c Pleiades Publishing, Ltd., 2013. ISSN 1063-7729, Astronomy Reports, 2013, Vol. 57, No. 10, pp. 778–785.  c B.P. Filippov, 2013, published in Astronomicheskii Zhurnal, 2013, Vol. 90, No. 10, pp. 848–856. Original Russian Text 

Height of a Solar Filament before Eruption B. P. Filippov* Institute of Terrestrial Magnetism, Ionosphere, and Radio-Wave Propagation, Russian Academy of Sciences, IZMIRAN, Troitsk, Moscow, 142190 Russia Received March 23, 2013; in final form, April 04, 2013

Abstract—The relationship between the height of a solar filament observed above the photosphere before the eruption on October 21, 2010, and the critical height of a stable equilibrium of magnetic flux ropes in the coronal magnetic field is analyzed. Data from the SDO, SOHO, and STEREO space observatories observing at different viewing angles makes it possible to deduce these parameters with high accuracy. It is shown that the filament height slowly increased over several days, with the eruption occuring when the height reached the critical value of 80 Mm. DOI: 10.1134/S1063772913100028

1. INTRODUCTION Solar filaments are dark, uneven stripes observed against the solar disk in filtergrams in strong chromospheric lines and in the extreme ultraviolet continuum. These stripes are narrow clouds of dense plasma suspended in the hot corona and heated nearly to the chromospheric temperature. The most remarkable feature of filaments is that they are extended along the lines of the polarity inversion of the photospheric magnetic field. The filaments scatter the photospheric radiation in strong chromospheric lines; therefore, the filaments are darker than the chromosphere, on average. These elements become brighter than the background on the limb, since the brightness of the corona is low in the visible light. In this case, filaments are called prominences. Prominences are well known from solar-eclipse observations obtained decades before the availability of the spectroheliographs and narrow-band filters that make possible observations of filaments, and the term prominence is associated with a wider class of phenomena, including flare and post-flare loops, arcades, ejections, and bright loop systems; nevertheless, we will treat “prominence” and “filament” primarily as synonyms identifying a single physical object, with possibly the only difference being the viewing angle, namely, at the limb or on the disk. The prominences of active regions only project slightly from the mean level of the chromosphere. The heights of these prominences are usually within 10 Mm [1, 2]. Quiescent prominences beyond active regions reach significantly higher altitudes, up to 200 Mm [3]. The sizes and heights of quiescent *

E-mail: [email protected]

prominences grow with time over their lifetimes. The total number of prominences exponentially decreases with height in the range 10–100 Mm [4]. The mean height of prominences is about 30 Mm [5, 6]. Variation in the annual average height of prominences in the course of the solar cycle was first presented in [7], and then verified in [8] by Kodaikanal data obtained from 1912 to 1974. The maximum prominence height is observed near or two years before the activity maximum. In addition, there is a systematic increase in the mean prominence height from 25 to 33 Mm over the period considered. A dependence of the mean prominence height on the cycle phase in the second half of the 21st cycle is also indicated in [9, 10], as well as two maxima in the prominence height distribution at 18 and 33 Mm. Considering only high-latitude prominences, which form an almost continuous polar crown around the polar cap, Markarov et al. [11] and Makarov [12] found that the prominence heights above the global neutral line decreased by a factor of two, from 35 to 16 Mm, as this line drifted toward the pole. Since it is most likely that the magnetic field keeps the cool, dense prominence material from falling into the chromosphere, the field properties should determine the filament equilibrium conditions, in particular, the filament height. Therefore, it is quite reasonable to try to treat the prominence height as a kind of magnetic characteristic [13, 14]. For example, it has been suggested that the prominence height reflects the strength of the background magnetic field [11], or that the prominence height distribution is related to the distribution of the horizontal gradient of the longitudinal magnetic field near the polarity-inversion line [15–18]. 778

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In traditional two-dimensional magnetic models of prominences, equilibrium is possible at any altitude; the question is in the stability of an equilibrium. The classical Kippenhahn–Schluter model [19] assumes an unlimited altitude extent for prominences. A limited vertical size results in inconsistent horizontal and vertical conditions for stability [20]. Models treating prominences as magnetic flux ropes determine the limiting height for stable equilibrium from the spatial features of the magnetic fields of photospheric sources [21, 22]. The transition of the flux rope from a stable to an unstable state represents a catastrophic loss of stability, and is believed to be the origin of sudden eruptions of prominences [23–27]. If we approximate the coronal magnetic field as a function of the height h within a certain interval using a power law B(h) = Ch−n ,

(1)

we can determine the index n for the critical stability of the equilibrium of magnetic flux ropes in such a field. For thin long flux ropes, which can be treated like linear electric currents, the critical index is nc = 1 [22, 23]. There is an additional electromagnetic force applied to curved flux ropes, outward from the curvature center [28]. In this case, the “toroidal” instability is excited when the field decrement is nc = 1.5 [29]. For flux ropes with appreciable cross sections, which in addition increase during eruptions, the critical index is in the interval 1.1–1.3 [30]. When the sources of the field are large-scale, the changes in the field near the photosphere are small (n ≈ 0). At large distances, the field should be nearly dipolar (n ≈ 3). Thus, for every given field, there is a critical height hc , at which the critical index nc is reached. Quiescent prominences should not exist above the critical height. Measurements of magnetic fields in the solar corona encounter extreme difficulties, and these measurements can currently be realized only occasionally, during test experiments [31]. We can estimate the magnetic field decay index dependence of altitude only by extrapolating the photospheric magnetic field to the corona in a potential or force-free approximation [20, 32–34]. Comparison of the height of a filament detected above the photosphere with the calculated limiting height provides information on the filament’s margin of stability, i.e. on the extent to which it is ready to erupt. However, the problem is that we must compare two quantities obtained through ground or near-Earth orbital observations, each of which is detected only when the other is unknown. A filament’s height is best detected at the limb, when the filament is observed as a prominence, but the magnetic field ASTRONOMY REPORTS

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N

Quiescent Eruptive

10

5

0

0.2

0.4

0.6

0.8

1.0

1.2 hp /hc

Fig. 1. Distributions of the ratio of the filament height to the critical height hp /hc for quiescent and eruptive prominences.

in the photosphere beneath the prominence is undetectable, since the photospheric surface is parallel to the line of sight. The most favorable conditions for magnetic measurements occur when the region is near the center of the disk, or at least near the central meridian; i.e., a quarter of the solar rotation, or about a week, later. We expect that the heights of quiescent prominences remains almost unchanged over this period. However, this assumption becomes invalid for prominences that are about to erupt. Some methods for estimating the height of a filament above the chromosphere when the filament is seen against the disk are available, which take into account changes in projection resulting from the solar rotation [35] and the deviation of the filament symmetry plane from the vertical [36, 37]. Though these methods have limited accuracy, the ratio of the filament height to the critical height hp /hc has been analyzed for quiescent and eruptive prominences [38]. Figure 1 shows that this parameter clearly divides prominences into two families, which is promising for predictions of eruptive phenomena. Here, we trace the pre-eruptive state and the eruption of a large filament observed from the ground on October 21, 2010 near the central solar meridian. The filament’s location on the disk was favorable for reliable measurements of the photospheric magnetic fields beneath the filament, while the two STEREO (Solar Terrestrial Relations Observatory) spacecraft observed this filament at the limb, supplying reliable information on the filament’s height above the photosphere. Thus, we can compare the real filament height with the critical characteristic of the coronal magnetic field calculated for precisely the same time.

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(a)

40 30 11117

11116

20 11113

11118 10 0 (b)

(c)

Fig. 2. (a) Position of the filament on the solar disk shortly before the eruption, (b) with respect to the photospheric magnetic ¨ fields, and (c) with respect to the polarity-reversal line at an altitude of 48 Mm. An Hα filtergram obtained at the Kanzelhohe Solar Observatory on October 21, 2010 at 07 : 50 UT and the SOHO/MDI magnetogram obtained at 08 : 02 UT were used. The angular size of fragments (b) and (c) is 880 × 645 .

2. FILAMENT ERUPTION OBSERVED ON OCTOBER 21, 2010 A well developed filament of the intermediate type located between the active regions AR 11113 and AR 11118 suddenly began to rise at about 13h UT on October 21, 2010. Shortly before the eruption, the filament was located almost precisely at the central meridian, in the latitude interval 15◦ N–35◦ N (see Fig. 2). The filament axis was directed about 30◦ to the meridian. The eruption was complete, and no filament traces were seen in Hα filtergrams obtained on ¨ October 22. Hα images obtained at the Kanzelhohe Solar Observatory present only the starting phase of the eruption observed before 14h UT. The entire event was traced by spacecraft observatories. The Solar Dynamic Observatory (SDO) observed the eruption on the solar disk in various extreme ultraviolet ranges using the Atmospheric Imaging Assembly (AIA) telescopes located in geostationary orbit [39]. Due to absorption in the UV continuum, the filament is visible in various channels of the AIA telescopes that select radiation from coronal plasma with temperatures from 1 to 10 MK, with the ˚ 304 Achannel detecting radiation from the transition region being most informative for the event observed (see Fig. 3). The telescopes of the STEREO spacecraft observatory are the first to observe the Sun simultaneously

at different angles [40, 41]. In the Autumn of 2010, the two STEREO spacecraft were almost facing each other at angle distances of 84◦ (STEREO A) and 80◦ (STEREO B) from the Earth along its orbit. At the beginning of the eruption, the filament was precisely at the limb for STEREO A and partially observed against the visible disk near the limb for STEREO B (see Fig. 3). The filament eruption resulted in a coronal mass ejection (see Fig. 4), which was observed by the STEREO/COR2 and SOHO/LASCO C2 (Solar and Heliospheric Observatory/Large Angle Spectrometric Coronagraph) coronagraphs [42]. The latter is located at the Lagrange point L1 on the Sun– Earth line. The fact that the ejection moved in the field of view of this instrument in the northwestern sector at an angle of about 45◦ with the equatorial plane indicates that the eruption occurred with a large westward deviation from the meridional plane. 3. FILAMENT HEIGHT ABOVE THE PHOTOSPHERE The filament height can first be estimated when the filament appears on the limb in the SDO/AIA images. The heights of the upper strands of the filament above the limb reached 22 Mm at 13h UT on October 13, 41 Mm at 14h UT on October 14, ASTRONOMY REPORTS

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STEREO A

13:36:15 UT

13:35:45 UT

15:36:15 UT

17:36:15 UT

19:26:15 UT

15:37:09 UT

17:30:45 UT

19:24:09 UT

14:16:15 UT

16:16:15 UT

18:16:15 UT

STEREO B

12:16:15 UT

Fig. 3. Filament eruption observed on October 21 in the 304 A˚ channel of the STEREO A/SECCHI EUVI (upper row of images), SDO/AIA (middle row), and STEREO B/SECCHI EUVI (lower row) telescopes. The angular size of each image is 650 × 650 .

STEREO A COR2

SOHO LASCO C2

02:09 UT

STEREO B COR2

02:12 UT

02:09 UT

Fig. 4. Coronal mass ejection observed on October 22 by the STEREO A/COR2 (left), SOHO/LASCO C2 (middle), and STEREO B/COR2 (right) coronagraphs.

and 45 Mm at 0h UT on October 15. Since the prominence was partially behind the limb on October 13 and 14, these heights must be corrected. Let us take the point at latitude 25◦ N as the middle of the filament. This point was on the central meridian at 5h UT on October 21. Taking into account the mean synodic angular velocity of the filament rotation at ASTRONOMY REPORTS

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this latitude of 13.38 degrees/day [35], we obtain the filament longitudes corresponding to the observation times λ = 102.6◦ E (behind the limb), λ = 88.7◦ E, and λ = 83.1◦ E, respectively. The true height H of the prominence above the photosphere (see Fig. 5) can be determined as follows. According to the law of sines for the trian-

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h, Mm 300 O R

250

λ

200

H β P

γ

150

h P' δ'

100 δ

50 S 0 12

Fig. 5. Schematic depiction of the observation of the prominence behind the limb.

gle

POP h+R H +R = , sin γ sin β

(2)

where R is the solar radius and h is the visible height of the prominence above the limb. Using the relations γ = π/2 + δ − δ and β = π − λ − δ, we obtain H = (h + R)

δ − δ − R. sin(λ + δ)

(3)

Since δ ≈ δ and δ  λ, the accuracy of the approximate formula 1 −R (4) H = (h + R) sin λ is quite adequate. The prominence heights are then 40, 41, and 50 Mm for October 13, 14, and 15, respectively. The filament heights closer to the beginning of the eruption can be estimated using the STEREO images. For the STEREO A spacecraft, the prominence appeared from behind the eastern limb at about 11h UT on October 19. Although the filament was observed against the disk by the STEREO B spacecraft on all the days preceding the eruption, the filament became fairly high-contrast and visible only when it approached the western limb on October 20. Figure 6 presents the prominence height above the photosphere, taking into account the projection (4). Before 0h UT on October 21, we determined the prominence height for the constant latitude 25◦ N, since the influence of the projections was high. After 0h UT on October 21, we traced the highest edge of the prominence observed above the limb. The edge location changed due to changes in the prominence shape during its activation and eruption. The abrupt

00

12 October 20

00

12 00h October 21 Time (UT)

Fig. 6. Height of the upper edge of the prominence above the photosphere as a function of time. The solid curve indicates the STEREO A/SECCHI EUVI 304 A˚ data and the circles indicate the STEREO B/SECCHI EUVI ˚ 304 A.

jumps in the curve are due to the appearance and disappearance of some filament elements resulting from internal motions of the prominence material. When we can use the images obtained by both STEREO spacecrafts (given the projection conditions and data availability), the heights coincide quite accurately. Thus, the prominence height determined during October 13–20 varied between 40 and 50 Mm, probably due to the appearance and disappearance of individual fine threads resulting from internal motions. During the last day preceding the eruption, the height increased to 75–80 Mm. After 13h UT on October 21, the prominence started to rise slowly with an acceleration below 1 m/s2 and reached a maximum velocity of about 40 km/s in the STEREO/SECCHI EUVI field of view. 4. MAGNETIC FIELD DECREMENT AND CRITICAL HEIGHT The potential magnetic field in the corona can be calculated by solving the outer Neumann boundary problem. Since we calculate the magnetic field at the prominence height, which is small compared to the solar radius, we can neglect the sphericity and use the well known solution for a half-space with a planar boundary (see, for example, [43])   Bn (x , y  , 0)r   1 dx dy , (5) B= 2π r3 S

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October 20 16:03 UT

October 21 12:51 UT

783 October 22 12:51 UT

h = 72 Mm

h = 72 Mm

h = 72 Mm

h = 60 Mm

h = 84 Mm

h = 81 Mm

h = 81 Mm

h = 72 Mm

h = 89 Mm

h = 84 Mm

h = 84 Mm

h = 81 Mm

h = 96 Mm

h = 96 Mm

h = 96 Mm

h = 84 Mm

Fig. 7. Decrements of the magnetic field at various altitudes during successive days. Regions with n > 1 are shown gray and regions with n < 1 light. The bold curves indicate the polarity-inversion lines at the altitudes presented. The short segments shown at an altitude of 72 Mm for October 21 indicate the filament ends visible in Hα filtergrams (see Fig. 2). The angular sizes of the fragments shown are somewhat different on different days; for the region observed on October 21, the size is 880 × 645 .

where Bn is the normal component of the magnetic field in the plane S, and r is the radius vector from a certain point on the surface to a given point in the corona. Real measurements of the solar magnetic field detect the mean line-of-sight component in a certain area determined by the magnetograph resolution. Within this small rectangular area, the integrals in (5) can be calculated analytically [44], and the field sought is the sum of contributions of all areas in the plane. We calculated the decrement for the horizontal component of the potential magnetic field Bt of the photospheric sources determining the vertical equilibrium of the magnetic flux rope, ∂ ln Bt , (6) n=− ∂ ln h using SOHO/MDI (Michelson Doppler Imager) data [45]. Extracting a rectangular area surrounding the ASTRONOMY REPORTS

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filament being analyzed from the magnetogram of the entire disk, we neglected the contributions of sources located beyond the area chosen. This is reasonable when the size of the area is much larger than the filament height, or if the main field sources (active regions) are within the area extracted. Although the filament was on the central meridian before the eruption, the filament latitude was responsible for the difference between the line of sight from the ground and the normal to the photospheric area beneath the filament. In the preceding days, the longitude deviation also contributed in this difference. We took the projection of the line-of-sight field onto the normal to be the normal field component necessary for the Neumann problem. The integration technique proposed in [44] requires a boundary array of cells with identical linear sizes. We must then project this grid onto the spherical surface of the photosphere and take the field of the nearest pixel in

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the magnetogram as the value for each cell. This yields a rectangular region with identical square cells, as if the region were located at the center of the solar disk. Figure 7 presents the decrements for the potential magnetic field at various heights within the region surrounding the filament during October 18–22 (there are no magnetograms for October 19). Regions with n > 1 are gray and regions with n < 1 are light. Polarity-inversion lines, on which stable magnetic flux ropes are expected, are calculated for each height. In reality, the filament was observed only on a certain section of the line, indicated by two short segments in the fragment for October 21 in the upper row of maps in Fig. 7. In this row, the segment with the filament is located in the region with n < 1 (light), where the magnetic flux rope is stable. As the altitudes grow, the gray areas broaden and cover the segment of the polarity-inversion line with the filament. The altitude where the edge of the unstable (gray) zone becomes tangent to the polarity-inversion line is the limiting critical height for the filament. Though the magnetic field did not strongly change near the filament during the days analyzed, this was indicated by the similarity between the field decrements and the shape of the polarity-inversion line, with the critical heights being somewhat different. The critical height was 89 Mm for October 18, 81 Mm for October 20 and 21, and 72 Mm for October 22 (see Fig. 7); thus, the photospheric magnetic field slowly changed, reducing the critical height for the stable equilibrium. At the same time, the actual height of the filament above the photosphere increased slightly in the days preceding the eruption. A comparison of Figs. 6 and 7 shows a good agreement between the critical height found for October 21 and the height from which the filament rose rapidly upwards. 5. CONCLUSIONS In a model in which most of the material of a prominence is contained in magnetic flux ropes, the height of a prominence above the photosphere characterizes the filament’s stability. This height must be compared with the limiting height for the stable equilibrium of the magnetic flux ropes in the coronal magnetic field, with the latter height being determined by the vertical gradient of the magnetic field. For flux ropes with straight or circular axes, the instability begins at the height where the decrement of the coronal field reaches the value n = 1−1.5. It is fairly difficult to verify this criterion for real solar filaments, using observations carried out only at one place (on the ground or in near-Earth orbit). The viewing angle required to determine the filament height above the limb is unsuitable for measurements

of the photospheric magnetic field beneath the filament and vice versa; i.e., the situation when the filament is located near the center of the solar disk, with the normal to the photosphere being directed along the line of sight, is favorable for magnetographic measurements, but makes measurements of the filament height almost impossible. The launch of the two identical STEREO spacecraft flying in opposite directions along the Earth’s orbit makes it possible to obtain UV solar images simultaneously for different viewing angles. The filaments on the solar disk are more poorly visible in UV lines than in Hα, but prominences at the limb are clearly visible, especially in the 304 A˚ channel. Therefore, the most favorable situation for the analysis of the onset of eruptions is when the filament erupts from region located near the center of the disk for ground observers. The filament eruption of October 21, 2010 occurred on the central meridian. This eruption was observed on the disk by the SDO and SOHO space observatories. The beginning of the eruption was also ¨ observed by the Kanzelhohe ground solar observatory. The filament was located at the limb for both STEREO spacecraft, and both instruments observed the eruption dynamics in detail. We have calculated the decrements for the potential field in the corona during October 18–22, as well as the critical height at which the n = 1 contour becomes tangential to the polarity-inversion line. This height slowly decreased from 89 to 72 Mm over five days. At the same time, the filament height above the photosphere gradually increased from 40 to 80 Mm during October 13–21. The eruption occurred when the filament height reached its critical value. In contrast to earlier studies, the filament height was detected with a high accuracy in our study, since observations were carried out at three viewing angles making right angles to each other. The magneticfield measurements are also reliable, since the region examined was located on the central meridian in the magnetogram. The height at which the filament started to rapidly rise agrees well with the critical height for straight magnetic ropes; this indicates that the force associated with the axis of curvature is essentially not important for the equilibrium of real flux ropes. The event studied shows that the ratio of the prominence height to the critical height is a useful parameter characterizing the stability of the filament equilibrium in the coronal magnetic field. This parameter can be monitored for predictions of eruptive phenomena. The realization of this method with high accuracy requires observations of the Sun at several viewing angles. ASTRONOMY REPORTS

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ACKNOWLEDGMENTS ¨ SoThe author thanks the staff of the Kanzelhohe lar Observatory and the STEREO, SOHO, and SDO scientific teams for the observational data provided. This work was supported by the Russian Foundation for Basic Research (projects 12-02-00008 and 1202-92692). REFERENCES 1. B. Schmieder, in Dynamics and Structure of Solar Prominences, Ed. by J. L. Ballester and E. R. Priest (Univ. de les Illes Balears, Mallorca, 1987), p. 5. 2. B. Schmieder, in Dynamics and Structure of Quiescent Solar Prominences, Ed. by E. R. Priest (Kluwer Academic, Dordrecht, 1989), p. 15. 3. B. Rompolt, Hvar Observ. Bull. 14, 37 (1990). 4. R. Ananthakrishnen, Astrophys. J. 133, 969 (1961). 5. S. O. Obashev, Izv. Krymsk. Astrofiz. Observ. 29, 118 (1963). 6. V. S. Dermendzhiev, Astrofiz. Issled. (Sofia), No. 2, 8 (1977). 7. A. M. Cantu, G. Godoli, and G. Poletto, Mem. Soc. Astron. Ital. 38, 367 (1968). 8. V. I. Makarov, Soln. Dannye, No. 4, 100 (1983). 9. I. S. Kim, V. Yu. Klepikov, S. Kuchmi, et al., Soln. Dannye, No. 1, 75 (1988). 10. I. S. Kim, V. Yu. Klepikov, S. Kuchmi, et al., Soln. Dannye, No. 5, 77 (1988). 11. V. I. Makarov, K. S. Tavastsherna, E. I. Davydova, and K. R. Sivaraman, Soln. Dannye, No. 3, 90 (1992). 12. V. I. Makarov, Solar Phys. 150, 359 (1994). 13. J. L. Leroy, V. Bommier, and S. Sahal-Brechot, Astron. Astrophys. 131, 33 (1984). 14. I. S. Kim and V. F. Uvakina, in Atmosphere of the Sun, the Interplanetary Medium, and Planetary Atmospheres, Collection of Articles, Ed. by R. A. Gulyaev (IZMIRAN, Moscow, 1989), p. 125 [in Russian]. 15. V. P. Maksimov and L. V. Ermakova, Sov. Astron. 29, 323 (1985). 16. S. A. Yazev and G. M. Khmyrov, Soln. Dannye, No. 12, 75 (1987). 17. B. A. Ioshpa and E. Kh. Kulikova, in Solar Magnetic Fields and the Corona, Collection of Articles, Ed. by R. B. Teplitskaya (Nauka, Novosibirsk, 1989), Vol. 1, p. 167 [in Russian]. 18. V. P. Maksimov and A. A. Prokop’ev, Astron. Rep. 37, 554 (1993). 19. R. Kippenhahn and A. Schluter, Zs. Astrophys. 43, 36 (1957). 20. B. P. Filippov and O. G. Den, Astron. Lett. 26, 322 (2000).

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21. M. Kuperus and M. A. Raadu, Astron. Astrophys. 31, 189 (1974). 22. W. van Tend and M. Kuperus, Solar Phys. 59, 115 (1978). 23. M. M. Molodenskii and B. P. Filippov, Sov. Astron. 31, 564 (1987). 24. E. R. Priest and T. G. Forbes, Solar Phys. 126, 319 (1990). 25. T. G. Forbes and P. A. Isenberg, Astrophys. J. 373, 294 (1991). 26. J. Lin, T. G. Forbes, P. A. Isenberg, and P. Demoulin, Astrophys. J. 504, 1006 (1998). 27. B. Schmieder, P. Demoulin, and G. Aulanier, Adv. Space Res. 51, 1967 (2013). 28. G. Bateman, MHD Instabilities (Massachusetts Inst. Technol., Cambridge, MA, 1978). ¨ ok, ¨ Phys. Rev. Lett. 96, 255002 29. B. Kliem and T. Tor (2006). 30. P. Demoulin and G. Aulanier, Astrophys. J. 718, 1388 (2010). 31. S. Tomczyk, in Science with Large Solar Telescopes, ID E2.14. http://www.arcetri.astro.it/IAUSpS6 32. B. P. Filippov and O. G. Den, J. Geophys. Res. 106, 25177 (2001). 33. B. Filippov and A. Zagnetko, J. Atmosph. Sol.-Terr. Phys. 70, 614 (2008). 34. A. Nindos, S. Patsourakos, and T. Wiegelmann, Astrophys. J. Lett. 748, L6 (2012). 35. B. Vrsnak, D. Rosa, H. Bozic, et al., Solar Phys. 185, 207 (1999). 36. M. d’Azambuja and L. d’Azambuja, Ann. Observ. Paris. Meudon 6, Fasc. VII (1948). 37. A. M. Zagnetko, B. P. Filippov, and O. G. Den, Astron. Rep. 49, 425 (2005). 38. B. P. Filippov, A. M. Zagnetko, A. Adzhabshirizade, and O. G. Den, Solar Syst. Res. 40, 319 (2006). 39. J. R. Lemen, A. M. Title, D. J. Akin, et al., Solar Phys. 275, 17 (2012). 40. J.-P. Wuelser, J. R. Lemen, T. D. Tarbell, et al., Proc. SPIE 5171, 111 (2004). 41. R. A. Howard, J. D. Moses, A. Vourlidas, et al., Space Sci. Rev. 136, 67 (2008). 42. G. E. Brueckner, R. A. Howard, M. J. Koomen, et al., Solar Phys. 162, 357 (1995). 43. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1972), p. 363 [in Russian]. 44. O. G. Den, O. E. Den, E. A. Kornitskaya, and M. M. Molodenskii, Soln. Dannye, No. 1, 97 (1979). 45. P. H. Scherrer, R. S. Bogart, R. I. Bush, et al., Solar Phys. 162, 129 (1995).

Translated by V. Badin