact with previous work b first writing this in terms of a dimensionless effective collisionalit parameter η c = ηc /4πc A,up δ k, getting η c = 1 S α/1+α) δk ) 1 α)/1+α). 6) This generalizes the result b Daughton et al. 9a) to include embedded effects B up,c instead of B asmp in η c ), guide field effects δ k instead of d i ), and the amended Loureiro et al. 7) Figure 5. From the model, the predicted ical field B up,c and quiet time τ quiet as a function of the exponent α in the secondar island generation model for coronal parameters emploed in the text. result in Equation 5). Solving for B up,c gives S α/1+α) B up,c B asmp S asmp LSP δ k ) /1+α). 7) This result is plotted as a function of α in Figure 5 using δ k = ρ s. For ver small α, B up,c > B asmp, so a flare would never occur α.5). For larger α, B up,c B asmp, so a flare would onset too easil to store energ α.5). The time it takes for this upstream field to convect into the diffusion region is a measure of the quiet time before a flare τ quiet. We estimate it using an analsis similar to Sha et al. 4); Cassak et al. 6). The time it takes to convect flow is τ quiet δ f dδ/v in, where the bounds of integration are the initial and final thicknesses. We write v in in terms of δ using conservation of mass v in c A,up δ/l. Then Equation 4) implies δ/l δ SP N 1/ /. Finall, using Equation 5) to eliminate N and B up B asmp δ/w s gives the integrand entirel in terms of δ and asmptotic sstem parameters. Integrating from the distance upstream that the field strength equals the ical field δ up,c B up,c /B asmp )W s to ver small kinetic scales δ f gives a quiet time τ quiet W s S α/1+α) 1 α c A,asmp LSP δ k ) 1 α)/1+α). 8) This is plotted in hours) in Figure 5 as a function of α. For.5 α.5, there are several hours for energ storage, which is a reasonable pre-flare storage time Priest & Forbes ). Since τ quiet is linear in W s, wider macroscopic sheets produce longer quiet times. We conclude that secondar islands do not invalidate recent models Cassak et al. 5, 6; Uzdensk 7; Cassak et al. 8) of energ storage provided α is in this range for the parameters and secondar island model emploed here). If α 1, the quiet time before a flare would be exceedingl small, so energ would not have time to accumulate. The Dreicer field. The Dreicer field is E D.43 πne Ze 3 k B T e ln Λ ) statvolt cm 1. 9)

5 L16 CASSAK & DRAKE Vol. 77 Using α = 3/8 and the asmptotic fields, E statvolt cm 1, which exceeds the Dreicer field. However, from Figure 5, the transition to collisionless reconnection occurs when B up 1 G. For these fields, E statvolts cm 1, comparable to the Dreicer field. In this scenario, the field during the quiet time is below the Dreicer field, so the description using a Spitzer resistivit is valid. 4. DISCUSSION The precise scaling of reconnection with secondar islands remains an open question. The model emploed here, for example, does not take into account possible hierarchical secondar island scenarios Shibata & Tanuma 1; Daughton et al. 9a; Bhattacharjee et al. 9). Regardless, the present results conclusivel show that embedded effects can be considerable, and need to be taken into account when modeling coronal reconnection. In addition to coronal reconnection, the present results ma be relevant to chromospheric reconnection Litvinenko 1999; Chae et al. 3; Litvinenko & Chae 9), where S The inferred reconnection rate exceeds the classical Sweet Parker prediction b a factor of a few. Using Equation 4) with α = 3/8 predicts an enhancement b a factor of about 5 for S asmp 1 8, which brings the observations into closer agreement with theor. Limitations to the present simulations include the absence of a guide field, the usage of a constant resistivit instead of a Spitzer resistivit, and the omission of Ohmic heating and viscosit; the latter of which ma be important for secondar island generation Park et al. 1984). Three-dimensional effects are omitted, but are potentiall important. The models emploed here treat onl self-generated secondar islands, not externall driven turbulence Lazarian & Vishniac 1999; Smith et al. 4; Kowal et al. 9; Loureiro et al. 9). The authors thank W. Daughton, A. A. 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