Simulation of Buoyant Flux Ropes in a Magnetized Solar ... - CiteSeerX

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Simulation of Buoyant Flux Ropes in a Magnetized Solar Atmosphere J. Krall, J. Chen Plasma Physics Division, Naval Research Laboratory, Washington, DC, 20375-5346 R. Santoro NRC/NRL Research Associate, Naval Research Laboratory, Washington, DC, 20375-5346 D. S. Spicer, S. T. Zalesak NASA Center for Computational Sciences, Goddard Space Flight Center, Greenbelt, MD, 20771 and P. J. Cargill Imperial College, London SW7 2BZ, United Kingdom

ABSTRACT Dynamics are investigated for buoyant ux ropes in a gravitationallystrati ed solar atmosphere using realistic chromosphere and corona parameters. Two-dimensional magnetohydrodynamic simulations of horizontally-oriented

ux ropes are performed under the approximation that the ux rope remains parallel to the solar surface as it rises. As expected, this motion tends to halt at the chromosphere-corona boundary, where the buoyancy force drops signi cantly. It is shown that fragmentation of the rising ux rope into multiple

ux bundles can be reduced if the ux rope eld lines are suciently twisted and that this is consistent with previous results. It is also shown that a weak, vertically-oriented ambient magnetic eld, (Ba=Btube )2  1, can prevent fragmentation of an untwisted ux tube. A vertical eld of sucient strength, but still \weak", can direct the upward buoyancy-driven ow so that the untwisted ux tube penetrates the chromosphere-corona boundary. Subject headings:

MHD { Sun: atmosphere { Sun:magnetic elds

{2{

1. Introduction The understanding of both the origin and the dynamics of magnetic eld structures in the solar atmosphere is one of the keys to understanding solar observations because such structures appear to be ubiquitous. For example, it is known (e.g., Solanki, 1993) that the photospheric magnetic ux of the Sun is concentrated into vertically-oriented magnetic ux bundles. As far back as the Skylab and Solar Maximum missions, it has been established that a signi cant fraction of the energy emitted by the corona is localized in magnetic structures. The presence of this eld in the corona has been evident in Yohkoh observations (Acton et al. 1992), where X-ray ares are observed to brighten along magnetic loops. In interplanetary space, in situ measurements suggest the occurrence of magnetic ux ropes, often referred to as \magnetic clouds" (Burlaga et al. 1981), which are strongly associated with prominence eruptions and coronal mass ejections at the Sun (Wilson and Hildner 1984, 1986). It is well known that magnetic clouds impinging on the magnetosphere of the Earth are closely associated with the occurrence of geomagnetic storms. (Wilson 1987, Zhang and Burlaga 1988). In this paper, we will focus on the process of ux rope emergence in the chromosphere and corona. At present, it is not known how (or if) a ux rope emerges though the photosphere to become a coronal loop. In one well-studied theoretical scenario (e.g., Browning and Priest 1984; Mikic et al. 1988) these loops are formed in the corona so that the emergence of a ux bundle through the photosphere is not required. In this scenario, the photospheric foot points of a magnetic eld structure, either a bundle of initially-untwisted eld lines or a magnetic arcade, are sheared by photospheric ows to form the twisted coronal loops that are often observed. In the sheared-magnetic-arcade case, large-scale reconnection is necessary. Observations suggest that ux loop emergence does, in fact, take place. For example, the occurrence of emerging bipolar regions (Zirin 1970) shows the transition of ux through the photosphere and chromosphere. More recent observations suggest that ux ropes can emerge through the photosphere in a pre-twisted state (Tanaka 1991, Lites et al. 1995, Leka et al. 1996). New instruments and techniques may add to our ability to detect ux emergence. For example, it may be possible for the Michelson Doppler Imager (Scherrer et al. 1995) to measure photospheric ow patterns that are characteristic of ux emergence. One mechanism of ux emergence is magnetic buoyancy (Parker 1955). Brie y, a region of high magnetic eld, such as a ux rope, in pressure and temperature equilibrium with a gravitationally strati ed plasma, tends to rise because the density is reduced in the high- eld region. In this paper, we will use \ ux rope" as a generic term for twisted or untwisted ux concentrations and will use \ ux tube" when speci cally referring to an

{3{ untwisted ux rope. Key features of the dynamics of buoyant magnetic ux ropes have been studied via two-dimensional (2D) magnetohydrodynamic (MHD) simulations by Schussler (1979) for parameters corresponding to the convection zone. These simulations model a horizontallyoriented ux rope under the assumption that it remains horizontal as it rises. Schussler (1979) found that vortices driven by the buoyant motion of an isolated, untwisted ux tube tend to distort the ux tube into an \umbrella shape", leading to fragmentation of the ux tube. A more recent simulation study by Longcope et al. (1996) showed that the fragmentation observed by Schussler occurs in accordance with the theory of buoyant thermals (Turner 1960, 1973). In particular, two parallel ux tube fragments are formed with equal and opposite vorticity. Each fragment has a net circulation which, when combined with its horizontal motion, provides a downward aerodynamic force, canceling the buoyant force. Thus, while this circulation persists, the rise of the ux tube fragment is halted. Longcope et al. (1996) suggested that a twisted ux rope would avoid this fragmentation. Moreno-Insertis and Emonet (1996) con rm this conjecture, with the result that the transverse magnetic eld becomes dynamically important when tan > (a=HP )1=2 where is the pitch angle of the magnetic eld, a is the ux rope radius, and HP is the pressure scale height. Matsumoto et al. (1993) have performed 2D and 3D MHD simulations of buoyant ux sheets and ux tubes in a strati ed atmosphere, intended to model the chromosphere and the corona. This work was an extension of a study of the magnetic buoyancy instability (Shibata et al. 1989, 1990), also known as the Parker instability (Parker 1955). As expected, ballooning of ux into the corona was observed in these simulations. However, the dynamics in the Matsumoto et al. (1993) simulations are such that the ux tube tends to distort and expand, with the expansion moving ux into the corona (see, e.g., Fig. 18 of Matsumoto et al. 1993). The eect of the buoyancy instability, which primarily distorts the ux tube, is distinct from the buoyancy force, which tends to move the entire ux tube upwards. In this paper, we present the results of 2D simulations in which the full compressible MHD equations are solved numerically. We revisit the 2D ux rope geometry of Schussler (1979), Longcope et al. (1996), and Moreno-Insertis and Emonet (1996), but place it in a strati ed chromosphere-corona atmosphere. We are studying this regime to investigate the dynamics of the buoyancy force (pressure gradient), which weakens signi cantly above the chromosphere-corona boundary, and because ux ropes above the photosphere can be observed directly, using a variety of instruments. The model atmosphere, which is initially in hydrodynamic equilibrium with gravity, is similar to that used by Matsumoto et al. (1993). However, while Matsumoto et al. (1993)

{4{ used Tcor =Tch = 25, where Tcor is the coronal temperature and Tch is the chromospheric temperature, we use a more realistic Tcor=Tch = 200. In addition, we consider the eect of a vertically-oriented ambient magnetic eld, which is diamagnetically excluded from the region of the ux rope, as discussed by Pneuman (1983). In the scenario in which an ambient magnetic eld is present but is diamagnetically excluded from the ux rope, it is assumed that the rope is formed before being introduced into a region with nonzero ambient eld. In this case the ambient eld, far from the ux rope, is assumed to be uniform. This is in contrast to the situation discused in Pneuman (1983), where the ambient ux is diverging and exerts a net force on the diamagnetic region containing the ux rope. The signi cance of this diamagnetic force, which does not appear in the present work, will be discussed in section 5 below.

2. Magnetohydrodynamic Simulation Model In this study, we perform 2D MHD simulations of ux ropes in a gravitationallystrati ed model solar atmosphere. In this geometry, shown in Fig. 1 (upper left), the axis of the ux rope is in the y-direction (into the paper) and the gravitational force is in the ?z-direction. Note that the remainder of Fig. 1 will be discussed in section 4 below. The simulations are 2-1/2 dimensional in the sense that all components of the electromagnetic and hydrodynamic elds are included, but no spatial variation is allowed in the y-direction. The code computes a numerical solution to the compressible MHD equations, which are presented here in conservative form: @ + r
(v) = 0 (1) @t @ (v) + r
(vv + IP ? BB ) = ?g^z (2) T @t 4 @ B + r
(vB ? Bv) = 0 (3) @t @E + r
[(E + P )v ? B( B
v )] = ?gv (4) T z @t 4 where  is the density, v is the uid velocity, B is the magnetic eld, I is the unit tensor, z^ is a unit vector in the z-direction, and g = 2:74  104 cm/s2 is the acceleration due to gravity which is assumed constant over the relatively small simulation region (zmax  R ). Here, PT is the total (thermal and magnetic) pressure

PT = p + B 2=8;

(5)

{5{ and E is the total energy

2 E = 21 v2 + ?p 1 + B8 :

(6)

Note that the energy equation uses a simple adiabatic law. In so doing, a number of potentially important eects, such as the non-locality of the radiative transfer physics, are ignored. It is dicult to incorporate such processes into our work, as this would require a 3D radiative MHD code incorporating the relevant solar emission and absorption lines. As the dynamics in this study are dominated by pressure forces and ows in the initially-realistic strati ed solar atmosphere and by the ambient and ux-rope magnetic elds, it is reasonable to proceed within the con nes of ideal MHD. In the present work, we use = 5=3 and assume that the plasma is composed of fully-ionized hydrogen, so that p = 2nkT , n = =mp is the proton number density, and mp is the proton mass. The simulation code uses ux corrected transport algorithms for both the uid quantities, based on the method developed by Zalesak (1979), and the magnetic eld, using the technique of DeVore (1991). In order to maintain r
B = 0 to high accuracy and avoid spurious eld-aligned ows, a staggered grid is used, with uid quantities de ned at the cell center, while the elds are de ned at the interfaces (Evans and Hawley 1988; DeVore 1991). For a uniform mesh, as is used in this study, the code is forth-order accurate in space and second-order accurate in time. Also, zero-derivative Neumann boundary conditions are used on all boundaries.

3. Initialization of a Buoyant Flux Rope The simulation starts with the ux rope in the lower chromosphere, in approximate horizontal total pressure balance with the surrounding plasma. The geometry is depicted in Fig. 1 (upper left), where contours of By are plotted in the (x; z) plane. The speci cation of the initial conditions begins with the determination of a strati ed solar atmosphere, in which the temperature has an assumed analytical pro le T = Tch + (Tcor ? Tch )[tanh( z w? zcor ) + 1]=2; (7) cor with zcor = 2:2  108 cm, Tch = 6000 K, Tcor = 1:2  106 K, and wcor = 2:5  107 cm. Note that this temperature pro le is consistent with observation (Vernazza et al. 1981, Athay 1976). Ambient values pa and a are then determined from a numerical solution of @pa=@z = ?ga, such that the initial atmosphere outside the ux rope is in hydrodynamic equilibrium. With pa = 100 dyne/cm2 and a = 10?10 g/cm3 (n = 6:0  1013 cm?3) speci ed at z = 0; the pro les shown in Fig. 2 are obtained. For the model atmosphere, the pressure

{6{ scale height varies from H = 2kT=mpg = 3:6  107 cm near z = 0 to H = 7:2  109 cm in the corona. Note that the boundary conditions at the top and bottom require g = 0: The eect of these transitions, which are made smoothly, can be seen in Fig. 2, where the pro les atten near the top and the bottom of the simulation region. Away from the boundaries, where the simulated dynamics occur, g = 2:74  104 cm/s2 is constant. In this study, we consider a longitudinal ux rope, with an small degree of twist, embedded in a strati ed atmosphere with a small ambient eld. Thus, the largest contribution to the ux rope eld is an axial component B = Bs (z)^y. In general, we include a force-free twisted eld component within the ux rope, parameterized by Bf , and a background eld outside the ux rope, which is B = Baz^ far from the ux rope. Here, Ba and Bf are constants and are \small" in the sense that (Bf =Bpeak )2  1 and (Ba=Bpeak )2  1, where the peak eld Bpeak occurs within the ux rope. These elds will be described in greater detail below. Inside the ux rope, which has a circular cross-section, we reduce the thermal pressure to maintain total horizontal pressure balance. The reduction in thermal pressure within the

ux rope is parameterized by specifying constant = 8p=B 2 < 1 just inside the ux rope boundary. This reduction in thermal pressure is accomplished by reducing the density, as the temperature remains speci ed as in Eq. (7) above. With pa(z), Bf , Ba, and given, we solve for the axial eld Bs(z) by specifying total pressure balance,

@ (p + B 2=8)=@x = 0;

(8)

across the ux rope boundary. In practice, we obtain total pressure balance to an approximate but adequate degree. The result of this is that, initially, net horizontal forces are very small (or zero), while @p=@z = ?g is not satis ed within the ux rope and the

ux rope is buoyant. Considering the simple case of an untwisted ux tube Btube = Bs(z)^y (Bf = 0) with no ambient magnetic eld (Ba = 0), it can be shown that the pressure-balance condition gives Bs(z) = [8pa(z)=(1 + )]1=2 and pin = By2=8 = pa=(1 + ) within the ux tube, where pin is the thermal pressure inside the ux tube and pa is the ambient pressure. We now consider the general case of a ux rope with radius a, centered at x = 0 and z = z0: In general, the ux rope magnetic eld (r < a) is a combination of a non-twisted \straight" eld Bs and a force-free twisted component Bf :

By (z) = Bs(z) + Bf [4 ? 2(r=a)2]1=2 B = Bf ar

(9) (10)

{7{ where the (r; ) coordinates have their origin at the ux rope center and Bf is a constant. Note that the force-free twisted component, which provides the azimuthal eld within the ux rope (B = Bf at r = a), also contributes to the axial eld. The ambient eld, which is Baz^ far from the ux rope, is assumed to be diamagnetically excluded from the area occupied by the ux rope. In other words, the eld lines of the ambient eld wrap around the ux rope without intersecting it. This eld is determined following the method of Pneuman (1983) for r > a: 2 (11) Ba;z = Ba(1 + ar4 [x2 ? (z ? z0)2]); Ba;x = ?2Bax(z ? z0)a2=r4 ; (12) where Ba is a constant. The pressure inside the ux rope (r < a) is determined as a function of Bs and Bf by specifying < 1 just inside the ux rope boundary: pin = (Bs2 + 3Bf2)=8: (13) Finally, Bs(z) is determined from approximate total horizontal pressure balance at the ux rope boundary: Bs2(z) = [8pa(z) + Ba2]=(1 + ) ? 3Bf2: (14) Equation (14) describes horizontal pressure balance between the dominant axial eld component Bs (z) inside the ux rope and the ambient pressure pa(z) outside the ux rope, including small contributions from Bf , Ba, and the thermal pressure inside the ux rope. For example, the horizontal By contour lines that can be seen in Fig. 1 (upper left) indicate strati cation of By within the ux rope, corresponding to strati cation of pa outside of the ux rope. With Bf , Ba, , and pa (z) speci ed, Bs(z) within the ux rope (r < a) is determined from Eq. (14). The pressure within the ux rope is then computed using Eq. (13). The density within the ux rope is n = pin =2kT , with T given by Eq. (7). This completes the speci cation of the initial conditions. Note that Eq. (14) only approximately satis es the total pressure balance condition, Eq. (8), because the variation of the ambient eld strength along the ux rope boundary has not been accounted for. In particular, we treat the background eld as if it contributes a uniform magnetic pressure Ba2=8 for r > 0. In making this approximation, which is appropriate for (Ba=Bpeak )2  1, we have avoided complicating the ux rope speci cation in such a way that the initial shape of the ux rope cross section is dependent on the value of Ba. For example, if one considers the force balance between only the force-free ux rope contribution (Bf ) and the ambient eld, one nds that the ux rope must have an elliptical (rather than circular) cross-section (Pneuman 1983, Cargill et al. 1996). Note that Eq. (8) is exactly satis ed for Ba = 0.

{8{ It should also be noted that the right-hand side of Eq. (14) can become negative for Bf suciently large, meaning that it is not possible to satisfy Eq. (8) for a ux rope with magnetic pressure greater than the ambient thermal pressure. While such a ux rope might occur in nature, it would tend to expand rapidly and violently. This situation will not be addressed in the present work. As discussed above, the algorithm used in the code maintains r
B = 0 to high accuracy. Thus, we require that the magnetic eld, which is de ned on a numerical grid, satisfy r
B = 0 on the grid at t = 0. This is accomplished by specifying the Bx and Bz eld components (recall that @=@y = 0) in terms of the vector potential A = Ay y^ : (15) Ay = 21 aBf [1 ? (r=a)2]; for r < a and Ay = Bax[1 ? (a=r)2]; (16) for r > a.

4. Dynamics of Buoyant Flux Ropes We consider a number of cases with varying ux rope and ambient eld parameters, as indicated in Table 1. As we are speci cally considering the eect of a transverse magnetic eld (B or Bz ) on the buoyant emergence of ux, we keep all input parameters constant across all simulations, except Bf (recall that B = Bf at r = a) and Ba (Bz = Ba for r  a). In particular, all runs presented here feature the same initial values of = 0:05,

ux rope center coordinate z0 = 0:5  108 cm, ux rope radius a = 0:25  108 cm (see Fig. 1), ambient solar atmosphere parameters (see Fig. 2), and numerical parameters (each run is performed on a 100 by 100 grid). Because the initialization consists primarily of specifying pressure balance between the dominant axial (^y) eld component Bs within the ux rope and the ambient pressure pa outside the ux rope [see Eq. (14)], and because pa and ux rope position z0 do not change from run to run, each simulation has a similar value of the spatially-averaged axial eld hBy i within the ux rope. Average values are in the range 35:2G  hBy i  46:1G as indicated in Table 1. Alfven velocity amplitudes are also similar from run to run, with vA ' 2  107 cm/s being a typical value. For these parameters, the sound speed is Cs ' 1:3  106 cm/s in the chromosphere and ' 1:8  107 cm/s in the corona. Thus, the 200-400 second duration of each of these simulations represents 75-150 Alfven transit times (2a=vA ) across the ux rope and 0.7-1.3 acoustic transit times (4  108 cm=Cs ) across the base of the simulation box.

{9{ We rst consider case 1, in which Ba = Bf = 0. Figure 1, which shows By contours and

ow vectors in the (x; z) plane at t = 0, 146, 218 and 401 sec, illustrates the fragmentation of the buoyantly rising untwisted ux tube. Note that the ow vector lines indicate the direction of ow away from the dot. This result, particularly Fig. 1, t = 146 s, is similar to the convection-zone results of Schussler (1979) and Longcope et al. (1996) and the interplanetary-medium results of Cargill et. al (1996). Figure 1 shows that two ux tubes, with opposite vorticity, are formed parallel to the original ux tube axis. The circulation in the parallel ux tubes provides a downward aerodynamic force resulting in a net horizontal motion (see Longcope et al. 1996, Turner 1960). However, there is an additional eect due to the chromosphere-corona transition at z = zcor = 2:2  108 cm. At this point the temperature increases and the pressure gradient, which drives the upward motion, drops dramatically. This is shown in Fig. 3 where rpa (solid line) is plotted versus z. Once the central part of the ux tube reaches zcor at t = 146 s, upward motion stops, and the ux tube begins to oscillate about an approximate equilibrium position. This oscillation, which cannot be discerned within the few frames of Fig. 1, is quite slow; it is just completing its rst cycle at the end of the simulation. Figure 1 also shows bulk motion of the ambient plasma in reaction to the ux rope. At t = 146 s, the plasma above the ux rope, which was initially pushed upwards by the ux rope, can be seen to be falling under the in uence of gravity. After the upward motion of the central portion of the ux tube has stopped (see Fig. 1), the parallel ux tube fragments continue to move outwards, albeit more slowly than before. At this time, the parallel ux tube fragments break up quite rapidly in a manner that has been previously observed in hydrodynamical studies of buoyant thermals (Turner 1960). Speci cally, the behavior of a buoyant thermal (or buoyant ux tube) depends on both its upward motion and its circulation pattern. In the case that the upward motion falls to zero while the circulation is still nonzero, as happens here, the vortices break up rapidly. At t = 401 s (see Fig. 1), the central portion of the ux tube is oscillating in z (it is in an upward phase), and the parallel ux tube fragmentsRshow signs of spreading and breaking up. At this time, 43% of the initial axial ux  = dxdzBy is contained within the central part of the ux tube (center=initial = 0:43). We now consider increasing the ux rope azimuthal eld B (through Bf ) in cases 2 and 3 (see Table 1). The dynamics in these cases are quite similar to case 1, but with the degree of fragmentation decreasing with increasing B . In fact cases 2 and 3 are so similar that we will concentrate our description on case 3, the results for which are shown in Fig.

{ 10 { 4, where By contours and ow vectors are plotted for comparison to Fig. 1. Results show that the presence of B tends to reduce the strength of the parallel ux rope fragments, in agreement with Moreno-Insertis and Emonet (1996). A measure of this is the fraction of the initial axial ux contained within the central part of the ux rope at the end of the simulation (this fraction is center=initial = 0:43 for case 1). In case 2, where Bf = 4 G, center=initial = 0:54 at t = 442 s. In case 3, Bf = 8 G, a greater percentage of the initial

ux remains in the central portion of the ux rope: center=initial = 0:65. Noting that B ; which has its peak value at the ux rope edge, reduces fragmentation of the ux rope in cases 2 and 3, we now consider the eect of an ambient vertical eld Ba on an untwisted ux tube. Recall that the ambient eld is B = Baz^ far from the ux tube. We use Ba = 4 G in case 4 (for correspondence to case 2) and Ba = 8 G in case 5 (for correspondence to case 3). Figure 5 shows side-by-side contour plots of By and Ay (with ow velocities) for case 4. Note that Ay contours represent approximate magnetic eld lines in the x-z plane. As in case 1 (Fig. 1), fragmentation produces parallel ux tubes which move horizontally away from the central ux tube. These parallel ux tube fragments displace the ambient vertical eld, reaching their greatest separation at t = 245 sec. The ambient eld, however, is strong enough to reverse this motion. At t = 309 s the longitudinal ux has been forced back into a small horizontal space. The By contours of the resulting ux tube are convoluted, but it has risen buoyantly without loss of ux. Figure 6 shows side-by-side contour plots of By and Ay (with ow velocities) for case 5. In this case the ambient eld not only contains the parallel ux tube fragments, it suppresses their formation. In addition, the ambient eld appears to direct or con ne the upward ow in such a way that the ux tube rises farther into the corona than in any of the previous cases. This example shows that that an untwisted ux tube can emerge buoyantly into the corona. Finally, we consider a combination of transverse elds with Ba = Bf = 4 G in case 6. The results, shown in Fig. 7 as side-by-side contour plots of By and Ay ; are remarkably similar to those of case 4 (Ba = 4 G, Bf = 0, see Fig. 5). This suggests that an ambient vertical eld has a greater eect than a twist eld of the same amplitude. Speci cally, the ambient eld provides an external barrier to the ux tube region which can contain the ux tube even in cases (see Figs. 5 and 7) where the hydrodynamic forces are strong enough to tear the ux tube apart. The primary dierence between Figs. 5 and 7 is the asymmetry due to a small degree of reconnection (in Fig. 7) that takes place between the ux rope and the ambient eld

{ 11 { in the region where the eld-line geometry is favorable (x > 0). To illustrate this, several closely-spaced contour lines have been included in the reconnection region. In Fig. 7 at t = 0 (top), open Ay contours represent vertical eld lines corresponding to Ba; and closed contours represent eld lines corresponding to B . Plots at later times, which were chosen for comparison to Fig. 5, show evidence of reconnection for x > 0. They also show that the

ux rope fragmentation in this region is not as well contained as it is for x < 0. This result is similar to those presented in Cargill et al. (1996), where reconnection eects are discussed in some detail for ux ropes in the interplanetary medium. Note that the \reconnection" observed here is enabled by the diusion of eld across the numerical grid; no attempt has been made to model reconnection in a physical way in these simulations.

5. Discussion 5.1. Flux Emergence into the Corona As noted above, the rise of a buoyant ux rope typically halts at the chromospherecorona boundary, where the buoyancy force (rpa) drops signi cantly. To further investigate, additional simulations were performed with smaller values of and larger values of z0 in an eort to observe buoyant emergence of ux into the corona without the assistance of an ambient eld. Because of computational diculties associated with solving the total energy equation, Eq. (4), at low , we were not able to consider < 0:005 within the ux rope. For this reason, these attempts were largely unsuccessful. It is reasonable to expect that, in 3D, after the ux rope rises to the chromospherecorona boundary and the buoyancy force becomes ineective, the buoyancy instability (Parker 1955, Shibata et al. 1989, 1990) might cause distortions of the ux rope eld, allowing a portion of the ux rope to penetrate the corona. Such a case was simulated by Matsumoto et al. (1993) for Tcor=Tch = 25. It is not clear, however, if their results apply to the model atmosphere of the present paper, in which Tcor =Tch = 200. In fact, our value of rpa falls so precipitously at zcor (see Fig. 3), that magnetic forces should dominate in this region for modest values of Ba. To support this statement, we consider the eect of a diverging ambient magnetic ux, which is diamagnetically excluded from the ux rope region (as in Pneuman 1983). Speci cally we imagine a uniform magnetic

ux normal to the solar surface, diverging with radius of curvature R : B = B0R2 [(R(R ++zz)2)^z++xx2]x^3=2 : (17) where R is the solar radius and B0 is the ambient eld value at the base of the simulation

{ 12 { region. Pneuman (1983) has shown that, because this ambient eld is excluded from the

ux rope region and because the value of the eld changes across the face of the ux rope, the ux rope experiences an average force approximately equal to r(B 2=8) which is analogous to rp. This force, which is nearly constant within the simulation box (z  R ), is compared to rpa in Fig. 3 for various values of B0. We nd that r(B 2=8) (dashed lines in Fig. 3) exceeds rpa for z > zcor for B0 as small as 20 G. For B0 = 50 G, which is too small to be detected by many magnetographs, this force could be quite signi cant. The combination of buoyant and diamagnetic forces discussed here and depicted in Fig. 3 appear to provide a mechanism of ux emergence involving only modest elds (< 100 G). If ux is emerging in this way, even at a rate large enough to eect the solar ux budget (Schriver and Harvey 1994), it may thus far have escaped our notice. Further simulations, in which the (Ba=Bpeak )2  1 approximation of the present work is dropped, would clearly be of interest.

5.2. Twisted Flux Ropes in Other Parameter Regimes It is interesting that the distortions of the ux rope shown in Fig. 4 resemble distortions previously simulated in very dierent regimes: in addition to the convection zone studies of Schussler (1979), Longcope et al. (1996), and Moreno-Insertis and Emonet (1996), which typically have  1; these distortions and their suppression by azimuthal eld lines were observed by Cargill et al. (1995, 1996) for interplanetary plasma parameters. In the latter case, the plasma was uniform with > 1 and the ux rope motion was driven by an applied force. The agreement with Cargill et al. (1995, 1996) may, in fact, be serendipitous. For a buoyant ux rope with a round cross section, the net-vertical buoyancy force is greatest for the vertical slice of the ux rope with the largest mass de cit, in comparison to the mass of an equivalent volume of the surrounding plasma. In the case of Cargill et al. (1995, 1996), the ux rope experiences an ad hoc force, which is uniform within the ux rope. As in the buoyancy case, the net force in the direction of motion is greatest in the center of the

ux rope. In both cases, because the net force is maximum at the center of the ux rope, the ux rope quickly takes on the characteristic umbrella shape. In the work of Cargill et al. (1995, 1996), these distortions of the ux rope are signi cant in that interplanetary

ux ropes (magnetic clouds) can be observed in situ; the numerical model can aid in the interpretation of these observations. In the present work, it appears that such distortions must be suppressed to enable buoyant emergence of ux ropes to produce the coherent loop-like structures that are observed in the corona.

{ 13 {

5.3. Heat Transport Across the Flux Rope Boundary With the use of the adiabatic approximation in Eq. (6) above, heat diusion is neglected. This approximation merits further comment. Speci cally, if the ux tube were able to heat up to the temperature of the surrounding plasma with sucient rapidity, it would remain buoyant throughout the simulation. Heat transport across the ux rope boundary in the 2D geometry can occur through collisions or radiation. We have estimated transport rates for case 5 due to these mechanisms. In this regime, where we have assumed a plasma of fully-ionized hydrogen, ion-ion collisions dominate other collisional processes. For case 5, the computed thermal transfer time th = jT (@T=@t)?1j > 103 seconds at all times, indicating that collisions can be safely ignored. Again assuming fully ionized hydrogen, a rough estimate of the radiative transport rate can be obtained by assuming that any given volume of plasma, such as a ux rope, emits and absorbs radiation like a blackbody (i.e., we neglect line radiation). Initially, the thermal transfer time for blackbody radiation rad = jT (@T=@t)?1j ' 10 seconds for the volume of plasma contained within the ux rope. As this volume rises and expands adiabatically, it tends to cool and rad  1=T 3 increases. While it is not clear which process (heat transport or adiabatic cooling) would win out, it can be said that our neglect of radiative heat transfer, which is required to make the computation numerically tractable, is not entirely justi ed on physical grounds. In particular, the inclusion of this physics might allow the ux tube to remain buoyant throughout the simulation. As discussed in section 5.1 above, however, the buoyancy force drops signi cantly above the transition region. Even if the ux tube were to remain buoyant, its upward motion is still expected to fall nearly to zero in this region. We have veri ed the insensitivity of the results to the thermodynamics by repeating case 5 with = 1:1 [see Eq. (6)] and nding results nearly identical to those of Fig. 6. Thus, we maintain that our interpretation of the simulations, in particular our description of the behavior of the ux ropes at the transition region, remains valid. This includes the slowing of the upward motion of the ux tube at the transition region, the tendency of the parallel ux tube fragments to break up rapidly at this transition, and the signi cant eect of the vertical magnetic eld.

5.4. Numerical Convergence The above simulation results were produced by numerically solving the MHD equations for (typically) 4000 time steps on a 100 by 100 grid. A typical simulation consumed

{ 14 { 37 minutes of Cray C-90 time and used 5 MW of computer memory. As a numerical convergence check, additional simulations were performed with reduced grid size and time step on a 200 by 200 grid (8000 steps, 5.1 hours, 19 MW) and with the same physical parameters as the above runs. Results were nearly identical to the 100 by 100 simulations, except in regions where the geometry becomes particularly convoluted, such as in the parallel ux tube fragments of Fig. 1. These are not signi cant dierences. In particular, all physical descriptions of the above results apply equally well to the 200 by 200 simulations. Also, several runs were performed with zmax = 6  108 cm, such that the upper boundary was placed farther from the chromosphere-corona transition. These runs showed that the position of the boundary did not in uence the results.

6. Conclusion In this study, we have investigated the dynamics of emerging magnetic structures, speci cally ux ropes, in a gravitationally-strati ed solar atmosphere using realistic chromospheric and coronal parameters. In two-dimensional magnetohydrodynamic simulations of horizontally-oriented ux ropes, performed under the approximation that the ux rope axis remains parallel to the solar surface as it rises, we have shown that (a) suciently twisted ux rope eld lines can reduce the tendency of a buoyant ux rope to fragment, generating parallel ux ropes, (b) the upward buoyant motion tends to halt at the chromosphere-corona transition, where the pressure gradient drops signi cantly, (c) when this upward motion stops, the parallel ux rope fragments break up rapidly in a manner consistent with the results of hydrodynamical studies of buoyant thermals (Turner 1960), (d) a weak, vertically-oriented, ambient magnetic eld, (Ba=Bpeak )2  1, can prevent fragmentation of an untwisted ux tube, and (e) that an ambient vertical eld of sucient strength (but still \weak") can direct the upward buoyancy-driven ow such that an untwisted ux tube can penetrate the chromosphere-corona boundary. Note that the rst result above, the suppression of ux rope fragmentation by twisted ux-rope eld lines, is consistent with recent studies by Moreno-Insertis and Emonet (1996) for convection-zone parameters. We feel that the latter results (b-e) are of greater signi cance for the study of

ux emergence in the chromosphere and corona. In particular, the use of realistic chromospheric and coronal temperature values is an important step towards direct comparisons between simulation and observation in this dicult regime. Previous studies have tended to feature smaller values of Tcor =Tch than has been used here, both for comparison to theory [Matsumoto et al. (1993) use Tcor =Tch = 25] and to avoid numerical diculties associated with large changes in density and pressure

{ 15 { in the simulation region. In the present simulations we have been careful to avoid initial conditions which lead to overly violent ows, as these can lead to a numerical instability which is characterized by an unphysically large (and growing) ow velocity in a region with an unphysically low density. The use of realistic chromospheric and coronal temperature values also allows the study of new physics. For example, one might imagine that the parallel ux tubes, which were seen to break o from the central ux tube in the convection-zone simulations of Schussler (1979) and Longcope et al. (1996), might themselves emerge into the corona as separate entities. Our results show that this is not likely to be the case. Speci cally, we nd that the parallel ux tube fragments tend to break up rapidly at the chromosphere-corona boundary. Finally, we have found that a weak (Ba=Bpeak )2  1, vertically-oriented, ambient magnetic eld can both prevent fragmentation of an untwisted ux tube and direct the upward buoyancy-driven ow such that the ux tube penetrates the chromosphere-corona boundary. In our example (see Fig. 6) a ux tube with a modest eld strength of 40 G is directed into the corona by a modest ambient vertical eld of 8 G. This example tends to support the view that signi cant amounts of ux may be emerging continuously, unspectacularly, and thus far without our notice. This work was supported by the Oce of Naval Research. This work was also supported in part by a grant of High-Performance Computing (HPC) time from the Department of Defense HPC Shared Resource Center, US Army Corps of Engineers Waterways Experiment Station.

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REFERENCES Acton, L., Tsuneta S., Ogawara, Y., Bently, R., Bruner, M., Can eld, R., Culhane, L., Doschek, G., Hiei, E., Hirayama, T., Hudson, H., Kosugi, T., Lang, J., Lemen, J., Nishimura, J., Makishima, K., Uchida, Y., Watanabe, T. 1992, Science, 258, 618 Athay, R.G. 1976, The Solar Chromosphere and Corona: Quiet Sun (D. Reidel, Dordrecht, Holland) Browning, P.K., and Priest, E.R. 1984, Sol. Phys., 92, 173 Burlaga, L. F., Sittler, E., Mariani, F., and Schwenn, R. 1981 J. Geophys. Res., 86, 6673 Cargill, P.J., Chen, J., Spicer, D.S. and Zalesak, S.T. 1995, Geophys. Res. Lett., 22, 647 Cargill, P.J., Chen, J., Spicer, D.S., and Zalesak, S.T. 1996, J. Geophys. Res., 101, 4855 DeVore, C.R. 1991, J. Comp. Phys., 92, 142 Evans, C.R. and Hawley, J.F. 1988, ApJ, 332, 659 Leka, K. D., Can eld, R. C., McClymont, A. N., and Driel-Gesztelyi, L. 1996, ApJ, 462, 547 Lites, B. W., Low, B. C., Pillet, V. M., Seagraves, P., Skumanich, A., Frank, Z. A., Shine, R. A., Tsuneta, S. 1995, ApJ, 446, 877 Longcope, D. W., Fisher, G. H., and Arendt, S. 1996, ApJ, 464, 999 Matsumoto, R., Tajima, T., Shibata, K., and Kaisig, M. 1993, ApJ, 414, 357 Mikic, Z., Barnes, D. C., and Schnack, D. D. 1988, ApJ, 328, 830 Moreno-Insertis, F., and Emonet, T. 1996, ApJ, 472, L53 Parker, E.N. 1955, ApJ, 121, 491 Pneuman, G.W. 1983, ApJ, 265, 468 Scherrer, P.H., Bogart, R.S., Bush, R.I., Hoeksema, J.T., Kosovichev, A.G., Schou, J., Rosenberg, W., Springer, L., Tarbell, T.D., Title, A., Wolfson, C.J., Zayer, I., and the MDI engineering team 1995, Sol. Phys., 162, 129 Schriver, C.J. and Harvey, K.L. 1994, Sol. Phys., 150, 1

{ 17 { Schussler, M. 1979, A&A, 71, 79 Shibata, K., Tajima, T., Matsumoto, R., Horiuchi, T., Hanawa, T., Rosner, R., and Uchida, Y. 1989, ApJ, 338, 471 Shibata, K., Tajima, T., and Matsumoto, R. 1990, Phys. Fluids B, 2, 1989 Solanki, S.K. 1993, Space Sci. Revs., 63, 1 Tanaka, K. 1991, Sol. Phys., 136, 133 Turner, J. 1960, J. Fluid Mech., 7, 419 Turner, J. 1973, Buoyancy Eects in Fluids (Cambridge: Cambridge University Press) Vernazza, J.E., Avrett, E.H., and Loeser, R. 1981, ApJS, 45, 635 Wilson, R.M., and Hildner, E. 1984, Sol. Phys., 91, 169 Wilson, R.M., and Hildner, E. 1986, J. Geophys. Res., 91, 5867 Wilson, R. M. 1987, Planet. Space Sci., 35, 329 Zalesak, S.T. 1979, J. Comp. Phys., 31, 335 Zhang, G. and Burlaga, L. F. 1988, J. Geophys. Res., 93, 2511 Zirin, H. 1970, Sol. Phys., 14, 328

This preprint was prepared with the AAS LATEX macros v4.0.

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Fig. 1.| Contours of By and ow vectors plotted in the (x; z) plane for case 1. Flow vector lines indicate the direction of ow away from the dot.

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Fig. 2.| Pro les versus z of ambient temperature T (K), pressure pa (dyne/cm2), and proton number density na = a=mp (cm?3).

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Fig. 3.| Pro le versus z of rpa (dyne/cm3). For comparison, r(B 2=8) pro les (dashed lines) are also plotted for B given by Eq. (17) and for various values of B0.

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Fig. 4.| Contours of By and ow vectors plotted in the (x; z) plane for case 3.

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Fig. 5.| Contours of By (left column) and Ay contours with ow vectors (right column) plotted in the (x; z) plane for case 4. Contours of Ay correspond to ambient magnetic eld lines.

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Fig. 6.| Contours of By (left column) and Ay contours with ow vectors (right column) plotted in the (x; z) plane for case 5.

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Fig. 7.| Contours of By (left column) and Ay (right column) plotted in the (x; z) plane for case 6. Note that the Ay contours are not evenly spaced; several closely-spaced contour lines have been included in the reconnection region.

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Table 1. Simulation Parameters Case

Bf (G)

Ba (G)

hBy i (G)

1 2 3 4 5 6

0 4 8 0 0 4

0 0 0 4 8 4

35.2 41.4 46.1 35.4 36.1 41.6