A. A. van Ballegooijen- Observations and Modeling of a Filament on the Sun

8/3/2019 A. A. van Ballegooijen- Observations and Modeling of a Filament on the Sun

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OBSERVATIONS AND MODELING OF A FILAMENT ON THE SUN

A. A. van Ballegooijen

Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS 15, Cambridge, MA 02138; [email protected]

Receivved 2004 February 21; accepted 2004 May 11

ABSTRACT

H observations of a filament were obtained at the Swedish Vacuum Solar Telescope in 1998 June. TheU-shaped filament has a prominent barb that exhibits interesting fine structure and internal motions. A three-dimensional magnetic model of the filament is presented. The model is based on a National Solar Observatory(Kitt Peak) magnetogram and is constructed by inserting a twisted flux rope into a potential field representing theoverlying coronal arcade; the flux rope has an axial flux of 3:4 ; 1019 Mx and poloidal flux of 3:7 ; 109 Mx cm1.Magnetofrictional relaxation is used to drive the configuration to a nonlinear force-free field. The shape of theresulting flux rope is distorted by neighboring network elements. The dips in the helical field lines reproduce theobserved filament barb, which is caused by a local distortion of the flux rope resulting from a weak-field extension($4 G) of a neighboring network element. The pitch of the helical field lines is larger than expected on the basis ofa model of flux rope formation. I suggest that this is due to magnetic diffusion within the flux rope. A simple modelof magnetic diffusion in a cylindrical flux rope is presented.

Subject headinggs: MHD Sun: corona Sun: filaments Sun: magnetic fields

1. INTRODUCTION

Filaments (aka prominences) have long been used as probesof the magnetic field in the solar atmosphere. Filaments lieabove polarity inversion lines (PILs) on the photosphere andoccur within filament channels, bands in which the chro-mospheric fibrils emanating from neighboring network ele-ments are aligned with the PIL (Gaizauskas et al. 1997, 1998;Martin 1998). This alignment indicates that the magnetic fieldin a filament channel has a strong axial component. Foukal

(1971) showed how the direction of the axial field within achannel can be deduced from the observed fibril pattern andknowledge of the magnetic polarities of the network elements.Martin et al. (1992, 1994) applied this technique to many fila-ment channels. They characterized filament channels as eitherdextral or sinistral depending on the direction of the axialfield as seen by a hypothetical observer standing on the positivepolarity side of the channel. This so-called chirality of filamentchannels is correlated with latitude on the Sun. Martin et al.(1994) showed that mid-latitude channels in the northernhemisphere are predominantly dextral, while those in the southare predominantly sinistral.

Low-lying filaments often have a dark spine that runsalong the top of the filament. Quiescent filaments have barbs

protruding from the side of the filament (Martin et al. 1992,1994; Martin 1998). When observed at the solar limb, thesebarbs are seen to extend down from the spine to the chromo-sphere below. When observed at disk center, the barbs are seento protrude at an acute angle with respect to the long axis of thefilament (like ramps off an elevated highway). Martin et al.(1992) classified filament barbs as either right-bearing orleft-bearing depending on the acute angle of the barb as seenfrom above. They found a strong correlation between the right-and left-bearing structure of the filaments and the chirality ofthe underlying channel; filaments in dextral channels haveright-bearing barbs, and those in sinistral channels have left- bearing barbs. Overlying these filaments are left-skewed orright-skewed coronal arcades ( Martin & McAllister 1996;

McAllister et al. 2002).

The ends of the barbs appear to be connected to weakmagnetic fields in between the network elements (Martin et al.1994). Martin & Echols (1994) proposed that the filamentbarbs are anchored in parasitic polarity elements, i.e., weakmagnetic fields with opposite polarity compared to the net-work elements on the side of the filament where the barb islocated. Wang (1999, 2001) showed that filament barbsoverlie small-scale PILs on the side of the filament. He pro-posed that flux cancellation between the parasitic polarity andthe neighboring dominant polarity plays a key role in the

formation of filament barbs. Flux cancellation is a process thatoccurs everywhere on the Sun (Livi et al. 1985; Martin et al.1985) but is particularly common at large-scale PILs on thequiet Sun where opposite polarity elements intermix and azone of mixed polarity is created. At the edge of such a mixed- polarity zone, parasitic flux elements are likely to cancelagainst dominant polarity elements. Wang (1999, 2001) ar-gued that magnetic reconnection accompanying photosphericflux cancellation is the dominant mechanism for injectingmass into quiescent prominences.

Many authors have suggested that filaments are supportedby helical flux ropes that lie horizontally above the PIL (e.g.,Kuperus & Raadu 1974; Pneuman 1983; Priest et al. 1989; vanBallegooijen & Martens 1989; Rust & Kumar 1994; Low &

Hundhausen 1995; Chae et al. 2001). Consistency with thedirection of the magnetic field in the overlying arcade requiresa left-helical flux rope in a dextral channel and a right-helicalflux rope in a sinistral channel. Rust & Kumar (1994) inter- preted the barbs as the lower parts of the helical windings.Aulanier & Demoulin (1998) proposed that barbs are located atbald patches, sites where the magnetic field is tangential tothe photosphere and curved upward. Such bald patches occurnaturally near parasitic polarities on the photosphere.

Zirker et al. (1998) observed counterstreaming of plasma infilaments. Using observations in the red and blue wings ofthe H line, they showed that matter flows along the spine ofthe filament in both directions. Similar counterstreamingoccurs within the barbs; in one wing of the line they observed

up-flows from the barbs into the spine, while in the other519

The Astrophysical Journal, 612:519529, 2004 September 1

# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.


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wing they saw flows in the opposite direction. Zirker et al.(1998) proposed that a filament barb consists of a set ofclosely spaced flow channels that are highly inclined with re-

spect to the vertical direction, with some channels having up-flows and others having down-flows. The velocities involvedare 510 km s1, much less than the free-fall velocity cor-responding to the filament height. Lin et al. (2003) foundsimilar velocities of counterstreaming in a large polar crownfilament.

Aulanier and collaborators developed three-dimensionalmagnetic models of filaments using extrapolation of observed photospheric magnetograms (Aulanier & Demoulin 1998;Aulanier et al. 1998, 1999, 2000; Aulanier & De moulin 2003).The coronal magnetic field B(r) is modeled using either linearforce-free fields (: < B k0B with k0 = constant) or linearmagnetohydrostatic fields (Aulanier et al. 1999). The mainbody of the filament is assumed to be supported by a helical flux

rope; the cool plasma is located at local dips in the helicalwindings. The authors show that there are dips in the field linesabove parasitic polarities some distance away from the main body of the filament, forming barblike extensions of the fila-ment. This model can explain many of the observed features ofbarbs, including their right- or left-bearing structure. However,linear force-free (or hydrostatic) models cannot accurately de-scribe theglobal structure of the magnetic field in which the fluxrope is embedded. Nonlinear force-free models are needed.

In this paper new high-resolution H observations of afilament are presented (x 2), and a three-dimensional magneticmodel of this filament is developed (x 3). The model is basedon a National Solar Observatory (Kitt Peak) (NSO/KP) lon-gitudinal magnetogram taken 1 hr before the Swedish Vacuum

Solar Telescope (SVST ) observations. In x 4 the effects of

magnetic diffusion on filament flux ropes are considered. Theconclusions are presented in x 5.

2. OBSERVATIONS

Figure 1 shows the southern part of a full-disk H imagetaken at the Big Bear Solar Observatory (BBSO) on 1998 June21 16:48 UT. The object of interest is theU-shaped filament onthe southern side of NOAA active region8249. High-resolutionH observations of this filament were obtained at the SVSTbetween 18:03 and 19:04 UT. The small rectangle in Figure 1indicates the SVST field of view (FOV) at the start of theobservations (the SVST FOV rotates by 23

in the course of the

1 hr observation). The filament erupted 6 hr after the SVSTobservations, starting at about June 22 00:30 UT. The eruptionwas observed with the Transition Region and Coronal Explorer(TRACE). The large rectangle in Figure 1 shows the TRACEFOVat the time of the eruption. Note that there is another,larger

filament farther south at the western edge of the TRACEFOV.The eruption of the U-shaped filament caused both filaments tobe ejected, resulting in a halo coronal mass ejection (CME), oneof several CMEs observed on 1998 June 22.

The formation of the U-shaped filament is illustrated inFigure 2, which shows a series of H images extracted fromthe BBSO full-disk images for the period 1998 June 1621.Note that on the days before the SVST observations there aretwo separate filaments that slowly approach each other andfinally merge into a single U-shaped structure. The triangularbarb observed on June 21 is located at the site where the twofilaments merged on the previous day.

The SVST observations were obtained with the Lockheedtunable filtergraph and consist of a time series of narrowband

filtergrams at the H line center (k6563) and the red and blue

Fig. 1.BBSO H image of the Sun for 1998 June 21. The target filament is the U-shaped feature at the center of the image. The rectangles show the TRACEandSVST fields of view.

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wings of the line (300 m8). The data were recorded with aKodak Megaplus 1.6 CCD camera (exposure time 105 ms atline center, 78 ms in the wings). To mitigate the effects ofseeing, real-time frame selection was used and only the bestframe acquired during a 5 s time interval was retained. This produced three interleaved image sequences at the threewavelengths, each consisting of 167 images with a cadence of21 s. The FOV of these images is 18700 ; 12500 , and the bestframes have a spatial resolution of about 0B5.

Data reduction involved several steps. After flat-field cor-

rection, 2;

2 pixel binning was applied to increase the signal-to-noise ratio of the images. This resulted in images with ascale of 4.4 pixels arcsec1. To correct for image trackingerrors, image shifts were measured using visual inspection andcomparison of images belonging to the same wavelength off-set. The rotation angle of each image was determined based ontime of observation and the known properties of the SVST.Finally, correlation tracking was used to correct for imagedistortions due to seeing. This correction is important forobtaining good Dopplergrams and movie sequences.

The following distortion correction procedure was used.First, each rigidly aligned, rotated image was divided into 9 ; 6subfields, and subfields in neighboring frames were cross-correlated to produce an array of vectors representing the

subfield displacement from one image to the next. Then the

cumulative displacement of each subfield was computed andhigh-passfiltered to prevent slow image drifts. These vectorswere interpolated onto the image to obtain the seeing distortionfor each pixel (these vectors also correct for any residual errorsin the rigid alignment). Finally, each original image wasdestretched using a single mapping that includes correctionsfor image tracking errors, image rotation, and seeing distortion.This produced a time series of corrected images in H

line

center (Ic) and in the red and blue wings (Ir and Ib). The meanvalues of Ir and Ib over an image are larger than that of Ic, asexpected for an absorption line profile. However, the relativeintensity calibration of the three channels is not accuratelyknown. Therefore, Ir and Ib were multiplied by factors of 0.92and 0.96, respectively, to make the mean values of Ir and Ibover an image approximately equal to that of Ic. This improvesthe resulting Dopplergram movies but removes informationabout the mean intensity as a function of wavelength.

Figures 3a3c show corrected filtergrams obtained at18:18:21 UT. In these figures solar north is toward the lowerright of the image ( cf. the small rectangle in Fig. 1). Figure 3 dshows the resulting Dopplergram, V (Ir Ib)=(Ir Ib),

which is a measure of motion along the line of sight (LOS;white represents motion toward the observer). Figure 3e showsthe map ofW 0:5(Ir Ib)=Ic, the ratio of line-wing and line-center intensities. These figures show the filament and thesurrounding chromosphere. Figure 3fshows the LOS magneticfield extracted from an NSO/KP full-disk magnetogram taken between 17:04:35 and 17:59:16 UT, before the start of theSVST observations. The magnetogram was aligned with theH images using the strong network elements. In Figures 3a3c the network elements appear as bright rosettes of chromo-spheric fibrils. The streaming direction of these fibrils (Foukal1971) indicates that the filament is located in a sinistralchannel.

The filament has a prominent barb that protrudes from the

side of theU

-shaped filament and points toward its center ofcurvature. The barb has a left-bearing structure (as defined byMartin et al. 1992), consistent with the sinistral channel. Thebarb has a triangular shape with one side of the triangle runningfrom left to right in Figure 3a along the dark spine of thefilament. The barb is asymmetric: its right edge (in the figure)is long and smooth, while the left edge is short and ragged. Inline-center images the middle of the barb appears diffuse andunstructured, but line-wing images show a collection of thinthreads that veer off the main path of the filament. Thesethreads are especially clear in Dopplergrams (e.g., Fig. 3d),indicating that different threads have different motions alongthe LOS. The threads terminate at the left edge of the barb,creating its ragged appearance. In contrast, the right edge of the

barb is parallel to the threads and appears to be a continuationof the base of the filament. It appears that the smooth (right)edge and tip of the barb lie just above the chromosphere,whereas the ragged edge is elevated above the chromosphere;i.e., the barb lies in a plane that is roughly perpendicular to theLOS but inclined with respect to the local vertical.

The fine threads within the barb presumably outline the localmagnetic field, but the structure of the field at the ragged edgeis unclear. I suggest that the field lines continue beyond theragged edge but that the plasma becomes invisible in H . Timesequences of images were used to study the motion of threadswithin the barb. During the first 10 minutes of observations,threads are seen to move upward along the ragged edge towardthe spine of the filament. This is illustrated in Figure 4, which

shows a time series of Dopplergrams and line-center images.

Fig. 2.Evolution of target filament over 6 days (1998 June 1621) priorto the SVST observations. These images are extracted from full-disk Himages obtained at BBSO.

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The dark thread seen in the line-center images moves towardthe spine with velocity 10:2 0:4 km s1 (projected onto the

plane of the sky). This motion is attributed to transversemotion of a magnetic flux tube within the barb and suggeststhat the barb is a dynamic structure with a magnetic field thatchanges in time. The Dopplergrams show a combination ofredshifts and blueshifts within the barb; however, because ofthe adjustment of line-wing intensities (see above), it is im-possible to assign accurate LOS velocities to these shifts. Thereis no evidence for the counter-streaming flows observed byZirker et al. (1998). This may be due to a combination of theshort duration of the observation (1 hr) and the unfavorableviewing angle (barb is perpendicular to LOS).

There are two positive polarity (white) network elementsnear the top of Figure 3fthat are located behind the filamentand therefore not visible in the H images. One of these ele-

ments is located directly behind the filament barb. Martin et al.

(1994) and Wang (2001) suggested that filament barbs may beconnected to weak, parasitic polarity fields in the photosphere.

The NSO/KP magnetogram shows weak fields almost every-where between the network elements, including near the tip ofthe barb. However, these fields are very weak, and it is doubtfulthat they have a significant effect on the magnetic structure ofthe filament. Full-disk magnetograms from the MichelsonDoppler Imager (MDI) on board the Solar and HeliosphericObservatory for June 20 and 21 (not shown) were also ana-lyzed. There is no evidence from these magnetograms that thestronger network elements near the barb are involved in fluxcancellation. The noise in these magnetograms is too large tosee the weak fields.

On the right-hand side of Figure 3a, the filament has twostraight, parallel threads that presumably outline the localmagnetic field. If the flux rope containing the filament were

strongly twisted, there would be only one straight field line,

Fig. 3.Swedish Vacuum Solar Telescope (SVST ) observations of target filament obtained on 1998 June 21 at 18:18:21 UT using the Lockheed tunablefiltergraph: (a) H blue wing, (b) H line center, (c) H red wing, (d) Dopplergram V, and (e) ratio Wof line-wing and line-center intensities. Panel ( f) shows a co-aligned magnetogram extracted from the full-disk NSO/Kitt Peak magnetogram. White indicates magnetic field directed toward from the observer.



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namely, the axis of the helical structure. The presence of theseparallel threads argues against strong twist in this part of thefilament.

3. MAGNETIC MODEL

In this section a three-dimensional magnetic model of thefilament is developed. The filament is assumed to be sup-ported against gravity by a helical flux rope. The flux rope issurrounded by a coronal arcade that prevents the flux ropefrom moving radially outward. I assume that the arcade can beapproximated using potential-field modeling. The flux ropehas both axial and poloidal field components that are spatiallyseparated from each other and distinct from the overlying

arcade. The axial flux axial (in Mx) and the poloidal flux Fpol

per unit length along the filament (in Mx cm1) are treated asfree parameters.

3.1. Model Construction

The computational domain is a rectangular box with itslower boundary on the photosphere (the curvature of the solarsurface is neglected). I define Cartesian coordinates (x; y; z),where z is the height above the photosphere. The box size is326 Mm in x and y and 145 Mm in height. The verticalmagnetic field on the photosphere, Bz(x; y; 0), is determined by extracting the relevant region from the full-disk magneto-gram and correcting for foreshortening effects (the observedfield is assumed to be radial at the photosphere). The magnetic

field B(r) is described in terms of the vector potential A(r)

Fig. 4.Time series illustrating the dynamics of fine structures within the barb. Images in H line center and Dopplergrams are shown at eight different times.



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(B : < A). The grid is uniform, consisting of cubic cells ofsize 1:45 Mm, corresponding to 200 on the Sun. The components of the vectors A and B are defined on different pointsof the grid: for example,Ax is defined at the middle of thex-ribs ofthe cells, Ay at the middle of the y-ribs, etc. (here rib refers tothe connection between two corner points of the cubic cells). Themagnetic field components are defined on the cell faces.

The first step in the construction of the three-dimensionalmagnetic model is to select a path x(s); y(s) along the PILwhere the flux rope will later be inserted (s measures distancealong the path). The path is shown in Figure 5 on a gray-scaleimage ofBz(x; y; 0). The circles at the two ends of the path aresites where the flux rope can be anchored in the photosphere.The net flux in each circle is measured and subtracted from thecentral cell to produce a modified flux distribution B0z(x; y; 0).This guarantees that the original flux distribution Bz(x; y; 0)will be recovered after the flux rope is inserted.

The second step is to compute the potential field associatedwith B0z(x; y; 0). The vector potential Ap of this field can bewritten as

Ap(x; y; z) @h

@yx

@h

@xy; 1

where h(x; y; z) is the solution of the Laplace equation ( 92

h 0). The boundary conditions are (1) @2h=@z2 Bz(x; y; 0) atz 0, (2) @h=@z 0 at z zmax 145 Mm, and (3) periodic boundary conditions in x and y. The fast Fourier transformmethod is used for solving the Laplace equation.

The third step is to create a field-free cavity along the se-lected path. This is achieved by setting A Ap everywherewithin the box, except in a volume surrounding the selectedpath. The volume is defined by z 6 (8.7 Mm) and d 4(5.8 Mm), where dis the horizontal distance to the path. Withinthis volume A(x; y; z) is set equal to Ap(x; y; 0), the value ofAp on the photosphere. This has the effect of pushing up thehorizontal field to a height of 8.7 Mm above the channel. SinceBz(x; y; 0) % 0 within the channel, this creates an elongated

cavity that is nearly field-free. Figure 6a shows the intersectionof this cavity with the plane y 13:8 Mm; the vectors atheight 810 Mm represent the surrounding arcade that ispushed up above the cavity.

The fourth step is to insert the flux rope into the cavity. Theaxial field is inserted by adding axial to the A-values on thex- andy-ribs crossed by the filament path in the planez 0, andin the next higher planes, z and z 2, where is thegrid spacing (the sign depends on whether a rib is crossed in thepositive of negative direction). The resulting tube of axial fluxis only a single cell wide and lies at a height z 2:5 3:63 Mm. The poloidal field is inserted by adding Fpol s to the

A-values on thex- andy-ribs that lie within a horizontal distance2 from the selected path and between heights z andz 4 (here s is the direction of the selected path). This causesthe poloidal field to be wrapped around the axial field (but stillwithin the cavity). Figure 6a shows the result foraxial 3:4 ; 1019 Mx and Fpol 3:7 ; 10

9 Mx cm1. The octagonal

Fig. 5.Filament path (white curve) on a gray-scale image of Bz(x; y; 0)(white for Bz> 0; black for Bz< 0; full white or black for jBzj 20 G).

Fig. 6.Cross section of the flux rope with the plane y 13:8 (Mm)during the early stages of magnetofrictional relaxation: (a) initial field; (b)field after 1000 iterations. The curves are contours of By(x; 13:8; z) withdashed curves for By > 0 and full curves forBy < 0. The peak values of By are(a) 400 G and (b) 38.2 G. The vectors show (Bx; Bz) with maximumvector length (a) 90 G and (b) 45 G.



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contours at x 32 Mm and z 3:5 Mm represent the axialfield, and the vectors immediately surrounding these contoursrepresent the poloidal field. The resulting magnetic field is farfrom equilibrium (strong Lorentz forces) but has the desiredmagnetic topology.

The final step is to allow the magnetic field to relax to aforce-free state. I use the magnetofrictional method of vanBallegooijen et al. (2000), which is an iterative relaxationmethod specifically designed for use with vector potentials. Inthe present application, the vector potentials on the photo-sphere, Ax(x; y; 0; t) and Ay(x; y; 0; t), are held fixed duringthe relaxation. The vector potential A(x; y; z; t) in the coronais evolved in accordance with the ideal-MHD induction equa-tion. The plasma velocity is assumed to be proportional to theLorentz force. The boundary conditions imply that the magneticflux through any vertical surface within the box remains un-changed; therefore, the flux rope is forced to remain within thebox. Furthermore, the connectivity of the coronal field lines ismore or less preserved, although numerical diffusion does causesome changes in connectivity at the edges of the flux rope.

The main effect of the magnetofrictional relaxation is the

lateral expansion of the flux rope until it touches the sur-rounding arcade. Figure 6b show the cross section of the fluxrope after 1000 iterations. At this stage the field still deviatessignificantly from a force-free field. During the iterative pro-cess, the flux rope expands until the magnetic pressure of theflux rope is balanced by magnetic tension in the surroundingarcade. To monitor the convergence to a force-free state, Icompute the sine of the angle between the vectors j and B:

jj < Bj

jjjjBj; 2

where j : < B is a measure of the coronal current density.The quantity is computed in a volume surrounding the flux

rope; the height and width of this volume are 25

, or 36.3 Mm.Figure 7 shows histograms of within this volume for fivedifferent iterations. Note that for later iterations the distributionis strongly peaked at % 0, as expected for a force-free field.However, there remains a small fraction of the volume in which> 0:5. This is attributed to numerical diffusion, which con-tinues to modify the connectivity of the field at the edge of theflux rope. In the next section I show the magnetic configurationafter 3000 iterations when the distribution of is close to itsfinal state. Far away from the flux rope the magnetic fieldremains close to the potential field at all times.

3.2. ModelinggResults and Comparison with Observvations

The magnetic field was computed for different combinations

of axial and poloidal fluxes of the flux rope. Figure 8 showsresults for axial 3:4 ; 1019 Mx and Fpol 3:7 ; 10

9 Mxcm1, which appears to give the best fit to the observations.Figure 8a shows magnetic field lines plotted on a gray-scaleimage of the photospheric field; the viewing angle is the sameas in Figure 3. Seven field lines inside the flux rope are shown;the overlying coronal arcade is not shown. Note that the fluxrope has a right-helical twist. The average width of the fluxrope is about 20 Mm. Figure 8b shows the same field lines seenfrom the side (viewing angle 1

from photospheric plane).

Figure 8c shows the location of dips in the field lines, i.e., siteswhere Bz 0 and B = :Bz ! 0. Presumably, the cool filament plasma collects in these dips (e.g., Kippenhahn & Schluter1957; Kuperus & Raadu 1974). The short line segments indi-

cate the direction of the horizontal field at the dips. Only dips

within 22 Mm of the filament path are shown. Figure 8dshowsa side view of the dips.

Comparison of Figures 8c and 3a3c shows that the distri-bution of field-line dips reproduces some features of the ob-served filament. Specifically, the model shows an extension ofdips away from the main body of the filament at the location ofthe observed filament barb. In the model this extension is dueto a distortion of the flux rope by weak positive flux located

directly below the barb (Bz $ 4 G). This weak flux is anextension of the positive-polarity network element that ispartially hidden behind the barb. The predicted horizontal fieldin the main body of the barb is consistent with the direction ofthe filament threads seen in Figure 4. However, the model doesnot accurately reproduce the observed direction of the threadsat the left (ragged) edge of the barb.

The observed filament has a dark spine that is presenteverywhere along the filament (see Figs. 3a3c). It seemsnatural to associate the spine with the flux rope axis. However,the height of the flux rope axis varies significantly with posi-tion along the filament (see Fig. 8b). In regions in which theflux rope is tilted upward or downward, the field-line dips arepresent only at lower heights, well below the height of the flux

rope axis (see Fig. 8d). Therefore, the filament spine is notlocated at dips in the field lines.If the magnetic field on the spine is not horizontal, why does

the plasma notslide down along the spine under the influence ofgravity? Clearly, there must be other forces at work that eitherpreventthe matter from sliding down, or continually redistributethe matter along the field lines. The nature of these forces is notknown. The counter-streaming flows observed in some fila-ments (Zirker et al. 1998; Lin et al. 2003) may also be a mani-festation of these forces. Although such flows have not beenobserved in the present case, their presence cannot be ruled out.

3.3. Avveragge Pitch of Helical Field Lines

The pitch l of a helical field line is defined as the axial

length of a section of the field line that encircles the axis once

Fig. 7.Histograms of, the sine of the angle between the vectors j and Bin a volume surrounding the flux rope.



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(Priest 1982). Approximating the flux rope as a cylindricallysymmetric structure, l 2rBz=B, where r is the distance tothe flux rope axis, Bz(r) is the axial field component, and B(r)is the poloidal component. The average pitch of a flux ropecan be defined as

L 2

RR0

rBzdrRR0

B(r) draxial

Fpol; 3

where R is the flux rope radius and axial and Fpol are the axialand poloidal fluxes. For the model shown in Figure 8, the ratioof axial and poloidal fluxes yields L 92 Mm. This is sig-nificantly larger than the flux rope radius (R $ 10 Mm).

4. DISCUSSION

The method used in x 3.1 for inserting a flux rope into thecoronal field is not intended to represent the true physicalprocess by which coronal magnetic flux ropes are formed on the

Sun. However, the result of x 3.3 can be compared with pre-dictions from theories of flux rope formation. Such theories canbe divided into two groups. Some authors propose that helicalflux ropes are created below the photosphere and emerge intothe solar atmosphere fully formed (e.g, Rust & Kumar 1994;Low & Hundhausen 1995). Others propose that helical fieldsare created in the corona by reconnection occurring within asheared coronal arcade afterthe field has emerged into the at-mosphere (e.g, Pneuman 1983; van Ballegooijen & Martens1989; Mackay et al. 2000; Mackay & Gaizauskas 2003). In thissection I adopt the second model ( formation by reconnection ina sheared arcade) and compute the initial pitch of the fluxrope as predicted by the reconnection model. Comparison withthe observed pitch (x 3.3) suggests that the pitch of the flux

rope increases with time. This leads us to consider the effects ofmagnetic diffusion on the evolution of coronal flux ropes.

Models of flux rope formation by reconnection in a shearedarcade (e.g., Pneuman 1983; van Ballegooijen & Martens1989) predict that helical field lines are wrapped around a long

straight field line that becomes the core of the growing fluxrope. According to such models, the average pitch L0 of theflux rope shortly after its formation equals the shear distance Dof the surrounding arcade. The latter is defined as the distancebetween the footpoints of an arcade loop as measured along thePIL. For the filament described in x 2 the shear distance D isunknown because the arcade was not directly observed.However, an upper limit on D can be obtained as follows. Thearcade apparently provides the magnetic tension forces nec-essary to hold down the flux rope in stable equilibrium formany hours. This suggests that the arcade loops cross the PILat a significant angle, say, within 45 of the direction perpen-dicular to the PIL. Therefore, the shearD must be less than thewidth of flux rope, D < 20 Mm. The reconnection model pre-dicts L0 D, which yields an upper limit on the initial pitch ofthe flux rope, L0 < 20 Mm. This upper limit is much smallerthan the observed pitch L 92 Mm found in x 3.3, suggestingthat the pitch L(t) has increased significantly with time t sincethe formation of the flux rope (here trefers to real time on theSun, not simulation time in the magnetofrictional model).

The increase in pitch length found above is likely due tomagnetic diffusion inside the coronal flux rope. If no newhelical field lines are formed at its outer edge, magnetic dif-fusion causes the poloidal flux Fpol to be annihilated at the fluxrope axis, while the axial flux axial is conserved. Therefore,the ratio L(t) axial=Fpol(t) increases with time t. The effectsof magnetic diffusion on force-free, cylindrical flux ropes wasstudied by Low (1973) and Reid & Laing (1979b); also see Reid

Fig. 8.Model of magnetic flux rope supporting the observed filament. (a) Magnetic field lines within the flux rope on a gray-scale image of Bz(x; y; 0), as seenfrom Earth. (b) Same field lines seen from the side of the filament. (c) Locations of dips in the magnetic field lines, as seen from Earth. Only the dips within 22 Mmof the filament are shown. (d) Same dips seen from the side.



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& Laing (1979a) for similar analyses in slab geometry. Theseauthors found that the force-free condition requires the presenceof a radial inflow of plasma that transports poloidal flux to thecylinder axis (where it is annihilated by diffusion). To demon-strate that magnetic diffusion leads to an increase in pitch

length, I now present a simple model of flux rope evolution.The model is similar to the one presented by Reid & Laing(1979b) but uses different outer boundary conditions. The mag-netic induction equation for a cylindrical flux rope is given by

@B@t

@

@rvrB

r

@

@rrB !

; 4

@Bz@t

1

r

@

@rrvrBz r

@Bz@r

!; 5

where r is the radial distance from the flux rope axis(0 < r


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results are presented for one case with constant diffusivity, 500 km2 s1, and the following Bz-profile:

Bz(r; 0) exp 10(r=R)2 5(r=R)4

; 8

where R 10 Mm is the outer radius of the flux rope. Theinitial plasma density (r; 0) is assumed to be constant. Fig-

ure 9 shows the resulting radial profiles of the axial field Bz, poloidal field B, radial velocity vr, helicity k, and plasmadensity for two instants of time. Note that the radial velocityis negative (toward the flux rope axis). The inward transport ofBz by the radial velocity slows the outward magnetic diffusion;the net transport of Bz is radially outward. In contrast, thepoloidal field B is transported inward both by diffusion and byadvection and is annihilated at the origin. The inward flow alsoleads to an enhancement of plasma density in the inner part ofthe flux rope; the observed filament spine may be a manifes-tation of this density enhancement. Figure 9fshows the averagepitch L(t) as function of time. Note that the pitch increases bya factor of 8 over a period of about 14 hr. This increase isconsistent with the above comparison of the observed pitch

(L 92 Mm) with the initial pitch predicted from the recon-nection model of flux rope formation (L0 < 20 Mm).

5. CONCLUSIONS

According to the present model, the observed sinistral filamentis supported by a right-helical flux rope. The flux rope is helddown by an overlying arcade that is anchored in the neighboringnetwork elements. The axial flux of the rope is estimated to beabout 3:4 ; 1019 Mx, and the poloidal flux is about 3:7 ; 109 Mxcm1, not including the flux from the surrounding arcade. Theuncertainty in these fluxes is at least a factor of 2.

The shape and thickness of the flux rope are strongly affected by the neighboring network elements. The observed filamentbarb can be explained in terms of dips in the field lines. The

extension of the dips away from the main path of the filament isdue to a distortion of the flux rope by the neighboring elements;similar effects were found by Aulanier & De moulin (2003). Inthe present case, the distortion of the flux rope is due to weakpositive flux located directly below the barb (Bz $ 4 G). It isnot known whether this weak positive flux is canceling against

nearby negative flux. The weak positive flux below the barb isan extension of the positive-polarity network element hiddenbehind the barb. In contrast, both Martin et al. (1994) and Wang(2001) associate barbs with weak parasitic polarities that aredistinct from network elements.

The three-dimensional magnetic model reproduces the ob-served direction of H threads within the main body of the barb but not at its left (ragged) edge. The height of the fluxrope axis varies along the filament from about 13 Mm at itshighest point to less than 3 Mm in regions in which the sur-rounding fields are very strong. The observed filament spine isbelieved to be located at the flux rope axis and is not located atdips in the field lines.

The pitch of the helical field lines is larger than expected onthe basis of the reconnection model of flux rope formation(Pneuman 1983; van Ballegooijen & Martens 1989). I suggestthis is due to magnetic diffusion within the filament flux rope.A simple model of magnetic diffusion in a cylindrical fluxrope was presented. Diffusion causes an increase of the pitchon a timescale of several hours. Therefore, magnetic recon-nection plays an important role not only in the formation of

the flux rope but also in its subsequent evolution.

I thank Guillaume Aulanier for valuable discussions duringthe early stages of this project. I also thank the referee, DuncanMackay, for providing detailed comments that significantlyimproved the presentation of this paper. The full-disk Himages used in this study were obtained at the Big Bear SolarObservatory, which is operated by the New Jersey Institute ofTechnology. The high-resolution H images were obtained atthe Swedish Vacuum Solar Telescope, which is operated by theSwedish Royal Academy of Sciences at the Spanish Obser-vatorio del Roque de los Muchachos of the Instituto de Astro-physca de Canarias (IAC). I thank Oddbjrn Engvold and the

SVST staff for their support during the observations, andLockheed Martin Solar and Astrophysics Laboratory for theuse of their tunable filter. The magnetic data used in the studywere produced cooperatively by NSF/NOAO, NASA/GSFC,NOAA/SEL, and NSO/Kitt Peak and were made publicly ac-cessible via the World Wide Web.

APPENDIX

MAGNETIC DIFFUSION IN A CYLINDRICAL FLUX ROPE

The force-free condition (7) can also be written as a balance between magnetic pressure and tension forces:

@

@r

B2 B2z

2B2

r

0; A1

which can be integrated to yield

(rB)2

Zr0

r2@(B2z)

@rdr: A2

By taking the time derivative of equation (A1) and inserting equations (4) and (5), a linear, second-order differential equation forthe radial velocity is obtained

B2z B2

r@2f

@r2 3B2z B

2

@f@r

2kBBzf g; A3

where f(r; t) vr=r and

g(r; t) B2z B2

@

@r

k2 2BBz

r

@

@r

k : A4



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This is a generalization of equation (8) of Reid & Laing (1979b) for the case in which the diffusivity (r) is a function of radius.The boundary conditions are vr(0; t) 0, @f=@r 0, vr(R; t) 0, and k(R; t) 0. The outer boundary conditions imply that thequantity in square brackets in equation (5) vanishes at r R; hence the axial magnetic flux in the model is strictly conserved,axial

RR0

Bz(r; t)2r dr constant. The poloidal flux is defined by Fpol(t) RR

0B(r; t) dr, which decreases with time because

the quantity (=r)@(rB)=@r in equation (4) generally does not vanish at r 0.The computational method is as follows. Given a spatial profile of Bz(r; t) at one instant of time, the poloidal field B(r; t) is

computed using equation (A2). At interior grid points k(r; t) is computed from the second part of equation (7), and vr(r; t) isdetermined by solving equation (A3) using tri-diagonal matrix inversion. Then the axial field B

z(r;

tt) and density

(r; tt) at the next time step are computed from equations (5) and (6), respectively.

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A. A. van Ballegooijen- Observations and Modeling of a Filament on the Sun