A SIMPLE DYNAMICAL MODEL FOR FILAMENT FORMATION IN THE SOLAR CORONA

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1 The Astrophysical Journal, 630: , 2005 September 1 # The American Astronomical Society. All rights reserved. Printed in U.S.A. A SIMPLE DYNAMICAL MODEL FOR FILAMENT FORMATION IN THE SOLAR CORONA Yuri E. Litvinenko Institute for the Study of Earth, Oceans, and Space, University of New Hampshire, Durham, NH ; yuri.litvinenko@unh.edu and M. S. Wheatland School of Physics, University of Sydney, Sydney, NSW 2006, Australia; m.wheatland@physics.usyd.edu.au Received 2005 March 29; accepted 2005 May 23 ABSTRACT Filament formation in the solar atmosphere is considered. In the limit of sub-alfvénic but supersonic motion, plasma flow in the solar corona is driven via the induction equation by a slow evolution of force-free magnetic fields. Methods for solving the relevant magnetohydrodynamic equations are presented and applied to filament modeling in two and three dimensions. An illustrative two-dimensional example is given, which is based on a potential magnetic field with a dip. The example describes the formation of a normal filament between two bipolar regions on the Sun. Next a detailed three-dimensional calculation is presented, which uses linear force-free magnetic fields. The boundary conditions are chosen to resemble the qualitative head-to-tail linkage model for the formation of filaments, suggested by Martens & Zwaan. Consistent with this model, dense formations, reminiscent of filament pillars, are shown to appear in the corona above the region of converging and canceling magnetic bipoles. The numerical results are consistent with the principal role of magnetic field in the dynamical processes of dense plasma accumulation and support in filaments, advocated by Martens & Zwaan. Subject headinggs: MHD Sun: corona Sun: magnetic fields Sun: prominences 1. INTRODUCTION Large-scale magnetic fields in solar filaments (prominences) can be assumed to be force-free with a good accuracy. Static force-free models have been successful in describing the threedimensional magnetic structure of filaments (e.g., Aulanier & Demoulin 1998; Mackay et al. 1999). A common feature of these models is the presence of magnetic dips, where the relatively cool and dense plasma is supported against gravity. Magnetic reconnection, manifested observationally as convergence and mutual cancellation of photospheric magnetic fragments of opposite polarity, appears to be a key dynamic process in the formation and maintenance of filaments (e.g., Litvinenko 1999, 2000; Litvinenko & Martin 1999; Wang 2001; Wood & Martens 2003; Chae et al. 2004). Reconnection creates the magnetic dips that can carry the photospheric material upward, thus supplying the filament mass. It is clearly of interest to extend the available static models and analyze the dynamics of plasma flows and mass accumulation in filaments, and that is the goal of this paper. It was suggested some time ago that the formation of dense structures in the solar corona is controlled by the evolution of a strong magnetic field (Platov et al. 1973). The strong-field assumption corresponds to both the sound speed and the typical plasma speed being sub-alfvénic. Observations of plasma flows in filaments indicate that this assumption is quite accurate (e.g., Zirker et al. 1998; Wang 1999). Therefore, the evolution of the coronal magnetic field is approximately a sequence of force-free states. Solutions to the strong-field magnetohydrodynamic (MHD) equations have been computed, describing the formation of filaments associated with magnetic reconnection ( Ivanov & Platov 1977). The results are of limited value: only a very simple twodimensional geometry of two magnetic bipoles with a single magnetic X line between them was considered, which resulted in an 587 unrealistically symmetric magnetic dip above the X line. The advantage of the technique, however, is that it can be applied to more general three-dimensional geometries. For three-dimensional calculations, it is important to adopt a realistic magnetic geometry specified by a model for filament formation and evolution. Nonlinear force-free fields provide the most accurate description of coronal magnetic fields, but unfortunately they are difficult to compute in the general case. Linear force-free fields provide a less realistic model, but they are easy to compute, for example, by Fourier methods (e.g., Alissandrakis 1981), and as a result they have been commonly used in filament modeling (e.g., Aulanier & Demoulin 1998; Mackay et al. 1999). As the model for the formation and geometry of solar filaments, we use a promising head-to-tail linkage model ( Martens & Zwaan 2001). This qualitative model, which is based on the observational conditions for filament formation ( Martin 1998), describes the formation of solar filaments by convergence and reconnection of magnetic bipoles. Flux convergence is assumed to drive reconnection, leading to the formation of looplike filament segments that later connect to form longer quiescent filaments. Eventually, further flux cancellation can cause filament eruption. The model naturally reproduces the fine structure of filaments, with barbs extending laterally from the filament axis. Notably, the interaction of magnetic bipoles has been shown to reproduce the observed hemispheric pattern of solar filaments ( Mackay & van Ballegooijen 2005). This result is consistent with the Martens-Zwaan model. This paper has a simple organization. Section 2 describes the method for computing the dynamics of a compressible plasma in the strong-field approximation. Section 3 gives an illustrative example of plasma accumulation in the dip of a two-dimensional magnetic field that describes a filament formed between two solar bipolar regions. Section 4 treats a more realistic three-dimensional

2 588 LITVINENKO & WHEATLAND Vol. 630 example involving linear force-free fields, motivated by the Martens- Zwaan model for filament formation. Section 5 discusses the implications of the numerical results. 2. DYNAMICAL MODELING OF THE STRONG-FIELD EVOLUTION The strong-field approximation describes a slowly evolving, force-free equilibrium of magnetic field in an ideal plasma. The approximation takes into account both the freezing-in condition and the momentum equation (Syrovatskii 1966, 1978). Formally, the approximation describes sub-alfvénic flows, corresponding to a small value of the parameter v 0 =v A T1; where v 0 is a typical flow speed and v A ¼ B 0 /4 ð 0 Þ 1/2 is the value of the Alfvén speed based on the reference values B 0 and 0 of the magnetic field and plasma density. In what follows, we assume that the gas pressure is small: 8P 0 =B 2 0 T1; where P 0 ¼ n 0 kt 0 is the reference pressure. Furthermore, we assume that the magnetic field is strong and the plasma is cold in the sense that T 2 T1: In other words, we consider plasma flows that are simultaneously sub-alfvénic and supersonic. As an example, we adopt as reference values the magnetic field B 0 ¼ 100 G, density 0 ¼ gcm 3, and temperature T 0 ¼ 10 4 K, which characterize solar filaments (e.g., Tandberg-Hanssen 1995). The resulting value of 10 4 is consistent with the assumption T 2 for the Alfvén speed of order v A ¼ 1000 km s 1 and the observed flow speeds of order v 0 30 km s 1 (e.g., Zirker et al. 1998), which imply The observed speeds tend to increase along the length of the filament (Wang 1999), improving the accuracy of the approximation T 2. We neglect the gravitational force g, although it could easily be incorporated into the calculation. The neglect is justified as long as the inverse Alfvén-Froude number, gl 0 /v 2 A,issmall compared with 2,wherel 0 is the filament height. This requirement is satisfied for the heights of order l cm, given the reference values above. The requirement is easier to satisfy forfasterflows. Nondimensionalizing the ideal MHD equations, dropping the pressure term, expanding all quantities in power series with respect to 2, and keeping only zero-order terms, we obtain the following equations for the zero-order quantities: ð1þ ð2þ ð3þ B < (: <B) ¼ 0; :=B ¼ 0; t B ¼ : <(v < t þ : =(v) ¼ 0; P ¼ : Here the full energy equation is replaced for simplicity by a polytropic equation of state with ¼ const. The plasma displacement parameter is defined as v 0 t 0 =l 0 ; ð5þ ð6þ ð7þ ð8þ where t 0 is the reference timescale (note that this timescale is not necessarily Alfvénic). Because the original equation of motion was replaced by the force-free equilibrium condition, the set of equations above is not complete. One more equation is needed to determine the plasma velocity component along the magnetic field. Taking the scalar product of the full equation of motion and the magnetic field B and dividing the result by 2 leads to another equation that contains only zero-order quantities: B = d t v ¼ 0: The resulting system of equations describes sub-alfvénically slow evolution of a force-free magnetic equilibrium (e.g., Syrovatskii 1978). Other limiting cases, such as subsonic flows with 2 T T1, can be treated in a similar manner. In general, the strong-field equations are not much easier to solve than the original MHD equations. The initial stage of an ideal MHD evolution, however, can be easily described in the limit of small plasma displacements. Since we are interested in identifying the geometry of a forming filament, rather than in details of the emerging density distribution, analysis of the linearized equations is sufficient for our purposes. Assuming the parameter ordering 2 TT1; we expand with respect to the zero-order terms in 2, B ¼ B 0 þ B 1 þ :::; v ¼ v 1 þ :::; ¼ 1 þ 1 þ :::; P ¼ 1þ P 1 þ :::; and obtain the following first-order equations: ð9þ ð10þ ð11þ ð12þ ð13þ ð14þ (: <B 0 ) < B 1 þ (: <B 1 ) < B 0 ¼ 0; ð15þ B 0 = x ¼ 0; B 1 ¼ :<(x < B 0 ); 1 ¼ :=x; ð16þ ð17þ ð18þ P 1 ¼ 1 ð19þ (Bobrova & Syrovatskii 1979), where x ¼ R t 0 v 1 dt is the Lagrangian displacement of the plasma. It is important to realize that the first-order equations, while based on the ideal MHD, can describe nonideal processes such as magnetic reconnection. The imposed initial B 0 and final B 0 þ B 1 magnetic fields are not necessarily topologically equivalent. This is why we can use the equations above to model flux cancellation and photospheric reconnection by imposing the initial and final magnetic geometries and calculating the dynamic response of the plasma. Both the plasma displacement x and the corresponding density perturbation 1 are defined by the initial force-free magnetic field B 0 and its perturbation B 1. For potential fields in two dimensions, the procedure is straightforward, and an example is described in x 3. In general, in three dimensions the induction equation (17) needs to be solved. We developed an iterative approach for this purpose, which is described below for the special case of linear force-free fields.

3 No. 1, 2005 FILAMENT FORMATION IN SOLAR CORONA 589 The procedure is as follows. We consider an initial magnetic field configuration B 0 and a perturbed magnetic field configuration B ¼ B 0 þ B 1, absorbing into B 1 for brevity. If B 0 and B 0 þ B 1 are both linear force-free fields with the force-free constant,thenb 1 is also a linear force-free field with constant, and equation (15) is satisfied. Substituting B 1 ¼ :<B 1 / into equation (17) results in an algebraic equation for the plasma displacement: 1 B 1 þ : ¼ x < B 0 ; ð20þ where : is an unknown curl-free term. Taking the cross product with B 0 gives x ¼ B 0 B 2 < 1 0 B 1 þ : : ð21þ In principle, the unknown scalar potential may be obtained by taking the dot product of equation (20) with B 0 and integrating the resulting first-order partial differential equation. However, in practice, it is desirable to avoid integrating along field lines. Taking the divergence of equation (20) leads to Poisson s equation for the scalar potential: : 2 ¼ :=(x < B 0 ): ð22þ If x were known, would be given by a solution to the Neumann problem for Poisson s equation. Specifically, assuming that the photosphere is represented by the plane z ¼ 0, we can consider the Neumann problem in the half-space z > 0, which has the formal solution (r) ¼ 1 Z G N (r; r 0 ): 0 = ½x(r 0 ) < B 0 (r 0 ) Šd 3 r 0 4 z 0 > 0 þ 1 Z G N (r; r 0 )ẑ =: 0 (r 0 ) 2 z 0 ¼ 0 dx0 dy 0 ; ð23þ z 0 ¼ 0 where G N (r; r 0 ) ¼ (x x 0 ) 2 þ ( y y 0 ) 2 þ (z z 0 ) 2 1=2 þ (x x 0 ) 2 þ ( y y 0 ) 2 þ (z þ z 0 ) 2 1=2 ð24þ is the appropriate Green function (e.g., Eyges 1972, p. 82). The boundary term in the second integral may be rewritten using equation (20) as ẑ =: ¼ ẑ = (x < B 0 ) 1 ẑ = B 1: ð25þ Hence, equations (23) (25) specify, provided x is known, and : is obtained by differentiating equation (23) with respect to the unprimed coordinates. However, x is unknown, except for boundary values that are included via equation (25). The problem suggests a Picard iteration procedure (e.g., Zwillinger 1989, p. 480) to obtain self-consistent values of : and x. The initial form : (0) ¼ B 0 = B 1 B 2 0 B 0 ð26þ is adopted for the gradient of the potential, which gives the correct component along B 0, according to equation (20). Equation (21) then allows x (0) to be computed in terms of, B 1,andB 0.Next x (0) is used in equation (23) together with the boundary values for x to determine a corrected gradient of the potential : (1), leading to a corrected displacement x (1) from equation (21). The procedure is repeated until the convergence criterion * + jb 0 = B 1 þ : ðþ i j T1 ð27þ B 0 B 1 is satisfied at some step, where the average in angle brackets is over the region of interest. Finally, the sought-after density perturbation, caused by the evolution of the magnetic field, follows from the linearized continuity equation (18). This completes the dynamical solution to the problem of a slowly evolving linear force-free magnetic field. In x 4 we apply this technique to model the formation of solar filaments. 3. AN ILLUSTRATIVE TWO-DIMENSIONAL EXAMPLE The equations of the previous section are particularly simple in two dimensions, in which case we can write B 0 ¼ :<A 0, where A 0 ¼ A 0 (x; y)ẑ is the vector potential. The perturbation B 1 is similarly expressed through A 1 (x; y). Because both x < B 0 and A 1 are directed along the z-axis, the induction equation is satisfied by setting the potential ¼ 0: A 1 ¼ x < B 0 : ð28þ Physically, the inductive electric field is directed along the z-axis, perpendicular to the magnetic field. We emphasize that 6¼ 0 in the general three-dimensional case, because the inductive electric t A 1 generally has a component along B 0, which has to be balanced by the potential electric t :. We consider the two-dimensional magnetic field containing a dip, used by Lepeltier & Aly (1994) to model a solar prominence. Figure 1 shows an example of this field configuration. The flux function A 0 is 1 A 0 (x; y) ¼ ( y þ h) (x þ d ) 2 þ( y þ h) 2 þ 1 (x d ) 2 þ( y þ h) 2 þ 2H (h þh )2 d 2 ½d 2 þ(h þ H ) 2 Š 2 log x 2 þ ( yþh ) 2 x 2 þ ( y H ) 2 : ð29þ The Lepeltier-Aly solution represents the structure of a filament formed between two bipolar regions on the Sun. The magnetic field in the model is created by two subphotospheric bipoles of moment, each located at depth h and separated by distance 2d from each other, and by a coronal line current I at height H and its image. The current magnitude I is chosen so that the coronal current is in equilibrium. Lepeltier & Aly (1994) argued that the coronal current not only creates a magnetic dip but also ensures the stability of the equilibrium with respect to ideal MHD perturbations. Although the model possesses two magnetic X lines, neither of them is located on the y-axis. Hence, the magnetic field approximates the structure of a normal filament. Once the perturbation A 1 (x; y)ẑ is prescribed, the equations of the previous section can be used to compute the plasma displacement and the corresponding density perturbation A 1 :A 0 1 ¼ :=x¼ := (:A 0 ) 2 : ð30þ

4 590 LITVINENKO & WHEATLAND Vol. 630 Fig. 1. Magnetic field structure of a normal filament with a dip in the Lepeltier-Aly model with d ¼ 1, h ¼ 0:8, and H ¼ 0:4. The magnetic X lines, indicated by X, are located at (x X ; y X ) ¼ (0:36; 0:29). Figure 2 gives the density distribution resulting from the upward motion of the bipoles, h! 0:9h, in the case d ¼ 1, h ¼ 0:8, and H ¼ 0:4. Darker regions correspond to increased density. Clearly, the plasma is accumulated at the location of the dip as a result of the flux emergence. Figure 2 also allows us to conclude that the plasma flow is driven primarily by the electric field at the two X lines of the magnetic field. This effect is easy to understand because a given electric field has to be balanced by the v < B term, where v is normal to B in our model. Hence, the flow is the fastest at the location of a minimum of the magnetic field. The principal role of the X lines is confirmed by computing the density perturbation when the electric field is uniform throughout the region. A uniform electric field E(t) applied in the region for a finite time corresponds to a constant perturbation of the flux function A 1 ¼ R Edt. Figure 3 shows the density distribution for A 1 ¼ 0:2. Comparison of Figures 2 and 3 confirms that virtually identical density distributions emerge as long as the flux function changes at the X line by the same amount. The solutions for x and 1 are finite everywhere except at the X line itself, where the linearization technique is not valid. Apart from the obvious limitation of restricted dimensionality, the weakness of this illustrative example is that the variation of the coronal line current I is dictated by the rate of flux emergence in the Lepeltier-Aly solution. Nevertheless, our analysis demonstrates both the generation of plasma flows, driven by an evolving magnetic field, and the accumulation of plasma at the location of the magnetic dip in the corona. 4. FILAMENT FORMATION IN AN EVOLVING FORCE-FREE MAGNETIC FIELD Turning to the more realistic three-dimensional problem, we need to specify photospheric boundary conditions for the initial and perturbed magnetic field configurations, as well as boundary conditions for the displacement x consistent with a transition between the two configurations. Our model for the magnetic field is a linear force-free field. In that case we need to specify two sets of boundary conditions on the vertical component of the field at the photosphere, B 0; z (x; y; 0) and B z (x; y; 0), as well as a value of the force-free constant. We consider a boundary region 0 x 1, 0 y 1 in normalized units. Figure 4 shows the chosen boundary field B 0; z (x; y; 0) as a gray-scale image, with white and black representing maximum and minimum values, respectively. The field is shown over a 100 ; 100 computational grid. The configuration may be thought of as a pair of magnetic bipoles, one in the upper left and the other in the lower right of the figure. The polarity inversion line is shown by the solid line in the figure. This configuration is intended to represent two magnetic regions on Fig. 2. Plasma density perturbation due to the motion of bipoles h! 0:9h in the Lepeltier-Aly model. Darker regions correspond to increased density. The six contours shown correspond to the density range j 1 j < 20. The parameters are the same as in Fig. 1.

5 No. 1, 2005 FILAMENT FORMATION IN SOLAR CORONA 591 Fig. 3. Plasma density perturbation due to a uniform electric field and A 1 ¼ 0:2 in the Lepeltier-Aly model. Darker regions correspond to increased density. The parameters are the same as in Fig. 1. the Sun that are initially unconnected. The specific functional form used is B 0; z (x; y; 0) ¼ X4 i ¼1 h i B 0i exp (x x i ) 2 =(2ix 2 ) ( y y i) 2 =(2iy 2 ) ; ð31þ where B 01 ¼ B 03 ¼ 1, B 02 ¼ B 04 ¼ 1, (x 1 ; y 1 ) ¼ (0:2; 0:6), (x 2 ; y 2 ) ¼ (0:5; 0:6), (x 3 ; y 3 ) ¼ (0:5; 0:4), (x 4 ; y 4 ) ¼ (0:8; 0:4), ( 1x ; 1y ) ¼ ( 4x ; 4y ) ¼ (0:1; 0:2), and ( 2x ; 2y ) ¼ ( 3x ; 3y ) ¼ (0:05; 0:1). We consider a perturbation in which the two smaller poles at the center of the figure are driven slightly closer together. The perturbed boundary conditions are taken to be B z (x; y; 0) ¼ X4 i ¼1 h i B 0i exp (x xi 0 )2 =(2ix 2 ) ( y y0 i )2 =(2iy 2 ) ; ð32þ where xi 0 ¼ x i for all i, y1 0 ¼ y 1, y2 0 ¼ y 2 y, y3 0 ¼ y 3 þ y, and y4 0 ¼ y 4,withy ¼ 0:01. In other words, only the vertical positions of the two central poles are changed. Figure 4 indicates the change in location of the two poles by two short vertical lines. It is also necessary to specify boundary conditions on the plasma displacement. Sophisticated methods are available for inferring a horizontal velocity from a sequence of photospheric magnetograms (e.g., Longcope 2004). Because the assumed magnetic sources are well localized, however, we make the following simple plausible choice: h i y (x; y; 0) ¼ yexp (x x 2 ) 2 =(22x 2 ) ( y y 2) 2 =(22y 2 ) h i þ y exp (x x 3 ) 2 =(23x 2 ) ( y y 3) 2 =(23y 2 ) ; ð33þ with the other components of x(x; y; 0) being zero. This displacement is localized around the central poles and consistent with the imposed change in the location of the poles. Finally, we need to specify the electric current at the boundary by the choice of the force-free constant, and we consider ¼ 0:25 (in units of the reciprocal of the length scale). Given the boundary conditions and the choice of, we can calculate linear force-free fields B 0 and B (and hence B 1 ¼ B B 0 )inthe region 0 x 1, 0 y 1, 0 z 0:5 ona100; 100 ; 50 computational grid, using the Fourier method (Alissandrakis 1981). Because the Fourier fields are periodic in x and y, calculation of B 0 and B is initially performed using boundary conditions prescribed over a 256 ; 256 grid, with zero boundary values beyond the 100 ; 100 region shown in Figure 4, to reduce the influence of the periodicity (e.g., Alissandrakis 1981). Figure 5 shows a top view of the final 100 ; 100 ; 50 computational domain, including a gray-scale image of the vertical component of the initial field B 0 in the lower boundary. Selected field lines of the initial field are also shown. It should be noted that the perturbed field has a very similar configuration, because the change in the boundary conditions is small (see Fig. 4). The magnetic field geometry and the assumed plasma displacements are suggested by the filament formation mechanism of Martens & Zwaan (2001). The idea is that a filament segment is created by convergence and cancellation of two originally unconnected magnetic bipoles (see Fig. 3 in Martens & Zwaan 2001). A filament is eventually formed through connection of Fig. 4. Boundary conditions for the three-dimensional calculation.

6 592 LITVINENKO & WHEATLAND Fig. 5. Top view of boundary values and field lines for the initial field B 0. several segments. Like several previous global models for filament formation (e.g., van Ballegooijen & Martens 1989), this mechanism relies on photospheric magnetic reconnection. An important difference, however, is that in the Martens-Zwaan model the flux concentrations on either side of the filament channel are initially not magnetically connected in the corona. It is not possible to exactly reproduce this aspect of the model with linear force-free fields. However, the choice of larger outer sources reduces (but does not eliminate) magnetic connection above the polarity inversion line, as illustrated in Figure 5. The imposed boundary displacements correspond to flux convergence, cancellation, and the resulting head-to-tail linkage of two initially unconnected bipoles in the qualitative Martens- Zwaan model. We attempt to quantify the model by simulating the interaction of a pair of initially unconnected magnetic bipoles, which is the basic ingredient of the Martens-Zwaan model. The question we ask is whether the formation of a filament segment, predicted in the model, can be confirmed quantitatively by solving the strong-field MHD equations. We use the iterative procedure outlined in x 2 to obtain the displacement of plasma and the emerging density structure in the corona, given the initial and perturbed magnetic fields. The calculation is performed on the 100 ; 100 ; 50 grid, with the integrals in equation (23) being evaluated using the trapezoidal rule. We find that the lefthand side of the convergence criterion (eq. [27]) is less than after 75 iterations, indicating that an accurate solution has been found. We have also checked that the induction equation (17) is satisfied by the solution. Figure 6 visualizes the results of the calculation. The figure shows the same boundary field and field lines as in Figure 5, but a transparent isosurface of the density 1 is also shown, at a level of 0.25 of the maximum density. A region of enhanced density is found in the corona above the inner magnetic sources. The density enhancement is largest near the lower boundary. Figure 7 shows a side view of the same structures seen in Figure 6. The density enhancement is observed to have a considerable vertical extent above the central sources. Visualization of the displacement field x shows motion into this region of enhanced density from the surrounding volume, i.e., from the corona. Observationally, solar filaments consist of the thin, bladelike bridge (also called spine), connected to the chromosphere by broader pillars. In the Martens & Zwaan (2001) picture, the pillars, located above the region of flux convergence and cancellation, appear when cool plasma is carried upward with the reconnected field lines. The observed filaments grow through

7 Fig. 6. Top view of the initial field, together with the region of density enhancement caused by the perturbation of the field.

8 594 LITVINENKO & WHEATLAND Vol. 630 Fig. 7. Side view of the initial field, together with the region of density enhancement caused by the perturbation of the field. linkage of filament segments. The plasma that becomes visible first in several pillars eventually reaches the connecting bridges. The computed density enhancement seen in Figures 6 and 7 appears to correspond to a single pillar, like that sketched in Figure 7 in Martens & Zwaan (2001). The numerical results therefore are consistent with the possibility of filament formation above the converging magnetic regions through the process of magnetic reconnection. Note for clarity that magnetic dips are not described explicitly in our calculation because only the initial and final configurations of the magnetic field appear in the linearized strong-field equations that we solve. At the same time, we emphasize that reconnection is an essential process in our model. If reconnection were forbidden, the imposed photospheric displacements would lead to the formation of tangential discontinuities in the corona. It is rapid reconnection that allows the transition from the initial to the final linear force-free magnetic field. Thus, the magnetic dips are implicitly present in our calculation. Finally, a couple of comments on the numerical method are appropriate. The need to evaluate a volume integral for each point in the volume (eq. [23]) means that the method is very slow specifically of order N 6,whereN is the number of gridpoints on one side of the computational volume. Hence, the method is limited to relatively small numbers of gridpoints (the calculations described here use a 100 ; 100 ; 50 grid). Some improvement in speed is afforded by parallelization, and the method has been implemented in parallel using OpenMP (Chandra et al. 2001), for use on shared-memory parallel computers. 5. DISCUSSION In this paper we explore a model for filament formation in the solar corona, proposed by Martens & Zwaan (2001). The central process in the model is the convergence and cancellation of a pair of initially unconnected magnetic bipoles, which is postulated to produce dense and thick pillars observed in solar filaments. We compute an MHD solution that quantitatively describes the basic process underlying the qualitative Martens- Zwaan model. Specifically, we numerically investigate the MHD motions produced by the convergence of a pair of magnetic bipoles. Our solution produces a coronal density enhancement in reasonable agreement with the theoretical prediction of the Martens-Zwaan model. This is the main conclusion of this paper. An important aspect of the Martens-Zwaan model is the role of magnetic dips in the dynamics of filament formation. The dips are formed by the magnetic field lines above the canceling magnetic fragments. The newly reconnected field lines scoop up the dense plasma from the chromosphere and carry it upward. In addition to the upward mass flux, our calculation also shows the transfer of coronal plasma toward the central region when the photospheric magnetic sources approach each other. This mass transfer occurs because the magnetic field is strong enough to be the primary driver of plasma flow throughout the coronal volume. As a result of the inflow, the local pillar-like density enhancement in the region of flux cancellation should be accompanied by the global density decrease in the surrounding corona. A low-density cavity that surrounds the filament is indeed usually observed (Tandberg-Hanssen 1995). Our calculation emphasizes the raking up of the coronal plasma by the moving magnetic field lines, which is generally distinct from the scooping up of the chromospheric plasma by the reconnected field lines. It would be important to discover whether either mechanism provides the bulk of the filament mass. We would have to incorporate a realistic density structure in our model in order to address this question. What can be said, however, is that even the mass of a quiescent filament can be a significant fraction of the coronal mass, and the observed flows in filaments imply that the mass of the entire corona is not enough to

9 No. 1, 2005 FILAMENT FORMATION IN SOLAR CORONA 595 sustain them. Hence, the upward mass flux generated by reconnection should play the dominant role in the mass balance of solar filaments. In contrast to previously constructed static models for the magnetic structure of filaments (e.g., Aulanier & Demoulin 1998; Mackay et al. 1999), we develop a simple numerical technique that makes it possible to address the fundamental question of how plasma flows are generated by an evolving force-free magnetic field in the solar corona. The MHD equations are simplified in the strong-field limit, which provides an efficient way to model the formation and structure of solar filaments in realistic threedimensional magnetic geometries, while retaining the essential physics of the problem. Furthermore, our approach permits determination of mass fluxes in the corona and the locations of plasma accumulation, which in turn identifies the developing geometry of filaments. The strong-field approximation takes into account the MHD equations of motion, induction, and compressibility. Once the magnetic field is known as a function of time and space, the induction equation gives the plasma displacement normal to the field, the equation of motion gives the displacement along the field, and the continuity equation gives the plasma density. The plasma dynamics is relatively simple in the strong-field limit. Because the gas pressure and gravity can be neglected, there is no plasma acceleration along the magnetic field lines, and the plasma velocity perpendicular to the magnetic field is defined by the evolution of the field. The linearized strong-field MHD equations also incorporate rapid magnetic reconnection into the model because reconnection allows the transition between two specified magnetic geometries with different topology. We believe that our computational technique, combined with the boundary conditions dictated by the Martens-Zwaan model, captures the essential physics of filament evolution in the corona. The results are therefore complementary to those obtained in numerical time-dependent simulations of the full MHD system (e.g., Aulanier et al. 2002). Finally, it is worth noting that our technique may be useful for studying other questions of filament dynamics and evolution. Analysis of the evolution of three-dimensional magnetic configurations is required to relate the theory of filament formation to solar observations of quiescent and erupting filaments. For example, by analyzing how the photospheric boundary conditions determine the behavior of coronal structures, it may be possible to clarify the role of flux emergence in filament eruption and coronal mass ejection initiation a question that has been hotly debated in the literature (Feynman & Martin 1995; Wang & Sheeley 1999; Jing et al. 2004). This work was supported by NSF grant ATM and by NASA grant NAG Y. E. L. thanks Donald B. Melrose of the Research Centre for Theoretical Astrophysics at the University of Sydney for warm hospitality and generous support of a visit to Sydney. M. S. W. acknowledges the support of an Australian Research Council QEII Fellowship. The authors are grateful to Petrus C. Martens and the anonymous referee for several useful suggestions. Alissandrakis, C. E. 1981, A&A, 100, 197 Aulanier, G., & Demoulin, P. 1998, A&A, 329, 1125 Aulanier, G., DeVore, C. R., & Antiochos, S. K. 2002, ApJ, 567, L97 Bobrova, N. A., & Syrovatskii, S. I. 1979, Sol. Phys., 61, 379 Chae, J., Moon, Y.-J., & Pevtsov, A. A. 2004, ApJ, 602, L65 Chandra, R., Dagum, L., Kohr, D., Maydan, D., McDonald, J., & Menon, R. 2001, Parallel Programming in OpenMP (San Francisco: Morgan Kaufmann) Eyges, L. 1972, The Classical Electromagnetic Field ( New York: Dover) Feynman, J., & Martin, S. F. 1995, J. Geophys. Res., 100, 3355 Ivanov, L. N., & Platov, Y. V. 1977, Sol. Phys., 54, 35 Jing, J., Yurchyshyn, V. B., Yang, G., Xu, Y., & Wang, H. 2004, ApJ, 614, 1054 Lepeltier, T., & Aly, J. J. 1994, Sol. Phys., 154, 393 Litvinenko, Y. E. 1999, ApJ, 515, , Sol. Phys., 196, 369 Litvinenko, Y. E., & Martin, S. F. 1999, Sol. Phys., 190, 45 Longcope, D. W. 2004, ApJ, 612, 1181 REFERENCES Mackay, D. H., Longbottom, A. W., & Priest, E. R. 1999, Sol. Phys., 185, 87 Mackay, D. H., & van Ballegooijen, A. A. 2005, ApJ, 621, L77 Martens, P. C., & Zwaan, C. 2001, ApJ, 558, 872 Martin, S. F. 1998, Sol. Phys., 182, 107 Platov, Y. V., Somov, B. V., & Syrovatskii, S. I. 1973, Sol. Phys., 30, 139 Syrovatskii, S. I. 1966, Soviet Phys. JETP, 23, , Ap&SS, 56, 3 Tandberg-Hanssen, E. 1995, The Nature of Solar Prominences ( Dordrecht: Kluwer), chap. 3 van Ballegooijen, A. A., & Martens, P. C. H. 1989, ApJ, 343, 971 Wang, Y.-M. 1999, ApJ, 520, L , ApJ, 560, 456 Wang, Y.-M., & Sheeley, N. R. 1999, ApJ, 510, L157 Wood, P., & Martens, P. C. 2003, Sol. Phys., 218, 123 Zirker, J. B., Engvold, O., & Martin, S. F. 1998, Nature, 396, 440 Zwillinger, D. 1989, Handbook of Differential Equations (San Diego: Academic)