MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE. 1. Introduction

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE G. AULANIER 1, P. DÉMOULIN 1, B. SCHMIEDER 1, C. FANG 2 and Y. H. TANG 2 Observatoire de Paris, DASOP, F-92195 Meudon, France; Department of Astronomy, Nanjing University, Nanjing 210093, China (Received 23 April 1998; accepted 30 July 1998) Abstract. On 18 May, 1994, a subflare was observed in AR 7722 in X-rays by Yohkoh/SXT and in Hα at National Astronomical Observatory of Japan. The associated brightenings are due to smallscale magnetic energy release, triggered by parasitic fluxes emerging and moving at the edge of leading sunspots. Using the magnetohydrostatic equations derived by Low (1992), we model the magnetic field configuration by extrapolation of the Kitt Peak photospheric field, taking into account the effects of pressure and gravity. Hα flare kernels are shown to be located at computed separatrices associated with field lines which are tangent to the photosphere, namely bald patches (BPs). This is evidence that BPs can be involved in flares, and that current sheets can be dissipated in low levels of the solar atmosphere. The presence of dense plasma which is supported against gravity in the magnetic dips above BPs is correlated to dark elongated features observed in Hα. Mass flows in these flat fibrils are discussed in the context of energy release in the BP separatrices. The effect of the plasma on the computed magnetic configuration is shown to be of secondary importance with respect to the topology of the field. 1. Introduction Several observational studies at different wavelengths show that flares, and even less intense coronal phenomena, are due to interactions between coronal magnetic structures (see, e.g., Shimizu et al., 1994; Hanaoka, 1995; van Driel-Gesztelyi et al., 1996). Detailed studies of the magnetic topology show that the location of the energy release site is associated with the locations of the coronal magnetic separatrices (e.g., Bagalá et al., 1995; Démoulin, Hénoux, and Mandrini, 1992; Mandrini et al., 1991, 1993, 1995; Parnell, Priest, and Golub, 1994). However, the presence of null-points in the coronal magnetic configuration is not a necessity (Démoulin, Hénoux, and Mandrini, 1994). Moreover in these works the photospheric field was extrapolated to the corona by using a series of subphotospheric magnetic sources. These works were extended to coronal magnetic fields obtained with a classical extrapolation by Démoulin et al. (1997). This requires the definition of so-called quasi-separatrix layers (QSLs) which are the generalization of separatrices to magnetic configurations with a non-vanishing magnetic field strength. By analysing several flares and an X-ray bright point, Mandrini et al. (1996), Démoulin et al. (1997), Schmieder et al. (1997), and Gaizauskas et al. (1998) have shown that Hα flare ribbons are located on the intersection of QSLs with the chromosphere Solar Physics 183: 369 388, 1998. 1998 Kluwer Academic Publishers. Printed in the Netherlands.

370 G. AULANIER ET AL. and that they are connected by magnetic field lines where intense soft X-rays are emitted. Apart from the presence of magnetic null in the corona, separatrices can only be present in a magnetic volume when some field lines are tangential to the boundary. This is a particular interesting case for QSLs which, so far, have not been looked at systematically in magnetic configurations derived from observations. A theoretical study of the conditions for the appearance of field lines that present a dip which is tangent to the photosphere has been completed by Titov, Priest, and Démoulin (1993). They named the line which is formed by such tangent points as a bald patch (BP). Bungey, Titov, and Priest (1996) have studied their 3-D topology, and have proposed that they could be related to low altitude (photospheric or chromospheric) reconnection, in the context of flares. This kind of topology has also been proposed in the context of X-ray bright points for converging opposite polarities (see Parnell, Priest, and Titov, 1994). In this article we study the first example where observed brightenings are related to BP separatrices deduced from observed magnetograms. The events are detected in Hα (National Astronomical Observatory of Japan) as well as in X-rays (Yohkoh/SXT). They are of small intensity, in particular they are not detectable by the GOES satellite. They have the characteristics of X-ray bright points or/and subflares. In the following we use the terminology subflares, but the reader may keep in mind also the terminology Xray bright points since we believe that the physics of these phenomena is basically the same (see references above). As the studied subflares are linked to BPs, let us first summarize their main topological properties from a theoretical point of view. The continuous set of field lines starting at a BP forms a separatrix surface which separates three topological regions (see Bungey, Titov, and Priest, 1996). It is known that strong currents can be generated at BP separatrices (see Low and Wolfson 1988; Vekstein, Priest, and Amari, 1991; Aly and Amari, 1997), that may lead to ohmic dissipation or reconnection, hence heating. Billinghurst, Craig, and Sneyd (1993) have studied the conditions of current sheet formation at BP separatrices, in the case of a symmetric 2 1 D-configuration. They found that the amount of current accumulated 2 along the separatrix strongly depends on the applied horizontal footpoint motions. Contrary to Low and Wolfson (1988), they have found little evidence of vertical current sheet formation at the BP when photospheric shearing motions are present. However Aly and Amari (1997) showed that a vertical current sheet can still be present above the BP, with the appearance of a neutral X-point when photospheric converging (or diverging) motions are applied. So a current sheet can appear at least in particular cases where there are no volume currents. The physics of current accumulation and dissipation is expected to be significantly different at BP locations than in the corona, mainly due to the high density of the plasma at the photospheric/chromospheric level. One of the known effects is to reduce the validity of line-tying at the BP. This point has been studied by Karpen, Antiachos, and DeVore, (1990, 1991) and further discussed by Billinghurst, Craig, and Sneyd

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 371 (1993). This effect is expected to broaden the BP separatrice, forming a QSL. The possible formation of a current layer in such configurations is still under debate, though Démoulin et al. (1996) have given some positive arguments for current accumulation at QSLs. The aim of this paper is to bring a first observational evidence of the possible role of BPs in flares and X-ray bright points. In Section 2 a brief description of the observed subflare is given, and the main magnetic polarities involved are described. In Section 3 the extrapolation method in magnetohydrostatics developed by Low (1992) is described, and the effects of pressure and gravity are discussed. In Section 4 the global topology of the magnetic field is studied in the region, where separatrices from three BPs are correlated to Hα brightenings. In Section 5 the computed magnetic dips at low heights (mainly above BPs) and the over-densities found from the extrapolation are correlated to dark features observed in Hα. In Section 6 we present our conclusions and discuss the many questions which are still open. 2. Observations of the Subflare On May 1994, an active region, AR 7722, appeared on the solar disk with the Carrington coordinates of N8, L122. It consisted of two leading sunspots of negative polarity, surrounded by weak positive polarities, which were followed by a plage of positive polarity. On May 18 and 19, two subflares located in the southern direction of the major sunspot were observed in Hα and soft X-rays, right above an emerging flux region, as AR 7722 was almost at the center disk. 2.1. Hα OBSERVATIONS Hα filtergrams in the line center as well as in the wings (± 0.6 Å) were obtained at the National Astronomical Observatory of Japan (Sakurai et al., 1992), covering a field of view of 345 323. The first group of brightenings was observed at about 03:52 UT and disappeared after 04:50 UT on 18 May 1994 (see Figure 1(b)). This region brightened again from 04:58:09 UT to 05:12 UT on 19 May 1994. This group of brightenings mainly appeared as two thin flare kernels, in the South-West part of the AR sunspots. During the brightenings, a large dark absorbing feature observed in the vicinity of the kernels (see Figure 1(b)) was distorted and showed horizontal motions (see Figure 2 for its shape evolution). However according to the Hα filtergrams obtained in Mitaka, this feature was not detectable in the Hα wings, and consequently did not show large Doppler velocities.

372 G. AULANIER ET AL. Figure 1. Small-scale field of view of the subflare. (a) shows a Yohkoh/SXT image of the subflare taken on 18 May 1994 at 04:37 UT, (b) shows an Hα observation from Mitaka taken on May 18 at 04:36 UT, (c) is an Hβ chromospheric magnetogram taken at Huairou on 18 May 1994 at 04:23 UT, and (d) is a Kitt Peak magnetogram taken on 17 May at 15:47 UT. (a) (d) are coaligned, solar north is up, and the unit of the axes shown in (c) and (d) is in Mm. In (c) and (d), thin full (dashed) lines represent isocontours of the longitudinal component of the magnetic field of 10, 50, 200, and 600 G positive (negative) values. The main polarities involved in the event are named in (d). Bald patches (BPs) have been computed using α = 6.3 10 3 Mm 1, a = 1, and H = 2Mm(seetextfor details). BPs are shown by the thickest lines; the intersection of their respective separatrices with the photosphere is represented by thick lines. Arrows (labeled a1 a4) in (a) (d) show the regions of bright flare ribbons. A fairly good correspondence is found between some observed brightenings and computed separatrices. 2.2. SOFT X-RAYS Yohkoh/SXT (see Tsuneta et al., 1991) obtained full-disk and partial frame soft X-ray images of the active region. AR 7722 was overlaid by a classic group of large loops, showing however a group of intermittent, thin and extended emissions at the south-west, temporally correlated with Hα brightenings (see Figure 1(a)). This event did not disturb much the large-scale appearance of coronal loops. This fact has already been interpreted as a first proof that this brightening was a local event (Tang et al., 1998). The radiated energy has been evaluated to be 10 28 10 29 ergs. No substantial radio emission was detected. Yohkoh/HXT did not register any hard X-ray emission.

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 373 2.3. MAGNETOGRAMS Two kinds of magnetograms were available for the study of AR 7722: chromospheric (in the Hβ line) magnetograms, obtained at Huairou (Figure 1(c)), and a photospheric one from Kitt Peak (Figure 1(d)). They showed an emerging flux region at the south of the two leading sunspots. Note that in Figure 1 the two leading spots are in the north-west corner and partially out of the represented field of view. Allthough the magnetic data from Huairou were used by Tang et al. (1998) for the time-variation of the magnetic flux in this emerging region, we use the Kitt Peak magnetogram for extrapolations. This choice comes from several reasons: (i) the Huairou magnetogram does not cover the whole AR, leading to a flux imbalance, which is artificial; (ii) the seeing was bad during the observations, which main effect is to erase lower intensity and small-scale magnetic polarities. (iii) Hβ line observations are complex to reduce and to understand, as Hβ is not an optically thin line, so that the magnetogram does not represent a thin cut of the chromosphere. A first attempt to understand the event in the magnetic configuration computed from the Hβ magnetograms has failed because of the above reasons. As Tang et al. (1998) have shown that the brightening events had a local magnetic origin, some weak polarities play an important role in their appearance. We name on the Kitt Peak magnetogram (Figure 1(d)) the polarities which we need for the description of the event. Let N (respectively P) be given for negative (respectively positive) magnetic polarities. So, the main polarities involved are: N1: leading sunspot group, N2: weak parasitic polarity between P1 and P2, at the south of N1, N3: extended low magnetic field region at the east of P1, P1: polarity at the south-east of N1, P2: emerging polarity at the south of N1, P3: weak parasitic polarity at the north of P1, embedded in N1, We locate the brightenings with respect to the observed polarities, using coalignments between the Kitt Peak magnetogram, the Mitaka Hα and Yohkoh/SXT observations. Brightenings were observed in Hα (see Figure 1(b)) between P2 and N2 (arrow a1), and between P2 and N1 (arrow a2). A faint elongated brightening was present at the west of P2 (arrow a4). A bright dot also appeared close to P3 (arrow a3). Only the brightenings pointed by arrows a1 and a3 were observed in soft X-rays (see Figure 1(a)). 3. Extrapolation of the Photospheric Field In order to model the flare events, the magnetic configuration above the active region has to be found. A linear force-free field (lfff) extrapolation for AR 7722 has already been achieved in Tang et al. (1998). This previous study has shown that the magnetic configuration was nearly potential. Furthermore, it has been shown from

374 G. AULANIER ET AL. the energy budget of the event that magnetic reconnection was a good candidate to explain the observed emission. Moreover, the lfff extrapolation did not show any large-scale field lines that would have been likely to reconnect with one another. Consequently the observed sub-flare is probably related to a local smaller scale magnetic configuration. This point is justified in Tang et al. (1998). Hence, the reconnection event probably takes place in the lower atmosphere, where the lfff approximation is not valid any more. The gravity and pressure effects may play an important role in the local 3-D structure of the magnetic field in these layers. In this section, we describe the magnetohydrostatic equations, which include the effect of the plasma. 3.1. ANALYTICAL SOLUTIONS FOR THE LINEAR MAGNETOHYDROSTATICS The equations governing magnetohydrostatic equilibrium are given by 1 ( B) B p ρgu z = 0, (1) µ 0 B = 0. (2) Let s use a cartesian system of coordinates, where z refers to the height and (x,y) to planes parallel to the photosphere. Low (1991) has solved these equations by writing the current density (j = B/µ 0 ) of Equation (1) with Euler potentials. For any given function of the altitude f(z),andfixingα to a constant (as in the lfff), the linear magnetohydrostatic (lmhs) equilibrium condition can be re-written as B = αb + f(z) B z u z. (3) From Equations (1) and (3), the plasma pressure and density can be expressed as follows: p = p 0 (z) δp = p 0 (z) f(z) B2 z 2µ 0, (4) ρ = ρ 0 (z) δρ = 1 g dp 0 dz + 1 [ ] 1 df µ 0 g 2 dz B2 z + f(b )B z, (5) where p 0 (z) is a function which is independent of the magnetic field, and which only varies with z. This defines a background pressure (and density). Only the depletions δp and δρ depend on the magnetic configuration. We choose f(z)to be an exponential decreasing with height, as Low (1992) did in order to compute the general properties of lmhs solutions: f(z)= a exp( z/h), (6)

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 375 where a and H are called the plasma parameters. Their physical meaning is explained in Section 3.3. The function f(z) roughly represents the strength of the interaction between the plasma effects (i.e., pressure and gravity) and the magnetic field. 3.2. LIMITS ON α AND a In the absence of plasma, the current density j is proportional to the magnetic field B: j(lfff) = αb. (7) The linearity of the equations permits us to decompose B in a series of harmonics. In a 3-D computational box of which the horizontal length is L, α has a maximum value of α(max) = 2π/L, if we make the reasonable assumption that the magnetic field strength decreases to zero at infinitely large heights (which leads to a finite magnetic energy). This property of lfff has been discussed by Aulanier and Démoulin (1998). The value for α(max) is still valid in the case of lmhs (Low, 1992). Moreover, it is important to know the physical limit on a which can be used. Low (1992) has shown that the horizontal Fourier transform B z of the vertical magnetic field component B z satisfies d 2 B [ z dz + α 2 + K 2( ( a exp z ) 1)] B 2 z = 0, (8) H where K = kx 2 + k2 y,andk x and k y being the wave numbers with respect to x and y. The solution of Equation (8) is in the form of Bessel functions, that are used in the extrapolation method: ( B z = B K J s [q exp z )], (9) H where q = 2KH a and s = 2H K 2 α 2 and where B K is the amplitude of the term associated to the wavenumber K. The argument of the Bessel function (Equation (9)) increases with the strength a of the plasma effects. Then there is a finite value for a, namely a(max), where B z changes its sign at z = 0. For a> a(max) the magnetic field amplitude oscillates in the lower part of the atmosphere, and this behaviour may not be physical. This can be seen in Equation (8) where the strongest limit on a is found for the largest wavenumbers K (so the small magnetic features), physically this is because the magnetic field is localized in a small vertical extension where gravity and pressure have their greatest values. These considerations imply that a(max) is found for z = 0 and at the limit of large K, where the solution of Equation (8) can be simplified as ) B z = B K exp ( α2 + K 2 (1 a) z. (10)

376 G. AULANIER ET AL. In order to get a decreasing B z with height, the term in the parenthesis has to be kept positive. For very large values of K, the shear α can be neglected so a(max) = 1. Low (1992) has used higher values (up to 4) because only large scale harmonics were considered, for which the constraint on a is less severe. When a too large value for a is used, with respect to the wave numbers that are taken into account, the Bessel functions oscillate with z, giving oscillations of the magnetic field with height. In order to avoid such effects, and as we want to investigate the maximum effects of the pressure and gravity on the magnetic field configurations, we choose to use a = 1 for the following extrapolations. 3.3. THE PLASMA EFFECTS ON THE MAGNETIC FIELD The influence of the plasma on the magnetic field is expressed by f(z) in Equation (6). Its effect is to create currents oriented in the (x,y) planes, parallel to the photosphere. In Equation (3), a is the measure of the intensity of these currents at z = 0, where it is also the ratio between the pressure depletion δp and the magnetic pressure B 2 z /2µ 0 (Equation (4)). The vertical extension of the plasma effects is given by the scale-height H. Low (1992) explained in detail how H can be in some cases equal or close to an isothermal pressure scale-height. A dense chromosphere where there is a significant interaction between the plasma and the magnetic field can then be modeled, using a value of H = 2 Mm (typical height of the chromosphere). This value is kept constant over this whole study. It is noteworthy that the horizontal currents induced by the plasma have a component parallel to the magnetic field when it is not vertical, so that the current density j parallel to the field is no longer equal to αb. As a consequence there is a slight non linearity in the relation between the current distribution and the magnetic field brought in by the plasma, especially where the plasma effects are the strongest (see Low, 1992). Though the lmhs method still differs from the non-linear magnetohydrostatics (nlmhs) where α = constant, which is why we keep the notation linear magnetohydrostatics (lmhs). From Equation (4), we see that the pressure depletion is larger if the magnetic field is more vertical. This is consistent with measurements of the pressure above strong fields, such as sunspots. A secondary effect is to locally confine the magnetic field in a more vertical structure, because B has to decrease in height less, in order to equilibrate the pressure depletion. The properties of the density depletion are more complex, due to its expression (see Equation (5)). We just point out here that the last term is related to the curvature of the magnetic field, and that the density is higher in the presence of magnetic dips than in the case of arcades. This is consistent with the support of dense plasma in magnetic dips, such as filaments (Aulanier and Démoulin, 1998) or dark fibrils (Tang et al., 1998). More details about the behavior of the pressure and density can be found in Low (1992).

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 377 Figure 2. Larger-scale observations of AR 7722 in Hα line center, showing the main dark fibril evolution, taken at Mitaka on 18 May 1994 at (a) 04:08 UT, (b) 04:36 UT (see correspondence with Figure 1(b)) and (c) 04:46 UT. 3.4. MAGNETIC FIELD EXTRAPOLATION IN lmhs The solutions of Equation (3) can be expressed as a series of harmonics of the three components of the magnetic field, as in the lfff case. As a consequence, any given magnetic field distribution in the lmhs case can also be expressed with the fast Fourier transformation (FFT, see Low, 1992). For the study of any particular active region, the boundary conditions B z (x,y,z = 0) (or equivalently the longitudinal component) need to be imposed from a magnetogram. The lmhs harmonics derived from the FFT impose the two other components of B at z = 0. B is then calculated at any z. More details on the extrapolation method and on the transformation of coordinates can be found in Démoulin et al. (1997). 4. Magnetic Field Topology of the Bald Patch Flare In this section, we show that in AR 7722, the locations of the Hα flare brightenings are correlated to the intersection of BP separatrices with the chromosphere. Consequently we deduce that these brightenings can be interpreted as Hα flare ribbons. A large number of low lying (z 4 Mm) dipped field lines has been found in AR 7722 from lfff extrapolations (see Tang et al., 1998). A careful look at their Figure 3 shows that some of the computed dipped field lines are located just in the vicinity of the brightenings observed in Hα. Consequently, energy release is not only expected to take place at low heights, but also in the vicinity of very low magnetic dips. In this study we show more precisely that the computed dips above BPs in the vicinity of the observed sites of energy release are correlated to Hα dark fibrils, where observed flows are induced by this energy release. We show that successive brightenings observed at different BP separatrices are triggered by a main flux emergence.

378 G. AULANIER ET AL. Figure 3. 3-D view of the field lines forming the separatrice of BP1. A multiplicative factor of 8 for vertical extension of the field lines is used for a better viewing of the configuration. On the base plane, thin full (dashed) lines represent isocontours of the longitudinal component of the magnetic field of 10, 50, 200, and 600 G positive (negative) values, BP1 is shown by the thickest line and the intersection of its separatrix curve with the photosphere is represented by the thick lines. The separatrice is formed by two asymmetric lobes. 4.1. ACTIVE BPS IN THE VICINITY TO FLUX EMERGENCE We extrapolate the magnetic field, using the same value of the shear that was found for the overlaying coronal loops in Tang et al. (1998): α 6.3 10 3 Mm 1. This very low shear value indicates that the global magnetic field is almost potential, so that it does not contain a lot of free energy. We remind the reader that the plasma parameters are chosen as a = a(max) = 1andH = 2Mm. The main criterion for a bald patch (or BP) at the point x,y,z = 0where B z = 0 is given by (Titov, Priest, and Démoulin, 1993) B z B x x + B B z y y > 0. (11) After extrapolating the magnetic field in lmhs, BP locations are naturally found on the whole magnetogram. The computation of BPs and their associated separatrices, in the region of the strongest emission observed in soft X-rays and Hα (see Figure 1), reveals the presence of a BP located at the west of N2. It lies on the local magnetic inversion

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 379 line between N2 and P2, and we will call it BP1 (see Figure 1(d)). One of the intersection of its associated separatrice with the z = 0 plane has a half-circle shape, which partially encloses N2. The other intersection has an elongated shape at the North of P2 (this portion is the brightest in soft X-rays and in Hα). Both intersections with the photosphere of the separatrix curve associated to BP1 as well as BP1 itself, have bright observational counterparts (in Figure 1, arrow a1). A 3-D view of the BP1 topology is shown on Figure 3. The BP1 separatrix curve shows two asymmetric lobes, the largest one joining the BP1 to the long northern ribbon which crosses N1. Yohkoh/SXT shows flare activity in the region of the largest lobe of BP1 (see saturation on Figure 1(a)), which implies that most of the energy was released there. A weaker X-ray emission is observed above the other separatrice lobe. This is because of its low altitude above the photosphere (see Table I). Hα observations also show that the flare ribbon is the brightest at the larger lobe location (see Figure 1(b)). This result suggests that as P2 emerges and moves, its magnetic flux impacts on very low lying field lines coming from the side of N2. Consequently, a current sheet is formed along the BP1 separatrix, and then dissipated. It can be said that by this process, the BP is activated. Thus, this event can then be considered as a bald patch flare. Billinghurst, Craig, and Sneyd (1993) have performed 2.5-D MHD simulations of BPs, and they have shown that the distribution of the currents in the current sheet in a BP separatrix greatly depends on the imposed photospheric velocity fields. But as their configurations were symmetric and invariant by translation, it is difficult to apply their results in the present observational context. From the observations, it can only be stated that energy dissipation (manifested by soft X-ray and Hα emission) is present along the whole separatrix with a maximum emissivity at the footpoint of the largest lobe. However we are unable to say if the energy release occurred mainly in a current sheet above the BP (and then is transported by thermal conduction and/or accelerated particles along field lines), or if a significant part of the energy release occurs due to the dissipation of a current sheet located along the whole BP separatrix. 4.2. MULTIPLE BP ACTIVATIONS It is noteworthy that from the lmhs extrapolation, three BP regions can be identified in the vicinity of the brightest Hα ribbon (pointed by arrow a1 in Figure 1). BP1 has been described in Section 4.1, BP2 is found between N3 and P1, and BP3 in the East of the small P3 polarity (see Figure 1(d)). Typical field lines of the separatrix curves for the computed BPs are drawn in 3-D in Figure 4(a) for BP1 and BP3 and in Figure 4(b) for BP2. For a better understanding of the topology, the three BPs and associated field lines are represented in Figure 4(c) from the observer s view. All the BP separatrices are formed by asymmetric lobes: a large one and a smaller one. The height of the apex of the lobes is low (see Table I).

380 G. AULANIER ET AL. Figure 4. Same as Figure 3, except that a larger scale is represented. A multiplicative factor of 8 for vertical extension of the field lines is used as in Figure 3. (a) The field lines forming the BP1 and BP3 separatrices, as well as their intersections with the photosphere. (b) The same for BP2. (c) Top view showing the three sets of field lines located on the BP separatrices. (d) The intersections of separatrices with the photosphere for a linear force-free field (lfff) extrapolation (i.e., a = 0). It has been shown in Section 4.1 that the footpoints of the BP1 separatrix were correlated with bright Hα ribbons. The separatrices associated to BP2 and BP3 also show enhanced Hα emission, but only at some of their footpoint locations (see Figure 1(b) arrows a2, a3, a4 and Table I). As a consequence the three BPs were activated. It can be clearly seen from Table I that the location of energy release is completely different from one BP to another. Following Billinghurst, Craig, and Sneyd (1993), this would imply that the kinds of horizontal motions of the involved magnetic polarities are different for each of those. Unfortunately no observed data

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 381 TABLE I Relative brightness of the footpoints, and maximum height of the BP separatrices, reported from Figure 1 and Figure 4 Bald patch Topological Height Hα Soft X-rays Arrow feature BP1 small lobe 0.3 Mm faint bright a1 bald patch faint bright a1 large lobe 4 Mm bright saturated a1 BP2 small lobe 1.2 Mm bald patch large lobe 11 Mm bright a2 large lobe 11 Mm faint a4 BP3 small lobe 0.1 Mm bright bright a3 bald patch bright bright a3 large lobe 2 Mm are available for a significant interpretation and no theoretical work has been done on 3-D configurations with BPs. 4.3. ARE THE THREE BPS TOPOLOGICALLY LINKED? The relation between the activation of the three BPs can be discussed, as their associated brightness occurred at the same time. Bungey, Titov, and Priest (1996) have proven the possible existence of a separator joining two BPs. Under this condition, they have proven that both BP separatrices should cut one another. As a consequence of the activation of one BP, the second one will be activated too. From our lmhs extrapolations, no topological link has been found between the three BPs, even varying the plasma parameters a and H in reasonable ranges. But it has been pointed out in Tang et al. (1998) that the emerging flux was probably more sheared than the surrounding magnetic field that was found to be almost potential. Varying the shear α in our lmhs extrapolations still does not provide the required separators. Maybe a nlmhs extrapolation, taking into account the fact that the shear is large near the emerging fluxes, and that it is almost potential everywhere else, would bring such separators. But such non-linear extrapolations cannot be performed as the required transverse field measurements are not available. From Figure 1(d), one can still realize that some of the footpoints of the BP separatrices almost touch each other at a distance of a few Mm. This can be found at the north of P2 (near the arrow a2), at the west of P3 (near arrow a3) and at the north of N3. From Figure 4(c) it is easy to identify to which BP each intersection

382 G. AULANIER ET AL. of the separatrix with the photosphere belongs to. BP2 and BP3 are then likely to be linked as their associated separatrices are very close to one another (less than 3 Mm, see Figure 1(d)). This may also be the case for BP1 and BP2 which show a close approach of their separatrices near arrow a2, but this is less conclusive since on the other side the BP1 separatrice extends only a little above the polarity P1, towards BP2. The absence of intersection between separatrices may be due to the underestimate of the magnetic flux in weak positive polarities (see Section 4.5). Also, nlmhs extrapolations may also slightly deform this already-close separatrices, forcing them to cross each other. Do the observations give some insight about the existence of separators in this region? From Figures 1(a) and 1(b) one can see that the brightness pointed by arrow a1 is more elongated towards P1, implying that the small lobe of the BP1 separatrix is probably more extended. Moreover, the strong brightness at the north of P2, which is the signature of a strong energy release, might be due to the presence of a separator linking BP1 and BP2. Finally at the present time, it cannot be proven that some separators are present in this regions, but some topological and observational facts are likely to give a positive answer to this question. 4.4. lmhs VERSUS LINEAR lfff EXTRAPOLATION This paper investigates an extrapolation method for analytical lmhs solutions derived by Low (1992). The main result concerning AR 7722 was the presence of BPs at flaring locations. It is a well known fact that dense plasma at the top of an arcade can bend the field lines, forming magnetic dips. This has been mainly studied in the context of prominences (e.g., Fiedler and Hood, 1993). Furthermore, as it is the first time such a bald patch flare is computed from observations, one might wonder whether the computed topology is due to the effects of pressure and gravity or are the dips (and the BPs) present in the absence of plasma effects. In order to answer that question, the magnetic field is extrapolated in the lfff approximation (i.e., a = 0) for the same shear α 6.3 10 3 Mm 1. It is noteworthy that BP1, BP2 and BP3 are all recovered at the same locations than in the lmhs case, even if the separatrix shapes are slightly different (compare Figures 4(c) and 4(d). This is closely related to some prominence models, where the dips in some field lines are expected to be present in the magnetic configurations without any help from pressure and gravity (e.g., Priest, 1989; Démoulin and Priest, 1989; Aulanier and Démoulin, 1998). This comparison has shown that the BPs and the dips above them are still present in the absence of pressure and gravity, which effects are only to deform slightly the dips. As a consequence the BP topology is reliable, as it is more stable than if it would have been due to the bending of the magnetic field by the plasma. It can finally be said that lfff extrapolations still give satisfactory results, and that

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 383 the effects of the plasma in lmhs extrapolations are of secondary order with respect to the magnetic field topology. 4.5. DISCUSSION ON THE RELIABILITY OF PARASITIC POLARITIES The presence of BPs in AR 7722 is due to the small parasitic polarities N2, N3 and P3, in the photospheric Kitt Peak magnetogram (see Figure 1d), though they are not observed in the Huairou magnetogram (see Figure 1c). This is not surprising, as the magnetic fields from Huairou are measured in the Hβ line, in the chromosphere, where small scale magnetic fields expand, decrease in amplitude and eventually become erased by stronger fields. Moreover, chromospheric magnetograms are less sharp than photospheric ones. These weak photospheric polarities have an average width of a few Mm, for a measured mean field intensity of 5 to 10 G. Such low flux values imply that they are in fact lost in the noise of the magnetogram, so that they are not significant for the observed events. But the presence of dark material in Hα above the computed BPs, as their associated flare ribbons (see Figure 1 and 2), strongly support their existence. As another comparison between observations and magnetic extrapolations we have computed the low lying magnetic dips. The bottom of the associated field lines is drawn on a depth of 0.3 Mm, which is the typical pressure scale-height of the chromosphere on which the plasma is likely to be dense enough to be observed in absorption in Hα (see Aulanier and Démoulin, 1998; and Tang et al., 1998, for the description of the method). The computed dips (see Figure 5(c)) do not extend far enough in P1 to explain completely the shape of the dark Hα feature, which was observed to be extended to the middle of P1 (see Figure1(b)). A possible explanation is the underestimation of the flux of N2, because of averaging with positive polarities on the spatial resolution size of the Kitt Peak magnetogram. When a parasitic polarity is present in a surrounding field of opposite polarity, the seeing has a dramatic effect: there is not only a usual spreading (like with intensity brightness) but a cancelling of the parasitic flux with the neighbouring flux in the resolution region. This effect can delay the appearance of one of the polarities of an emerging bipole more than one day (a particular example can be found in Mandrini et al., 1993). Consequently the parasitic polarities in AR 7722 (N2, N3, and P3) are certainly much stronger. We think that this underestimate of the parasitic polarities is the major limitation of present study since its effect on the BP (and associated separatrix) extension can easily be much larger than the effect of the parameters α, a and H.Itmaybe responsible for the absence of intersections between the computed separatrices of the three BPs.

384 G. AULANIER ET AL. Figure 5. Same field of view as in Figure 1. In (a) and (b), full (dashed) lines show isocontours of the plasma density variations of δρ = 5 10 11,5 10 12,5 10 13,5 10 14 cm 3 positive (negative) values in the chromosphere at the different heights of (a) z = 1Mmand(b)z = 2Mm.TheBPs locations are well correlated to overlaying over-densities. (c) shows the dipped portion of field lines computed between z = 0andz = 4 Mm, from the linear magnetohydrostatics lmhs; their vertical extension is 0.3 Mm. The dips are well correlated to excess densities ((a) and (b)) and Hα dark fibrils (Figure 1(b)). 5. The Hα Dark Fibrils 5.1. BPS DENSITY DEPLETIONS As magnetic dips are present above BP1 in the vicinity of N2 (see Section 4.5 and Figure 5c), dense plasma can be supported there against gravity (from Equation (5)). This may explain the dark elongated Hα feature observed before the subflare. It can be considered as a dark Hα fibril, supported in magnetic dips (see Tang et al., 1998).

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 385 Some of the basic properties that have been discussed in Low (1992) and in Section 3.3 are recovered in the extrapolation of AR 7222. An overdensity is present at low heights in regions of magnetic dips, and an under-density is found above regions of strong vertical field, as expected from Equation (5). All of the density fine structures are smoothed with height, as the plasma effects decrease, and as high order harmonics of the magnetic field become dominated by lower order ones (see Figures 5(a) and 5(b)). The strength δρ and the vertical, as well as horizontal, extension of the excessdensities mainly depend on the plasma parameters a and H. Even though H can be reasonably fixed for a typical chromosphere, the choice for a is unfortunately free, as no measurements of the transverse fields or plasma density are available. Although the variations with a for a given H are not so large for such low values of α, the tendency to create over-densities is still enhanced as a becomes larger. One still has to take with caution the physical values for the maximum excess-densities, that can vary by 10 12 δρ 5 10 12 cm 3 in the range of 0.5 a 1atthe height z = 2Mm. 5.2. ASSOCIATED DARK FIBRIL MOTIONS The darkest fibril which lies above the BP1 region close to N2 shows transverse mass motion during the subflares. A first look at the observations suggests that this moving absorbing feature in Hα could be a surge, but no significant upflows/downflows were observed: according to the Hα filtergrams, this feature was not detectable in the wings (Hα ± 0.6Å). This means that the Doppler shifts do not reach 40 km s 1, which is the standard velocity of surges. However the computed dipped field lines above BP1 indicate that this structure can be interpreted as a horizontal fibril (see Figure 5(c)). Material was observed moving along the structure, alternating between both directions (see Figure 2). From the magnetic topology in this region, and from physical considerations, a preliminary explanation for these flows can be given here. The energy released at the separatrix locations increases the plasma pressure and the previous dark material can be forced to move away (to the east, so to P1). When the sub-flare stops, some of the dark material comes back to its original place, because of the cooling which creates a pressure depletion. Moreover, the plasma is observed to go further to the north of P2, following field lines close to the separatrix. Then the kinetic energy gained was probably enough to bring part of the material on the other side of the dips, to the west. This is consistent with the computed field lines above the BP at N2, whose dip morphology shows an asymmetry (the western field lines are longer than the southern one).

386 G. AULANIER ET AL. 6. Conclusion In this paper we have investigated the magnetic topology of a subflare in the active region AR 7722, that was observed on 18 May 1994, at the National Astronomical Observatory of Japan at Mitaka in Hα as well as with Yohkoh/SXT in soft X-rays. We used the linear magnetohydrostatic extrapolation (lmhs) method developed by Low (1992); this method introduces the plasma effects (pressure and gravity) on the magnetic field. The boundary conditions for the magnetic field have been given by a photospheric magnetogram from Kitt Peak. The confrontation of the computed magnetic configuration with the observations brings new lights on the origin of the subflares and of the chromospheric fibrils. The extrapolation has revealed that three bald patches (BPs) were involved in the subflares. Some parts of their associated separatrix were shown to be correlated to soft X-ray brightenings while the separatrix footpoints were well correlated with the Hα brightenings. These have then naturally been interpreted as Hα flare ribbons. As a consequence, the energy release occurs at BP separatrices, and, this event can be considered as a bald-patch flare. The relation between the activation of the three BPs has been discussed. Bungey, Titov, and Priest (1996) have proven that in some cases, a separator can join two BPs, but none has been found in our case. On the other hand some parts of the BP separatrices have been found to be close (less than 3 Mm), so that if the observed polarities were slightly modified, they could cross each other, forming a separator. The validity of the measurement of the observed magnetic parasitic polarities involved in the BPs have been discussed, and it revealed that their flux was probably higher than what was measured. It implies that the presence of at least one separator is highly probable. The presence of BPs implies the existence of field lines whose curvature is upwards (i.e., dips), just above them. The computation of the density variation from the lmhs equations have revealed the presence of over-densities in the regions of dipped field lines, as expected. Furthermore the orientation and the location of the computed dipped field lines is in agreement with the observed dark elongated features in Hα. This confirms what has been proposed in Tang et al. (1998) and Aulanier and Démoulin (1998), where some chromospheric dark fibrils have been successfully related to dense plasma trapped in magnetic dips. These quasi-horizontal fibrils are supported against gravity in the same way as filaments. Most studies of ARs with respect to magnetic field extrapolations deal with the linear force-free (lfff) method, however in this paper we have investigated the effects of the plasma via a lmhs method. We have shown in this case that lmhs extrapolations do not differ much from lfff ones (i.e., a = 0) with respect to magnetic field topology, location of separatrices and presence of dipped field lines. As a consequence the lfff assumption in the solar atmosphere is fairly well justified. We point out that dipped field lines are still present without any plasma effects, so that they are not created by gas pressure and gravity, but by an increase of the magnetic pressure with height.

MAGNETOHYDROSTATIC MODEL OF A BALD-PATCH FLARE 387 The present results on AR 7722 are expected to have a much broder implications for the following reasons: the BPs, the associated separatrices and dips are linked to the presence of small parasitic polarities at the border of sunspots. Such small polarities are often observed in the vicinity of sunspots where soft X-rays, UV and Hα brightenings are present (e.g., Golub, Zirin, and Wang, 1994; Shimizu, 1994; Nitta et al., 1998) just like in AR 7722. One classical interpretation of these parasitic polarities is the emergence of U-loops through the sea-serpent process (Harvey and Harvey, 1973; Parker, 1984; Spruit, Title, and van Ballegooijen, 1987). The magnetic configuration that we have deduced from the magnetograms is fully compatible with such a scenario. We believe that the case of AR 7722 is indeed often present in active regions; we then propose that many of these bright points observed in soft X-rays, or blinkers in UV (see Harrison et al., 1997) can be interpreted as bald-patch flares as in AR 7722. Acknowledgements The authors thank T. Amari and V. S. Titov for fruitful discussions. We would like to thank the National Astronomical Observatory of Japan for kindly providing the chromospheric magnetic field and soft X-ray data. We also thank K. Harvey for providing the photospheric magnetogram. C.F. and Y.H.T. would like to express their sincere gratitude to CNRS of France for supporting their stay at Paris- Meudon Observatory. This work was partly supported by NSFC under a Major Project 19791090 and by a key project from the National Science and Technology Committee of China. References Aly, J. J. and Amari, T.: 1997, Astron. Astrophys. 319, 699. Aulanier, G. and Démoulin, P.: 1998, Astron. Astrophys. 329, 1125. Bagalá, L. G., Mandrini, C. H., Démoulin, P., Rovira, M. G., and Hénoux, J. C.: 1995, Solar Phys. 161, 103. Billinghurst, M. N., Craig, I. J. D., and Sneyd, A. D.: 1993, Astron. Astrophys. 279, 589. Bungey, T. N., Titov, V. S., and Priest, E. R.: 1996 Astron. Astrophys. 308, 223. Démoulin, P. and Priest, E. R.: 1989, Astron. Astrophys. 214, 360. Démoulin, P., Hénoux, J. C., and Mandrini, C. H.: 1992, Solar Phys. 139, 105. Démoulin, P., Hénoux, J. C., and Mandrini, C. H.: 1994, Astron. Astrophys. 285, 1023. Démoulin, P., Hénoux, J. C., Priest E. R., and Mandrini, C. H.: 1996, Astron. Astrophys. 308, 643. Démoulin, P., Bagalá, L. G., Mandrini, C. H., Hénoux, J. C., and Rovira, M. G.: 1997, Astron. Astrophys. 325, 305. Fiedler, R. A. S. and Hood, A. W.: 1993, Solar Phys. 146, 297. Gaizauskas, V., Mandrini, C. H., Démoulin, P., Luoni, M. L., and Rovira, M. G.: 1998, Astron. Astrophys. 332, 353. Golub, L., Zirin, H., and Wang, H.: 1994, Solar Phys. 153, 179. Hanaoka, Y.: 1995, Solar Phys. 165, 275.

388 G. AULANIER ET AL. Harrison, R. A. et al.: 1997, Solar Phys. 175, 467. Harvey, K. and Harvey, J.: 1973, Solar Phys. 28, 61. Karpen, J. T., Antiochos, S. K., and DeVore, C. R.: 1990, Astrophys. J. 356, L67. Karpen, J. T., Antiochos, S. K., and DeVore, C. R.: 1991, Astrophys. J. 382, 327. Low, B. C.: 1991, Astrophys. J. 370, 427. Low, B. C.: 1992, Astrophys. J. 399, 300. Low, B. C.: 1996, Solar Phys. 167, 217. Low, B. C. and Wolfson, R.: 1988, Astrophys. J. 324, 574. Mandrini, C. H., Démoulin, P., Hénoux, J. C., and Machado, M. E.: 1991, Astron. Astrophys. 250, 541. Mandrini, C. H., Rovira, M. G., Démoulin, P., Hénoux, J. C., Machado, M. E., and Wilkinson, L. K.: 1993, Astron. Astrophys. 272, 609. Mandrini, C. H., Démoulin, P., Rovira, M. G., de la Beaujardière, J. F., and Hénoux, J. C.: 1995, Astron. Astrophys. 303, 927. Mandrini, C. H., Démoulin, P., van Driel-Gesztelyi, L., Schmieder, B., Cauzzi, G., and Hofmann, A.: 1996, Solar Phys. 168, 115. Nitta, N., van Driel-Gesztelyi, L., Leka, K. D., Sakurai, T., and Ichimoto, T.: 1998, Solar Phys., submitted. Parker, E. N.: 1984, Astrophys. J. 283, 343. Parnell, C. E., Priest, E. R., and Golub, L.: 1994a, Solar Phys. 151, 57. Parnell, C. E., Priest, E. R., and Titov, V. S.: 1994b, Solar Phys. 153, 217. Priest, E. R.: 1989, Dynamics and Structure of Quiescent Solar Prominences, Kluwer Academic Publishers, Dordrecht, Holland. Sakurai, T., Ichimoto, E., Hiei, E., Irie, M., Kumagai, K., Miyashita, M., Nishino, Y., Yamaguchi, K., Fang, G., Kambry, Zhao, Z., and Shinoda, K.: 1992, Publ. Astron. Soc. Japan 44, L7. Schmieder, B., Aulanier, G., Démoulin, P., van Driel-Gesztelyi, L., Roudier, T., Nitta, N., and Cauzzi, G.: 1997 Astron. Astrophys. 325, 1213. Shimizu, T.: 1994, in T. Sakurai, T. Hirayama, and G. Ai (eds.), Proc. of the 2nd Japan-China Seminar on Solar Phys., Nat. Astron. Obs. of Japan, p. 193. Shimizu, T., Tsuneta, S., Acton, L.W., Lemen, J.R., Ogawara, Y., and Uchida, Y.: 1994, Astrophys. J. 422, 906. Spruit, H. C., Title, A. M., and van Ballegooijen, A. A.: 1987, Solar Phys. 110, 115. Tang, Y. H., Li, Y. N., Schmieder, B., Aulanier, G., Démoulin, P., Sakurai, T., and Fang, C.: 1998, Astrophys. J. submitted. Titov, V. S., Priest, E. R., and Démoulin, P.: 1993, Astron. Astrophys. 276, 564. Tsuneta, S., Acton, L., Bruner, M., Lemen, J., Brown, W., Caravalho, R., Freeland, S., Jurcevich, B., Morrison, M., Ogawara, Y., Hirayama, T., and Owens, J.: 1991, Solar Phys. 136, 37. van Driel-Gesztelyi, L., Schmieder, B., Cauzzi, G., Mein, N., Hofmann, A., Nitta, N., Kurokawa, H., Mein, P., and Staiger, J.: 1996, Solar Phys. 163, 145. Vekstein, G. E., Priest, E. R., and Amari, T.: 1991, Astron. Astrophys. 243, 492.