MAGNETIC ENERGY STORAGE IN THE TWO HYDROMAGNETIC TYPES OF SOLAR PROMINENCES

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1 The Astrophysical Journal, 600:043 05, 2004 January 0 # The American Astronomical Society. All rights reserved. Printed in U.S.A. MAGNETIC ENERGY STORAGE IN THE TWO HYDROMAGNETIC TYPES OF SOLAR PROMINENCES M. hang,2 and B. C. Low 2 Received 2003 July 3; accepted 2003 September 22 ABSTRACT We present analytical solutions that describe the hydromagnetic support of solar prominences in two characteristic configurations, called normal and inverse. We model the corona as axisymmetric outside a unit sphere and treat the prominence as a distributed cold plasma inside a purely azimuthal magnetic flux rope, held in equilibrium by the prominence weight and by an external poloidal magnetic field rigidly anchored to the base of the modeled corona. We focus on the storage of magnetic energy, in particular its potential for driving solar coronal mass ejections (CMEs). Our calculations indicate that both characteristic magnetic configurations are capable of storing enough magnetic energy to overcome the Aly limit for opening up an initially closed magnetic field. These calculations also indicate that magnetic topology is an important influence in magnetic energy storage. Fields with a normal configuration are more likely to attain energetic states leading to CME-type expulsions than those with an inverse configuration, a property we use to explain Leroy s observations of the height distributions of the two types of solar prominences. Subject headings: MHD Sun: corona Sun: coronal mass ejections (CMEs) Sun: magnetic fields Sun: prominences. INTRODUCTION efforts have focused on the inverse prominence field and its Prominences are cool, dense condensations suspended in the relationship to CME-type expulsions (Low & Hundhausen large-scale magnetic fields of the million-degree solar corona 995; Lin, Forbes, & Isenberg. 200; Fong et al. 2002). In the (Tandberg-Hanssen 995; Priest 982). Vector magnetic field recent work by Low & hang (2002), the point was made that measurements suggest that the prominence field is nearly the distinct field topologies of normal and inverse prominences imply observable differences in the manner that horizontal, leading to the interpretation that the prominence mass is suspended at the gravitationally lowest points of the associated CMEs accelerate out of the corona. The consistency of the theory with recent observations (hang et al. locally bow-shaped magnetic lines of force. Statistically, prominences can be classified into two hydromagnetic types 2002; hang & Golub 2003) further encourages us to consider (Leroy 989; Bommier 998). The prominence is called a the topological distinction between the normal and inverse normal prominence if its magnetic field threads across the configurations. prominence in the same direction as the photospheric magnetic In this paper, we first build the normal and inverse prominence fields in a parallel way (x 2) and then consider their field underneath. If the prominence field points in the opposite direction to the underlying photospheric field, the prominence magnetic energy storage for driving CME-type expulsions is then called an inverse prominence. (x 3).Weaddressthedifferencebetweenthetwotypesof Coronal mass ejections (CMEs) are a major form of solar fields associated with the two types of prominences (x 4) and activity in the corona. They are found to take place in make an estimation of the mass necessary for confining the association with prominence eruptions about 75% of the time flux rope (x 5). Finally, a brief summary and discussion is (Munro et al. 979; Webb & Hundhausen 987). Observations also suggest that CMEs may involve the complete given in x 6. opening up of a closed magnetic field (Hundhausen 988). 2. HYDROMAGNETIC PROMINENCE MODELS So, if we treat the field around a prominence as the preeruption state producing a CME, we would then expect such a We treat the Sun as a unit sphere in spherical coordinates IN AXISYMMETRY pre-eruption state to store enough magnetic energy for the and consider a static, axisymmetric atmosphere in which the field to open up. This has been discussed by several authors in Lorentz force is balanced against the pressure gradient force terms of whether the stored energy may exceed the Aly limit, and solar gravity described by the equation the threshold for completely opening up an initially closed magnetic field (Aly 984, 99; Sturrock 99; Low & Smith 993; Low 200, hang & Low 200; Fong, Low, & Fan ð 4 H B Þ B Hp GM ˆr r 2 ¼ 0; ðþ 2002). However, because a majority of observed prominences so where B, p,,andm are the magnetic field, pressure, density, far are inverse prominences (Leroy 989), most theoretical and the solar mass, respectively. Express the magnetic field in terms of a flux function A and an associated scalar function Q National AstronomicalObservatory,Chinese Academy ofsciences, Beijing B ¼ 2 HighAltitudeObservatory,National Centerfor AtmosphericResearch,P.O. ˆr r â þ Qĵ : ð2þ Box 3000, Boulder, CO

2 044 HANG & LOW Vol. 600 Then, if jhaj 6¼ 0, equation () requires that Qðr;Þ¼QðAÞ and reduces to a pair of equations for force balance in the r- plane. They are LA þ Q dq da þ 4r2 sin rþ ¼ 0 and þ GM r 2 ¼ 0; ð4þ L 2 r 2 with ¼ cos. Two points are relevant to make before discussing the details of field constructions. First, since the energy involved in opening up a closed magnetic field in the Cartesian plane is unbounded (Low 990), the large-scale considerations of CME-related phenomena naturally require the use of spherical geometry rather than the Cartesian plane. Second, solving the partial differential equations above would confront us with the mathematical difficulties of solving nonlinear equations. Given a form of Q(A), we have a set of two equations with three unknowns: A, p, and. In general, if an equation of state is added, the system is then closed. For the unbounded domain outside a unit sphere, the admissible free function Q(A) generally leads to a nonlinear source term in equation (3) (Flyer et al. 2003; Wolfson 2003). Thus, any physically reasonable formulation of the boundary value problem for A based on equation (3) in the unbounded domain requires a treatment of nonlinear partial differential equations. In order to keep the physics simple and the mathematics tractable, we take the following first approach to address the physical question of how the mass of a prominence of the two hydromagnetic types may trap magnetic flux and magnetic energy in equilibrium in the unbounded atmosphere. We take the field to be composed of two separate fields found in two complementary regions denoted by and 0. The region is bounded by a flux surface, everywhere within which we set jhaj ¼0andQ 6¼ 0, representing a rope of purely azimuthal magnetic flux. The complementary region 0 contains a purely poloidal field with Q ¼ 0. By assuming that the atmosphere does not interact with the magnetic field in 0, the field in this region is then potential. This construction is not trivial because the potential field in 0 is due to currents inside the unit sphere as well as currents located within. In the following parts of this section, we will use such an approach to build global solutions by matching the potential field in 0 with the flux rope in across their common boundary. 2.. Global Structure In constructing global solutions, we first treat the potential field in 0 by setting for this region Q ¼ 0 and taking the atmosphere to be negligible, i.e., p ¼ ¼ 0. Equation (3) then reduces to the Laplace equation: LA ¼ 0: The potential field we need must account for both a prescribed boundary flux at the unit sphere and the presence of ð5þ ð6þ currents in defined in some suitable manner. Since potential fields can be linearly superposed, we look for a superposition of two potential fields, one accounting for the flux distribution on the unit sphere and the other contributing no flux at the unit sphere but accounting for the currents in. It serves the purpose of this paper to prescribe the normal flux distribution on the unit sphere to be that of the classical dipole; then the first potential field for the superposition becomes the potential dipolar field with the stream function sin 2 A dipole ¼ B 0 ; ð7þ r where B 0 is a constant field amplitude. For the second potential field, we assume a net azimuthal current of intensity I 0 flowing within some region in the r- plane yet to be defined. This second potential field is prescribed to contribute no flux across the unit sphere, since the flux at that boundary is prescribed to be entirely accounted for by the potential dipolar field in equation (7). The second potential field is thus the field due to the azimuthal current in combined with the image of that current under inversion about the unit sphere. A simple solution of this kind can be constructed as follows. As we know, an axisymmetric magnetic field B with a stream function A can also be written in terms of a vector potential A, where B ¼ H A and A ¼½A= ðr sin ÞŠˆ. So, if we take the vector potential of a circular line current of intensity I 0 lying in the equator with radius a 0 > and centered at the origin, from Jackson (975), a simple substitution will give the associated stream function A I in the form of cos d A I ðr;þ¼2i 0 a 0 r sin 0 a 2 0 þ =2 : ð8þ r2 2ar sin cos Figure a gives an example of this kind of current loop field for the case of I 0 ¼ 2:5 anda 0 ¼ 2:0. The image field A image of A I is given by the transformation ra I (/r, ), which yields r cos d A image ðr;þ¼2i 0 a 0 sin 0 ða 2 r 2 þ 2a 0 r sin cos Þ : =2 ð9þ This image field has the same flux distribution at the solar surface r ¼ as the original current loop field. By subtracting the image field from the original current loop field, the subtracted field (A I A image ) will have no flux at the solar surface. By adding the above dipolar field, we then fix the normal field at the boundary. The combined field, A ¼ A I A image þ A dipole cos d ¼ 2I 0 a 0 r sin 0 a 2 0 þ =2 r2 2ar sin cos r cos d þ 2I 0 a 0 sin 0 ða 2 r 2 þ 2a 0 r sin cos Þ þ B =2 0 sin 2 ; r ð0þ is still potential, satisfying the Laplace equation (6), except at the line current located at r ¼ a 0, but the normal field

3 No. 2, 2004 MAGNETIC ENERGY STORAGE IN SOLAR PROMINENCES 045 the superposed flux function A in 0. The constructed potential field A satisfies the prescribed dipolar flux distribution at the unit sphere, and any one of the flux surfaces of constant A may be used to exclude a region containing the line current at r ¼ a 0, which we define to be the flux rope region. Theglobalfieldwehaveconstructedhasthreefreeparameters: I 0, a 0,andB 0.ThetermB 0 determines the flux distribution at r ¼, and we fix it at B 0 ¼ 0. Here and elsewhere, we use suitable nondimensional units for all physical quantities, unless otherwise mentioned. The sign of I 0 will influence the topology of the global magnetic field. When I 0 is positive, that is, when I 0 is of the same sign of B 0, the solution will produce a field containing an inverse prominence configuration, as shown in Figure 2a. WhenI 0 is negative, that is, when I 0 is of the opposite sign of B 0, the solution will produce a field containing a normal prominence configuration, as shown in Figure 2b. By normal or inverse prominence configuration, we mean that the horizontal magnetic field at the bottom of the flux rope (domain ) is in the same or opposite direction, respectively, compared to the underlying photospheric magnetic field (Leroy 989). These can be seen from the arrows in Figures 2a and 2b where the directions of magnetic fields are shown. Here we need to mention that, with the further development of the model construction in this paper, we will remove all the poloidal fields in the flux rope and replace them with purely azimuthal fields held by the weight of cold plasma distributed inside the rope. In so doing, the flux rope will have no transverse fields anymore. However, the difference in the global field topologies is still there, and we regard this difference as the fundamental property that is implied by the observed normal and inverse prominences. In x 2.2, we will construct the atmosphere in with a purely azimuthal magnetic field for a flux rope. This conveniently removes the field singularity along the circular line current. Fig.. Examples of magnetic configurations of fields, which are used to construct the global structures in this paper. (a) Current loop field with I 0 ¼ 2:5 anda 0 ¼ 2:0; (b) Image field of the current loop field in (a); (c) Field having no flux at the solar surface, obtained by subtracting the field in (b)from the field in (a); and (d) Dipolar field, used to control the boundary flux at the surface. See text for details. distribution at r ¼ is then uniquely determined by B 0.By taking B 0 at a fixed value, we may construct a whole family of potential fields in 0 by varying the constant parameters I 0 and a 0, all having the same boundary flux distribution. In so doing, the magnetic energy necessary for the fields to open up to infinity (Aly 984), that is, the Aly limit, will be the same for all fields we construct here. For each of these potential fields, we can then define, a posteriori, the boundary of to be a flux surface of constant A ¼ A I A image þ A dipole, which encloses the line current I 0. Depending on how we construct the equilibrium atmosphere inside, we would then have a global solution filling up the unbounded region outside the unit sphere. The internal structure of the region self-consistently accounts for the total current I 0 that contributes to the potential field A in 0.Figureb shows the image field of the current loop field in Figure a, Figure c shows the subtracted field (A I A image ), and Figure d shows the dipolar field that we have added to obtain 2.2. The Solution of the Flux Rope in Before we use the method in Fong et al. (2002) to construct a flux rope solution in, we motivate our construction with an explanation of the physical issues we are addressing. The free magnetic energy stored in the atmosphere outside the unit sphere is due to the presence of the total atmospheric current I 0. Although the field is potential in 0, this field shows the presence of that total current flowing in the azimuthal direction. In so far as the region 0 is concerned, that current is equivalent to a line current of intensity I 0 located at r ¼ a 0. By selecting a flux surface of constant A to define, we need to build a volumetric atmosphere in this region that has the same effect on the magnetic field in 0 as the line current at r ¼ a 0. Current systems in the absence of rigid walls tend, by the hydromagnetic virial relationships, to expand to fill all space (Parker 979; Low 200). Whatever atmosphere we will be able to construct in to account for that total current I 0 will involve confinement against the current system expanding outward. The surrounding potential field in 0 plays a role in the confinement by the magnetic pressure it exerts on. Depending on what is constructed inside, plasma pressure and weight generally must play a role in the force balance that determines the equilibrium state in 0. Pressure inside is not a good confinement agent because, being positive definite, it drives an expansion into the region 0. The only agent available for additional confinement of at its prescribed location is gravity. Hence, we seek the simplest model in which the weight of a cold gas serves to confine a rope of azimuthal flux in.

4 046 HANG & LOW Vol. 600 Fig. 2a Fig. 2b Fig. 2. Examples of the two types of the fields constructed in this paper. (a) Field showing inverse prominence configuration, with parameters I 0 ¼ 0:7, a 0 ¼ 2:0, and B 0 ¼ 0; and (b) Field showing normal prominence configuration, with parameters I 0 ¼ 0:7, a 0 ¼ :5, and B 0 ¼ 0. The heavy curve in each panel indicates the flux surface that separates detached fields () from external anchored fields ( 0 ). Note that we have replaced the field inside each curve () withpurely azimuthal fields, held in equilibrium by the weight of the internally distributed plasma, to represent flux ropes. Such an atmosphere in is required to be in equilibrium with the atmosphere in 0 under the condition of the continuity of the magnetic pressure across their common boundary. Since the field is poloidal in 0 and azimuthal in, this equilibrium surface is a magnetic tangential discontinuity. Suppose the boundary between and 0 is the flux surface A ¼ A 0, A 0 being a constant. Then, setting A ¼ A 0 uniformly in to neglect the poloidal field component of B in equation (2), the equilibrium equation () for a cold plasma with p ¼ 0 reduces to 4r 2 sin 2 QHQ þ GM ˆr r 2 ¼ 0: ðþ Here, in this degenerate case, the previous requirement of Q to be a function of the flux function A will not arise because jhaj ¼0. Equation () then requires Q to be a purely radial function, and the density is determined by ¼ 8GM sin 2 dq 2 ðrþ : ð2þ dr For a fixed r, the magnitude of the density ( ) increases away from the equator, diverging at the polar axes, which means that this solution can only describe a localized structure away from the polar axes. Now we take the following steps to construct the global solution in r >. The potential field given by equation (0) is made up of a detached field surrounded by fields anchored to r ¼ (see Fig. 2). We define the region to occupy the entire region of the detached field. Therefore, the common boundary

5 No. 2, 2004 MAGNETIC ENERGY STORAGE IN SOLAR PROMINENCES separating and 0 is the flux surfaces marked by the thick lines in the two subfigures for the two cases displayed in Figure 2. We take the flux rope solution described by equations () and (2) to occupy the region. Since@ is a boundary between purely poloidal and purely toroidal fields, it is a magnetic tangential discontinuity, which, under equilibrium, requires ¼ QðrÞ: ð3þ This continuity condition determines Q(r) inequation(2)in terms of the potential field in 0. In principle, any of the nested magnetic flux surfaces within our choice of the region couldbeusedtoredefineasmaller, a sophistication of the model that we avoided to keep the mathematical model simple. This approach suits our purpose in this paper, which is to capture the property of trapping a magnetic flux rope in equilibrium, taken in its simplest form of a pure azimuthal flux, by the mass in the rope, taken again in the simplest form of a cold plasma. The top and bottom panels of Figure 3 give the variation of Q 2 (r) for the field presented in Figures 2a and 2b, respectively. We can see that, for normal prominence solutions, the form of jhaj gives a monotonically decreasing Q 2 (r) and, hence, is associated with a positive definite density in. For the inverse prominence solutions, Q 2 (r) increases with r first and then decreases. So the mass distribution in our inverse prominence solutions will have a cavity (that is, negative density in the solution) in the lower part of the flux rope and mass loading (that is, positive density in the solution) at the upper part of the flux rope. One way to understand the negative density in the solution is to interpret it as a departure, relative to a global spherically symmetric atmosphere with the density D 0 (r) and pressure P 0 (r) occupying the entire region r >, governed by the equation dp 0 dr þ D 0GM r 2 ¼ 0: ð4þ This is the atmosphere, which, by assumption, does not interact with the magnetic field itself and is added to our constructed field to make the net plasma density inside the flux rope, which is D 0 ðrþþðr;þ now, not negative. The latter, (r, ), is the density distribution determined by equation (2), which, for our inverse prominence cases, becomes negative in the lower parts of the rope. By adding the atmosphere described by equation (4), the plasma pressure and density outside the flux rope become P 0 and D 0, respectively, instead of the p ¼ ¼ 0 assumption we made before. Now the plasma outside the flux rope has a temperature T 0ðrÞ ¼P 0 ðrþ=d 0 ðrþ, up to a multiplicative factor from the ideal gas law. Inside the flux rope, the plasma has a pressure P 0 (r) and density D 0 ðrþþðr;þ governed by the force-balanced equation, ð 4 H B Þ B HP 0 ðd 0 þ ÞGM ˆr r 2 ¼ 0: ð5þ The plasma inside the rope is then not cold (T ¼ 0) anymore. It has a temperature T ðr;þ¼p 0 ðrþ=½d 0 ðrþþðr;þš; up to a multiplicative factor. Note where ðr;þ < 0, T ðr;þ > T 0ðrÞ at the same radial distance. Our solution suggests that the commonly seen cavity structures associated with streamers and CMEs are more likely to be associated with inverse prominences (Illing & Hundhausen Fig. 3. Variation of the Q 2 parameter of the inverse prominence field shown in Fig. 2a (top); and of the normal prominence field shown in Fig. 2b (bottom). 986; Low & Hundhausen 995). The flux rope is then characterized by a temperature T(r, ), which is higher than the external temperature at the same height. If the temperature in the flux rope is indeed higher than that in the external field as modeled by our simple solution, this may explain why cavity structures are often clearly seen in white-light observations (Hundhausen 999) while they are only rarely detected by soft X-ray observations. We suggest that this is because the whitelight emission is only sensitive to the plasma densities while the soft X-ray emission is sensitive to both the plasma density and the temperature. The way we have defined in Figures 2 and 3 is to to be the boundary separating the magnetic flux in r > anchored to r ¼ from that which is not anchored to r ¼. By this definition there also exist solutions for the inverse flux rope that are topologically distinct from the one shown in Figure 2a, the latter characterized with an X-type magnetic neutral point. If, for some suitably large I 0, we set a 0 sufficiently close to unity, the inverse flux rope magnetic topology takes the form in Figure 4 with no X-type magnetic neutral point. A variation of a 0 down toward unity will show that the global magnetic topology evolves parametrically from onewithanx-typeneutralpointsuchasshowninfigure2a to one without an X-type neutral point such as shown in Figure 4. In the latter, the magnetic pressure is monotonically decreasing with radial distance, and the rope is anchored by a mass density that is everywhere positive inside the flux rope. In other words, if there is sufficient mass to serve as anchor, an inverse flux rope can be confined to the base of the low corona without any cavity-like depletion within the rope. However, as the solution in Figures 2a and 3 (top) show, when such a rope

6 048 HANG & LOW Vol. 600 with continuity of the total pressure across them (Low & Smith 993). Thus, the magnetic energy in r > for our flux rope solution can be written as E ¼ B 2 dv 8 r> ¼ B 2 r 4 GM B2 sin d þ dv; ð7þ r r¼ taking note that is not zero only in and that B ¼ 0along r ¼. We present diagrams of the magnetic energies stored in the inverse and normal prominence fields in the top and bottom panels of Figure 5, respectively, plotted against the magnitude of the current density I 0. In each diagram, the dashed line and triangle symbols show the case of a 0 ¼ 2:0 fields; the solid line and diamond symbols show the case of a 0 ¼ :8 fields; and the dotted line and the plus symbols show the case of a 0 ¼ :5 fields. For fields having the same a 0 value, we can see that increasing the magnitude of I 0 will increase the magnetic energy stored. If the boundary flux is of a dipolar form with the field intensity 0 in the dimensionless unit we use, the corresponding Aly limit is Figure 5 tells us that, Fig. 4. Same as in Fig. 2a, but for an inverse prominence field of the type that has no X-type neutral points in the field. See text for details. Here I 0 ¼ 3:0, a 0 ¼ :, and B 0 ¼ 0. rises to greater equilibrium heights, the density depletion becomes a natural feature of these equilibrium states. 3. MAGNETIC ENERGY STORAGES Following Fong et al. (2002), we recall the virial theorem of Chandrasekhar (96): 8 V B 2 dv ¼ 8 þ B 2 r ds 2ðB r GM r dv; ÞðB dsþ ð6þ which gives the total magnetic energy in domain V in terms of the values of B evaluated on the and the total potential energy of the cold plasma (Low 999). This virial equation remains valid even if the hydromagnetic equilibrium contains magnetic tangential discontinuities in force balance Fig. 5. Plots of the magnetic energy stored in each field against the magnitude of the current intensity I 0 for different a 0 values of fields with inverse prominence configurations (top) and of fields with normal prominence configurations (bottom). In each plot, the dashed line and triangles are for a 0 ¼ 2:0 fields; the solid line and diamonds are for a 0 ¼ :8 fields; the dotted line and plus signs are for a 0 ¼ :5 fields. The straight dotted line indicates the energy threshold, that is, the Aly limit, for fields constructed in this paper.

7 No. 2, 2004 MAGNETIC ENERGY STORAGE IN SOLAR PROMINENCES 049 for suitable values of I 0 and a 0, both inverse and normal prominence fields can store enough magnetic energy to exceed the Aly limit, that is, these fields are capable of opening up to infinity as CME-type expulsions. The capability of opening up to infinity is taken in the sense of Low & Smith (993) that the magnetic energy is sufficient to account for stretching the field to open up, with an excess to drive the part of the mass that is to be ejected in the CME. Not all the mass that holds the flux rope in equilibrium in our formal model is to be expelled with the free energy. In some cases, the excess magnetic energy above the Aly limit may not be sufficient to do the work of lifting all of the anchoring cold mass in the flux rope equilibrium state (see the discussion in Fong et al. 2002). 4. THE DIFFERENCE BETWEEN THE TWO TYPES OF FIELD Figure 5 shows contrast in the ways normal and inverse prominence fields store magnetic energy. Fields with the inverse prominence topology tend to need larger current intensities (I 0 ) to overcome the Aly limit. To better understand this, we change the modeling parameter I 0 to the azimuthal flux of the flux rope as a free parameter, F ¼ B ds ¼ B rdrd: ð8þ The top and bottom panels of Figure 6 show, respectively, the magnetic energies of the inverse and normal prominence fields plotted against the azimuthal flux of the flux rope (F ), instead of the magnitude of the current density I 0, for those cases displayed in Figure 5. We can see that, of the two field topologies, the inverse flux rope generally requires a greater amountofazimuthalfluxtobeinastateofenergyinexcessof the Aly limit. Recall that both the normal and inverse prominence fields we built have the same boundary flux distribution at r ¼ and share the same Aly limit. This difference is due to the interplay between force balance and field topology, a property that may be conveniently understood in terms of Biot-Savart current interactions as we describe below. In each solution, we can identify three azimuthal currents whose mutual static interactions account for the forces acting on the prominence flux rope. The background dipolar field given by equation (7) is due to a positive current in the azimuthal direction associated with the dipolar moment of this potential field. In the atmosphere, there is the total current I 0 representing the azimuthal current flowing within the flux rope. This current is associated with its image current of the opposite sign located in r <. The latter is induced to flow so that r ¼ conserved its flux distribution when a flux rope movesintooriscreatedintheatmospherer >. Therefore, the weight of the plasma in the flux rope is balanced by the summed attractive or repulsive forces between these current systems acting on the flux rope. Similar considerations of current attraction and repulsion have also been carried out for straight line currents or distributed current systems in the study of inverse and normal prominences in two-dimensional Cartesian models (van Tend & Kuperus 978; Amari & Aly 989, 990, 992; Low & Hundhausen 995). Here we have a more complex situation in spherical geometry. Whereas a straight line current exerts no Biot- Savart force on itself, its equivalent in axisymmetric spherical geometry, the circular current loop, exerts a hoop force on Fig. 6. Same as Fig. 5, but here the magnetic energy stored in each field is plotted against the azimuthal flux of the rope in each field. This plot shows that the fields with the normal prominence topology need less azimuthal flux of the rope to overcome the Aly limit. itself that always tends to increase the area of the hoop (Landau & Lifschitz 960; Fong et al. 2002). Nevertheless, the basic interactions are similar and we shall see that, in our solutions, there is a direct connection between the sign of I 0 and the nature of force balance in the inverse and normal prominence fields. For normal prominence fields, the flux rope current I 0 is negative, whereas its image current in r < and the current of the potential dipolar field are both positive. The flux rope is thus repelled by both these currents in r <, and the radial outward force is capable of being in equilibrium only by loading a sufficient amount of mass. This means that, inherently in this type of topology, the flux rope is already in a state ready to be expelled radially outward. If the mass is unloaded by prominence drainage, for example, the flux rope will be forcefully expelled to infinity as a CME. This may explain why, compared to inverse flux ropes, normal equilibrium flux ropes with relatively less azimuthal flux may have a greater amount of energy stored, as indicated by Figures 5 and 6. For the inverse flux rope, I 0 > 0, its image current in r < is negative, but the current of the dipolar potential field is positive. The flux rope is repelled by the image current but attracted by the potential-field current. The intensity of attractive or repulsive force between the two currents is, to first order, proportional to the product of the current intensities.

8 050 HANG & LOW Vol. 600 If I 0 is too small in magnitude, the mutual repulsion between the flux rope current cannot dominate, and the net force on the flux rope is then radially inward. For cases of small I 0,the calculated total mass required to hold the flux rope in equilibrium is negative, so the inward Lorentz force is balanced by an outward, mathematically defined, plasma weight. In such a solution, the inverse flux rope is then physically interpreted to be partially evacuated of mass and is subject to an outward buoyancy force that balances the inward Lorentz force. In contrast, an inward radial force on the flux rope cannot be found in the normal configuration for any current intensity I 0. For the inverse flux rope, only when I 0 is large enough for the mutual repulsion between the flux rope current and its image to dominate does the field become ready for expulsion, that is, the radial Lorentz force on the rope is directed outward, and, if it is strong enough, the rope has enough energy to exceed the Aly limit. Our calculations give a physical insight into the height distribution of the two types of prominences observed by Leroy (989). When using coronagraphs to observe the magnetic fields of prominences, those prominences accessible to observation must lie above the height of 0 4 km above the solar limb. Of such prominences, Leroy (989) found that the number of the inverse prominences is larger than the number of the normal prominences by a factor of about 3 and that the positions of the observed normal prominences are statistically lower in heights than those of the observed inverse prominences. We suggest that the flux ropes are formed in the solar interior by dynamo actions and levitated into the corona by passing through the photosphere to re-form in the corona (Low 996). Let us assume that flux ropes emerging into the corona have equal probabilities to form normal or inverse prominences, a simplifying assumption for the purpose of making a physical point. Then we expect an equal number of normal and inverse prominences at the time of flux emergence. However, since ropes associated with normal prominences are always subject to an outward-directed Lorentz force and need comparatively less flux to overcome the Aly limit, once they get into the corona, more of them will have the tendency to erupt with expulsion out of the corona than those of the inverse types. Hence, after a length of time, there will be more inverse prominences left over to be observed as quiescent prominences as observed by Leroy, and more inverse prominences will get a chance to levitate themselves to greater heights in the corona and be observed. This may qualitatively account for Leroy s observations that more inverse prominences are observed and are observed at a relatively greater height. For those ropes of inverse prominence type that hang in the corona, we suggest that they will experience a relatively long lifetime. In this duration, continuous flux emergence through the photosphere underneath may bring additional flux and magnetic helicity to add to the flux rope by magnetic reconnections (hang & Low 2003). Also, the Aly limit applying at a given time may fall below the total energy as the result of the continuous changing of boundary. Ultimately, these inverse prominence ropes may still explode as CMEs but at a later time when the above two evolutionary processes have brought about the conditions for eruptions. 5. DISCUSSION Since both normal and inverse prominence fields can store magnetic energies in excess of the Aly limit, several additional questions become relevant: How much mass is needed to anchor such energetic fields? Are the required masses close to the observed masses of solar prominences? Is the difference in the physics of magnetic energy storage in normal and inverse prominence fields reflected in the masses required to anchor the fields? Fong et al. (2002) and Low et al. (2003) have suggested that, for inverse prominences, prominence masses play a significant role in the trapping of magnetic flux ropes as a mechanism of storing magnetic energy to drive CME expulsions. We have estimated the masses necessary for trapping magnetic flux ropes based on our solutions. For a comparison with these earlier estimates, we adopt, for our dipolar background field, the field intensity 2B 0 at the poles of the unit sphere to be in the range 5 0 G. Then, we pick two cases from our solutions, one in a normal and the other in an inverse configuration, both with the same a 0 ¼ :5 value, but with values of I 0 such that they have roughly the same magnetic energy just exceeding the Aly limit. We find that the total mass needed for anchoring the flux rope associated with the selected inverse prominence lies in the range ð3 2Þ0 6 gforthe adopted background field intensity range. These numbers are roughly the same as those estimated by Low et al. (2003) for their models of inverse prominences, the latter treating the prominence as a cold sheet of mass rather than a distribution of cold mass as found in our model. A smaller mass for the prominence is obtained if we scale down the circumnavigating prominence in the axisymmetric model to a prominence length characteristic of real prominences. Subject to observational uncertainties in inferred prominence masses, as discussed in Low et al. (2003), our estimated prominence masses, based on simple idealized solutions, are consistent with observations. Our estimate for the selected normal prominence field gives a total mass in the range ð2 8Þ0 7 g, which is 6 7 times of the corresponding mass of the selected inverse prominence field. This suggests that more mass is needed for trapping the flux ropes of the normal prominence type. We imagine that if the corona fails to provide enough plasma for a flux rope to be resupplied against the observed, ever present, slow drainage of plasma out of a prominence (Tandberg-Hanssen 995), the flux rope would break confinement and rise through the corona. Therefore, the greater amount of mass needed to anchor a flux rope of the normal type, compared to that of the inverse type, suggests a general tendency for the former to more readily break loose and be expelled out of the corona. This effect might further enhance the disparity in the populations of the two types of prominences that Leroy and his collaborators have observed, as we have discussed in x SUMMARY A family of magnetostatic solutions describing the normal and inverse prominence topologies has been presented in this paper. The prominence is modeled as a flux rope of purely azimuthal magnetic field, anchored by the surrounding global poloidal field of an axisymmetric atmosphere, combined with an appropriate amount of cold plasma weight inside the flux rope. We find that both the normal and inverse prominence fields are capable of storing enough magnetic energy to overcome the Aly limit, but the normal prominence fields are more ready than the inverse prominence fields to lead to expulsion. In particular, the normal flux rope is subject to a Lorentz force that is always radially outward, and it needs

9 No. 2, 2004 MAGNETIC ENERGY STORAGE IN SOLAR PROMINENCES 05 comparatively less azimuthal flux in the rope for the field to store enough energy to overcome the Aly limit. This difference in the manner of magnetic energy storage in the two types of prominences implies physical properties that may account for the height distribution and ratio of the two types of prominences observed by Leroy (989). Our models are among the simplest hydromagnetic solutions demonstrating the trapping of magnetic flux ropes and free magnetic energy in a global atmosphere. They, of course, do not model the realistic hydromagnetic environments in the solar corona, but we do expect that they capture the basic physics of how quiescent prominences may trap magnetic flux ropes in the low corona. As pointed out by the referee of this paper, our modeled equilibrium states have an inherent instability. The field is expected to be unstable with respect to -dependent perturbations. We can also imagine the Newcomb type (Newcomb 96) of interchange instability operating here, in which heavy fluids sink and light fluids rise, with the displacement of the frozen-in azimuthal fields taking place by contracting or expanding the circular flux lines without breaking axisymmetry. As an example of a more realistic model than the ones we have presented, we could allow for a flux rope endowed with both poloidal and azimuthal fields to produce helical fields in the flux rope, still within the idealization of axisymmetry. In such a more realistic model, the stability might be improved, but the basic physics of holding down the flux rope by its embedded mass and the surrounding fields would still apply. Our solutions are therefore useful for the demonstration of this basic physics in the theoretically simplest system. Finally, we need to point out that having a magnetic energy in excess of the Aly limit is only a necessary condition, and, more specifically, a necessary condition for the magnetic field to give up additional energy above the Aly limit to drive a mass expulsion. Furthermore, we cannot be assured that the dynamics of an expulsion, involving conservation laws other than that of the energy, may, in fact, drive an expulsion. These are all left for future numerical studies of the dynamical evolution to explore. However, if the expulsion does happen dynamically, then, according to the theory of Low & hang (2002), the field topologies of these two hydromagnetic types of flux ropes have implications for the manner in which a flux rope identified with a CME may be expelled. The interplay between magnetic reconnection and expulsion of the flux rope by its hoop force in the two contrasting magnetic topologies suggests that the normal flux rope will undergo an impulsive acceleration early in the expulsion process, while the inverse flux rope will tend to accelerate more gradually as it travels out of the corona. This theory is corroborated by the force balance in the equilibrium states of the two hydromagnetic types of prominences, as we have illustrated, and has also been shown to be consistent with observations in observational studies (hang et al. 2002; hang & Golub 2003). An important next step to take in testing and developing this theory is that of MHD numerical simulations of the expulsion process, for which our family of solutions will be useful as initial states. We thank Peter Gilman and the anonymous referee for helpful comments. 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