Interactions between nested sunspots

Astron. Astrophys. 332, 353–366 (1998)

ASTRONOMY AND ASTROPHYSICS

Interactions between nested sunspots II. A confined X1 flare in a delta-type sunspot V. Gaizauskas1 , C.H. Mandrini2 , P. D´emoulin3 , M.L. Luoni2 , and M.G. Rovira2 1

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Herzberg Institute of Astrophysics, National Research Council of Canada, 100 Sussex Drive, Ottawa K1A 0R6, Canada ([email protected]) Instituto de Astronom´ıa y F´ısica del Espacio, IAFE-CONICET, CC.67, Suc.28, 1428 Buenos Aires, Argentina? ([email protected]; [email protected]) Observatoire de Paris, DASOP, URA 2080 (CNRS), F-92195 Meudon Cedex, France ([email protected])

Received 22 August 1997 / Accepted 11 November 1997

Abstract. We study the flaring activity in a nest of sunspots in which two bipolar regions emerge inside a third one. These bipolar regions belong to a large complex of activity (McMath 15314) formed by five bipoles on its May 1978 rotation. The usual spreading action during the growth of the bipoles leads to the formation of a δ-configuration: the preceding and following spots of the two interior regions overlap (p-f collision) into a single penumbra. While δ-configurations created in this way normally favor strong flaring activity, only very small flares occur during 5 days. Only when the following umbra in the δ-spot breaks into pieces, accompanied by rapid photospheric motions, do intense flares occur. The largest and best observed one in this sequence, a class 1B/X1 flare on 28 May 1978, is remarkable for the absence of ejecta and for the concentration of its emission in three widely spaced sites, a pattern which holds in general over two days for lesser flares. We take this pattern as evidence that the flare is confined to the low corona. We first compute the coronal magnetic field using subphotospheric sources to model the observed magnetic data and derive the location of separatrices. In this case the magnetic field topology is defined by the link between these discrete sources. The relevant generalization of separatrices in any kind of magnetic configuration are ‘quasi-separatrix layers’ (QSLs). We calculate them using the previous model, but also for a model obtained with a more classical extrapolation technique based on the fast Fourier transform method. We show, with both approaches, that the plage brightenings during the quiescent phase of the region and the flare kernels are located at the intersection of separatrices and QSLs with the photosphere. Moreover, they are magnetically linked. Bright and dark ‘post’-flare loops which form in the maximum and gradual phases of the 1B/X1 flare also highlight the location of the separatrices and the QSLs. This confirms previous studies on the importance of the magnetic field topology for Send offprint requests to: V. Gaizauskas ? C.H.M. and M.G.R. are Members of the Carrera del Investigador Cient´ıfico, CONICET

flares and, with this study, we further constrain the underlying physical mechanism. We draw some conclusions about the role of magnetic reconnection in the solar corona; depending on the photospheric conditions that we identify, reconnection can lead to steady heating or flaring. Key words: magnetic fields – MHD – Sun: corona – Sun: flares – Sun: magnetic fields

1. Introduction It is a common tendency of active regions (ARs) to emerge near, or even within, existing ones (see e.g. Gaizauskas et al. 1983). Such new flux emergence frequently leads to the formation of complicated magnetic configurations (complexes of activity); they are known to be associated with enhanced activity since the new flux tends to reconnect with pre-existing fields. Several observational studies in different wavelengths show that flares, and even less intense coronal phenomena, are due to interactions between coronal magnetic structures (see eg. Machado et al. 1988; Shimizu et al. 1994; Hanaoka 1995; van Driel-Gesztelyi et al. 1996); while recent topological studies (see Mandrini et al. 1996; D´emoulin et al. 1997 and references therein) show that this interaction takes place via 3 dimensional (3-D) magnetic reconnection at places where the field-line connectivity changes rapidly. The energy needed to power flares, and also to heat the solar corona, is thought to come from the coronal magnetic field and several flare models have been proposed. For example, it has been argued that current sheets can store enough magnetic energy to power a flare (Somov 1986, 1992 and references therein). At some point in the evolution, this current sheet becomes unstable and turbulence develops increasing the plasma resistivity.

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Fig. 1. Photospheric magnetogram (NSO-KP) obtained on May 27 1978. The 8’ x 8’ box in the upper panel is seen close up below with the different bipoles of the complex outlined with dashed curves. Letters designate each bipole in order of emergence, suffixes -p and -f are attached to the letters to signify preceding and following polarity, respectively. This and all subsequent figures are oriented with North at the top, West to the right.

Then, the stored energy is rapidly released as a flare (e.g. Heyvaerts et al. 1977). It has instead been argued that reconnection tends to occur at a rate imposed by the evolution of the magnetic field (e.g. Priest & Forbes 1992b). In this latter case, the current in the sheet is always small and magnetic energy is instead stored in smooth field-aligned currents, such as a twisted flux tube, at a spatial scale (length typically 1-10 Mm) that gives a negligible role to the resistive term. In this evolution an ideal instability or non-equilibrium occurs forcing reconnection to take place at the separator (e.g. Priest & Forbes 1990). Another possibility is that the field-aligned currents are formed by photospheric or convective motions and then carried towards the locations of current sheets where the stored energy can be rapidly released. This list of possible solar flare models is far from being complete, but it shows instead that we still need to combine a large set of observations with adequate modeling of the magnetic field in a search about hints on the energy release process. The 1B/X1 flare, which occurred on 28 May 1978 inside a collinear nest of three bipolar regions in McMath 15314 (CMP 27 May 1978), is a very interesting case as the global evolution of the region and the flare itself were well-observed. The main observational points that support this statement are: 1. Build-up was followed for many successive days at high spatial resolution in Hα . The complex of activity in which the flare occurred evolved slowly and without peculiar internal motions: this makes it easier to model and to compare the roles of emerging flux and sheared magnetic fields than in other more complex and rapidly evolving cases. Key steps

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in the energy storage and release process can be identified. A δ-spot formed when adjacent expanding bipoles, already emerged, forced facing p- and f-polarity spots into contact (Gaizauskas et al. 1994; hereafter Paper I). A high degree of spatial symmetry developed in the magnetic pattern (three nested collinear bipoles). A direct test can be made of the classic Sweet model for the formation of an X-type neutral point by approaching sunspots (Sweet 1958). For the 1B/X1 flare there is good evidence that pre-heating plays a role for about 2 minutes before a very sharp flash phase. The Hα morphology suggests that the energy release mechanism works in different ways at different stages during this event. The 1B/X1 flare is a classic example of a confined flare (see Sect. 2.5). Bright and dark ‘post’-flare loops (Schmieder 1992 and references therein), formed as transient features in the maximum and gradual phases of this flare, highlight some of the field lines belonging to the separatrices or QSLs derived for this activity complex. Three flares (two of X-ray class M1) homologous to the 1B/X1 flare preceded it in a 4-hour interval; at least two more homologues followed it in a 27-hour interval. These examples (see Sect. 2.4) conflict with models proposing energy storage for flares in neutral current sheets.

Therefore, the central aim of this paper is to use a large set of photospheric and chromospheric observations, combined with

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Fig. 2. Scheme of the splitting of Cf on May 28. The 2 fragments have larger velocities (≈ 100 m s−1 ) in the directions indicated by the arrows than the velocity of the whole spot before (≈ 30 m s−1 ) the splitting. The dotted zone corresponds to the smallest cupola defined by the intersection of separatrices (see Fig. 11).

a model of the coronal field, to gain some insight to the buildup and release of energy in flares. We describe the long-term evolution and the quiescent state of McMath 15314 in Sects. 2.2 and 2.3, while in Sect. 2.4 we describe in general its flaring state. After this, we analyze in detail the different phases of the largest flare occurring in the region (Sect. 2.5). We compute the magnetic topology of the nested bipoles using two methods: the source method (SM, D´emoulin et al. 1992) and the quasiseparatrix layer method (QSLM, D´emoulin et al. 1996). Both methods are discussed and compared in Sect. 3.1 and applied, in Sects. 3.2 and 3.3, to the magnetic field observations obtained on May 27 and 28 respectively. Our results show that the energy release sites, both during the quiescent and flaring phases, are related to the topological structures that we identify. Combining the observations with a magnetic field model, we discuss in Sect. 4 what we have learned about the storage and release of magnetic energy in an active region. 2. Observational evidence 2.1. Data The data used in this paper include daily magnetograms of the line of sight component of the field obtained at the National Solar Observatory at Kitt Peak (NSO-KP) and chromospheric filtergrams photographed using a 25-cm refractor at the Ottawa River Solar Observatory (ORSO). Filtergrams were taken every 2.5 s at 15 wavelength positions across the central portion of Hα ˚ The data were obtained during the third passage out to ± 1.4 A. of McMath Region 15314 across the solar disk, from 21 May to 3 June 1978. In this study we will deal mainly with the events occurring during 27 and 28 May 1978. More details about the characteristics of the data can be found in Paper I. 2.2. Evolution of the complex of activity: clues on the storage and release of magnetic energy This so-called ‘Great Complex of Activity’ (Gaizauskas et al. 1983) on its third rotation consisted of five magnetic bipoles, labeled in chronological order of emergence in Fig. 1. Bipole A survived from the previous rotation; B and C were formed just

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before and just as the complex made its eastern limb passage, respectively. Bipoles D and E were born on the visible disk on 22 and 26 May, respectively. Concerning the evolution of the nested collinear sunspot groups, the penumbra of spot Df was partially formed on 23 May (Fig. 4b of Paper I). The formation of a δ-spot then began on 24 May with the approach of spots Cf and Dp (Figs. 4b and 6 of Paper I). It is characterized by a growing overlap of the penumbra of these spots; they reach their closest approach on 27 May (see Sect. 2.3). A detailed study of the nested spots is presented in Paper I. A δ-type sunspot is often associated with rapidly changing, complex magnetic patterns. Here, throughout the slow creation of this δ-spot, ordinary evolutionary processes were at work on ordinary sunspots. For nearly 100 hours, until 28 May 1978, there were no twisting or shearing motions by the colliding components inside the δ-spot. During this continuous approach there was no flaring activity until 27 May when very small flares erupted over Dp and Cf (see Sect. 2.3). During this period, energy was being released in the steadily compressed magnetic fields of the colliding bipolar regions. The system seems a real version of the textbook example proposed initially by Sweet (1958) in his neutral point theory of solar flares. In reality, as described in the next paragraph, the big energy release does not happen through continued steady merging of adjacent fields. Other events intervene. On 28 May, a new relative motion of the umbrae in the δspot replaced the strictly converging motion of the preceding 5 days. The critical events happened in a 15-hour interval when there were no observations at high spatial resolution. In this period the Cf-umbra split into 2 fragments which flew apart in new directions shearing past the now distorted Dp-umbra at ≈ 100 m s−1 (see Fig. 5 of Paper I and the scheme in Fig. 2). In fact, the time of the Cf splitting can be circumscribed by a 6 hour gap between a full disk white light photograph obtained at Kodaikanal Observatory (7:36 UT) and the first ORSO image on 28 May (13:37 UT). A possible scenario, based on observations from just before to just after this blank window, was developed in Paper I to account for the disrupting f-spot. It invokes the fluting instability to drive the disruption and the suppression of the vertical escape of sub-surface heat flux between the colliding spots to trigger it. There is serious flaring only after the splitting of Cf: two Mclass flares and a subflare in the 4 hours preceding the 1B/X1 flare. The energy build-up for the 1B/X1 event takes no longer than 90 minutes since the peak of the preceding class SN/M flare; even then it is interrupted about mid-way by a minor energy release in a small subflare (Fig. 3a). The shearing motion is then sustained for the remainder of the lifetime of the δ-spot. Its onset coincides with flare activity on an unprecedented scale in this activity complex during Carrington Rotation 1668. It must therefore be a key ingredient to the energy release process. Since field lines are rooted in the diverging fragments of the Cf-umbra, they will slip through each other and create current sheets. The process is a gradual one, because some of the kernels in the succession of flares early on May 28 recur close to the same location.

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Fig. 3a–f. Stages of the 1B/X1 flare on May 28 as seen in raw (undigitized) off-band Hα images from ORSO. a Kernels of a prior subflare at its maximum phase. b Onset of the 1B/X1 flare. It coincides with the onset of GOES X-ray burst, but precedes the onset of 10 cm burst by 1 minute (see Fig. 7). c Beginning of the impulsive phase of the 1B/X1 flare in Site 1. d Beginning of another impulsive phase at remote Site 2. The time delay with respect to the brightening of Site 1 is ≈ 77 s. e Beginning of a third impulsive phase at remote Site 3. Although closer to Site 1, Site 3 is energized ≈ 76 s after Site 2. f A fourth remote flaring site shows a single kernel when the other sites approach maximum size and brightness.

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Fig. 4. Umbral dynamics (upper 6 frames, digitally enhanced to show umbral details) and miniflare activity (lower 3 frames) on 27 May 1978 in the δ-spot formed by colliding p- and f-spots. Numerals in the upper left indicate the offset in Angstrom units from the center of Hα, UT is shown in the lower right of each frame. Frame size is 48”x 27”.

2.3. Flaring activity just before disruption of Cf On the day preceding the 1X/1B flare, 27 May 1978, the approaching umbrae were actually closer together than they were just before or just after this event; at the narrowest part of the gap between umbrae the distance was ≈ 3200 km (see Table 2 in Paper I). Thirteen flare-like events were observed over the δ-spot in the ORSO data for that day, they were so small that only one of them was noted by the flare patrols reporting to Solar Geophysical Data (SGD) (Fig. 14c in Paper I shows one of these events). At the same time, and at the photospheric level, the colliding p-f umbrae were undergoing dynamic changes in their internal structure. The 6 upper panels in Fig. 4 show that two bright fingers traversed the Cf-umbra in almost 2 hours, and that one corner of the Dp-umbra was covered by intruding penumbra in about the same time. We interpret this as evidence for reconfiguration of the compressed sunspot fields on a small scale. The numerous miniflares (three different ones are shown in the lower panels of Fig. 4) can be taken as evidence for the formation of current sheets during the reconfiguration. The total energy released by these transient events is negligible compared to the 1B/X1 flare. 2.4. Flaring activity following disruption of Cf 2.4.1. Flares on 28 May 1978 ˚ channel), Referring to the GOES satellite soft X-ray data (1 - 8 A serious flare activity probably began around 10:47 UT on 28

May when a class M1 event was attributed to this region from a normal subflare. Another class M1 event peaked at 13:15 UT. It was observed in progress at ORSO at 13:37 UT to have two flare kernels, one each at the edges of the umbra of Cf and Dp (not shown). A short-lived subflare with no recorded GOES event erupted at 14:12 UT with several kernels, again on the edges of each of the same two umbrae and in the gap between them (Fig. 3a). The next flare, the most powerful from this region, peaked at 15:02 UT as a class 1B/X1 event (see Sect. 2.5). The subflare and the dying phase of the second M-class flare were observed at ORSO to be homologous with the X1 flare in the location of off-band kernels in the gap between Cf and Dp; the same was true of the first class M1 flare observed at Catania (F. Zuccarello, private communication). A subflare erupting five hours after the 1B/X1 flare was a much weaker homologue with Hα ribbons at sites 1, 2, and 3. 2.4.2. Flares on 29 May and 30 May 1978 Two more flares (not shown) erupted in McMath 15314 during an 8-hour interval observed at ORSO on 29 May. The earlier subflare was unrelated to any of the sites of the 1B/X1 flare; the second erupted 2.5 hours later as a slowly rising 1B/C1 flare with ribbons occupying locations similar to sites 1, 2, and 3 of the 1B/X1 flare. Despite large disparities in the speeds of onset in Hα and in the magnitudes of soft X-ray flux, we infer that the 1B flares of 28 and 29 May were triggered by similar mechanisms confined to the polarity inversion in the δ-spot and the energy

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Fig. 5a–f. Enlarged, digitally enhanced views of ˚ from: a the onset, b the 1B/X1 flare at Hα + 1.0 A growth of additional kernels, c impulsive brightening of new kernel at the arrow, d filling in of ribbons and eruption of S-shaped structure at T, e maximum phase with first sign of emitting loops crossing between flare ribbons, f to further spreading apart of the flare ribbons and development of bright emitting loops. The polarity inversion line is shown as a full trace and it is calculated from the coaligned NSO/KP magnetogram for May 28. Each frame is 40”x 81”.

was transported to the two remote sites over a magnetic topology which did not alter significantly during 27 hours.

2.5. The 1B/X1 flare on 28 May 1978 2.5.1. Characteristics of the event

By 30 May, however, subflares were triggered inside the δspot of McMath 15314 in entirely different locations (described by Gaizauskas 1986). After two days, therefore, the magnetic topology changed significantly due to the rotation and shrinking of the spot pair Cf-Dp and to the rapid expansion of a major new bipole on the leading edge of the activity complex comprinsing McMath 15314.

The ORSO images of the flare taken in the core of the Hα line are so close to saturation that we rely on the off-band images, mostly in the red wing, to describe important properties of the flare (see Fig. 3). During the whole event there was no filament ejection, nor any Type II, III, IV, or V radio bursts. Yet the soft X-rays (GOES) reached a peak flux of 1.3 10−4 watts m−2 and a class 3 solar interplanetary disturbance was reported, making this a major X-ray event. The absence of any signature of ejecta disqualifies this flare from being classed as an eruptive event. ‘Confined’ is a better term to apply to this flare than ‘compact’

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˚ channel of the GOES satellite for Fig. 6. Soft X-ray flux in the 1 - 8 A the 1B/X1 flare of May 28 1978.

because its four remote flaring sites are spread over a region of 100 Mm width (see below). During the onset phase, starting at 14:56:13 UT, the flare consists of a string of bright kernels - 2 sets of 3 kernels located at Site 1, one set per umbral rim of the colliding p-f pair - aligned obliquely across the polarity inversion line at the photosphere (Fig. 3b and Fig. 5a). Their number grows and the space between them fills with emission; the kernels progressively transform to ribbons (Fig. 3c,d and Fig. 5b,c). The observations lack the temporal resolution to decide whether the kernels light up together or in succession, if they move along the string or stay put. This stage lasts 2 minutes, up to the time of a very large spike in soft X-rays at 14:58:21 UT (Fig. 6). Thereafter, the morphological character of the emitting ribbons changes (compare Fig. 5c to Fig. 5d). The flash phase occurs at 14:58:21 UT in an interval comparable to the 3-sec sampling time of GOES, within which the flux shoots up to 18 times the preflare threshold (Fig. 6). The flux drops back to the rising background in 9 s. The Hα responds at the same time with a brilliant flash centered on Site 1 (Fig. 3e and 5d). Prior to the flash, a kernel brightens at the North end of Cf (Fig. 5c, at the arrow). At 13 s before the X-ray flash, the kernel grows into a S-shaped structure aligned along the ribbon overlaying Cf (Fig. 5d, at T). Immediately after the flash (Fig. 5d) the S-shaped structure shifts away from the flare ˚ X-ray flux reaches its ribbon over Cf (Fig. 5e, f). The 1 - 8 A maximum at 15:02:20 UT (Fig. 6). By this time the Hα ribbons are clearly separated, but are joined by bright strands traversing the polarity inversion line (Fig. 5e,f). The development of the bright loops is presented in Sect. 2.5.2. Hα emission also builds very rapidly in other kernels located at several remote sites spread over the activity complex: Site 2, at a plage with pores of p-polarity South of Cp and Site 3, at the spots of f-polarity in D (Fig. 3). Finally, Site 4 appears later between the colliding Cp and Bp spots (Fig. 3f). As discussed in Sect. 3.3 we believe that Site 4 does not brighten as a result of the

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Fig. 7. Microwave burst at 10-cm wavelength associated with the 1B/X1 flare from Algonquin Radio Observatory.

same energy release process that produces the other sites, even though the change caused in the global magnetic configuration of the activity complex after the big flare could have caused its brightening. The surge-like feature above Cp (Figs. 3) is in fact one of many episodes of downflow in a large absorbing arch spanning most of the activity complex (Paper I, Fig. 2). This episode began ≈ 14:56 UT and faded out by 15:26 UT on May 28. It may be a signature of large-scale ‘post’flare loops connecting Df to Cp (see Sect. 2.5.2). Concerning the emission in radio wavelengths during this flare, the peak flux density of single-frequency radio events increased with increasing frequency (SGD, No.411, Part II, p.48). We reproduce in Fig. 7 a copy of the original record made by the 10-cm flux patrol at Algonquin Radio Observatory. The peak emission at 15:01.3 UT is a mere 34 solar flux units (sfu), one to two orders of magnitude smaller than many flares of similar optical and X-ray radiative output. There is no signature in this 10-cm flux record coincident with the soft X-ray spike at 14:58:21 UT. Other stations with higher frequency receivers reported higher peak emission for this event (e.g. 378 sfu at 3.4 cm wavelength). From this, we infer that energy at microwave wavelengths was released at a very low height of the atmosphere in at least 6 impulsive episodes over a time span of 3 minutes or more (see Fig. 7). 2.5.2. Bright and dark ‘post’-flare loops Several systems of loops appear during the maximum and gradual phases of the 1B/X1 flare. A system of short (≈ 7600 km) ˚ beginning bright loops, visible in the red wing at Hα + 1.0 A ≈ 15:01 UT, links the rapidly spreading flare ribbons obliquely across the polarity inversion at Site 1 (Fig. 5e,f). They are initially invisible in the blue wing of Hα and in the core of the line where small-scale features are obscured against a background saturated with flare emission. They are best seen ≈ 15:06 UT in the red wing (Fig. 5f) and then fade beginning ≈ 15:09 UT, losing all visibility at that wavelength by ≈ 15:20 UT. They remain

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Fig. 8a–h. Development of hot and cool loops at Site 1 of the 1B/X1 flare of 28 May 1978. a Preflare status. b Short bright loops indicated between arrows separating flare ribbons 1e and 1w. c Dark loops (between arrows) replacing bright loops between the separated flare ribbons at Site 1. d 4.7 hours after flare onset a short dark loop connects sunspots Dp (lower left contour) and Cf (upper right contour) at Site 1. e Red-shifted loops link Sites 2 and 3 and Site 3 with spot Dp. Note enhanced absorption at the footpoints. f Cool loops blanket the entire area between Sites 2 and 3 and between Sites 1 and 3 as fine dark threads. g Connection of some cool loops anchored at Site 2 switched from Site 3 to spot Dp at Site 1. h Cool loop anchored at Site 2 traverses spot Dp. A shorter loop system links Dp with Site 3. Figures in the upper left corners correspond to the offset (Angstroms) from the center of Hα. Each panel is 198”x 143”.

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visible in wavelengths near the core of the line until they are replaced entirely by dark loops ≈ 15:24 UT (Fig. 8c, between the arrows), about 5500 km in length. With time the nice alignment of the short dark loops gives way to disordered dark fine structure (Fig. 8f), and then hours later to a simple dark structure connecting Dp with Cf (Fig. 8d, between the contoured spots). Bright and dark loops at Site 1 proceed from an initially simple geometry of lines linking opposite polarities (Fig. 9a) to a very complex one (Fig. 9b) before returning to a simple linkage (Fig. 9c). This suggests that the field lines in the vicinity of the separator (Sect. 3.3) are violently perturbed during the flare and take hours to return to a stable equilibrium.

A system of longer cool loops blankets the flaring region for a short time during the gradual phase; it is best seen in a movie made from ORSO data in the core of Hα and kindly provided by Sara Martin. The entire episode lasts a half-hour, from ≈ 15:35 UT until ≈ 16:05 UT. The development can be followed in Fig. 8, where panels A, B, and C show an improving regularity in the alignment of fibrils between flare ribbon 1e (panel B) and ribbon 3. The alignment is most striking ≈ 15:41 UT (Fig. 8f), by which time flare ribbon 1e has faded considerably. The loops consist of many fine dark threads (≤ 1”) (Fig. 8e,g h); and since they are invisible in the blue wing of Hα, we can take their heavily absorbing footpoints in the red wing as evidence of strong downflows. The visibility of different loop systems changes quickly with time. Initially, one long set connects directly from Site 2 to Site 3 (Figs. 8e,f), while another shorter set connects Site 1e to Site 3 (Figs. 8c,e,f). Within a few minutes the linkage between Sites 2 and 3 fade and other loops between Sites 2 and 1e become the dominant features (Figs. 8g,h).

We take the change and the downflow in the individual loops connecting Site 2 to Site 3 as evidence for plasma condensing via radiative cooling of hot loops connecting the flare ribbons as usual in ‘post’-flare loops (Schmieder et al. 1995). They correspond to the second set of reconnected loops (the first one being the usual ‘post’-flare loops at Site 1; see Sect. 3.3). The presence of other loops connecting Sites 1e to 3 and 1e to 2 is more difficult to understand because these loops seam to outline the initial magnetic connection (before reconnection!). A strong compression of the plasma is unlikely to occur before the reconnection process because the velocities are sub-Alfvenic there, and because the volume of a flux-tube going to the reconnection region R increases (the volume per unit of magnetic flux is given by ds/B which is maximum at the QSLs in such quadrupolar configuration: see Fig. 3b of D´emoulin et al. 1996). One possibility is that energy is transported from the reconnected loops to the pre-reconnected loops by radiative transfer in the lower (denser) part of the configuration. This may induce a gentle evaporation in the pre-reconnected loops. This mechanism is most efficient if the evolution is slower than the radiative-time scale, so we expect qualitatively to have it at work at the end of the main phase, as observed.

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2.5.3. Energy transport mechanism Prior to the 1B/X1 flare there were no signs of excess energy being stored at Sites 2 and 3. We deduce this from the direction of chromospheric fibrils that we have used in all cases as proxy tracers of the horizontal component of the field, since no vector magnetograms are available for these days. We then conclude that Sites 2 and 3 result from sudden deposition of energy coming from a remote region that we have located at a low coronal level above Site 1 from the magnetic computations (see Sect. 3.3). Assuming that just one initial release of energy occurs there, we can estimate the speed of the disturbances traveling to the remote sites. We measure the delays (see Fig. 3) from the moments when impulsive brightenings of kernels are detected at each of the sites in red-wing filtergrams. The timing for these impulses are: 14:56:13 UT at Site 1; 14:57:30 UT at Site 2; and 14:58:46 UT at Site 3. Due to our short sampling interval between photographs and the impulsiveness of the phenomenon, our timing estimates are in error by no more than 20 s. We estimate the distance between these sites in two different ways, what gives us a velocity range for the traveling disturbance. Using the projected distance on the images, we find speeds of about 250 km s−1 for a disturbance traveling to remote Site 2, but only 140 km s−1 to the closer Site 3. On the other hand, taking a line integral along field lines connecting the relevant sites, we find speeds of 1200 km s−1 to Site 2 and 660 km s−1 to Site 3. The values we have found point to a mechanism of energy transport by a thermal-conduction front (see Bagal´a et al. 1995 and references therein for a justification). In the present event, the slower velocity corresponding to the disturbance traveling to Site 3 may be due to the fact that the lower set of magnetic field lines have a cooler and denser plasma trapped, in comparison to the higher field line connections to the West towards Site 2 (see Sect. 3.3). 3. Model and magnetic field topology 3.1. Our view on 3-D magnetic reconnection The 3 dimensional (3-D) characteristics of magnetic reconnection are highly complex and are only just beginning to be understood (Schindler et al. 1988; Priest & Forbes 1992a; Lau & Finn 1990; Priest & Titov 1996). From a classical point of view magnetic reconnection is closely related to the existence of separatrices or of their intersection (the separator) and, so, to magnetic null points. That is to say, in 2-D and 2.5-D approaches magnetic reconnection is thought to occur at places where the field-line mapping is discontinuous. When an observed 3-D magnetic configuration is modeled by a series of sources, the 2-D picture of magnetic reconnection can be directly generalized. Separatrix surfaces divide the magnetic volume into topologically distinct regions, in the sense that any of them contains only field lines that start at a particular source and end up at another particular source; when magnetic reconnection occurs, magnetic flux is transferred from one region to another. Baum & Bratenahl (1980) were the first to calculate the topology of a potential configuration formed by four

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propose that magnetic reconnection may occur in 3-D in the absence of null points at quasi-separatrix layers (QSLs), which are regions where there is a rapid change in the mapping of field lines from one boundary to another of a given magnetic volume. QSLs are defined in terms of a dimensionless function that we call N . If we integrate, over a distance s in both directions, a field line passing at a point P(x, y, z) of the corona, the end points of coordinates (x0 , y 0 , z 0 ) and (x00 , y 00 , z 00 ), define a vector D(x, y, z) = {X1 , X2 , X3 } = {x00 − x0 , y 00 − y 0 , z 00 − z 0 }. A rapid change in field-line linkage means that for a slight shift of point P(x, y, z), D(x, y, z) varies greatly. In the solar case, the distance s to be used is the distance to the photosphere (z 0 = z 00 = 0) and the expression for N (x, y) is: v " u 2  2 # uX ∂Xi ∂Xi t N (x, y) = + . (1) ∂x ∂y i=1,2

Fig. 9. Close-up view of dark loops at flare Site 1 during the gradual phase. Each panel is 28” x 78”.

magnetic poles; then, Gorbachev & Somov (1988, 1989) applied the same idea to an observed active region (AR). The next step was to introduce many subphotospheric sources (Mandrini et al. 1991, 1993) and to determine their positions and intensities by a least-square fitting of the computed magnetic field to the observed one (D´emoulin et al. 1994b). Using this Source Method (SM) to determine the magnetic topology, we have found that Hα flare brightenings and photospheric current concentrations are located on the intersection of separatrices with the photosphere and that they can be connected by field lines (Mandrini et al. 1991, 1993, 1995; D´emoulin et al. 1993, 1994b; van DrielGesztelyi et al. 1994; Bagal´a et al. 1995). Furthermore, flare kernels do not extend all along separatrices but lie on portions of them where the connectivity of field lines changes rapidly from one side of the separatrix to the other. For some of the studied ARs we have found a magnetic null point in the extrapolated coronal field, mainly when an almost oppositely oriented bipole emerged between the two main polarities of a group of spots (see Sect. 3.2). However, several ARs contain no such null points (D´emoulin et al. 1994a); that is to say, no discontinuity is present in the coronal linkage (see Sect. 3.3). The so-defined SM has two main limitations: first, one needs to integrate below the photosphere along a few thousands kilometers, where a magnetic field model is not available; second, it cannot be used with other extrapolation techniques because it intrinsically needs sources to define the connectivity. These limitations, together with our findings described in the previous paragraph, pointed out the need for a method that could take into account only the field-line linkage above the photosphere. This method came together with new developments in 3-D magnetic reconnection (Priest & D´emoulin 1995). In this work we

This function is evaluated on the boundary and represents the norm of the displacement gradient tensor defined when mapping, by field lines, points on one boundary to the other of a given magnetic volume; both boundaries correspond to different sections of the photosphere in our particular case. The locations of the high values of N (x, y) characterize the field lines involved in the QSLs; following these lines we can locate the coronal portion of these layers. QSLs are open layers that behave physically like separatrices when their thickness is small enough so that the resistive term in the induction equation becomes important for the magnetic field evolution. A discussion on the properties of N (x, y) and the basic characteristics of QSLs, together with a description of the QSLM, can be found in D´emoulin et al. (1996). The QSLM requires only a model of the coronal magnetic field. Thus, we can either use a model with sources or extrapolate the photospheric longitudinal field (Bl ) to the corona. In the last case we use the discrete fast Fourier transform (FFT) method under the linear force-free field (∇ × B = αB, with α a constant) hypothesis, as proposed by Alissandrakis (1981). 3.2. Quiescent state of McMath 15314: 27 May 1978 Fig. 10a shows a portion, centered in the δ-spot, of the NSO-KP magnetogram obtained on 27 May at 13:47 UT. Since no vector magnetograms are available for this and the following day, we will model the observed Bl only in the potential approach. The results of the SM using a model with 32 subphotospheric sources together with the plage brightenings observed at ORSO at 18:14 UT, can be seen in Fig. 10b. These brightenings are located on or close to the computed separatrices; it is remarkable how they follow the round shape of the small separatrix to the East. We have also derived the locations of QSLs at the photosphere for the same model of the field (not shown); comparing both results we have found that the sections of QSLs, where the plage brightenings lie, are located along portions of separatrices as happens for other studied flares (see Mandrini et al. 1997). Fig. 10c corresponds to the QSLM for a FFT extrapolation of the observed field. This representation of the field is more realistic than a model with sources, where we only keep the

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Fig. 10a–d. Magnetic topology for McMath 15314 on 27 May 1978. a Observed longitudinal field. Three isocontour levels of Bl (± 300, 1000, 1500 G) are shown with positive and negative values drawn with solid and dashed lines, respectively. All distances are in Mm. b Intersection of the separatrices (thick lines) with the photosphere for a model with 32 magnetic sources (as shown by the Bz isocontours having the same values as those of Bl ). Hα plage brightenings are represented as hatched regions. Local x and y point to the West and North, respectively. c Intersection of the QSLs with the photosphere (thick isocontour lines of N = 20) for a potential extrapolation of Bz using the fast Fourier transform method. Four field lines showing the different kinds of connectivity have been added (see text). d Lateral view of Fig. 10c (viewed from the West). The four kinds of field lines meet in a region with a top height of ≈ 8 Mm. Each division corresponds to 2 Mm.

main flux concentrations; and the location of QSLs should be more precise in this case and not necessarily coincident with those derived from a model with sources. Comparing Figs. 10b and c, it can be seen that QSLs and separatrices coincide or lie very close to the location of plage brightenings. In this last figure we have also drawn four field lines, issued form both sides of QSLs at the photospheric level, corresponding to the different kinds of connectivities: Dp - Cf and Df - Cp are shown with continuous style, while Df - Dp and Cf - Cp with dotted and dash-dotted lines respectively. A lateral view of these field lines is shown in Fig. 10d, it can be seen that they meet at a height of ≈ 8 Mm in the corona. Their shape is typical of the shape of lines around an X-type neutral point; in fact we found for the model with sources (using the method described by D´emoulin et al. 1994b) that a magnetic null point exists in the configuration at that location. With the FFT extrapolation, we found QSLs so thin (as small as the computer resolution (10−15 Mm !) – see D´emoulin et al. (1996) for a definition of the thickness of QSLs) that a null point is also present. Therefore, magnetic field reconnection seems to be at work at QSLs in a steady and gentle regime during this quiescent state induced by the convergence

of Dp and Cf (see Sect. 2.2); the energy release site lies towards the East of the polarity inversion line of the p-f colliding pair very low in the corona (Fig. 10d). Besides, as energy release during this state stays at a very low value, we do not expect to find remote brightenings as is the case with Sites 2 and 3 for the 1B/X1 flare. 3.3. Flaring state of Mc Math 15314: 28 May, 1978 We apply here the SM and the QSLM to the magnetic field observations obtained at NSO-KP on 28 May 1978 at 13:38 UT. The result of the FFT extrapolation, corresponding to the portion of the magnetogram comprising the same region as in Fig. 10a, is shown in Figs. 10c and d together with QSLs. No remarkable changes can be observed when comparing Figs. 10 and 11 at the places of high magnetic intensity, except an extension towards the West in Cf; by this time this spot had already broken. In Figs. 11a and b we show the result of the SM for a model with 32 sources representing again B, C and D bipoles. The computed separatrices have been overlaid in Fig. 11a to the off-band Hα image obtained at flare onset, 14:57:13 UT (see Fig. 3b). It can

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Fig. 11a–e. Magnetic topology for McMath 15314 on 28 May 1978. The drawing convention is the same as in Fig. 10. a, b Intersection of separatrices (thick lines) with the photosphere for a model with 32 magnetic sources overlaid with the Hα kernels at the onset and maximum of the 1B/X1 flare corresponding respectively to Figs. 3b and 3f. c, d QSLs computed at the photosphere (thick isocontour lines of N = 20) for a potential extrapolation using the fast Fourier transform method. The coronal linkage at the borders of QSLs is shown with field lines connecting Cf to Cp and Df to Dp in (c), and Dp to Cf and Df to Cp in (d). e Boundaries of separatrices plotted onto image with cool flare ‘post’-flare loop visible in the gradual phase at 15:46:07 for the 1B/X1 flare of 28 May 1978.

be seen that the brightest part of Hα kernels lie exactly on the boundaries of separatrices enclosing regions C and D. Fig. 11b shows the overlay of separatrices with an off-band Hα image obtained at 15:00:03 UT (see Fig. 3f). In this case the fit is less good, chiefly at Site 3, and the emission has spread beyond the separatrix boundaries. The lack of agreement is mainly due to the fact that the magnetic modeling cannot take into account the shearing motions affecting the field lines during this period. Separatrices and QSLs lie also in this case very close to each other, except at low field regions (Figs. 11c,d). Notice also that, in spite of some changes in the field between 27 and 28 May, the magnetic topology of the complex stays the same; this is because the locations of QSLs depend on the global properties of the configuration which remain the same from one day to the next. We have calculated the value of the thickness of QSLs at the location of Hα kernels, as for 27 May (see Sect. 3.2); it stays as small as the computer precision. The separator is located

Fig. 11a–e. (continued)

towards spot Dp (as for May 27, see Fig. 10d) at an estimated height of 10 Mm, very low in the corona. The likeliest place for reconnection to occur is along this region, since all the field lines along QSLs, including the ones tied to the moving fragment of the f-umbra, thread through it. The character of the radio bursts (Sect. 2.5) is also consistent with a low height for the energy release site in this flare. For the first time in a study of the influence of the magnetic field topology on flares, we can also show a close connection between the location of cool ‘post’-flare loops and magnetic separatrices. As the flare evolves and the flare ribbons at Site 1 spread apart, the ‘post’-flare loops joining them lengthen (Fig. 5). The orientation and location of the bright strands matches the modeled field lines in Fig. 11c and d. We have also identified the other set of reconnected loops as the large scale dark loops with

V. Gaizauskas et al.: Interactions between nested sunspots. II

Fig. 12a and b. Perspective view of Fig. 11 showing the coronal linkage at the borders of QSLs with field lines drawn as surfaces (for aesthetics we enhanced the vertical scale by a factor 1.5 compared to the horizontal one and we have chosen a back-side point of view). a Corresponds to Fig. 14 c); the dark gray (light gray) ribbon shows the connection between Cf and Cp (Df and Dp). b Corresponds to Fig. 14 d); the light gray (dark gray) ribbon shows the connection between Dp and Cf (Df and Cp).

downward flows (Fig. 8). We plot the boundaries of the separatrices in Fig. 11e on the relevant portion of panel H in Fig. 8, taken in the gradual phase of the 1B/X1 flare. The prominent red-shifted loop is anchored in the photosphere on the boundary of a separatrix; it follows the boundary so closely that we have omitted a portion of it to avoid obscuring the loop. 4. Summary and conclusion We have focused this study on the flares occurring in a complex of activity where a large set of observations is available, in particular in Hα. The long-term evolution of the region has been extensively studied before (Gaizauskas et al. 1994). A δspot formed when adjacent expanding bipoles, already emerged, forced facing p- and f-polarity spots into contact. In the present paper, we computed the separatrices and QSLs using a potential extrapolation of the photospheric magnetogram (using sub-photospheric sources or a FFT method). The different approaches show coherently that flare kernels are located on the computed separatrices and QSLs. They can be linked, two by two, by the computed field lines. The magnetic configuration is a classical quadrupolar one with energy release on the separatrices, that is partially conducted down to the chromosphere to form the observed Hα ribbons (Fig. 12). This confirms previous results obtained for other flaring regions (D´emoulin et al. 1997) and makes it evident that magnetic reconnection is at the origin of the studied flares. For the first time in a study of the influence of the magnetic topology on flares we can also demonstrate a close connection

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between the location of cool ‘post’-flare loops and magnetic separatrices. The ‘post’-flare loops are reconnected magnetic loops linking the ribbons at flaring Site 1. This indicates the way magnetic reconnection proceeds during this flare: field lines connecting initially the two independent bipoles Df-Dp and CfCp, reconnect and form the new connectivity Dp-Cf and Df-Cp. In a now classical model, the plasma is first compressed and heated, then ejected both up and below the reconnection region. The transport of energy to the chromosphere drives evaporation which further increases the reconnected-loop density. The increase of density induces enhanced radiative losses which trigger a thermal instability, then dense and cool loops are formed (Forbes & Malherbe 1986, 1991). This model is usually invoked to explain the observed lower reconnected loops only; the ones above the reconnection site, while formed by the same mechanism, are usually not observed. The main differences between the reconnected loops up and below the reconnection region are their lengths and volumes. Even with the same energy input, these differences imply a much lower temperature and density (in particular, because of a much smaller evaporation) for the upper loops. However, these large-scale reconnected loops have been observed in some X-ray events ranging from bright points (e.g. Mandrini et al. 1996) to eruptive flares (e.g. Manoharan et al. 1996). In Hα, flare observations are usually limited to the lower loops giving the false impression that magnetic configurations where flares occur are simple arcades. In Mc Math 15314 region the large set of Hα observations allows us to observe in part the large scale loops as elongated dark features with downward flows. As explained above, their formation is similar to that of the classical ‘post’-flare loops, in particular they show the same plasma dynamic: downward motions of the dense and cold plasma. Because this active region was very well observed during weeks we can infer how the build-up and release of magnetic energy occurred. The usual spreading action during the growth of the young bipoles leads to the formation of a δ-configuration: the preceding and following spots of the two inner regions overlap (p-f collision) into a single penumbra. While δ-configurations created in this way normally favor strong flaring activity, only very small flares occur during 5 days. Only when the following umbra in the δ-spot breaks into pieces, accompanied by rapid photospheric motions, do intense flares occur. The largest and best observed one, a class 1B/X1 flare on 28 May 1978, is a clear example showing the need of magnetic shear before a large flare can occur. The computed magnetic topology shows that the flaring part of the region is quadrupolar, implying the interaction of two, nearly aligned, bipoles (called C and D). The starting configuration is simple: four aligned spots with a potential coronal field. With only converging motions, as observed before 28 May, an ideal-MHD evolution keeps the magnetic field potential except at the separator location where a current sheet is formed (e.g Sweet 1958). The observations show that the energy is not stored in this configuration, but that it is almost continously released over two days producing brightenings as homologous flares at the chromospheric level where separatrices are located. This

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study is an example against the storage of magnetic energy in neutral current-sheets (see e.g. Somov 1992). Because of the very small scales involved in the current-sheet formed, the resistive term is locally important so that reconnection proceeds at the speed at which the photospheric motions drive the system (see Priest & Forbes 1992b). When shearing motions appear, due to the splitting of spot Cp, observations show that magnetic storage can then take place and large flares occur. What are the basic differences that shearing motions bring compared to converging ones? First, with shearing motions, a non-neutral current sheet is formed; not the three components of the magnetic field, but only two reverse when crossing the current sheet. Then added magnetic shear introduces a longitudinal component of the field to the current sheet that may stabilize it (Somov & Titov 1985). Second, the current sheet is formed, not only at the separator, but also all along the separatrices of the 2 21 D configurations (e.g. Low & Wolfson 1988; Finn & Lau 1991; Vekstein & Priest 1992). This can be generalized in 3-D configurations to the formation of high current densities at the locations of QSLs (D´emoulin et al. 1996). The results of this evolution can be a larger storage of energy and a dissipation spread along separatrices, but this needs further theoretical development. Third, shearing motions are also able to store magnetic energy in the whole magnetic volume. The associated electric currents are distributed in the large scale and, therefore, they are weakly dissipated. Storage of energy can proceed until the configuration becomes globally unstable. Then, as reconnection proceeds, these currents are progressively pushed towards the QSLs and a flare occurs. This process can repeat itself in a series of homologous flares until the contribution to the overall topology by existing flux sources vanishes or is dominated by a new bipolar source arising elsewhere inside the activity complex. In summary, the present observations put further constraints on the flare mechanism: neutral current sheets are unlikely to store significant energy. They rather favor the storage of magnetic energy distributed in the magnetic configuration, though non-neutral current sheets are not excluded by these observations. Acknowledgements. We thank Dr. Karen L. Harvey for transferring the magnetic field observations to us from the data archive produced cooperatively by NSF/NOAO, NASA/GSFC, and NOAA/SEC. We thank Sara F. Martin for producing a video movie of the 1B/X1 flare, and D. Wilkinson of NOAA, Boulder, for providing the high time resolution data from the GOES archive. P.D., C.H.M and M.G.R. acknowledge financial support from the CONICET (Argentina) and CNRS (France) through their Argentina/France cooperative science program.

References Alissandrakis C.E., 1981, A&A 100, 197 Bagal´a L.G., Mandrini C.H., Rovira M.G., D´emoulin, P., H´enoux, J.C. 1995, Solar Phys. 161, 103 Baum P., Bratenahl A., 1980, Solar Phys. 67, 245 D´emoulin P., H´enoux J.C., Mandrini C.H., 1992, Solar Phys. 139, 105 D´emoulin P., van Driel-Gesztelyi, L., Schmieder, B., et al., 1993, A&A 271, 292 D´emoulin P., H´enoux J.C., Mandrini C.H., 1994a, A&A 285, 1023

D´emoulin P., Mandrini C.H., Rovira M.G., H´enoux, J.C., Machado M.E. 1994b, Solar Phys. 150, 221 D´emoulin P., H´enoux J.C., Priest E.R., Mandrini C.H., 1996, A&A 308, 643 D´emoulin P., Bagal´a L.G., Mandrini C.H., H´enoux, J.C., Rovira, M.G. 1997, A&A 325, 305 Finn J.M., Lau Y.T., 1991, Phys. Fluids, B3, 2675 Forbes T., Malherbe J.M., 1986, ApJ 302, L67 Forbes T., Malherbe J.M., 1991, Solar Phys. 135, 361 Gaizauskas V., 1986, in D. F. Neidig (ed.), The Lower Atmosphere of Solar Flares, NSO/Solar Maximum Mission Symposium (Sunspot:NSO), p. 37. Gaizauskas V., 1990, in C.T. Russell, E.R. Priest, C. Lee (eds.), Physics of Magnetic Flux Ropes, Geophysical Monograph 58 (Washington:AGU), p. 331. Gaizauskas V., Harvey J., Harvey K.L., Zwaan K., 1983, ApJ 265, 1056 Gaizauskas V., Harvey K.L., Proulx M., 1994, ApJ 422, 883 (Paper I) Gorbachev V.S., Somov B.V., 1988, Solar Phys. 117, 77 Gorbachev V.S., Somov, B.V., 1989, SvA 33, 1, 57 Hanaoka Y., 1995, Solar Phys. 165, 275 Heyvaerts J., Priest E.R., Rust D.M., 1977, ApJ 216, 123 Lau Y.T., Finn J.M., 1990, ApJ 350, 672 Low B.C., Wolfson R., 1988, ApJ 324, 574 Machado M.E., Moore R.L., Hern´andez A.M., et al., 1988, ApJ 326, 425 Mandrini C.H., D´emoulin P., H´enoux J.C., Machado M.E., 1991, A&A 250, 541 Mandrini C.H., Rovira M.G., D´emoulin P., et al., 1993, A&A 272, 609 Mandrini C.H., D´emoulin, P., Rovira M.G., de la Beaujardi´ere J.F., H´enoux J.C., 1995, A&A 303, 927 Mandrini C.H., D´emoulin P., van Driel-Gesztelyi L., et al., 1996, Solar Phys. 168, 115 Mandrini C.H., D´emoulin P., Bagal´a L.G., et al., 1997, Solar Phys. 174, 229 Manoharan P.K., van Driel-Gesztelyi L., Pick M., D´emoulin P., 1996, ApJ 468, L73 Priest E.R., D´emoulin P., 1995, JGR 100, A12, 23443 Priest E.R., Forbes T.G., 1990, Solar Phys. 126, 319 Priest E.R., Forbes T.G., 1992a, JGR 97, A2, 1521 Priest E.R., Forbes T.G., 1992b, JGR 97, A11, 16757 Priest E.R., Titov V.S., 1996, Phil. Trans. R. Soc. Lond. A, 354, 2951 Schmieder B., 1992, in Z. Svestka, B.V. Jackson, M.E. Machado (ed.), Eruptive Solar Flares, Springer Verlag, p. 124 Schmieder B., Heinzel P., Wiik J.E., et al., 1995, Solar Phys. 156, 337 Schindler K., Hesse M., Birn J., 1988, JGR 93, A6, 5547 Shimizu T., Tsuneta S., Acton L.W., et al., 1994, ApJ 422, 906 Somov B.V., 1986, A&A 163, 210 Somov B.V., 1992, Solar Flares, Springer-Verlag Somov B.V., Titov V.S., 1985, Solar Phys. 102, 79 Sweet P.A., 1958, in IAU Symposium 6, Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnert (Cambridge U. Press), p. 123. van Driel-Gesztelyi L., Hofmann A., D´emoulin P., Schmieder B., Csepura G., 1994, Solar Phys. 149, 309 van Driel-Gesztelyi L., Schmieder B., Cauzzi G., et al., 1996, Solar Phys. 163, 145 Vekstein G.E., Priest E.R., 1992, ApJ 384, 333