The Astrophysical Journal, 624:1072–1079, 2005 May 10 # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.
MAGNETIC HELICITY INJECTION AND SIGMOIDAL CORONAL LOOPS Tetsuya T. Yamamoto Department of Astronomy, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan;
[email protected]
K. Kusano Earth Simulator Center, Japan Agency for Marine-Earth Science and Technology, 3173-25 Showa-machi, Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan
T. Maeshiro Graduate School of Advanced Sciences of Matter, Hiroshima University, 1-3-1 Kagamiyama, Higashi-hiroshima, Hiroshima 739-8530, Japan
T. Yokoyama Department of Earth and Planetary Physics, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
and T. Sakurai1 National Astronomical Observatory, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Received 2004 September 8; accepted 2005 February 1
ABSTRACT We studied the relationship between magnetic helicity injection and the formation of sigmoidal loops. We analyzed seven active regions: three regions showed coronal loops similar to the potential field, and four regions showed the sigmoidal loops. The magnetic helicity injection rate was evaluated using the method proposed by Kusano et al. In order to compare the helicity of regions of various sizes, we defined the normalized helicity injection rate as the magnetic helicity injection rate divided by the magnetic flux squared. We found that the sigmoidal regions and nonsigmoidal regions have comparable normalized helicity injection rates. Next, we calculated the magnetic helicity content of the sigmoidal loops by using the magnetic flux tube model (Longcope & Welsch) and compared it with the magnetic helicity injected from around the footpoints of three sigmoidal loops. For two sigmoidal loops, it is found that these values are comparable. Another loop showed significant disagreement between helicity injection rate and its magnetic helicity content. Excluding this region on the basis of its complexity (perhaps multiple loops forming a sigmoidal loop), we can conclude that geometric twist of the sigmoidal loops is consistent with the magnetic helicity injected from around the footpoints of the sigmoidal loops. Subject headings: Sun: corona — Sun: flares — Sun: photosphere — Sun: X-rays, gamma rays
1. INTRODUCTION
of the sigmoidal loop comprehensively: types, footpoint separation, lifetime, and other properties. Leamon et al. (2003) studied the relationship between the frequency of the flare eruption and the shear angle of the sigmoidal loop. They found that the range of sigmoidal shear angles is limited and that the frequency of the flare eruption shows no clear dependence on the sigmoidal characteristics. The solar corona has high electrical conductivity (Spitzer 1956) so that the coronal loops trace the magnetic field lines. Then the sigmoidal loops are regarded as S- (or inverse S-) shaped helical magnetic fields. The helical nature of the magnetic field R is quantitatively expressed by the magnetic helicity HM (¼ A = B dV ), which is a very important physical parameter related to magnetic free energy. Our motivation for this study is that magnetic helicity should be the cause of the sigmoidal loop. However, we cannot evaluate the magnetic helicity from the observed two-dimensional magnetic data because it requires three-dimensional distribution of the magnetic field and the vector potential. Recently several methods have been proposed in order to calculate the magnetic helicity injection rate from the photosphere. Kusano et al. (2002) and De`moulin & Berger (2003) used the (vector-) magnetograph data. Pevtsov et al. (2003) used the time variation of the linear force-free field with parameter ,
S- or inverse S-shaped coronal loops (Acton et al. 1992; Sakurai et al. 1992) have been observed by the Yohkoh soft X-ray telescope (SXT; Tsuneta et al. 1991) since its launch. Rust & Kumar (1996) called these phenomena ‘‘sigmoid.’’ Sigmoidal loops look more twisted than nonsigmoidal coronal loops and are thought to have a large amount of stored free energy. Sigmoidal loops were often observed with other solar active phenomena; flares (e.g., Canfield et al. 1999), coronal dimmings (e.g., Sterling & Hudson 1997), coronal mass ejections (CMEs), and geomagnetic storms (e.g., Leamon et al. 2002). There are earlier papers discussing the ‘‘sinuous loop’’ found in Skylab observations (Sheeley et al. 1975). Although the sigmoidal loop is defined from morphology, there are some papers studying physical parameters. Pevtsov et al. (1997) investigated the linear force-free parameter derived from vector magnetograms and from the shape of the sigmoidal loops, and they concluded that the -parameters of these data are well correlated. Pevtsov (2002) reported the observed properties 1
Also at Department of Astronomy, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
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SIGMOIDAL CORONAL LOOP FORMATION TABLE 1 Analyzed Regions
NOAA
Sigmoid
Observed Periodb ( Longitude)
8011.................................. 8014.................................. 8015.................................. 8036.................................. 8038.................................. 8056.................................. 8059..................................
Yes No Yes No Yes No (Yes) No (Yes)
Jan 15 (E31)–Jan 18 (W13) Jan 24 (E72)–Jan 28 (E13) Jan 28 (E56)–Feb 7 ( W81) Apr 26 ( W22)–Apr 29 ( W79) May 5 (E83)–May 17 ( W73) Jun 25 ( W06)–Jun 29 ( W69) Jul 1 ( E45)–Jul 10 ( W63)
a b c
a
Latitudec
Analyzed Period
S06 S12 N05 S19 N22 N18 S30
Jan 14 (04:03)–Jan 16 (21:36) Jan 26 (00:49)–Jan 29 (04:03) Jan 30 (02:27)–Feb 4 (12:03) Apr 26 (00:52)–Apr 28 (00:52) May 8 (00:52)–May 12 (04:04) Jun 26 (00:48)–Jun 27 (05:36) Jul 1 (11:12)–Jul 7 (02:24)
( Yes) is according to Canfield et al. (1999). Month and date in 1997. Latitude at the solar meridian.
which best fit the EUV image and the magnetogram. In this paper we use the method proposed by Kusano et al. (2002). Pevtsov et al. (2003) studied the coronal magnetic helicity of active regions with the magnetic flux tube model of Longcope & Welsch (2000). We also use this model to evaluate the magnetic helicity of the coronal loop. In this paper we study the magnetic helicity injection as the cause of the sigmoidal loop formation. In x 2 our data and the method of analysis are presented, and we introduce the normalized magnetic helicity injection. In x 3 individual active regions we analyzed are described. In x 4 we investigate two topics. One is the comparison of the normalized magnetic helicity injection rate in sigmoidal and nonsigmoidal active regions. Another is the relation between the magnetic helicity content of the sigmoidal loops and the magnetic helicity injection from the footpoints of sigmoidal loops. Discussion and summary are given in x 5. 2. MAGNETIC HELICITY INJECTION RATE AND NORMALIZED HELICITY The injection rate of relative magnetic helicity from the boundary (photosphere) to the region under consideration (corona) is represented as (Berger & Field 1984; Berger 1999) @HR ¼2 @t
Z
½(AP = V )B (AP = B)V = n dS;
ð1Þ
where HR is the relative magnetic helicity, AP is the vector potential of the vacuum ( potential ) field derived from the normal component of magnetic field on the boundary, B is the magnetic field, V is the velocity vector, and n is the unit normal vector on the boundary directed into the region. The first term, H˙ t , is due to the shearing motion, and the second term, H˙ n , is due to the emerging motion. Hereafter the relative magnetic helicity is simply called the magnetic helicity. We used the vector magnetograms observed by the Solar Flare Telescope (SFT; Sakurai et al. 1995) for the transverse magnetic field and the magnetograms by the Michelson Doppler Imager (MDI; Scherrer et al. 1995) for the longitudinal magnetic field. Because SFT is a ground-based observation, we interpolated the data gaps due to night or bad weather. The noise level of the transverse field is 150 G, and that of the longitudinal field is 30 G. The spatial resolution of these data is about 200 . The time cadence is 96 minutes (time cadence of the MDI magnetograms). The vector potential is calculated from the longitudinal magnetic field (Chae 2001). The transverse velocity field (of the magnetic
patterns) is determined with the local correlation tracking method (November & Simon 1988). The longitudinal velocity field is calculated by solving the inversion problem of the induction equation. It is noted that this velocity field is virtual and not the real plasma velocity, because any velocity component parallel to the magnetic field can be added without altering the correlationtracking velocity ( Kusano et al. 2002, 2005). This uncertainty does not affect the values of boundary electric field and magnetic helicity injection rate, but the partitioning into H˙ n and H˙ t is not unique. In this paper, we introduce the ‘‘normalized helicity.’’ The magnitude of the magnetic helicity is proportional to the square of the magnetic flux (Berger & Field 1984). Because individual active regions have different magnetic fluxes, we cannot directly compare the magnetic helicity injection rate among active regions unless proper normalization is made. Longcope & Welsch (2000) investigated the magnetic flux tube model analytically and derived HR ¼
qL2 ; 2
ð2Þ
where q is the field line pitch (radian /length), L is the length of a magnetic flux tube, and is the magnetic flux. Magnetic helicity is proportional to the magnetic flux squared, if q and L are independent of . We define the normalized helicity as H ¼ HR =2 ¼ qL=2;
H˙ ¼ H˙ R =2 :
ð3Þ
Using this parameter, we can compare the magnitude of helicity among active regions having different magnetic fluxes. 3. ACTIVE REGIONS We selected sigmoidal active regions from the list of Canfield et al. (1999), which contained regions of 1993 and 1997. The active regions are expected to have persistent observation by the SFT near activity minimum when there were a few active regions on the solar surface, because SFT has a limited field of view. We therefore selected the active regions in 1997 having the SFT data on at least two days and a data gap at less than two days. Nonsigmoidal active regions were also selected in order to be compared with sigmoidal regions. All analyzed regions (four sigmoidal regions and three nonsigmoidal regions) are shown in Table 1. According to Canfield et al. (1999), NOAA AR 8056 was classified as a sigmoidal region. However, we could not
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Fig. 1.—SXT images, MDI magnetograms, and maps of magnetic helicity injection in four sigmoidal regions. Rows A–D represent AR 8011, AR 8015, AR 8038, and AR 8059, respectively. Images in col. (1) are SXT AlMg images. Col. (2) shows MDI longitudinal magnetic field images. White ( black) indicates positive (negative) polarity. Col. (3) shows the magnetic helicity injection rate. White ( black) indicates positive (negative) magnetic helicity injection rate.
identify a sigmoidal loop in AR 8056 in the SXT data. Therefore we designate AR 8056 as a nonsigmoidal region. 3.1. Sigmoidal Regions AR 8011 appeared on 1997 January 14 and disappeared around January 20. Figure 1 (panels A1–A3) shows the SXT image, the longitudinal magnetic field, and the distribution of the magnetic helicity injection rate, respectively. There was no sigmoidal loop
at 15:46 UT on January 16, and a sigmoidal loop was shown around 16:12 UT on January 16. According to Solar-Geophysical Data, this region produced a B1 flare around 16:05 UT and an A5 flare around 19:30 UT on January 16. After the A5 flare, this coronal loop was similar to the potential field. The SXT image indicates that the sigmoidal loop had positive magnetic helicity. Figure 2 ( panels A1–A3) shows the time profiles of the magnetic flux, the magnetic helicity injection rate, and the
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Fig. 2.—Time profiles of magnetic flux, magnetic helicity injection, and time-integrated normalized helicity. Four groups of panels (A–D) correspond to four sigmoidal regions shown in Fig. 1. In each group, subset 1 shows the magnetic flux profile. The thin (thick) line indicates positive (negative) polarity. The dotted lines indicate the analyzed period. Subset 2 shows the magnetic helicity injection. Subset 3 shows the time-integrated normalized helicity. The dotted (dashed) line represents the helicity injection by shearing (emerging) motions. The solid line indicates the total (shear plus emerging) magnetic helicity. Open circles indicate the SFT data for which the time interval between the target time and the observation is shorter than 1 hr.
R time-integrated normalized helicity [H(t) ¼ H˙ R (t) dt= (t)2 ] summed over this region, respectively. We note a similarity between Figure 2 ( panel A2) and the simulation results of Magara & Longcope (2003), who studied the emerging magnetic flux and calculated the magnetic helicity injection. In their simula-
tion, first H˙ n increases, and next, H˙ n decreases and H˙ t increases (see their Fig. 7d ). AR 8011 showed a similar trend but with an additional negative spike in H˙ t . AR 8015 appeared on 1997 January 28 and disappeared on February 7. This region showed no sigmoidal loop before 18:23 UT
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Fig. 3.—Normalized helicity injection rates H˙ (dotted lines) and their time average H˜ (thick lines) of seven active regions. Sigmoidal regions are on the left column, and nonsigmoidal regions are on the right column. The unit of the plots is day1.
on February 1 and showed a sigmoidal loop around 19:11 UT on February 1. Then this sigmoidal loop disappeared at 21:16 UT. Panels B1–B3 of Figures 1 and 2 show the images and physical parameters of this region. The SXT image indicates that the sigmoidal loop had positive magnetic helicity. AR 8038 appeared on 1997 May 5 and disappeared on May 17. There was no sigmoidal loop at 21:32 UT on May 11, and this region showed a sigmoidal loop around 23:11 UT on May 11. The sigmoidal loop gradually decayed, and a C1 flare occurred around 04:45 UT on May 12 (see Fig. 1 of Leamon et al. 2002). One sigmoidal loop was seen at 14:29 UT on May 10, but we did not analyze this loop, because it had very faint footpoints. Panels C1–C3 of Figures 1 and 2 show the images and physical parameters of this region. The SXT image indicates that the sigmoidal loop had negative magnetic helicity. AR 8059 appeared on 1997 July 1 and disappeared on July 10. Panels D1–D3 of Figures 1 and 2 show the images and physical parameters of this region. The SXT image indicates that this region consisted of multiple coronal loops. This region might look like a single sigmoidal loop in long-exposure images. In this paper we define the sigmoid as a single S(or inverse-S) shaped loop. Therefore we regarded AR 8059 as a nonsigmoidal region. Glover et al. (2000) used the same treatment. Note here that sigmoidal loops formed within less than a few hours. Although time cadence of SXT data is no good for these
events, we confirmed that three sigmoidal regions showed a sigmoidal loop in about 12 –2 hr. According to Pevtsov et al. (1996), their sigmoidal loop also formed within about 30 minutes. We use this formation timescale in the following section. 4. RESULTS 4.1. Normalized Helicity Injection and Active Regions Because the sigmoidal regions show clearly helical loops, we naturally expect that the sigmoidal regions have larger normalized helicity injection than the nonsigmoidal regions. We compared the normalized helicity injection rate between the sigmoidal and nonsigmoidal regions. The dotted lines of Figure 3 show the normalized helicity injection rate summed over each region. These injection rates show oscillatory variations with a timescale of 1 hr or so. This tendency is also seen in Figure 2b of Chae (2001). Therefore we used the time-averaged values to compare longer timescale variations. The thick lines in Figure 3 show the averaged values ˜ Each data point of the normalized helicity injection rate, H. on the thick lines represents the average over 15 sample data points ( MDI obtains 15 magnetograms day1). In column (2) of Table 2, the mean and deviation of time-averaged normalized helicity injection rate H˜ are listed. It is shown in Figure 3 and Table 2 that the sigmoidal regions have the normalized helicity injection rate nearly as much as the nonsigmoidal regions,
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TABLE 2 Means and Oscillatory Variations in the Helicity Injection Rates NOAA (1) 8011.......................... 8014.......................... 8015.......................... 8036.......................... 8038.......................... 8056.......................... 8059..........................
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hH˜ i (2) 0.0119 0.0066 0.0238 0.0148 0.0274 0.0226 0.0069
0.0083 0.0037 0.0103 0.0031 0.0056 0.0018 0.0054
hH˙ H˜ i (3)
H˙ (4)
0.032 0.010 0.013 0.023 0.011 0.009 0.009
0.0062 0.0019 0.0033 0.0054 0.0047 0.0041 0.0025
contrary to expectation. The normalized helicity injection rate of the sigmoidal regions may be a little larger than that of the nonsigmoidal regions. In any case, however, the magnitude of the normalized helicity injection rate is 0.01–0.02 day1. This injection rate looks too small for the sigmoidal loop formation, for the following reasons. Here we estimate the normalized helicity of a magnetic flux tube with equation (3). It is assumed that L is nearly equal to the separation of the magnetic polarities. In the flux tube model, the pitch q is written in terms of the pitch angle and the radius of the flux tube $, q¼
tan : $
ð4Þ
H¼
L tan : 2$
ð5Þ
Finally we obtain
Leamon et al. (2003) indicate that most sigmoidal loops show the average shear angle 30 . The normalized helicity is 0.1 for ¼ 30, if we assume $ ¼ 1010 and L ¼ 1010 cm. It would take 5 days to reach the level of H ¼ 0:1 with H˙ ¼ 0:02. As noted in x 3.1, however, the sigmoidal loops were formed within a few hours in our observations. This result is therefore inconsistent with the observation. For the formation of sigmoidal loops, a larger amount of helicity must be injected in a short timescale. Considering that (1) sigmoidal loops are portion of active regions and (2) there is temporal activation of the helicity injection of both signs in helicity injection maps, we look next into the localized magnetic helicity injection around the footpoints of the sigmoidal loops. Is the magnetic helicity injection around the footpoints large enough to make the sigmoidal loop? 4.2. Localized Magnetic Helicity Injection and Sigmoidal Loop Formation In this section, we investigate the relationship between the magnetic helicity injection from the footpoints of the sigmoidal loops and the magnetic helicity content of the sigmoidal loops estimated from geometry. In order to estimate the magnetic helicity of the sigmoidal loops HS , we use, obtained from equations (2) and (4), HS ¼ HR ¼
L2 tan : 2$
ð6Þ
The flux tube length L is nearly equal to the separation of the footpoints. The footpoints of the sigmoidal loops are selected
TABLE 3 Physical Parameters of Sigmoidal Loops WSa (arcsec2)
(10 Wb) 11
HS (10 Wb2) 22
HI b (10 Wb2) 22
AR 8011: L ¼ 5:1 ; 109 cm, L=$ ¼ 4, ¼ 14 6 ; 6 .................... 10 ; 10 ................ 14 ; 14 ................
4.25 0.80 11.3 1.73 20.9 2.65
2.97 1.08 20.9 6.24 70.2 17.7
3.38 0.95 8.55 1.73 14.9 2.52
AR 8015: L ¼ 6:1 ; 109 cm, L=$ ¼ 6, ¼ 30 6 ; 6 .................... 10 ; 10 ................ 14 ; 14 ................
3.16 0.55 7.55 0.89 12.4 1.42
5.68 1.87 31.9 7.48 85.9 19.8
0.17 0.81 1.04 2.21 2.66 3.02
AR 8038: L ¼ 8:2 ; 109 cm, L=$ ¼ 4, ¼ 26 6 ; 6 .................... 10 ; 10 ................ 14 ; 14 ................ a b
2.31 0.71 6.36 1.71 12.7 3.22
1.81 1.11 13.4 7.49 53.1 27.7
4.24 1.44 8.52 2.27 9.83 4.39
Window size. Integration time is 96 minutes.
based on the SXT images and the longitudinal magnetograms. The shear angle is defined ( Pevtsov et al. 1997) as the angle between the two lines as follows: one line connecting the footpoints of the sigmoidal loop, and another line tangent to the sigmoidal loop at the loop midpoint. The radius $ of the flux tube is identified as the distance between the farthest excursion of the sigmoidal loop and the line connecting the footpoints. The magnetic flux of the footpoint region has been calculated for the window sizes of 6 ; 6, 10 ; 10, and 14 ; 14 arcsec2. Here 1000 was taken as a typical width of these sigmoidal loops, and we used other window sizes to investigate the uncertainty of the loop size. The location of the footpoints was chosen with eye estimate. Shifting each window from the footpoints horizontally and vertically between 200 (1 pixel ), we obtained nine values of magnetic flux. Then we finally calculate the mean magnetic flux and its standard deviation for each loop. To compare with the magnetic helicity content of a sigmoidal loop, the magnetic helicity injection rate was evaluated from the final data before the appearance of the sigmoidal loop. The magnetic helicity injection rate was integrated over the same window as where the magnetic flux of the loop was evaluated, and we calculated the mean value and its standard deviation. The magnetic helicity injection rate multiplied by an integration time gives the injected magnetic helicity, HI . The integration time is 96 minutes, which is the cadence of the MDI magnetograms and magnetic helicity injection rate. Since the timescale of the sigmoidal loop formation is of the order of hours, as was indicated in x 3.1 and Pevtsov et al. (1996), we thought that 96 minutes is appropriate. Table 3 shows the physical parameters derived for the sigmoidal loops. As the window size increases, magnetic flux, , HS , and HI also increase. In AR 8015, the magnetic helicity contents of the sigmoidal loop are significantly larger than the injected magnetic helicities in all window sizes. This mismatch could be because this sigmoidal loop is probably made of two loops. In the other two regions, the magnetic helicity contents of the sigmoidal loops are comparable to the injected magnetic helicities in the 6 ; 6 and 10 ; 10 arcsec2 windows, and they are larger than those in the 14 ; 14 arcsec2 window. The size of 1400
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may be too large for the loop size on the photosphere. It is promising that the geometric twist of the sigmoidal loops agrees with the helicity of the coronal magnetic fields for 600 –1000 window size. 5. DISCUSSIONS AND CONCLUSIONS 5.1. Contributions from the Flux Emergence In some of the previous studies (Chae 2001; Moon et al. 2002), the magnetic helicity injection rate due to the emerging motion has been neglected. This may be appropriate for the regions of low emerging flux activity. In panels B2–D2 and B3–D3 of Figure 2, the magnetic helicity injection rate due to the emerging motion (H˙ n , dashed lines) is generally less than that of the shearing motion (H˙ t , dotted lines). From panels A1 and A2 (AR 8011), however, it is clearly shown that H˙ n is comparable to H˙ t when the magnetic flux is increased. As shown by Magara & Longcope (2003), we argue that in a flux emergence activity, the emerging motions are mostly responsible for the magnetic helicity injection, and after the flux emergence, the shearing motions provide most of the magnetic helicity injection. Although there is no strong magnetic helicity injection due to the emerging motion at the beginning of the flux emergence event in panels B2 and B3 (AR 8015), we think this is because of the use of interpolated magnetic field data Bt . For AR 8011 there are vector magnetogram data in the flux emergence period, but for AR 8015 vector magnetograms are not available in the flux emergence period. Because these data are interpolated between, before, and after the flux emergence event, the transverse magnetic fields as well as the magnetic helicity injection rate due to emerging motion may be underestimated. It is also cautioned that the magnetic helicity injection rate by emerging motion is generally underestimated because of the high noise level of the transverse magnetic field measurements (150 G in our case). 5.2. Estimation of Errors As reported by Chae (2001), the magnetic (normalized) helicity injection rate often shows oscillatory variations in Figure 3. ˜ gives the mean of unsigned Column (3) of Table 2 (hH˙ Hi) differences between the normalized helicity injection rates and their time averages. In the following, we argue that these oscillations are not due to noise. As was discussed above, H˙ n is probably strong when there is flux emergence, and usually H˙ t is dominant. Therefore we investigate the oscillation source of H˙ t . The MDI magnetograms are unlikely to be a noise source. The correlation tracking method (CTM ) may be a noise source. In our CTM, the resolution of the velocity component is about 0.061 km s1. To estimate the effect of this uncertainty, we randomly gave deviations of 0.061 km s1 to the velocity components, calculated the helicity injection rate, and evaluated the standard deviation of 100 runs, H˙ ( Table 2, col. [4]). We found that generally H˙ is about 20%–40% of hH˙ H˜ i. Therefore we claim that these oscillations are the result of real motion of magnetic fields on the photosphere. The cause of the oscillations is not clear at this moment and will be investigated in a future paper. Here we mention two sources of uncertainty in evaluating the injected magnetic helicity, HI , and the magnetic helicity content of the sigmoidal loops, HS . One uncertainty is how to specify the integration time of the magnetic helicity injection rate. In this study the helicity integration time is 96 minutes, because our data cadence is 96 minutes and the timescale of the sigmoidal
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loop formation is of the order of a few hours. Our sigmoidal regions showed a persistent sign of magnetic helicity injection around the footpoints for a few hours before the appearance of a sigmoidal loop. For this reason, HI would increase as the integration time is increased by a few hours. Another uncertainty is in the evaluation of the magnetic helicity content of the sigmoidal loops. We evaluated HS by applying a magnetic flux tube model. Using this model and SXT images, we could obtain the writhe of the sigmoidal loops, but not the twist of those. Writhe represents the helicity of the axis of a magnetic flux tube, and twist is the helicity of a bundle of field lines about the axis of the tube ( Berger 1999). The magnetic helicity is the sum of writhe and twist. Therefore, if we could evaluate the twist of the sigmoidal loops, HS will increase. These uncertainties may probably increase HI and HS by a factor of 2 or so. 5.3. Relation to CMEs In this study we have shown that the normalized helicity injection rates of sigmoidal and nonsigmoidal regions are 0.01– 0.02 day1. These helicities must escape to the interplanetary space, some portion of them as CMEs. If not so, then twisted loops should be formed in all of the regions we studied. Many authors have studied the relation between the magnetic helicity injection rate and the magnetic helicity of CMEs (e.g., De`moulin et al. 2002; Green et al. 2002; Nindos & Zhang 2002). Our mean helicity injection rate is similar to the helicity ejection rate estimated by De`moulin et al. (2002), although their AR 7978 is different from our active regions. They reported that a mean CME frequency of AR 7978 is 0.6 day1, the magnetic helicity of a CME (magnetic cloud ) is 2 ; 1042 Mx2, and the magnetic flux of AR 7978 is 1022 Mx. From these values we can estimate that the normalized helicity of a CME is 0.02 and that the helicity ejection rate is 0.012 day1. This rate is nearly equal to our injection rate. However, note that this ejection rate may be larger by a factor of 4 or more, depending on the CME count rate and the assumptions on the calculation of CME helicity (De`moulin et al. 2002). 5.4. Formation of Sigmoids In the previous section, we showed that the magnetic helicity content of the sigmoidal loops and the magnetic helicity injected around the footpoints are comparable in two sigmoidal regions for the 600 –1000 window size (see Table 3). The normalized helicity injected into the sigmoidal loops is of the order of 0.1 (HS /2 HI /2 ) per 96 minutes. This value is larger than the normalized helicity injected into entire sigmoidal and nonsigmoidal regions, 0.01–0.02 day1 (see Fig. 3 and Table 2). The simplest scenario for the sigmoidal loop formation is that a single loop will become a sigmoidal loop because of strong magnetic helicity injection from the photosphere. In this scenario, it is also important to consider the helicity of other loops enclosing the sigmoidal loop. From the conservation of helicity, if the helicity injected to a sigmoidal loop is larger than the helicity injected into the entire active region, then it is inevitable that the helicity of opposite sign must be injected into the other loops in the region. If a sigmoidal loop and the surrounding loops have large enough helicity of opposite signs, then the system can become energetically unstable and may lead to a flare ( Kusano et al. 2004). Some other scenarios are also possible. One of the scenarios is ‘‘tether-cutting’’ scenario ( Moore et al. 2001), in which a long, unstable loop is created from several short loops by magnetic reconnection. It is true that in some of our regions there were several loops before the sigmoidal loop formation.
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Before we ‘‘confirm’’ the sigmoidal loop, magnetic reconnection might have happened between some loops. Therefore, a sigmoidal loop could be formed from a single loop or several short loops having enough magnetic helicity. Sigmoidal loops are due to large magnetic helicity injection, but how about the reverse? Does strong magnetic helicity injection always make the sigmoidal loop? For this analysis, we need more precise and higher cadence data, and more information on the relation between the coronal loop and photospheric magnetic field. Multiwavelength observations (e.g., Gibson et al. 2002) and numerical simulations (e.g., To¨ro¨k & Kliem 2003; Kusano 2005) are useful to clarify the process of sigmoidal loop formation. More precise and persistent observations will be possible in the forthcoming Solar-B mission (Shimizu et al. 2002). 5.5. Conclusions We analyzed the process of magnetic helicity injection in seven active regions. In order to compare the helicity of regions with different sizes, we introduced the normalized helicity injection. We found that the sigmoidal regions as a whole have the normalized helicity injection rate comparable to the nonsigmoidal regions. These normalized helicity injection rates are
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comparable to the normalized helicity ejection rate of AR 7978, derived from CME data. ( De`moulin et al. 2002). Using the magnetic flux tube model, we compared the magnetic helicity injected through the footpoints with the magnetic helicity content of the sigmoidal loops. For two sigmoidal regions, the injected magnetic helicity was found to be consistent with the magnetic helicity content of the sigmoidal loops. One sigmoidal loop showed disagreement between helicity injection rate and its magnetic helicity content. In this region, the sigmoidal loop may consist of multiple loops. Excluding this region, it could be concluded that the twist of these sigmoidal loops is due to the magnetic helicity injected from the footpoints.
We acknowledge an anonymous referee for his/ her useful comments and discussions. We are grateful to M. Hagino for his comments. The present work is supported by the National Astronomical Observatory of Japan and the Nobeyama Solar Radio Observatory. Yohkoh is a mission of the Institute of Space and Astronomical Science (Japan), with participation from the US and the UK. We thank the MDI consortia for providing data. SOHO is a mission of international cooperation between ESA and NASA.
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