SIMULATION STUDY OF THE FORMATION MECHANISM OF SIGMOIDAL STRUCTURE IN THE SOLAR CORONA

Transcription

1 The Astrophysical Journal, 631: , 2005 October 1 # The American Astronomical Society. All rights reserved. Printed in U.S.A. SIMULATION STUDY OF THE FORMATION MECHANISM OF SIGMOIDAL STRUCTURE IN THE SOLAR CORONA K. Kusano Earth Simulator Center, Japan Agency for Marine-Earth Science and Technology, Showa-machi, Kanazawa-ku, Yokohama, Kanagawa , Japan; kusano@jamstec.go.jp Received 2005 February 16; accepted 2005 May 28 ABSTRACT The formation mechanism of sigmoidal structure in the solar coronal magnetic field is studied using the threedimensional magnetohydrodynamic numerical simulations, based on the so-called reversed-shear flare model recently proposed by Kusano et al. The simulation results clearly indicate that magnetic reconnection driven by the resistive tearing mode instability growing on the magnetic shear inversion layer can cause the spontaneous formation of sigmoidal structure. Furthermore, it is also numerically demonstrated that the formation of the sigmoids can be followed by the explosive energy liberation, if the sigmoids contain sufficient magnetic flux. This implies that the reversed-shear flare model can provide a self-consistent explanation for the formation of sigmoids as well as for the onset of eruption, which is driven by magnetic reconnection above sigmoids. The geometric relationship between the sigmoidal structure and the minimum energy state predicted by J. B. Taylor in 1974 is examined. The result suggests that the sigmoidal formation could be understood as a manifestation of the minimum energy state, which has the excess magnetic helicity compared to the bifurcation criterion of the linear force-free field. The consistency with the observations of magnetic helicity is also discussed. Subject headinggs: MHD Sun: activity Sun: flares Sun: magnetic fields 1. INTRODUCTION The observations by the Yohkoh soft X-ray telescope (SXT; Tsuneta et al. 1991) found that a typical structure of forward-s or inverse-s shape often appears in the soft X-ray images of the solar corona. This typical morphology was named sigmoid first by Rust & Kumar (1996). Because sigmoids tend to appear associated with several eruptive events (Canfield et al. 1999), they are widely believed to be some precursor phenomena for eruptive flares. Several papers have been published to explain the formation mechanism of sigmoids so far. The most widely believed idea, which was originally proposed by Rust & Kumar (1996), is based on the ideal kink instability, which can grow in a flux tube twisted stronger than some criterion, in principle more than one turn. Recently, Kliem et al. (2004) performed the three-dimensional ideal-magnetohydrodynamic ( MHD) simulation of the kink mode instability of a twisted flux rope, and they clearly showed that a sigmoid-like current sheet can be formed at the interface with the surrounding medium. Magara & Longcope (2001) and Fan & Gibson (2004) also demonstrated by numerical simulations that the emerging of a twisted flux tube can form a sigmoidal structure in a magnetic field. However, it is still debatable whether the kink mode instability is indeed responsible for the formation of sigmoids. In fact, Leamon et al. (2003) found that there is no evidence of eruption for values of large-scale total twist approaching the threshold for the kink instability, although they sought 191 X-ray sigmoids. If sigmoids and subsequent eruptions could arise in spite of insufficient twist of magnetic field for the kink mode instability, it is worthwhile to keep asking whether and how sigmoidal structure can be created by some other mechanism than the kink mode instability. When we seek the answer to the question, it is crucially important to understand the causal relationship between the photospheric activity and the magnetohydrodynamics in the solar 1260 corona. In particular, the magnetic helicity injection into the solar coronal magnetic field from the photosphere must be a primary source for the sigmoidal activity. Recently, several new methods have been developed to measure the magnetic helicity flux out of the photosphere into the solar corona (Chae 2001; Kusano et al. 2002, 2003a; Démoulin & Berger 2003; Welsch et al. 2004; Longcope 2004). In particular, Kusano et al. (2002) and Yokoyama et al. (2003) revealed, using the method consistent with both the magnetograph observations and the induction equation ( Kusano et al. 2004b), that the helicity injection process in flare productive active regions was highly complicated in both time and space, and that even the sign of the helicity injection often changed. Furthermore, Maeshiro et al. (2005) recently clarified through the data analyses for the seven different active regions that the soft X-ray radiance from the active regions is almost proportional to the product between the intensity of the magnetic helicity injection and the probability of sign reversal of the magnetic shear on the photosphere. These results imply that the solar coronal activity is sensitive not only to the magnetic helicity injection but also to the complexity in the magnetic structure, particularly to the reversal of the magnetic shear angle. Motivated by the observations of magnetic helicity, Kusano et al. (2004a) investigated the nonlinear dynamics of a magnetic arcade, which is subject to reversal of the magnetic shear, in terms of the three-dimensional numerical simulations, and clarified that the shear reversal is able to cause a large-scale eruption of the magnetic arcade. On the basis of these results, they proposed the so-called reversed-shear flare model to explain the trigger mechanism of flares (Kusano et al. 2004a). The physical process of their model can be understood as a consequence of the annihilation of magnetic helicity ( Kusano et al. 2003b). The reversed-shear flare model predicts that the very early activity in the preflare phase should be initiated by the reversal of magnetic shear. Not only the shearing motion on the photosphere, but also the flux emerging, can make the reversed-shear configuration, if the orientation of the emerging magnetic field

2 FORMATION MECHANISM OF SIGMOIDS 1261 opposes the shear component of preexisting field. Recently, Wang et al. (2004) indeed found that the shear reversal caused by the flux emerging was associated with the coronal mass ejections. If the magnetic shear is steeply reversed, then the electric current flowing on the shear inversion layer could become unstable to the resistive tearing mode. The growth of the instability can commence the second phase of the reversed-shear flare model, in which the oppositely sheared fields are subject to magnetic reconnection. Here the first phase (shear reversal) and the second phase (tearing instability) compete with each other, because reconnection of the tearing instability works to eliminate the reversed-shear field. However, if the shear inversion layer can be maintained for longer than the tearing mode timescale, the instability can grow even nonlinearly. Then the third phase starts, in which the nonlinear coupling between different Fourier modes dominates the structural evolution, and magnetic reconnection substantially proceeds mainly at a few preferential points. Reconnection in this phase strongly annihilates the oppositely directed axial magnetic fluxes (AA 0 and BB 0 in Fig. 1a) and ejects shearless fields (CC 0 and DD 0 ) horizontally along the magnetic neutral line. The annihilation of the axial fluxes by reconnection must be followed by the inward collapse of magnetic arcade, because the magnetic pressure in the arcade core is reduced by the flux annihilation. As a result, the sheared fields (EE 0 and FF 0 in Fig. 1b) carried by the inward flows (f 3 and f 3 0 in the figure) must collide above the shear inversion layer, and another reconnection starts. It triggers the fourth phase, in which the field GG 0 ejected by the second reconnection collides with the oppositely sheared field II 0 on the shear inversion layer. In this process, the downward jet (f 4 0 ) strongly drives the original reconnection on the shear inversion layer. This is a mutual excitation of magnetic reconnections, in which one reconnection further drives the other reconnection. After this process, only the shearless fields (IG 0 and GI 0, not shown in Fig. 1) remain above the solar surface, and the sheared field (HH 0 ) must be ejected into the upper corona and the interplanetary space. This model has the advantage that it can naturally explain the explosive onset of solar flares by the mutual excitation of the double reconnections. Therefore, the onset timescale of flares and of arcade eruptions can be shortened extremely compared to the growth time of the resistive tearing mode instability. The reversed-shear flare model predicts that the downflow into the shear inversion layer (f 1 0 in Fig. 1a) should be created before the eruptive phase. It is consistent with the recent observations, which found that the down-flow appeared already in the impulsive phase of the flares (Asai et al. 2004). Another prediction of the reversed-shear flare model is that solar flares could occur near the shear inversion layer. It is consistent with the fact that the complexity in magnetic shear correlates well with the solar coronal activity ( Maeshiro et al. 2005). Moreover, we recently found, in the various flare events, that the preflare brightening observed by the Transition Region and Coronal Explorer (TRACE) image (Handy et al. 1999) was located on the shear inversion line, which was detected from vector magnetograms ( T. Maeshiro et al. 2005, in preparation). These results strongly support the validity of the reversed-shear flare model, even though the entire process from the shear reversal to the onset of flares was not yet clearly observed, because of the limitation in the resolution and the duration time of vector magnetic field measurement. In general, it is an important benchmark for flare models to explain the formation of sigmoids and the physical relationship between sigmoids and the eruptive events. The primary objective Fig. 1. Schematic diagram of the reversed-shear flare model. Solid curves and thick arrows represent the typical structure of magnetic field lines and plasma flows, respectively. (a) Relaxation phase: Magnetic reconnection between the oppositely sheared field lines annihilate the axial magnetic flux along the magnetic neutral line. (b) Eruptive phase: Annihilation of the axial field causes the collapse of magnetic arcade and leads to another reconnection between sheared lines EE 0 and FF 0. The mutual excitation between the original reconnection of the reversed-shear fields (GG 0 and II 0 ) and the new reconnection of the collapsing fields (EE 0 and FF 0 ) can trigger the explosive growth of the liberation of magnetic energy. of this paper is to investigate whether the reversed-shear flare model can account for the formation mechanism of sigmoidal structure. Moreover, we attempt to reveal the causal relationship between the appearance of sigmoids and the onset of solar flares and try to understand why sigmoid can play as a precursor of eruptive events. For that, we carried out the numerical simulations under various conditions for the shear reversal, and investigated the structural evolution of magnetic field in more detail. This paper is organized as follows. The numerical model and the simulation results are described in xx 2 and 3, respectively. On the basis of the simulation results, we discuss mainly the physical mechanism of the sigmoid formation in x 4. Important conclusions and remaining subjects are summarized in x SIMULATION MODEL The numerical model used in this paper is basically that same as the model developed in the previous papers (Kusano 2002; Kusano et al. 2004a). The simulation box spans a rectangular region (0 < x < L x,0< y < L y,0< z < L z ) above the photospheric surface, where L x : L y : L z ¼ 1 : 1 : 10 and the bottom

3 1262 KUSANO Vol. 631 Fig. 2. Four different snapshots of the two-dimensional simulation at (a) t ¼ 12:3, (b)17.7,(c)23.3,and(d) They correspond to the initial equilibria of the three-dimensional simulation for cases A D, respectively. The contours are for the magnetic flux and the gray-scale represents the axial magnetic field B x. The horizontal distributions of B x at the bottom (z ¼ 0) and the vertical variation at the center ( y ¼ 0:5) are plotted in the top and right panels, respectively, of each sidebar. plane (z ¼ 0) corresponds to the photosphere. The bottom boundary condition of the normal magnetic field is given by B z ( y) ¼ B 0 sin k y L y =2 ; where k ¼ 3/L y, and thus the magnetic neutral line is located on the box center ( y ¼ L y /2). At the top (z ¼ L z ), bottom (z ¼ 0), and lateral ( y ¼ 0, L y ) boundaries, the rigid and perfectly conducting wall condition (V ¼ 0, E < n ¼ 0) is imposed, where V, E,andn are velocity, electric field, and the unit normal vector on the boundary, respectively. On the other hand, the periodic condition is adopted along the magnetic neutral line (x-axis). The wall condition at the lateral boundaries certainly stabilizes the magnetic arcade system, in contrast to that the periodic system is easily destabilized when the magnetic shear exceeds a criterion (Kusano et al. 1995; Kusano & Nishikawa 1996). The basic equations in nondimensional form are ¼ V ð =: ÞV þ J < B þ : 2 ¼ :< ðv < B JÞ; ð2þ and J ¼ :<B, in which the pressure gradient force and the density variation are neglected based on the approximation that the plasma is generally very small in the solar corona. The nondimensional unit for length, velocity and time are given by L y, V A (Alfvén speed defined by B 0 ), and A ¼ L y /V A, respectively. The three-dimensional simulation is initiated by adding a small perturbation to a quasi static equilibrium, in which the magnetic shear is reversed near the magnetic neutral line. The quasi-static equilibrium state with the reversed shear was numerically generated using the two-dimensional simulation model, in which the translational symmetry along the x-coordinate (@/@x ¼ 0) is imposed. For the details of the two-dimensional calculation, refer to Kusano et al. (2004a), in which the initial state is given by the solution of the linear force-free field (LFFF) equation :<B ¼ B; where ¼ 9:2L 1 y and the slow shear motion (0.05V A at the maximum) is imposed only near the magnetic neutral line [(0:5 s)l y < y < (0:5 þ s)l y for s ¼ 0:07]. From the solution of the LFFF (Heyvaerts & Priest 1984), the magnetic shear ð3þ

4 No. 2, 2005 FORMATION MECHANISM OF SIGMOIDS 1263 angle, that is, the azimuth from the potential field, is determined by tan ¼ (k 2 2 ) 0:5 : In our case ( ¼ 9:2/L y and k ¼ 3/L y ), ¼ 77N4. This value is comparable to strong shear, which was observed in typical delta-spots (Hagyard et al. 1984). As explained in Kusano et al. (2004a), in the two-dimensional system, the shear reversal process is fairly stable and any eruptive dynamics is not observed at least until the calculation is finished at t ¼ 80 A, in which the maximum displacement between the field line foot points reaches 4L y. Moreover, since the footpoint motion is very sub-alfvénic, the magnetic field may evolve keeping the quasi-equilibrium condition. Figure 2 shows the four different snapshots of the two-dimensional simulation at t ¼ 12:3, 17.7, 23.3, and 28.9, where the magnetic flux and the axial magnetic field B x are plotted on the vertical ( y-z) plane. We can see that as time is elapsed, the reversed-shear region, which is seen as the black area above the neutral line ( y ¼ 0:5), is elongated almost vertically. The snapshots shown in Figure 2 are used also as the initial equilibria in the three-dimensional simulations for cases A, B, C, and D, respectively. Therefore, in this paper, we focus on the dynamics after the reversed-shear state is formed, rather than the formation process of the reversed shear. As mentioned in x 1, the formation of the reversed-shear state is a competitive process between the shear reversing due to the photospheric activity and the tearing mode instability growing on the reversed-shear region. However, since the growth of the tearing mode is sufficiently slow in the solar corona, the reversed-shear states may survive at least for a couple of hours, even though it is unstable (see x 4 for more detailed discussion). The linear stability of the quasi-equilibria was numerically analyzed by the same method as Kusano et al. (2004a), and it was confirmed that any states shown in Figure 2 are linearly unstable to several Fourier modes along the arcade (x) axis. They are the resistive tearing mode instability, which grows mainly on the shear inversion layer where B x ¼ 0. The most unstable equilibrium in them is case A, and the growth rate is reduced as the reversed-shear region is elongated from cases A to D. The reduction of the growth rate can be explained by the fact that the growth rate of the tearing mode instability is mainly influenced by the electric current profile at the resonant surface, which corresponds to the shear inversion layer in our model. As seen in the right sidebar of each panel in Figure 2, as the shear inversion moves to the higher portion (from A to D), the steepness of the magnetic shear reversal is weakened, and the electric current density is reduced. The resistivity is modeled as a function of the electric current density using the same model as that used in Kusano et al. (2004a); i.e., 8 >< 0 for J < J c ; ¼ J J 2 c >: 0 þ 1 for J J c ; J c where J ¼jJj, 0 ¼ 10 5, 1 ¼ 5 ; 10 4, and J c ¼ 3 ; 10 2 in nondimensional units, respectively. The nondimensional viscosity is given by a constant 5 ; The calculations were carried out in terms of the finite difference scheme. The grid number is 256 ; 256 ; 512 for each ð4þ dimension (x, y, and z). Since, in our simulation, the instability first grows on the shear inversion layer, the grid points are highly concentrated near the center ( y ¼ 0:5) in the lower portion z < 0:4, where the finest grid sizes are x ¼ 3:9 ; 10 3, y ¼ 1:0 ; 10 3, and z ¼ 1:0 ; The spatial differentials are approximated by the second-order accuracy finite difference using three grid points stencils, and the time integration is performed by the Runge-Kutta-Gill method of fourth-order accuracy. 3. SIMULATION RESULTS 3.1. Energy Evolution The time evolution of the magnetic and kinetic energies are plotted in Figures 3a and 3b, respectively, for the four different cases. In the early phase (t < 20), all cases exhibit the typical behavior for the MHD relaxation process, in which the fast decay of magnetic energy and the growth of kinetic energy proceed simultaneously. The more reversed the magnetic shear, the more delayed the initiation of the MHD relaxation phase is, as clearly shown by the small segments plotted in Figure 3a. The result can be explained by the difference in the linear growth rate of the resistive tearing mode instability, because the tearing mode instability works as the driver of the MHD energy relaxation. For instance, the magnetic energy in case A is quickly reduced earliest in the four cases, because it is the most unstable in them. In each case, the energy relaxation phase lasts for several A, and after that the decay rate of magnetic energy is slowing down. We should note that in the late phase of cases C and D, the eruption occurs at t ¼ 30 and 60, respectively, in which kinetic energy grows explosively, and magnetic energy is quickly reduced Fig. 3. Time evolution of the total magnetic and kinetic energies, E M and E K, for the four different cases. Thin and short segments in (a) indicate the start time of the MHD relaxation phase.

5 1264 KUSANO Vol. 631 Fig. 4. Top view of the magnetic field lines in case D at various times. The radius of the field line cross section is proportional to the intensity of the electric current averaged along each field lines. Color shading on the bottom plane represents the axial magnetic field B x, where red and blue indicate positive and negative values, respectively. The white line ( y ¼ 0) indicates the magnetic neutral line where B z ¼ 0: again. The onset of the eruptive phase is triggered by the second reconnection (see Fig. 1b), which is caused by the flux annihilation on the shear inversion layer and the inward collapse of the magnetic arcade, as first found by Kusano et al. (2004a). On the other hand, the eruptive phase does not appear in cases A and B, while only a weak enhancement of kinetic energy is seen at t ¼ 28 in case B. The result that the eruptive phase arises only in the weakly unstable cases (cases C and D) implies that the onset of the eruption does not directly relate to the linear stability growth rate, and that the geometric condition is more important to trigger the eruption Structure of Magnetic Field Lines Figure 4 represents the structural evolution of the magnetic field lines, which are traced from points collocated on several lines aligned above the bottom surface in case D. The results clearly indicate that the sigmoidal structure is spontaneously formed through the relaxation phase until t ¼ 27. The formation of sigmoid is performed through magnetic reconnection between the field lines sheared oppositely, and thus both the negative and positive components of the axial magnetic field are partially mixed in the sigmoidal field lines. For instance, in the panel for t ¼ 27:0, we can see that the sigmoid feet (R1 and R2) are located in the red area where B x > 0, although the main part of the sigmoidal field passes through the reversed-shear region, where B x < 0. The radius of the cross section of field lines plotted in Figure 4 is proportional to the intensity of the averaged electric current density along each field line, that is Z L hi¼ j L 1 jdl; where l isthearc-coordinatealongafieldlineandl is the arclength of that. The fact that thick field lines are concentrated in the sigmoidal region indicates that the current density is 0

6 No. 2, 2005 FORMATION MECHANISM OF SIGMOIDS 1265 Fig. 5. Three-dimensional structure of the magnetic field lines in the sigmoidal region for case D at t ¼ 30:1 (top) andt ¼ 58:7 (bottom). The color shading on the bottom plane indicates the same as Fig. 4. considerably higher in the sigmoidal region compared to the other region. Therefore, it is likely that field lines forming the sigmoid are more visible than the other field lines because of the Ohmic heating effect, although the thermal pressure profile is not taken into account in this simulation model. Figure 5 (top) represents the three-dimensional structure of magnetic field lines in the sigmoidal region at t ¼ 30:1 in case D. It is noted that the field lines in this region consist of three substructures. One is the main part of sigmoid, in which field lines form a helical shape as shown by red strings in the figure. Note that they are concave in the central part and raised in two offcenter regions. It is a common feature of the sigmoidal fields, which are seen not only in case D but also in all the other cases. The second substructure is configured of field lines passing over the main sigmoidal fields, as shown by green strings in the figure. The reversed-shear field embedded in the initial state above the magnetic neutral line survives in this region without being subject to magnetic reconnection. The other structure is the largescale overlaying field shown by blue lines, which has positive shear as same as the ambient field in the initial state. Thus, there is a current layer between the second and the third substructures, where the tangential magnetic field is discontinuous. Once the sigmoid is formed, the quasi-stable phase starts, in which the field line structure is very slowly modified from t ¼ 27:0 to The slow modification in this phase is a consequence of the long current sheet reconnection, which gradually cancels the axial fields between the second and the third substructures described above. During this phase, the sigmoid center sits in the reversed-shear region, which exists underneath the reconnection point between II 0 and GG 0 in Figure 1b. However, after the substantial axial fluxes of the opposite signs (negative inside the sigmoid, and positive outside that) are annihilated, the eruption is triggered ultimately at t ¼ 60. Figure 5 (bottom) represents the three-dimensional structure of field lines just prior to the onset of the eruption. We can easily see that the overlaying magnetic arcade collapses into the point just above the shear reversal, and a cusp structure is generated above the sigmoid. Since a new reconnection proceeds at the X-point in the collapsing arcade, the strong jets are emitted both upward and downward from the reconnection point. The downward jet strongly accelerates the original reconnection with the field lines of negative shear. Eventually, the reconnection point intrudes into the sigmoidal region, and some field lines (L1 and L2 in Fig. 4) in the sigmoid are connected to the outside, forming the almost shear-free field. Even after the eruption (t ¼ 69:5), the reversed axial field and the weak sigmoidal feature can survive near the photospheric boundary, because the second reconnection point (between EE 0 and FF 0 in Fig. 1) moves to rather higher portion in the late phase. In the center of the sigmoidal region, the magnetic shear is partially relaxed, and some snaky structure is generated after the eruption (L3 in Fig. 4). Here we should point out again that the cusp structure, which could correspond to a postflare loop, exists above the sigmoid and the sigmoid-remnant (see Fig. 5, bottom). It means that the sigmoid itself is not erupted, but it stays underneath the cusp structure. However, the field twist in the sigmoid is unraveled by magnetic reconnection. This is an important prediction of the reversed-shear flare model. Sterling et al. (2000) examined the morphological properties of the evolution of sigmoids into cusps and arcades for the four sigmoidal regions associated with soft X-ray flares observed by SXT and the EUV Imaging Telescope (EIT) onthesolar and Heliospheric Observatory (SOHO) satellite, and they concluded that the cusp-producing fields may be overlying the sigmoid fields in the preflare phase. Our prediction is well consistent with their observation, and can naturally explain the overlying process of the cusp structure. The spontaneous formation of the sigmoidal structure can be seen not only in case D but also in the all other cases, no matter whether the eruption occurs or not, as shown in Figure 6. It implies that the appearance of the sigmoid-like structure does not necessarily lead to the onset of the large-scale eruption. However, notable difference of the sigmoids among cases A to D is in the size of them. As the reversed-shear region in the initial state is extended from cases A to D, also the sigmoid size is enlarged. In contrast to case D, in which only a single sigmoid appears, two

7 1266 KUSANO Vol. 631 Fig. 7. Normalized power spectrum of magnetic field for the Fourier mode of cases A D. The variation of the number of sigmoids can be detected also in Figure 7, which represents the normalized power spectrum of the Fourier mode for magnetic field, i.e., where E m Ẽ m ¼ P ; E m Z Z E m ¼ Z B m ¼ L 1 x m>0 B m 2 dy dz; B expð ik m xþdx; Fig. 6. Top view of the magnetic field lines in cases A D. Solid ellipsoids represent the sigmoidal regions. The format is the same as Fig. 4. sigmoids are generated along the magnetic neutral line in the all other cases. In case A, since the energy relaxation is initiated in the very early phase, the magnetic shear in the region surrounded by the dotted ellipsoid in Figure 6 has been relaxed before this snapshot, and the two small sigmoids remain after the relaxation phase. In case B, the size of the two sigmoids is almost same, while in case C, there is some imbalance between them, in which the right one is rather longer than the left one. In cases B and C, the regional eruption occurs above the left sigmoid at t ¼ 28 and 15, respectively, and then the left sigmoid is partially raveled first. The regional eruption cannot lead to the large-scale eruption, whereas the kinetic energy is slightly enhanced corresponding to the regional events, as seen in Figure 3b. In case B, however, even after the regional eruption, the other sigmoid (right in the figure) survives without making the large-scale eruption until the calculation is finished at t ¼ 72. On the other hand, in case C, the large eruption arises above the larger sigmoid at t ¼ 30, as shown in Figure 3. k m ¼ 2mL 1 x,andi is the imaginary unit. It clearly indicates that the dominant Fourier mode for the coordinate x is switched from 2 to 1 only in case D. Moreover, we should point out that the contribution of m ¼ 3 is reduced as the reversed-shear region is enlarged from cases A to D. This results clearly demonstrate that the sigmoidal structure is sensitive to the shape of the reversed-shear region. This point is discussed in the following section. 4. DISCUSSION Since sigmoids should be a manifestation of magnetic helicity, the shear parameter, which is usually defined as the ratio between 9;B and B, might be a relevant parameter to characterize the sigmoidal structure. By definition, the helicity-free field gives ¼ 0, but the field of positive (negative) generates the righthanded ( left-handed) helicity. The absolute value of represents the helical wave number in magnetic field. In Figure 8, the vertical variation of the averaged value, ¼ B =:< B B 2 ; on the center ( y ¼ 0) is plotted as a function of z, wherethe bar denotes the mean field along the arcade, Z B ¼ L 1 x B dx: In the initial state, is steeply reversed at z ¼ 0:25 and has a large negative peak ( ¼ 82) at z ¼ 0:22. After the sigmoid is

8 No. 2, 2005 FORMATION MECHANISM OF SIGMOIDS 1267 that the shear inversion surface configures the relevant boundary, because magnetic reconnection on the shear inversion layer works as the driver of the energy relaxation. In order to examine the geometric effect, the lowest eigenvalue 1 is numerically calculated in a model geometry, which forms a rectangular tube of width w and height h. In order to model the reversed-shear region with the tube geometry, the periodic condition of the period L x is imposed along the tube axis in common with the simulation model. The eigenvalue in the tube was calculated based on the method developed by Finn et al. (1994). They derived the solution of equation (3) in the three-dimensional geometry as an asymptotic solution of the initial value problem governed by the equation Fig. 8. Vertical variation of the averaged shear parameter for case D at t ¼ 0(dashed line) andt ¼ 30:1 (solid line). The height of the shear inversion layer at t ¼ 30:1isdenotedbyh, and the corresponding wavenumber (2/h) is plotted by the dot-dashed line. formed (t ¼ 30:1), however, the negative peak of disappears, and in the lower part (z < 0:1) is almost flattened. Uniformity in indicates that the magnetic energy in this region tends to relax toward the Taylor-type minimum energy state (Taylor 1974, 1986), which can be described by the LFFF (3). According to the Taylor minimum energy theory, the LFFF bifurcates into multiple states when is larger than the lowest eigenvalue 1 of the curl operator (Taylor 1986; Kusano et al. 1995; Kusano & Nishikawa 1996). One solution of the bifurcated LFFF is called the coupled state, and the other is the mixed state. When the magnetic helicity is larger than the criterion for the bifurcation, the minimum energy state is switched from the former to the latter. Magnetic helicity in the coupled state is totally generated by the coupling with the field connecting to the boundary condition, but that in the mixed state is given by the mixing of the eigenfield corresponding to 1 and the field connecting to the boundary condition. Thus, even though magnetic helicity is further supplied exceeding the bifurcation criterion, of the minimum energy state is unchanged from 1, and only the magnitude of the eigenfield is amplified. The eigenvalue 1 approximately corresponds to the wavenumber (2k 1 ) of the characteristic scale k of the system, although it is more or less modified by the geometric factor depending on the boundary shape. For instance, 1 ¼ 3:1R 1 in the cylinder of radius R (Taylor 1986). Notice in Figure 8 that the wavenumber 2h 1 corresponding to the height h of the shear inversion layer (dot-dashed line) correlates well with the value of in the lower part. The result strongly suggests that magnetic field in the sigmoidal region can be approximated by the Taylor minimum energy state, although the spatial variation in remains mainly in the upper part of the sigmoid. If the sigmoidal structure can be explained as the Taylor-type minimum energy state, which corresponds to the mixed state, the sigmoid size (the wavelength along the arcade axis) must be determined by the geometry of the domain, which is subject to the energy relaxation, rather than by the amount of the contained magnetic helicity. Although the boundary of the relaxed state is not clearly defined in open space like the solar corona, it ¼ 92 A þ B; ð5þ where B ¼ :<A. The solution of equation (5) converges to the vector potential of the LFFF, which satisfies the Coulomb gauge condition :=A ¼ 0. Let us define the ith eigenfield b i by and the boundary condition :<b i ¼ i b i n = b i ¼ 0; where n is the unit vector normal to the boundary and 0 < i < iþ1 for i ¼ 1; 2; ::: is assumed for the sake of convenience. Since the eigenfields b i make a complete set for any solenoidal fields (Yoshida 1997), magnetic field with the boundary condition in equation (6) can be described by B ¼ X c i b i : ð7þ i If we substitute equation (7) into equation (5), the growth rate of the amplitude c i is given by ¼ i ( i ), and thus c i () ¼ c i (0) exp (): It indicates that the instability arises when is larger than 1, while any solution decays if jj < 1.Onthe other hand, if and only if jj matches 1, is the steady solution, which corresponds to b 1, asymptotically obtained. We derived the lowest eigenvalue 1 using the Newton method as follows. First, equation (5) is transformed to the Fourier series where y z k2 m Ãm ð6þ þ : 2 < à m ; for m ¼ 0; 1; 2; :::; ð8þ Z à m ( y; z) ¼ L 1 x A(x; y; z)expð ik m xþdx; 9 2 ¼ ik m y z : Here the boundary condition à m < n ¼ 0 is imposed, and the initial state is given by some trial function. Second, equation (8)

9 1268 KUSANO Vol. 631 Fig. 9. Three lowest normalized eigenvalues of the curl operator w i (i ¼ 1, 2, and 3) in the rectangle tube geometry of the width w and the height h vs. the aspect ratio (h/w). They correspond to the eigenvalues for the three lowest Fourier modes m ¼ 1, 2, and 3, which are plotted by solid, dot-dashed, and dashed lines, respectively. Dotted segments indicate the aspect ratio of the reversed-shear region in cases A D. is integrated by the pseudo-time using the Runge-Kutta-Gill method over a period respectively for ¼ 0 and 0 þ : Then, the growth rate is derived by () ¼ 1 à m = à D m E 2 ; where à m is the increment of à m for, angle brackets denote the average over the y-z plane, and a superscript asterisk represents the complex conjugate. Third, using the Newtonmethod, the value of 0 is changed to 0 à m ( 0 ) ( 0 þ ) ( 0 ) : The second and third stages above are repeated until becomes smaller than a small criterion (1 ; ). Figure 9 indicates the dependency of the normalized eigenvalue w i on the aspect ratio h/w for the three lowest modes m ¼ 1, 2, and 3, which correspond to the three lowest eigenvalues i ¼ 1, 2, and 3 in order of magnitude. There is a general tendency for any modes that the normalized eigenvalue decreases as the aspect ratio increases. In particular, for the lowest mode (m ¼ 1), the slope of the curve is steeper than the other modes, so that the lowest eigenvalue is switched from m ¼ 2to1 when the aspect ratio exceeds 1.4. At least in the range surveyed in Figure 9, the eigenvalue for m ¼ 3 is always larger than that for m ¼ 2. In the figure, also the ratio between the height and the width of the reversed-shear region in cases A to D is marked by dotted segments. We should note that only case D exceeds the critical aspect ratio, above which m ¼ 1 mode gives the lowest eigenvalue. The result is well consistent with the fact that the single sigmoid is generated by the nonlinear simulation only in case D, but two sigmoids appear in the other cases, because the eigenfield of the lowest eigenvalue constitutes the minimum energy state according to the minimum energy principle (Taylor 1986). On the basis of the results above, we propose that sigmoidal structure in the solar corona is a manifestation of the so-called mixed state, which is one branch of the Taylor-type minimum energy state. It could be spontaneously generated when the absolute value of the shear parameter reaches the eigenvalue 1. This proposition predicts that sigmoids should be generated near thin current layers, which are unstable to the tearing mode instability, and thus that the geometry of the unstable current layer, rather than the amount of magnetic helicity, is responsible for the wavelength of the sigmoid. Although both signs of magnetic shear could coexist across the shear inversion layer, the morphology of the sigmoid (S or inverse-s) can be decided by which sign of the magnetic shear is on the magnetic neutral line. For instance, in our simulation, the sigmoids of inverse-s are generated in any case, because the negative shear region is embedded in the arcade center. On the other hand, the amplitude of the S-shape is relevant to the excess magnetic helicity included in the eigenfield component. Recently, Yamamoto et al. (2005) investigated the relationship between sigmoids and the magnetic helicity injection in seven different active regions and found that the morphology of sigmoids (S or inverse-s) was consistent with the magnetic helicity injected into the sigmoidal region. However, they found that there is no big difference in the injection of magnetic helicity between the sigmoidal regions and the nonsigmoidal regions. Our prediction accounts well for this result, because our model implies that unless there were any current sheets unstable to the tearing mode, even strong injection of magnetic helicity may not generate sigmoids. Moreover, their study also indicated that the magnetic helicity injection tends to change the sign on the foot region of sigmoidal structure (see Fig. 1 of their paper). The result supports our prediction in part. In our simulation results, several times longer than the tearing timescale must elapse for sigmoid to form in the relaxation phase. Recently, Nandy et al. (2003) observed that the distribution of the shear parameter in active regions tend to be flattened indeed. Their observation suggested that the timescale of this Taylor-type relaxation is about several days in the solar corona. If we assume that their observation is consistent with our simulation results, the growth time of the tearing mode instability should be as long as a day. In the solar corona of temperature T ¼ 10 6 K, and plasma density n ¼ m 3, the growth time of the most unstable tearing mode tearing ¼ ( A ) 0:5 is given by 4 ; 10 5 ( 3 /B) 0:5 s on the current layer of the width in meters and the magnetic field B in teslas, where A and indicate the Alfvén transit time and the resistive diffusion time, respectively. If tearing is about a day (10 4 s), and if B ¼ 3 ; 10 2 T, the width should be about 500 km. It might be observed as a steep discontinuity of the tangential magnetic field by the present vector magnetographs. In fact, the detailed observation of vector magnetic field suggests that the thin current layer existed in active regions (Solanki et al. 2003), although the spatial resolution is limited to about 1B5, which corresponds to 1000 km on the solar surface. It is very likely that such a thin current sheet becomes unstable to the tearing mode instability. However, for the Taylor-type relaxation to generate the sigmoidal structure, magnetic helicity contained in the domain has to be larger than the criterion, which is determined by the eigenvalue and the geometry of current sheets. Thus, in order to examine our model, it is required to reproduce the force-free state, which is consistent with both the observations of vector magnetic field and the injection of magnetic helicity. It will be an important subject in the future. 5. SUMMARY In this paper, we demonstrated that the formation of sigmoidal structure is able to be explained as the MHD energy relaxation. The resistive tearing mode instability growing on the shear

10 No. 2, 2005 FORMATION MECHANISM OF SIGMOIDS 1269 inversion layer can work as the driver of the energy relaxation process. It should be reminded that the energy relaxation through the tearing mode instability is common phenomena in the experimental plasmas, e.g., the reversed-field pinch experiments (Taylor 1986; Kusano & Sato 1990). Once the quasi-minimum energy state is generated, the sigmoidal structure can be sustained at least for the tearing timescale. However, if substantial amount of the magnetic fluxes are annihilated between the sigmoidal region and the environmental field through magnetic reconnection, the magnetic arcade collapses and the other reconnection may start above the sigmoidal region. Since the down-flow from the new reconnection strongly drives the original reconnection at the shear inversion layer, the two magnetic reconnections mutually excite each other, and lead to the explosive growth of the energy liberation. This mutual excitation process of multiple reconnections, which was first proposed by Kusano et al. (2004a), may naturally explain the causal relationship between sigmoids and eruptive events. It predicts that not the sigmoid itself but the overlaying arcade above the sigmoid should erupt in flare process (see Fig. 5), and that the field lines in the sigmoid could be untwisted by magnetic reconnection associated with the eruption, as seen in Figure 4. In this paper, we showed that sigmoids can be generated by the tearing mode instability rather than by the kink mode instability, and that the spontaneous formation of sigmoids is able to be explained in the framework of the reversed-shear flare model (Kusano et al. 2004a). Our model can account for the observational result that magnetic helicity contained in sigmoids might be insufficient to destabilize the kink mode instability ( Leamon et al. 2003; Yamamoto et al. 2005). However, our model is still based on the hypothesis that there is a thin current sheet, where the horizontal magnetic field vector changes the direction in a scale as short as about 500 km near a sigmoid footpoint, although the current sheet does not have been directly observed. Moreover, the limitations in the numerical model, such as the zero approximation and the artificial boundary conditions, should be improved to examine the applicability of the model onto more realistic situation. Furthermore, in order to judge the validity of this model, we should study the spatial correlation between the vector magnetogram and the sigmoid feature in more detail. The realistic datadriven simulations, in which vector magnetograms are used as the photospheric boundary condition, will be a powerful tool to understand the three-dimensional structure of the coronal magnetic field in future works. The author acknowledges the Japan Society for the Promotion of Science for the financial supports. 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