the astrophysical journal 939,2000august1 2000...

THE ASTROPHYSICAL JOURNAL, 538 :932È939, 2000 August 12000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

MODELING THE 1994 APRIL 14 POLAR CROWN SXR ARCADE USING THREE-DIMENSIONALMAGNETOHYDROSTATIC EQUILIBRIUM SOLUTIONS

X. P. ZHAO, J. T. HOEKSEMA, AND P. H. SCHERRER

W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305Received 1999 November 16 ; accepted 2000 March 16

ABSTRACTThe coronal magnetic Ðeld in the polar crown soft X-ray (SXR) arcade of 1994 April 14 has been

computed using the global photospheric magnetic Ðeld distribution constructed for the time of interestand the three-dimensional magnetohydrostatic equilibrium solutions. In the solutions there are two freeparameters, a and a, respectively, characterizing the e†ect of the Ðeld-aligned and horizontal electric cur-rents. Magnetic Ðeld conÐgurations have been determined for various combinations of a and a. Themagnetic arcade computed with a ] 0 and a \ 0.9 agrees best in shape and size with the well-developedpolar crown SXR arcade. This suggests that the well-developed SXR arcade has a potential-like magneticconÐguration with nonÈÐeld-aligned currents. The computed loci of the apexes of lines of force that closeabove 1.3 match the shape of the bright axial features superposed on the well-developed SXR arcade.R

_This suggests that newly opened lines of force may have reconnected only above 1.3 and conse-R_

,quently the pre-eruption closed lines of force may not be completely opened up during the eruptionphase. The SXT images before the formation of the SXR loop show the occurrence of dimmings, sup-porting the inference of partially ““ transient ÏÏ coronal opening.Subject headings : Sun: corona È Sun: magnetic Ðelds È Sun: X-rays, gamma rays

1. INTRODUCTION

The soft X-ray (SXR) loop arcade of 1994 April 14occurred over a Ðlament channel with few visible Ðlamentsegments. The Ðlament channel extended parallel to thesouthern polar polarity inversion line. The SXR arcadea†ected an area of D150¡ in longitude and 30¡È40¡ in lati-tude. It is one of largest SXR arcades observed by Yohkoh/SXT. Because of its association with an interplanetarydisturbance observed by Ulysses near 60¡ S latitude(Gosling 1994) and a geomagnetic storm at Earth(McAllister et al. 1994), this event has been considered to bea good sample for coronal mass ejections (CMEs) on thesolar disk (Webb et al. 1997).

The April 14 event was observed almost exclusively byYohkoh/SXT. There was no Ha Ñare, no signiÐcant X-rayÑare, and no signiÐcant Ha Ðlament disappearance associ-ated with the event (McAllister et al. 1996). The Mauna LoaMark III image at D2019 UT on April 13, about 5 hoursbefore the SXR arcade began to form, showed a wide regionof nearly radial structure over the SE limb, positioned overthe latitudinal center of the subsequent SXR arcade. Basedon this image and other images observed on April 9, 14, and21, McAllister et al. (1996) suggested that a CME occurredbetween D2019 UT on April 13 and D0126 UT on April14, and that the SXR arcade formation was the aftermath ofthe CME (Hiei et al. 1993). He 10,830 images showed theformation of large ““ ribbons ÏÏ along the footpoints of thecoronal arcade and gave conÐrmation that a large-scaleCME had probably occurred (Alexander et al. 1994).

Studying of the magnetic Ðeld conÐguration over thearcade area is necessary to understand the cause of theevent. SXR loops are usually regarded as tracers of large-scale magnetic lines of force. However, it is not easy todetermine whether the magnetic Ðeld in SXR arcades ispotential or nonpotential using SXT images alone, espe-cially if the magnetic Ðeld is not far from potential. It is evenmore difficult to know the magnetic Ðeld conÐguration inthe region before the formation of the SXR arcade, espe-

cially whether the lines of force are fully or partially openedup, since no loops were detectable prior to the event.

We previously modeled the magnetic arcade using amonthly synoptic chart of the photospheric magnetic Ðeldand the potential-Ðeld source-surface model (Alexander etal. 1996). The goal of this e†ort is to obtain the magneticÐeld conÐguration in the region of the well-developed SXRarcade by extrapolating the ““ instantaneous ÏÏ global photo-spheric magnetic Ðeld distribution into the corona.

Models commonly used in global extrapolations arepotential-like magnetic Ðeld models (Zhao & Hoeksema1995 and the reference therein). Low (1991) found a generalclass of three-dimensional magnetohydrostatic equilibriasolutions that includes the e†ect of two electric currentsystems, one following the magnetic Ðeld and the other per-pendicular to the gravity Ðeld. All these solutions admit anarbitrary prescription of the magnetic Ðeld at the innerboundary of the domain of interest. Based on this formula-tion Neukirch (1995) has derived a set of spatial solutionsfor the case of a plasma around a spherical self-gravitatingbody. We use this solution to extrapolate the observedphotospheric magnetic Ðeld into the corona. To distinguishthis magnetic Ðeld solution from the sheared magnetic con-Ðguration produced by force-free magnetic Ðeld models, wecall it the nonÈforce-free sheared magnetic Ðeld model.

Section 2 presents the nonÈforce-free sheared Ðeld modeland describes its boundary condition. In ° 3 we show obser-vations of the SXR formation and the photospheric mag-netic Ðeld and computations of the coronal magnetic Ðeld inthe SXR arcade. We conclude with discussion of the resultsin the last section.

2. THE LARGE-SCALE NONÈFORCE-FREE SHEARED

MAGNETIC FIELD MODEL

Well-developed large-scale SXR arcades typically changetheir size with a characteristic velocity of a few kilometersper second. Both the expansion velocity in open regionsbelow the height of cusp points and the characteristic veloc-

932

MODELING POLAR CROWN SXR ARCADE 933

ity of evolving large-scale structures are much less than thelocal velocity. Thus the evolution of these coronalAlfve� nstructures can be approximately treated as a quasi-staticprocess. The magnetic Ðeld in the set of spatial solutionsderived by Neukirch (1995) in the spherical polar coordi-nate system (r, h, /) satisÐes the following equations :

$ Â B \ aB ]C 1r2[ 1

(r ] a)2D¾(r Æ B)] e

r, (1)

where the free parameters a and a are constants that charac-terize Ðeld-aligned and horizontal electric current systems,respectively (Low 1991). The two current systems areseparately conserved (Low 1993). Equation (1) is reduced tothe current-free magnetic Ðeld when both a \ 0 and a \ 0 ;it describes a purely horizontal-current magnetic Ðeld whenonly a \ 0 (Bogdan & Low 1986 ; Zhao & Hoeksema 1994),and a constant a force-free magnetic Ðeld when a \ 0 (e.g.,Nakagawa 1973).

The nonÈforce-free sheared magnetic Ðeld can beexpressed as (Neukirch 1995)

B \ ;n/1

N;

m/0

n;k/1

2 Gn(n ] 1)

gn(k)(r)r2 P

nm(h)G

nm(k)(/)e

r

]Ca

gn(k)(r)r2

Pnm(h)

sin hdG

nm(k)(/)d/

] 1r

dgn(k)(r)dr

dPnm(h)dh

Gnm(k)(/)

Deh

]C1r

dgn(k)(r)dr

Pnm(h)

sin hdG

nm(k)(/)d/

[ agn(k)(r)r2

dPnm(h)dh

Gnm(k)(/)

DeÕH

,

(2)

where N denotes the principal degree, relates to thegn(k)(r)

spherical Bessel function of the Ðrst kind as follows

gn(k)(r)\ 4

5600

Jr ] aNn`(1@2)[a(r ] a]) , k \ 1 ,

Jr ] aJn`(1@2)[a(r ] a)] , k \ 2 .

(3)

is the associated Legendre function with Schmidt nor-Pnm(h)

malization, and

Gnm(k)(/)\ g

nm(k) cos m/] h

nm(k) sin m/ . (4)

There are two sets of unknown expansion coefficientsand to be determined by inner and outer(g

nm(1), h

nm(1) g

nm(2), h

nm(2))

boundary conditions for the nonÈforce-free sheared Ðeldsolution. As in many solar applications, the normal mag-netic Ñux distribution on the solar surface is given and themagnetic Ðeld is taken to vanish at inÐnity high above thesolar surface (e.g., Bogdan & Low 1986 ; Low 1991 ; Neu-kirch 1995). Because the two independent spherical Besselfunctions vanish at inÐnity, each of them is also admissiblefor the system. For the pure horizontal-current magneticÐeld model, the boundary condition at inÐnity requiresomitting one set of expansion coefficients (Bogdan & Low1986). The solution here should have the appropriateasymptotic behavior in the radial coordinates in the limita ] 0. We thus set and The boundaryg

nm(2)\ 0 h

nm(2)\ 0.

values of on the solar surface translate into the innerBrboundary condition

<t>

gnm(1)

hnm(1)=t?

\2n ] 1

IJ;i/1

I;j/1

JC

nB

rijP

nm(h)

(t:

cos m/j

sin m/j

)t;

, (5)

where

Cn\ r

o2

n(n ] 1)Jro] aN

n`(1@2)[a(ro] a)]

. (6)

Here in (5) denotes the global distribution of the radialBrij

component of the photospheric magnetic Ðeld, and I and Jare the number of grid points in latitude and longitude.Corresponding to the spatial resolution of the magneto-gram at the Wilcox Solar Observatory, I\ 30 and J \ 72.The symbol in equation (6) denotes the heliocentric dis-r

otance of the solar surface.It should be noted that the solutions of equation (2) have

a unbounded energy, even though the nonÈforce-freesheared magnetic Ðelds can be made to vanish at inÐnity.This basic problematic property is caused by the slow ratewith which the spherical Bessel functions go to zero at inÐn-ity. As pointed out above, the large-scale coronal magneticÐeld in the well-developed SXR arcade can be treated as aquasi-static Ðeld actually only in the domain below theheight of cusp points of helmet streamers. In other words,the solution may be representative of the solar magneticÐeld only in the near region below the height of cusp points.The difficulty is thus alleviated.

The radial component of the magnetic Ðeld over theentire solar surface is usually derived from a ““monthly ÏÏsynoptic chart (Wang & Sheeley 1992 ; Zhao & Hoeksema1993). The synoptic chart combines central meridian datafrom a series of magnetograms observed during a wholesolar rotation. This approach implicitly assumes that thesurface magnetic Ðeld does not vary appreciably in time.Such maps cease to be useful on the timescale of such evolv-ing large-scale structures as the well-developed SXR arcade,though they have been successfully used in modeling stableglobal-scale structures such as the coronal streamer belt(Hoeksema 1991) and polar coronal holes (Wang & Sheeley1992). Numerical experiments (Zhao et al. 1999) show that anewly emerging bipolar magnetic region near the diskcenter in the unseen hemisphere has a negligible e†ect onthe coronal magnetic Ðeld in the visible hemisphere, to Ðrstorder. This suggests that we can use more timely data fromthe visible disk to construct a more timely global photo-spheric magnetic Ðeld distribution called synoptic frame.We begin with standard synoptic data including 360¡ oflongitude range centered at the time when the coronalstructure of interest is observed. Then we replace the com-posite synoptic chart values in the region of interest withdata from a single full-disk magnetogram. The synopticframe is corrected to make the net magnetic Ñux over entiresolar surface zero (see Zhao et al. 1999 for details). Thisprovides our best estimate of the global solar magnetic Ðeldat the time of the event.

Figure 1 shows how the computed magnetic Ðeld conÐgu-ration changes as the horizontal and Ðeld-aligned currentsystems change. The top-middle panel shows a purelydipole magnetic Ðeld with no currents. The left column dis-plays the e†ects of changing the horizontal current param-eter, a. By decreasing a from 0 to [0.4 (top left panel) themagnetic lines of force contract with respect to the dipoleÐeld ; as a increases from [0.4 to 4.0 the magnetic lines offorce monotonically expand. The middle column showshow the magnetic Ðeld deviates from the dipole magneticÐeld because of the change of Ðeld-aligned currents. When aincreases the magnetic lines of force become more and more

934 ZHAO ET AL. Vol. 538

FIG. 1.ÈThe e†ect of a horizontal current system and a Ðeld-aligned current system on a dipole Ðeld. The free parameters a and a characterize the twocurrent system, respectively. The closed magnetic lines of force expand (left panels) when the horizontal current parameter a is changed. The shear increases(middle panels) as the Ðeld aligned current parameter, a, increases. The Ðeld expands and becomes twisted (right panels) when the two current systems acttogether.

sheared as well as expanding slightly. When both a and aincrease, the magnetic lines of force are expanded andtwisted simultaneously (right column).

3. OBSERVATION AND COMPUTATION

Figure 2 displays eight SXT images observed between1711 UT on April 12 and 1815 UT on April 15. The obser-vation time is shown at the top of each image. The April 14SXR loop arcade Ðrst appeared between 0125 UT and 0246

UT on April 14. The SXT image at 0246 UT showed asingle, sharply cusped loop at DE45S45. Over the next 8È10hours bright arcade loops appeared to the west and to theeast, until the visible structure reached its full extent. The1426 UT image shows the well-developed SXR arcade.After that the clearly deÐned arcade started to fade and theloop arcade became a faint structure with di†use com-ponent. The 0415.1815 UT image shows the faded well-developed SXR arcade. The magnetic conÐguration of the

No. 2, 2000 MODELING POLAR CROWN SXR ARCADE 935

FIG. 2.ÈYohkoh SXT images show the coronal structures before (left column), during, and after the development of the large loop arcade over the southernpolar crown on 1994 April 14.

faint structure looks the same as in the SXR loop arcade.The images in the left column show changes of coronalstructures before the SXR arcade formed.

Figure 3 shows four daily magnetograms measured at theWilcox Solar Observatory between April 12 and 15. Thedashed lines denote the polarity reversal lines, i.e., the zerocontour of the line-of-sight component of the photosphericmagnetic Ðeld. The solid and dotted lines denote contoursof ^2, ^5, ^10, ^20, and ^50 Gauss. All coronal struc-tures in Figure 2, both arcade and faint, are located overpolarity reversal lines in the southern hemisphere. There isonly a slight change in the photospheric magnetic Ðeldbelow the arcade before and after the arcade formation(between 1994 :04 :12–21 :18 UT and 1994 :04 :14–18 :17UT). This is not surprising because Ñux concentrations donot emerge near the polar crown in the declining phase ofsolar activity.

To model the coronal magnetic Ðeld in the well-developed SXR arcade we begin with the appropriate““ monthly ÏÏ synoptic chart centered at 1994 :04 :14–18 :17UT. We then use the data from the 1994 :04 :14–18 :17 UTWSO magnetogram to replace the composite synoptic chartvalues within ^65¡ latitude and ^55¡ longitude of thecenter. This synoptic frame is used to determine the expan-sion coefficients and compute the coronal magnetic Ðeld.

The spherical harmonic order N in equation (2) is taken as 9in all computations. Figure 4 shows the magnetic lines offorce over the southern polar polarity reversal line (thickdashed line) computed with various combinations of param-eters a and a, as shown on the top of each panel. Amongthese combinations, the middle-left panel agrees best inboth shape and size with the well-developed SXR arcade.This shows that Ðeld-aligned currents may be absent in thewell-developed SXR arcade, though there are horizontalcurrents.

There are bright east-west features superposed on thewell-developed SXR arcade, as shown in the 0414.1426 UTimage of Figure 2. The bright features lasted for tens ofhours in the well-developed state. These features areassumed to be formed by heating at the sites of magneticreconnection of newly-opened magnetic lines of force(Koop & Pneuman 1976). A simple projection analysislocates the bright axial features at the apexes of the SXRloops (McAllister et al. 1994). By matching the loci of theapexes of computed closed magnetic lines of force with thebright axial features, the range of heights where the brightfeatures are located may be inferred. Because the magneticÐeld during reconnection is dynamically changing (say,from cusplike to inverse UÈlike) the bright features in thewell-developed state are probably below the site of recon-


FIG. 3.ÈDaily magnetograms measured at the Stanford Wilcox Solar Observatory

nection at the time reconnection is going on. Accordingly,the inferred lowest apex may indicate whether there is alower limit for the real site of reconnection and thuswhether the pre-eruption closed lines of force open up com-pletely during the eruption.

Figure 5 shows four magnetic arcades identical to themiddle-left panel of Figure 4. Filled circles in panels ofFigure 5 denote apexes of the magnetic lines of force. Thephrase ““ lowest brightening height ÏÏ on the top of each panelrefers to the lowest level of all displayed apexes in the panel.The loci of the apexes appear to be similar in thickness tothe observed bright features when the lowest brighteningheight is 1.3 and they are wider than the thickness ofR

_,

the observed bright axial features if it is lower than 1.3 R_

.This may suggest that the magnetic reconnection of newlyopened lines of force started above 1.3 and thus that theR

_pre-eruption closed lines of force are not fully opened upduring the eruption.

In order to search for observational evidence of theopening up of pre-eruption closed lines of force, we closelyexamine the SXT images before the SXR loop formation.The faint di†use structure in the 0412.1711 UT and0413.0224 UT images (see the left column of Fig. 2) is

similar to the di†use component in the 0415.1815 UTimage, implying that they consist of large-scale closed linesof force, even though no loop structure detected in theseimages. The faint structure faded continuously as time wenton. This may be caused by cooling or density decreasingbecause of a helmet streamer distending before eruption(Illing & Hundhausen 1986 ; McAllister et al. 1996). Thefaint structure at 0413.1717 was basically the same in thick-ness as the faint structure at 0414.0125 UT. These faintstructures are much thinner than the structure at 0413.0224UT, suggesting that signiÐcant dimmings occur between0413.0224 UT and 0414.0125 UT. To search for the onsettime of the dimmings, we made a movie using 258 SXTimages between 940412.101728 UT and 940416.230927 UT.The movie clearly shows that the transition of the faintstructure from thick to thin took place near 0413.0935 UTand that before the transition its thickness was decreasingslightly with time. The dimming is believed to be associatedwith opening up of initially closed lines of force and thusassociated with the ejection of coronal mass. The existenceof the thin faint structure suggests the opening is not com-pletely. The wide region of nearly radial structure over theSE limb in the Mauna Loa white-light image observed at


FIG. 4.È1994 :04 :14–18 :17 UT magnetic arcade computed over the southern polar polarity reversal line using various combinations of the two freeparameters (a and a) of the nonÈforce-free sheared Ðeld model.

0413.2019 UT showed the coronal mass ejection just beforethe arcade loop formation, instead of the helmet-streamerdistending before the CME as McAllister et al. (1996) sug-gested. Because of the similar shape and size of the faintstructures observed at 0414.0125 UT and 0414.0246 UT (seeFig. 2) the reconnection may have started at roughly thealtitude indicated by the cusp of the Ðrst arcade loop. Basedon the position of the cusp point we estimate the altitude tobe 1.5 solar radii.

4. SUMMARY AND DISCUSSION

We used the ““ instantaneous ÏÏ global photospheric mag-netic Ðeld distribution with the nonÈforce-free sheared Ðeldmodel to compute the large-scale coronal magnetic Ðeld inthe well-developed large-scale SXR loop arcades observedon 1994 April 14. The model contains e†ects of both hori-zontal and Ðeld-aligned currents. Various magnetic conÐgu-ration may be computed on the basis of observations of the


FIG. 5.ÈThe loci of apexes of closed magnetic lines of force computed to have a lowest brightening heights above the indicated altitude. The computedmagnetic arcade is the same as middle-left panel of Fig. 4.

photospheric magnetic Ðeld. The well-developed SXRarcade is found to be best produced with a ] 0 and a \ 0.9.This suggests that the magnetic arcade is basicallypotential-like without Ðeld-aligned currents. How can thisbe explained? Suppose a nonpotential Ðeld structure opensand then recloses. The shear or twist on initially closedmagnetic lines of force would propagate along the newlyopen magnetic lines of force out of the corona as Alfve� nwaves ; in this way the Ðeld structure rapidly becomes poloi-dal with zero magnetic helicity. On the other hand, thelarge-scale coronal Ðeld must not be current-free, eventhough it looks nearly potential, since if it were current-freethe coronal plasma would be spherical-symmetrically dis-tributed. The homogeneous horizontal current assumedhere is, of course, an simpliÐed approximation for under-standing the real coronal plasma distribution.

The method may be used to model the large-scale coronalmagnetic Ðeld in other structures evolving on timescale of aday or less. It should be noted, however, that all magneticlines of force computed using the nonÈforce-free shearedÐeld model are useful only when their apexes are lower thanthe cusp point of helmet streamers, since the solar windvelocity beyond the cusp point is no longer negligible andthe quasi-static approximation cease to be valid. If a \ 0,the ““ source surface ÏÏ technique that assumes all magneticÐeld at the source surface are radial may be used. In thatcase one can determine the another set of unknown coeffi-

cients in equations (2) and (4) and roughly mimic the e†ectof the interaction between the solar wind and coronal mag-netic Ðeld (Zhao & Hoeksema 1994, 1995).

The loci of computed magnetic arcade apexes above 1.3solar radii have similar shape and size to those of the brightaxial features over the well-developed SXR arcade. Sincethe bright features in the well-developed SXR arcade stateare probably lower than the sites of reconnection at timereconnection is going on, the heliocentric distance of 1.3solar radii inferred here may suggest that the pre-eruptionclosed lines of force do not open up completely during theeruption. The SXT images do show partially ““ transient ÏÏcoronal openings before the appearance of the Ðrst SXRarcade loop. The dimmings began near 940413.0935 UTand lasted for 15 hours. If the onset of the CME occurred atthe time when signiÐcant dimmings started to appear andits ending time is deÐned at the time when coronal massceases to eject, the SXR arcade formation is the aftermath ofthe CME. The delay between the onset of CME and theonset of SXR arcade formation must be considered in pre-dictions of geoe†ectiveness when taking the SXR arcade asthe signature of solar disturbances.

CMEs are generally believed to be a physical processinvolving the release of magnetic energy stored in coronalelectric currents. During the free energy release, the mag-netic Ðeld must be reconÐgured into a lower energy state.The reconÐgured Ðeld must be fully or partially opened to


provide an avenue for the frozen-in plasma to escape theSun. It is usually assumed that initially closed lines of forceare fully opened up during eruption (e.g., Low 1996 ; Anti-ochos, Devore, & Kllimchuk 1999). For a bipolar magneticsystem, force-free magnetic Ðelds with all lines of forceanchored to the coronal base cannot have an energy greaterthan any fully open Ðeld having the same Ñux distributionat the coronal base (Aly 1991 ; Sturrock 1991). Therefore,unless the magnetic system is multipolar (Antiochos et al.1999), the free magnetic energy must be stored in cross-Ðeldcurrents as well as Ðeld-aligned currents (Low & Smith1993). In particular, a detached magnetic Ñux rope helddown by anchored magnetic Ðelds and the weight of plasmamay have sufficient energy (Low 1996). Based on this idea,Wu, Guo, & Andrews (1997) have successfully simulated astreamer coronal mass ejection with a magnetohydro-dynamic model of the interaction of a helmet streamer andan emerging Ñux rope.

The 1994 April 14 event suggests another kind of CME,in which the initially closed lines of force open only partiallyduring eruption. It seems unlikely that changes in thepolarity reversal line under the SXR arcade caused theevent (McAllister et al. 1996). Nor was any emerging Ñuxrope observed before the eruption. The characteristics of theinterplanetary counterpart of the event detected by Ulysses(Gosling 1994) suggests that the event is indeed a massejection, rather than a mass expansion as Antiochos et al.(1999) proposed for the cause of high-latitude CMEs.

The CME of April 13 and the subsequent appearance ofthe SXR loop arcade occurred between two coronal holes ofopposite polarity : the large negative-polarity south polarhole and a signiÐcant equatorial hole of positive polarity.After the event, the southern boundary of the equatorial

hole had moved northward and the northern boundary ofthe polar hole had moved southward (see Fig. 1 of McAllis-ter et al 1996). This implies that some pre-eruption openlines of force became closed during the development of theSXR arcade. Before the sudden dimmings associated withthe eruption, there was slight expansion of the coronal holeboundary corresponding to a slight decrease of the faintstructures. Slight changes of a coronal hole boundary arefrequently observed and may be caused by global physicalprocesses other than CMEs (Zhao & Hoeksema 1999).

The corona is controlled by a competition between twoe†ects : the tendency of the million-degree corona to expandinto the solar wind and the opposing tendency of a bipolarmagnetic Ðeld in a highly conducting plasma to seek aclosed conÐguration and resist the solar-wind Ñow. Inhelmet streamer arcades the tension force of closed mag-netic Ðelds conÐnes high-density plasmas in quasi-staticequilibrium (Low 1990). It may be interesting to study thepossibility that changes of the coronal holesÏ boundariesreduce the tension force of the closed magnetic Ðeld andthus trigger the CME.

We would like to thank Thomas Neukirch for his sugges-tion of using the nonÈforce-free sheared Ðeld model and hishelpful comments on the manuscript. Figure 2 is producedfrom SXT data provided by Eva Alam at GSFC. All com-putations are based on magnetic observations from theWilcox Solar Observatory. This work was supported by theNational Aeronautics and Space Administration undergrants NAGW 2502 and NAG 5-3077, by the AtmosphericSciences Section of the National Science Foundation underGrant ATM 9400298, and by the Office of Naval Researchunder Grants N00014-89-J-1024 and N0014-97-1-0129.

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