Two-dimensional singular points in an observed transverse field in

Astron. Astrophys. 342, 854–862 (1999)

ASTRONOMY AND ASTROPHYSICS

Two-dimensional singular points in an observed transverse field in solar active region NOAA 7321 Wang Tongjiang1 , Wang Huaning1 , and Qiu Jiong2 1 2

Beijing Astronomical Observatory, Chinese Academy of Sciences, Beijing 100012, P.R. China (e-mail: [email protected]) Department of Astronomy, Nanjing University, Nanjing 210093, P.R. China

Received 14 March 1997 / Accepted 20 November 1998

Abstract. In the present paper, we give a realistic diagnosis to 2D singular points in the observed transverse field in AR 7321 on October 27, 1991. These singular points are compared with those in the extrapolated field, inferred from the observed longitudinal field boundary on a linear force-free field model. The extrapolated 3D magnetic connectivity patterns are taken to determine the involved separatrices. In order to understand the heating mechanism of flaring loops in the corona, we compare singular features in the photosphere with SXR observations from Y ohkoh. The major results include: (a) The filtering technique is proven efficient in excluding the false candidate points due to the measurement noise. (b) The 2D topology of the observed field indicates more interesting characters than those the linear force-free field can interpret, which are involved in the evolution of magnetic structures. (c) Two kinds of 2D cells, differed by 2D magnetic connectivity patterns, are associated with magnetic loops of different heights and shapes. (d) Some events releasing energy show that coronal loops are interacting at regions, which are involved in the separatrices. Key words: Sun: activity – Sun: flares – Sun: magnetic fields – Sun: X-rays, gamma rays

1. Introduction In recent years, there has been increasing interest, in both theoretical work and observations, in the study of magnetic topology in terms of separatrices and separators. A separatrix is the singular surface that divides topologically distinct magnetic fluxes into different cells, and the intersection between two separatrices is the separator. In a realistic active region, the presence of separatrices and/or separators is expected, since the mixture of opposite magnetic polarities readily produces the separatrices in three dimensional geometry (Baum & Bratenahl 1980). Under the frozen-in condition, current sheets can be created at separatrices by photospheric motions or flux emergence (e.g. Machado et al. 1983; Vekstein, Priest, & Amari 1991). As the strong current gets concentrated in a thin layer, resistive instability may occur which triggers reconnection. When this happens, Send offprint requests to: T. Wang

magnetic field lines of distinct connectivity cells exchange their topological link at the separator and the free energy stored in the current sheet is rapidly released (Priest 1982; Priest & Forbes 1990; Aly & Amari 1997). The energy release may not be confined to the localized area where the flare is initiated by the current sheet dissipation but could cover an extended region bounded by the separatrices in a highly sheared magnetic field configuration (H´enoux & Somov 1987; Low & Wolfson 1988). Indeed, this is a favorable scenario to explain the observations that the Hα flare brightenings are often seen to occur simultaneously at different sites over a global active region (Gorbachev & Somov 1988, 1989; Mandrini et al. 1991, 1993, 1995; D´emoulin et al. 1993, 1994a; Van Driel et al. 1994; Bagal´a et al. 1995). In these studies, the sites of flare brightening are found at the intersection of the separatrices with the chromosphere and associated with each other by magnetic field lines in the region around the separator. Such observation lends a strong support to the idea that energy release in solar flares takes place via reconnection at either the separator region or separatrices. In a 2D field, magnetic topology is studied in terms of discontinuities of the field-line linkage, so the location of separatrices can be easily found. In the 3D case, however, field discontinuities are usually referred to as null points or quasi-singular lines associated with the global property of the magnetic topology in a complex way (Seehafer 1986; Lau 1993). This is why, to date, only a handful numbers of approaches are developed to explore the correlation between the flare and the magnetic topology. A convenient way called source method (SM) has been devised by D´emoulin et al. (1992) to describe the 3D field by simulating the observed photospheric magnetograms with charges or dipoles. The connectivity of the field lines is then defined by grouping the sources which produce the field lines. With this method, however, one may lose the accuracy in recovering the real magnetic field which can be achieved by today’s computing methods. A generalization of the idea of separatrices and separators leads to the introduction of quasi-separatrixlayers (QSLs; D´emoulin et al. 1996). It can be applied to the 3D field extrapolated from the photospheric information while the strict restriction set for the case of separators is not necessary. Yet another different way was proposed by Wang & Wang (1996) to search for the singular points in a 2D field. By tracing the transverse field features, they can display the magnetic

T. Wang et al.: Two-dimensional singular points in an observed transverse field

lanes separating the magnetic lines of distinct connectivities. The method enables to investigate the properties of the extrapolated magnetic field in the plane parallel to the photosphere for a specific model such as potential or force-free field model (e.g. Nakagawa & Raadu 1972; Chiu & Hilton 1977; Sakurai 1981; Wu et al. 1990; Yan et al. 1991), and illustrate the spatial correlation between the flares and the X-points (or saddle points) and magnetic lanes (Wang & Wang 1996; Wang 1997). In this paper, we shall give a realistic diagnosis to 2D singular points in the observed transverse field in AR 7321 on October 27, 1991. In Sect. 2 some reasons are presented to prove that 2D observed field can be used to trace magnetic topology in certain conditions. In Sect. 3 observations and data reductions are introduced. In Sect. 4 the distinctive singular features in the observed field are explored in comparison with those in modeling field, and these features are further compared with the flare morphology in Hβ and SXR in order to understand the heating mechanism of flaring loops in the corona. Finally, we come to the conclusion in Sect. 5. 2. Saddle point and separator Since structures of the field near the separatrices or separators are rather peculiar, some observational characters such as the arrangement of the chromospheric fibril or the photospheric transverse field can serve to search for these structures. One example is demonstrated by Filippov (1995) by virtue of the Hα filtergrams, in which the stable saddle points and the imprint of separatrices, looking like a fir-tree branch, were clearly outlined by the large-scale fibrils. In another example, using the Poincar´e index, Wang & Wang (1996) determined the saddle points and magnetic lanes in the computed transverse field in the chromospheric plane. Their results encourage us to further explore such topological features directly from the observed photospheric transverse field, which may bear non-potential features favoring the occurrence of flares. In general, for a 2D field, if magnetic poles are assumed to be singular points of node type, the geometrical properties near singular points can be mathematically described with Poincar´e index with the value of 1 or −1. The value of −1 indicates the case of a saddle point (Arnold 1973; Bogoliubov & Mitropolsky 1961). Poincar´e index of an isolated singular point is defined as the number of rotations of the 2D field vector along a closed curve surrounding the point with a finite diameter, i.e. I 1 1dθ, (1) I= 2π L where θ = tan−1 (By /Bx ). It should be pointed out that the topology of a 3D field cannot always be determined merely with the 2D information (D´emoulin et al. 1996; Amari 1997). That is to say, if a special structure around a certain point is found in a 2D field, it may be only due to the perspective effect of a 3D field, but does not necessarily indicate the 3D singularity at this point (say, a separator passing through it). Such cases are shown by Priest & Forbes (1989) and Filippov (1995). Therefore, we should be

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much careful in the study when the 2D topological properties are referred to as the 3D ones. Nevertheless, when we discuss the properties of the 2D field which are confined on the photospheric or chromospheric surface, the validity of the 2D method may be justified in studying the relation of magnetic topology to energy release in solar flares. The reasons are: Firstly, when the vertical field component at the saddle point is zero, the saddle point of 2D field gives the true position of a separator intersecting with a horizontal plane; in another word, the saddle point corresponds to a 3D null (Sweet 1958, 1969; Baum & Bratenahl 1980; H´enoux & Somov 1987). In this case, the origin of the saddle point associated with the separator does not need to lie strictly in a plane normal to the separator, but can be in any plane passing through or near the null parallel to its eigen field lines (Lau & Finn 1990). These properties can be used to set up a set of procedures to confirm the relation of the saddle points to the location of the separators (Wang 1997). Secondly, from the model of magnetic sources, we can deduce that the saddle points are close to the possible intersections of the separators with the photospheric (or chromospheric) surface. It is known that observations are in agreement with the idea that magnetic flux converges at sub-photospheric layers. Consequently, the observed field concentrations can be modeled by the magnetic sources, which are situated below the photosphere usually in a depth much smaller than the horizontal size of the active region. Using 4 magnetic charges, D´emoulin et al. (1994b) modeled 10 datasets for 5 active regions, giving the average depth of 0.13±0.06 for the main bipole and 0.06±0.02 for the parasitic bipole if the size of the main bipole is normalized to unity. They found that the nulls were also present below the photosphere (with the depths less than 0.1) for most cases (nearly 90%). In other words, all the magnetic charges and the major nulls are supposed to be situated in a thin horizontal layer. With this assumption, the separatrices and separators are nearly vertical near this thin layer (Baum & Bratenahl 1980; Gorbachev & Somov 1988; Filippov 1995), thus we believe that the 2D magnetic lanes or the saddle points are not far from the ‘true’ location of the separatrices or separators in the photospheric (or chromospheric) surface. Thirdly, since the separators or the separatrices are preferable places for energy release and at the same time are sites of structural stability, we may expect them to be morphologically associated with the observed manifestations such as homogeneous flares or repeated occurrences of energy releases. This also provides us an important referable evidence for the relation between the 2D topology and the separators or separatrices. 3. Observations 3.1. Active region NOAA 7321 The active region NOAA 7321 was a new emerging flux region (EFR), passing across the solar disk from October 24 to November 1 in 1992. During this period, it evolved from an explicit main bipole, nearly resembling the potential field, into a complicated configuration with strong shear around the mag-

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T. Wang et al.: Two-dimensional singular points in an observed transverse field

netic neutral line when several new satellite spots continuously emerged from between the two main poles (Zhang 1995). On October 27, the active region (then located at S23W18) was fully developed, and a series of flares occurred in the flux-emerging area characterized by strong shear (Wang, Qiu & Zhang 1998). We hence take the observed fields on that day to investigate the 2D topological properties. The photospheric magnetic field in AR7321 was observed by a vector magnetograph system (Ai & Hu 1986) installed at the Huairou Solar Observing Station (HSOS) of Beijing Astronomical Observatory. To reduce the noise, and to match the practical spatial resolution, the 200 × 200 smoothing on the Stokes parame◦ ters V, Q and U is made. After the 180 ambiguity is resolved for the transverse field components with a multistep method (Wang et al. 1994), the vector magnetograms are transformed into the heliospheric plane. In addition, we also obtained Hβ filtergrams for a 1N/M1.1 flare on Oct. 27 from HSOS and observations of the SXR emission during 00:48UT–07:58UT from Yohkoh (Tsuneta et al. 1991). 3.2. Data reduction In order to accurately determine 2D singular points in the observed transverse field, we take a conventional lowpass filter technique to process the raw data, reducing the influence of the measurement noise. In the Cartesian coordinate system, Bs0 (x, y) and ˆs0 (u, v)(s = x, y, z) represent the observed field components B and their Fourier components, respectively. Then the filtered ∗ (x, y), are components, Bs0 XX ∗ ˆs0 (u, v)e2πi(xu+yv) , (x, y) = F (u, v)B (2) Bs0 u

v

where F (u, v) is a filter function. We take F (u, v) as the butterworth form, i.e. 1 , F (u, v) = 1 + k(u, v)/kc where k(u, v) = 2π(u2 + v 2 )1/2 , and kc is the lowpass cutoff frequency. After the filtering, the spatial resolution of magnetograms is changed as: Lx Ly , h∗y = ¯ , 2k¯c 2kc where k¯c is the relative cutoff frequency, defined as

h∗x =

k¯c =

(3)

kc q , 2π 1/L2x + 1/L2y

Lx and Ly are typical sizes of the active region. The loss of signals in the filtering can be estimated with a parameter, es ,   12 Ny Nx X X 1 ∗ (Bs0 (x, y) − Bs0 (x, y))2  , (4) es =  Nx Ny x=1 y=1 (s=l or t)

Table 1. Comparisons among δs , ∆δs and es for the data at 01:44UT on October 27 in the raw case and the filtered cases, k¯c =20, 10 and 5. All parameters are in units of Gauss. k¯c δl ∆δl el δt ∆δt et

no filter

20

10

5

−18.1 121.2 – 186.2 118.5 –

−17.8 78.8 98.3 126.2 79.2 107.2

−17.3 56.9 160.2 98.2 64.3 164.1

−17.2 36.2 235.4 72.4 47.6 240.4

where |Bs0 (x, y)| ≥ 3max(|δs |, ∆δs ). δs and ∆δs (s=l or t) are the average value and standard deviation of the measurement noise for longitudinal and transverse field components. Table 1 lists the values of δs , ∆δs and es in different filtered cases. According to Expr. (3), the cutoff frequencies, k¯c =20, 10 and 5, correspond to the effective resolutions of 4×3, 8×6 and 16×11 pixels. The comparisons show that for the raw data the noise level of Bt is nearly 10 times larger than that of Bl , but their deviations are similar; whereas for the filtered data (k¯c =5), the deviations of both Bt and Bl are reduced by about 2/3, but the maximum of the field loss (es ) increases by less than the noise levels (3∆δs ) for the raw data. The results indicate that such a filtering changes only a bit of fine structures, hence will not distort the 2D magnetic topology on large scale. 4. Analyses 4.1. 2D singular points in the observed transverse field We take the vector magnetogram observed at 01:44UT on October 27 as an example for determining singular points in the observed transverse field. We make use of Eq. (1) to calculate Poincar´e index of singular points from the transverse field. The distributions of computed singular points are shown in Fig. 1, in which figures (a) and (b) are obtained from the raw data, while figures (c) and (d) from the data treated with the lowpass filter technique in the cases, k¯c =10 and 5. The singular points with I = 1 are marked by the symbol, ‘ ’, and those with I = −1 are marked by ‘♦’. For the raw data, a large number of candidate points are presented, we thus have to tell apart which are real singular points. To start with, we neglect candidates in the weak-field region, where the average strength of the observed transverse field is lower than 2σ (see Fig. 1b). Secondly, we apply the lowpass filter technique to exclude the candidates in the rest, dubiously attributed to the small-scale structures or the noise fluctuation. Thirdly, we confirm the final ones by the 2D field-line pattern, which are traced from the directions of the filtered observed transverse field. Fig. 1c shows that all singular points are well correlated with special field-line patterns; those with I = 1 correspond to convergence patterns, while those with I = −1 saddle patterns. In this way, we then obtain the distributions of 2D singular points in the observed transverse fields, taken at 01:44UT and

T. Wang et al.: Two-dimensional singular points in an observed transverse field

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Fig. 1a–d. Illustration of the procedures to determine 2D singular points, defined by the Poincar´e index, in the observed transverse field taken at 01:44UT on October 27, 1991 in Active Region NOAA 7321. The singular points with I = 1 are marked by the symbol ‘ ’ and those with I = −1 are marked by ‘♦’. a, b The cases for the raw data. In b the false points due to the measurement noise are excluded. c, d The cases for the filtered data with ¯c =10 and 5. The 2D field lines are k traced from the transverse field directions; specifically, in c those with the grey color approach to their corresponding lanes which demarcate the 2D cells. In each case the solid (dashed) contours show the longitudinal field (isocontours of ± 500, 1500, 2500 G), and the thick solid lines indicate magnetic neutral lines. The field of view is 17000 ×17000 .

Fig. 2a and b. The 2D singular points derived from the observed transverse fields in the photosphere. The magnetogram in a was taken at 01:44UT, and the data were filtered ¯c =10. with the cutoff frequency k The magnetogram in b was taken at 04:12UT, and the data were fil¯c =20. ‘ ’ indicates the tered with k location of the node points and ‘♦’ that of the saddle points. The bold curves linking these singular points are called the magnetic lanes, which demarcate 2D magnetic cells of the different field-line connectivities.

04:12UT on Oct. 27 (Fig. 2). For both cases, there are six nodes (C1, C2, · · ·, and C6) and three saddles (P1, P2, and P3), which are linked by the magnetic lanes demarcating different 2D cells. All cells can be divided into two classes, L and H, based on the difference in the 2D magnetic connectivity patterns. For L cells 2D field lines connect two nodes located in opposite polarity

regions, while for H cells 2D field lines converge at a node. This classification will be explained in Sect. 4.3. 4.2. Comparison with the inferred linear force-free fields In the previous studies (Wang & Wang 1996; Wang 1997), 2D singular points and magnetic lanes are computed from extrap-

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T. Wang et al.: Two-dimensional singular points in an observed transverse field

Fig. 3a–d. The 2D singular points derived from the linear force-free field model, in which the observed longitudinal field at 01:44UT was taken as the boundary condition on the photospheric surface. a α=0; b α=0.03; c α=0.06; d α=0.09. The unit of α is 0.69 Mm−1 . All symbols are the same as those in Fig. 2.

olated transverse fields based on a specific model such as potential or force-free field model. In this section, we compare these 2D singular features in the computed field with those in the observed field. Using the Fourier Transform Method on a linear force-free field model (Nakagawa & Raadu 1972; Alissandrakis 1981), we compute transverse fields in the photospheric plane for four cases, α =0, 0.03, 0.06, and 0.09 in units of 0.69Mm−1 . The boundary condition takes as the longitudinal magnetic field at 01:44UT. The way to determine singular points and other features is the same as in Sect. 4.1. For all cases, the lowpass filtering takes the cutoff frequency, k¯c =10. Fig. 3 shows that the 2D topology of the computed field, compared with that of the observed field, is distinctly different. For example, in the observed field, the saddle P3 separates the emerging pole C3 from the main pole C2, while in the computed field P3 disappears, which results in the cells L2 and L3 merging into one cell, L23. In 2D magnetic connectivity patterns, the nodes C1, C4 and C6 are linked to the node C3 in the observed field, while those are linked to the node C2 in the computed field for the cases α=0 and 0.03. We can evaluate the force-free factor, α, for the field in 2D cells, from comparisons with 2D magnetic connectivity patterns

of the computed fields. The cells H1 and H2 in the main bipole are almost potential (α=0). The cells H3, H5 and L4 are of 0< α 0.09). The 2D topology of the observed field bears more characters than those the constant-α field can interpret, which is related to the evolution of magnetic fluxes with different origins. So this offers more chances approaching the real singular structures favorable to the energy release in the flares. 4.3. The 3D modeling field Since field lines close to a separator or separatrix, which initially close to one another, separate widely in a distance (D´emoulin et al. 1996), we can locate intersections of separatrices by tracing these field lines. Fig. 4 shows the connectivity patterns of 3D field lines in the computed fields in the cases, α=0 and α=0.06. Some special field lines, highlighted in thick solid style, are seen discontinuous at the saddles where they are separated widely. We suppose that the loci of their starting points, portrayed by the dashed lines, represent the intersections of separatrices in the

T. Wang et al.: Two-dimensional singular points in an observed transverse field

859

Fig. 4a and b. The top view of 3D field-line connectivity patterns extrapolated from the linear forcefree field models with different values of α. a α=0 and b α=0.06. The special field lines separating widely near the saddles are drawn in the boldface, the starting points of these lines depict the intersections of separatrices with the photosphere (the dashed lines). In the down-left of the panel a, the arrow indicates the view angles in Figs. 5a and 5b.

Fig. 5a and b. The side view of the special field lines related to saddle points. a α=0 and b α=0.06. The view angle, φ, is ◦ 20 with respect to y-axis (see the arrow in Fig. 4a).

Fig. 6a and b. The side view of magnetic loops. ◦ a α=0 and b α=0.06. The view angle, φ, is 90 .

photosphere. For both cases, these intersections are in good agreement with portions of the 2D magnetic lanes in the places close to the saddles. In some places, however, such as region A, they are shown divergent. Fig. 5 shows a side view of these special 3D field lines. The ‘cavity’ structures around the saddles suggest the associated nulls beneath the photosphere. Since the magnetic connectivities involved in the lanes separating L cell from H cell show no discontinuity (Fig. 4), these lanes do not seem related to separatrices. Fig. 6, however, shows that such lanes demarcate two kinds of magnetic loops different

in height and shape. The low loops are confined only in a L cell, while the high ones have long span between H cells. Table 2 lists height and the average ratio between height and horizontal extension of low and high loops. Moreover, the alignments of the low loops coincide well with those of transverse field components on the photospheric surface (Fig. 4). This may account for the validity of the way we explore the field topology in the low atmosphere in terms of connectivity patterns of the 2D field lines in the photosphere.

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T. Wang et al.: Two-dimensional singular points in an observed transverse field

Fig. 7a and b. The coalignment of SXR images (gray map), taken by Yohkoh SXT from the filter Al.1, with 2D singular points in the observed transverse field. a The comparison among SXR features, singular points and Hβ flare kernels at 01:55:26UT (plotted with thick black contours). b The comparison between SXR features and singular points. The white, dashed lines in a and b indicate magnetic neutral lines. Table 2. Comparisons between two kinds of magnetic loops in the height (Ztop ) and the average ratio between height and horizontal extension (Ztop /Lh ) for the cases, α=0 and 0.06. α=0 00

α=0.06 00

(H)

Ztop Ztop /Lh

89 − 140 1.4

7500 − 10300 1.0

(L)

Ztop Ztop /Lh

1800 − 5600 0.5

1500 − 2200 0.33

4.4. The heating of flaring loops A 1N/M1.1 flare occurred at 01:44UT on October 27. Based on a comprehensive study of the magnetic configuration from the coordinated observations, Wang et al. (1998) found that the event originated from the interaction between two strongly sheared loops. Fig. 7 shows several flaring loops in SXR are spatially related to the 2D singular features, and all the bright kernels in Hβ are located on the footpoints of these loops. The triggering of this flare was on the top (D) of a small loop, but further energy release took place along the loop, especially at its foot regions of A and B (Wang et al. 1998), which may be involved in the separatrices. Fig. 8 shows a series of small events associated with these singular structures, providing the additional evidence. At 01:28:13UT and 02:33:27UT, a glowing streak was just lying along segment P1P3 of the magnetic lanes (Figs. 8a and 8c). At 01:44:33UT, 02:53:15UT and 03:12:13UT, the SXR emissions

disclose the intermittent interaction of two loops at region A (Figs. 8b, 8d and 8e). In Fig. 8f the asterisk (*) marks the position of bright point A in Fig. 8e, which was identified as the interaction site. This fact may account for a puzzle in Qiu et al. (1997) who found the mixed thermal and nonthermal features of coronal plasma at region A at the onset of the 1N/M1.1 flare. We conjecture that the nonthermal component came from the depositing of nonthermal electrons produced at region D, while the thermal component due to the local heating by reconnection (Fig. 7a or Fig. 8b). The last example to show the heating of flaring loops involved in the separatrices is illustrated in Fig. 8c. A set of bright loops, commonly connecting an emitting SXR locus (G), are coincided with the magnetic lane (P2P3). The reconnection may contribute to the heating process and injected hot plasma into the field loops. Our results agree with those of Parnell, Priest, and Golub (1994) as they show that X-ray bright loops can be considered as being reconnected magnetic loops, and support the viewpoint that magnetic reconnection represents an elementary heating source in various coronal phenomena (Priest, Parnell, & Martin 1994; Mandrini et al. 1997).

5. Conclusion In this paper, we investigate the 2D singular points and their relevant features in an emerging flux region (NOAA 7321) directly from the observed transverse field on October 27, 1991. Based on a constant-α field model, we compare the 2D singular

T. Wang et al.: Two-dimensional singular points in an observed transverse field

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Fig. 8a–f. The SXR images taken at 01:00UT-05:00UT on Oct. 27 by Yohkoh SXT from the filter Al.1. The field of view is 15700 ×15700 . In order to illustrate clearly the structures at different times, the SXR images are not shown in the same grey scale.

features in the observed field with those in the computed field. The 3D extrapolated field-line connectivity patterns are taken to confirm the relationship between these 2D features and the separatrices. With these results, we explore the role of the inferred separatrices in the heating of SXR loops in the coronal events. Our main conclusions are summarized as follows: The filtering technique is proven efficient in removing the effect of measurement noises on the determination of 2D singular points, but trivial in changing the large-scale magnetic structures of interest. Moreover, the 2D field line patterns of connectivity can serve as a reference to confirm the 2D singular features. The distributions of singular points in the observed field and the involved lanes are distinctly different from those in the inferred field. The 2D topology of the observed field shows more characters than those the linear force-free field can interpret. The cells bearing the different α values suggest that they are involved in the magnetic fluxes of different origins. The 2D cells can be grouped into L and H types in the light of the difference of 2D field-line connectivity patterns. They are respectively associated with two kinds of magnetic loops of the different heights and shapes. An 1N/M1.1 flare and some small events show that the SXR bright loops are interacting where the inferred separatrices are

located. This implies that magnetic reconnection may play a role in the heating of these SXR loops. Acknowledgements. The authors thank Dr. T. Amari for his valuable comments and suggestions on this paper. W.T.J. is grateful to Prof. Ai Guoxiang for his considerations, and to Mrs. Paula Fulmer for her help in English language. This work was supported by NSFC grants 19573012 and 19791090.

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