Catastrophe of Coronal Magnetic Rope in Partly Open Multipolar Magnetic Field**
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1 ELSEVIER Chinese Astronomy and Astrophysics 29 (2005) CHINESE ASTRONOMY AND ASTROPHYSICS Catastrophe of Coronal Magnetic Rope in Partly Open Multipolar Magnetic Field** PENG Zhong HU You-qiu School of Earth and Space Sciences, University of Science and Technology of China, Hefei Abstract Catastrophe of coronal magnetic rope embedded in a partly open nmltipolar background magnetic field is studied by using a 2-dimensional, 3- component ideal MtID model in spherical coordinates. The background field is composed of three closed bipolar fields of a coronal streamer and an open field with an equatorial current sheet. The magnetic rope lies below the central bipolar field, and it is characterized by its annular and axial magnetic fluxes. For a given annual flux, there is a critical value of the axial flux, and for a given axial flux, there is a critical value of annual flux such that, below the critical value, the magnetic rope is attached to the solar surface and the system stays in equilibrium, but when the critical value is exceeded, the magnetic rope breaks free and erupts upward. This implies that catastrophe can occur in a coronal magnetic rope embedded in a partly open multipolar background magnetic field. Our computation gives a threshold value of magnetic energy that is about 15% greater than tile energy of the partly open magnetic field (the central bipolar field open and the fields on either side closed). The excess energy may serve as source for solar explosions such as coronal mass ejections. Key words: magnetohydrodynamics (MILD) sun: flares 1. INTRODUCTION Magnetic rope is typical of the structure of the solar corona, and is very closely related to solar active phenomena like coronal mass ejections [1'2]. A large amount of research has been Supported by National Natural Science Foundation and Ministry of Science and Technology Received ; revised version * A translation of Chin. g. Space Science Vol.25, No.2, pp.81 85, ~05~S-see front matter 2005 Elsevier B. V All rights reserved. DOI: /j.chinastron
2 PENG Zhong et al. / Chznese Astronomy and Astrophysics 29 (2005) devoted to the equilibrium characteristics and catastrophic phenomena of coronal magnetic rope systems [a-14]. These studies have shown that, under some specific conditions, catas- trophe may occur in such systems. For a closed background magnetic field in rectangular coordinates, the system parameters vary continuously with the magnetic rope, and no catastrophe occurs [7'sl. It is only when the background field is partly or completely open that catastrophe may take place [9-12]. Moreover, for a bipolar background field in spherical coordinates, no matter whether the field is partly open or completely closed, catastrophic phenomena always exist in magnetic rope systems. When investigating the equilibrium properties of coronal magnetic ropes, an important question is to judge whether or not a catastrophe can occur in the system and whether there is an energy threshold in the catastrophe. The solution of this question might provide physical interpretation for the formation mechanism and energy source of solar explosive phenomena, such as coronal mass ejections. Now, the magnetic fields in solar active regions are close to force-free fields and Aly [15] proposed the following conjecture for the energy of a force-free field: for a given normal component of the bottom boundary magnetic field, when at least one end of the magnetic lines is bound to the bottom, then the energy of the force- free field cannot exceed that of the corresponding completely open field. This conjecture has been supported by a series of analytical studies [16-18] as well as numerical simulations [19-22l. Hu [2a] analyzed the energy storage of the multipolar force-free field induced by photospheric shear and suggested extending Aly's conjecture to partly open fields to the effect that the magnetic energy of the force-free field resulting from a shearing of some portion of the magnetic field cannot exceed that of the corresponding partly open field (where only the sheared portion becomes open, while the un-sheared portion still remains closed). While the correctness of Aly's conjecture still awaits strict justification, we can now meaningfully investigate whether the magnetic energy of the force-free field of floating mag- netic lines can exceed that of the corresponding completely or partly open field. As pointed out by Hu et al. [131, the magnetic energy of the 2-D field of a coronal magnetic rope can indeed exceed that of a completely open field. Furthermore, Li and Hu [14] found the energy excess amount to 8%. The background field adopted in that work is a bipolar field. In the present paper, we consider a partly open multipolar field and whether catastrophe of coronal magnetic rope can occur, and, if occur, the relations between the energy threshold of catastrophe and the annular and axial fluxes of the magnetic rope, and between the energy threshold of catastrophe and the magnetic energy of the relevant partly open field. 2. BASIC EQUATIONS AND INITIAL CONDITIONS For the 2-dimensional questions in spherical coordinates, the magnetic flux function ~(t, r, 0) may be introduced. Then the magnetic field can be expressed as: B = V (~b) + B~, S~ = B~. (1) Hence the 2-dimensional and 3-component ideal MHD equations on the meridian plane can
3 398 PENG Zhon9 et al. / Chinese Astronomy and Astrophysics,99 (2005) be reduced to the following dimensiolfless form: Op a~ + V. (pv) = O, (2) ~+v.vv-i-~vy+ --/3T ~-o Vp+ P 1 [L~Vm+, B~ x (V B~)] + ~V~'. 1 (V X B~)@-t- r~ff~ = O, 0 O~ +~v* 0, (4) (3) at + rsin0v ( )+[V~ x V( )]{ =0, (5) Here OT O~- +v VT+(7-1)TV.v 0. (6) 1 (02~ 1 0~ cot00 ) Lgb - r 2 sin2~, Or S + r " 2 ~-~, (7) g is the dimensionless gravitational acceleration on solar surface, 7 is the polytropic index, /3 is the characteristic value of the ratio between gas pressure and magnetic pressure, = 2#poRToR4/D~, (8) # is the magnetic permeability in vacuum, R is the atmospheric constant, Po and 7 b are, respectively, tile density and temperature at the bottom, Ro is the solar radius, and %bo is the characteristic value of magnetic flux function. In the following calculations, we take Po = 1.67x 10-1akg-m -3, To = 2 x 106K, -~ = 1.05, and ~q = Then according to Eq.(8) we get %b0 = 5.69 x 1014 Wb, B0 = wo/r x 10-3T, vo Bo/~x/'~6 = 2.57 x 103 km.s -1, TA = Ro/vo s, where WA is the transit time of Alfven wave. The primary atmosphere is in isothermal hydrostatic equilibrium with dimensionless temperature and distribution of dimensionless density given by T = 1 and p = exp[29(1 - r)/t3r]. To determine the dimensionless magnetic flux function of the primary background magnetic field, we can start from tile following magnetic flux function [24]: where Z = Z1 - c~z3, (9) Z r(1 -- 1'2)[(] -- II 2) tan(i-) --?/,1 U Tra 2 sin " + 2a(rl - 1), (10)
4 a PENG Zhong et al. / Chinese Astronomy and Astrophysics 29 (2005) Z 3 = 1 ~r(1 - v2)(5v 2-1)[-3(1 + u2)(5u 2 + 1)aretan(~-) + 15u u] + /t 457ra 4 sin e 0(5 cos 2 0-1) 97ca 2 sin ~ a(z I - 1), 8r 3 2r (11) 1 a 2 u 2 = 1[(1 _ 5)2 a2 + --~-- 4a2 COS 2 011/2-3(1- ~), (12) 4a2 2,,2 = 1[(1 -- ~)~',,.~ + ~ cos 1(1 _ 2 0] 1/2 + ~ ~7), (13) 4r2 2 1 r 2 ) zl 2= 1[(1- r232+-~cos 0] 1/2+ (1--- (14) a 2 J 2 a 2 a = 3 is thc inner radius of the equatorial current sheet and c~ is a positive constant. In the following we take c~ = 1/36, and the corresponding magnetic field is composed of three closed bipolar fields of coronal streamer and an open field with an equatorial neutral current sheet. See Fig.l(a). The arrowheads indicate the directions of magnetic field lines in the various districts, and the thick solid line represents the equatorial current sheet. Fig.l(b) illustrates the configuration of the partly open magnetic field corresponding to the background field. The central bipolar field is completely open, while the bipolar fields on the two sides remain closed. There are three current sheets (represented by thick solid lines). We start from the Z given by Eq.(9) and choose tile initial magnetic flux function ~, = Z/Zm, where Zm = max(izi). On the polar axis (0 -- 0) it is equal to zero. It takes minimum values of -1 at the centers of the flank bipolar fields and the maximum of 0.56 on the equator (7" - 1 and 0 - n/2). Hence the characteristic value ~b0 of magnetic flux function figured above is the absolute value of the magnetic flux function at the centers of the two side bipolar fields. Following the procedure stated in Ref.[13], we let a magnetic rope emerge from the bottom of the central bipolar field. Then after a time-varying simulation, the magnetic rope attains equilibrium; the magnetic configuration in the equilibrium state is displayed in Fig.l(c), in which the thick solid curve represents the transverse current sheet caused by the pressing of tile magnetic rope. Tile equatorial current sheet rises beyond 4R0, so it does not appear in the figure. The axial magnetic flux of magnetic rope is (I)~0 = 5.16 x 1013Wb, and the annular magnetic flux in radians is q)p0 = Wb.rad RESULTS OF COMPUTATION The region of solution is (1 < 7" < 30, 0 < 0 < 77/2). Eqs.(2)--(6) are solved with the multistep implicit scheme The physical quantities at the bottom are always kept at their initial values, and for tile top the potential field conditions are adopted. For the relevant details see Ref.[13]. Starting from the above equilibrium state of the magnetic rope,
5 400 PENG Zhong et al. / Chinese Astronomy and Astrophysics 29 (2005) 396 ~ ,-i,% rfro ~/Ro Fig. 1 Magnetic configurations of (a) the initial magnetic field, (b) the corresponding partly open field and (c) the field containing a coronal magnetic rope we gradually increase the axial flux O~ and annular flux Op, to the values qs, = a~qs~0 (a~ _> 1) and Op = c~pqsp0 (ap > 1). The system then arrives by time-varying simulation at a new equilibrium state. For strict realization of the conditions of force-free field in the numerical simulation, we used the method of Ref.[23]: at every time step we restore the initial values of temperature and density, so that the condition of static equilibrium in the gravitational field is always satisfied. Thus, the equilibrium magnetic field obtained is force- free, and is independent of the value of ~. To avoid numerical magnetic reconneetion across the current sheets, we follow Refs.[13] gz [23] and revise the value of at each time step, so maintaining the ~p values in the original equatorial current sheet and in the newly born current sheets, including a transverse one between the central bipolar field and the external background field of the coronal streamer, formed under the pressure of the magnetic rope, and a vertical one pulled out from below by the magnetic rope that breaks free from the solar surface. When the annular magnetic flux is fixed (av = 1) and the axial magnetic flux is increased (a~ > 1), or when the axiat magnetic flux is fixed (a~, = 1) and the annual magnetic flux is increased (ap > 1), it is found that both a~ and C~p pass through certain threshold values. The cz~o threshold lies between 1.15 and 1.16, and tile ap threshold is between 1.12 and When the threshold value is exceeded, catastrophe occurs: the magnetic rope escapes from the solar surface and erupts upward. Fig. 2 displays the magnetic configuration at various times in the cases of ap = 1, a~ = 1.15 and c~ v = 1, a~ The thick solid lines represent the transverse current sheet produced by the extrusion of the magnetic rope and the vertical current sheet pulled out from below by the escaping rope. For ap = 1 and a~ = 1.15, the magnetic rope is attached to the solar surface and the system is in a state of equilibrium (Figs.2(a)--(c)). For ap = 1 and a~ = 1.16, on the other hand, the magnetic rope escapes and erupts upward (Figs.2(d)--(f)), with velocity times the Alfven velocity. We find a similar behavior when the axial magnetic flux is fixed and the annular magnetic flux is increased. For the equilibrium state thus obtained, we may calculate the magnetic energy of the system with the following expression:
6 PENG Zhong et al. / Chznese Astronomy and Astrophysics 29 (2005) (a) t=0 ra (b) t=200 TA (C) t=400 r A (d) t=o ra (~) t=5 ~A (f) t- 10 ~, rl~ rlro r/ro Fig. 2 The temporal evolution of magnetic rope before and after catastrophe. (a)-(c) The magnetic rope is attached to solar surface and is in equilibrium (~p -- if%0 and q~ ~varphiO). (d)-(f) The magnetic rope escapes from the solar surface and erupts upward (~p = ~po and q~ ~o). 30 Ir/2 ~r/2 Wm= ~ dr B2r2sinOdO + ~- (B 2 - B~)sinOdO, (15) in which the first term represents the energy within the range of computation, and tile second term, the energy outside the top (r = 30), both normalized with 47rW0 (W0 = BgRg/~ = 3.71 x 1025j). The system's magnetic energy depends on ~b~ and ~5p. The results are listed in Table 1. The upper half shows how the magnetic energy varies with O~ at a fixed ~p = ~p0, and the lower half, its variation with ~p at a fixed q~o. As the table shows, we have monotonic increase in either case. At the point of catastrophe (the last column of the table), the magnetic energy corresponds to the energy threshold of catastrophe. For the two cases the thresholds obtained are close to each other, being respectively 1.73 and Once the magnetic rope breaks free, the central bipolar field becomes open, while the bipolar fields on the two sides remain closed, and we have the partly open field displayed in Fig.l(b). The energy of the partly open magnetic field is Compared to this, the energy threshold of catastrophe is higher by 15%. This implies that in the vicinity of the point of catastrophe and after the partial opening of the magnetic field, a portion of the magnetic energy still remains stored in the system, and the excess may provide the energy for such events as coronal mass ejections. To probe the relation between the energy threshold of catastrophe and the properties of magnetic rope, we made calculations for the following two cases: (1) as function of ~ while ~5p is fixed at 0.8~5p0 and (2) as function of ~p while q~ is fixed at 0.8~5~0. It is found that the catastrophic phenomenon again exists in the system, but the energy threshold of catastrophe is approximately 1.70 (this is by about 13% greater than the energy of the corresponding partly open magnetic field). This means that there is certain relation between the energy threshold of catastrophe and the properties of the magnetic rope. A detailed and
7 402 PENG Zhong et al. / Chinese Astronomy and Astrophysics 29 (2005) Table 1 Magnetic energy of the system (W.~) as functions of the axial magnetic flux (qb~), and the annular magnetic flux (qbp) of the magnetic rope ~p ~- Op0 ~/d)~ Wm/(4?rWo) ~a : ~o0 ~llip/~i~p Wm/(47rWo) quantitative analysis of this relation will be undertaken in our next study. 4. CONCLUSIONS In this work, the 2-dimensional and 3-component ideal MHD model is adopted to study the catastrophic phenomena of coronal magnetic ropes in partly open multipolar background fields. Our results reveal that catastrophic phenomena do exist, as in the case of a bipolar background field. Tile examples of computation described in this paper show that the catas- trophe energy threshold is about 15% larger than tile magnetic energy of the corresponding partly open field. Once the catastrophe occurs, a portion of the system's energy is consumed in the opening of the central bipolar field of coronal magnetic rope and the remaining por- tion, amounting to nearly 15% of the magnetic energy of the corresponding partly open field, will be released to heat and accelerate the plasma. In other words, the catastrophic phenomenon that occurs in the multipolar magnetic field of an emergent magnetic rope can provide energy for solar explosions like coronal mass ejections. The 2-dimensional approximation is used in this work, and the magnetic rope is taken to bca closed circular ring encircling the sun and hangs in the solar corona. In reality such a "hanging" magnetic rope should ultimately fall down and plant itself on the solar surface. This will certainly have some effect on our calculated values of the energy threshold of catastrophe. The effect could be clarified quantitatively with a 3-dimensional MHD simulation. However, for a sufficiently long magnetic rope, the results of the simulation described in this article can approximately reflect the dynamical properties of the middle portion of the magnetic rope away from the footpoints, that is, when the system's magnetic energy reaches a certain level, the middle part of the magnetic rope suddenly breaks free of the solar surface and erupts upward. This leads to the formation of current sheets below. Meanwhile, magnetic reconnection across the current sheets will play a promoting role for the eruption and finally gives rise to the eruption of the el,tire magnetic rope.
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