\cma\a. T r. NAL REPORT (Limited Distribution) 1. INTRODUCTION

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1 '. ttional Atomic Energy Agency and Educational Scientific and Cultural Organization IATION AL CENTRE FOR THEORETICAL PHYSICS -r-rt \cma\a NAL REPORT (Limited Distribution) NUMERICAL SIMULATION OF A MAGNETIC RECONNECTION IN THE SOLAR CORONA* Shui Wang" International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT In this paper, a kind of magnetic reconnection process caused by the local heating in the closed magnetic field region over the equator in the solar corona is discussed. At first, we use a method in combination with analytical and numerical methods to obtain a coronal magnetic configuration with X-type neutral point. Furthermore we solve numerically the magnetohydrodynamic equations by using the full-implicit continuous Eulerian scheme in the spherical coordinates. The results of numerical simulation show that three different processes of magnetic reconncction are existed. On the other hand, the upward plasma velocity is small when the effect of solar gravitational field is considered. Hence, we suggest that the magnetic reconnection caused by the pure resistivity tearing mode may be unimportant for die coronal mass ejections. MIRAMARE - TRIESTE August 1990 To be submitted for publication. Permanent address: Department of Earth and Space Science, University of Science and Technology of China, Hefei, Anhui , People's Republic of China. 1. INTRODUCTION Magnetic reconnection is a phenomenon of considerable importance in solar system plasmas. In the solar corona, it results in the rapid release to the plasma of energy stored in large-scale magnetic configurations which become unstable, resulting in solar flares, while small-scale reconnection may play a role in heating the coronal plasma which leads to the outflow of the solar wind. Daytime reconnection in the magnetopause is believed to be an example of an externally driven reconnection, and it leads to efficient coupling of solar wind momentum into the magnetosphere. In the extended tail of the magnetosphere, the onset of rapid reconnection between the tail lobes can produce large-scale dynamical plasma-field reconfigurations which are associated with magnetospheric substorms on the earth and structure in the plasma tails of comets. The magnetic reconnection process also most probably plays an important role in astrophysical plasma systems such as accretion discs, and in various current sheet interface regions formed in interstellar and intergalacu'c space. For all these reasons the theory of magnetic reconnection has been studied by many authors, and remains an important and active research field to the present day (Cowley 1985; Hones 1984; Priest 1985, Shivamoggi 1985). The dynamical behaviour of plasma in die vicinity of X-type magnetic neutral line for a two-dimensional configuration were studied by Dungey (1953), Sweet (19S8) and Parker (1957). But the reconnection rate was too small to account for the observed energy release in solar flare, because the magnetic energy was degraded primarily through resistivity which ii very small in solar corona. Then Fetschek (1964) proposed that only a smalt part of die total exchange of field and flow energies occur in the central resistive region, the main part occurring across the Alfve'n waves. Sonnerup (1970) considered three uniform regions of flow and magnetisation separated by a leading Alfven discontinuity and a trailing slow shock. Yeh and Axford (1970) set up a similarity field flow solution for the convecttve region surrounding the neutral point Yeh and Dryer (1973) generalised Sonnerup *s solution (Sonnerup, 1970) to the case of a compressible plasma. Recently, the steady magnetic reconnection in three dimensions has been discussed by some authors (Greene, 1988; Hesse and Schindler, 1988; Schindkr et al., 1988; Sonnerup 1988; Steele and Priest, 1989). The steady-state reconnection models can provide an effective mechanism for the observed acceleration of plasma particles to high energies through shocks. However, the steady-state models are not intended to explain how a solar flare starts or how it stops. Hence we must consider nonsteady reconnection processes by using numerical methods. Tsudaet al. (1973,1977) studied the temporal dynamics of a current sheet in an initially quiescent plasma. Biskamp (1982, 1984) made a numerical simulation of the temporal evolution of plasma around a X -type magnetic neutral point. Forbes and Priest (1982,1983). Forbes (1986) and Forbes et al. (1989) numerically solved the resistive MHD equations describing the two-dimensional tearing mode in a current sheet which is anchored or line-tied at once end to a stationary surface. They found that the nonlinear tearing mode gave rise to multiple magnetic island which grew in size and coalesced, and the latter process was conjectured to lead to a particle acceleration. Lee and Fu (1986) studied the multi-x-line magnetic reconnection at the magnetopause by using a numerical method. T r

2 The purpose of this paper is to simulate numerically a land of magnetic icconnection in the solar corona. We limit ourselves to discuss the basic figures of the evolution of magnetic field configuration in a rcconnectiort process by solving MHD equations numerically. In Sec. II, we use a method in combination with analytical and numerical methods to obtain a coronal magnetic configuration with Jf -type neutral point in the spherical coordinates. Using this model of magnetic field, we discuss the evolution of magnetic field configuration and the dynamical response in the solar corona caused by a local heating in the closed magnetic region over the equator. The numerical results show that three different processes of the magnetic icconnection are existed (Sec. III). We conclude in Sec. IV with a summary of the results and a discussion of their physical implications. magnetostatic equilibrium, Eqs.(2) and (6) may be represented as follows A 1 Hm. * I flf i)i) II D T 1 * 4w (^ r or T _ (7r cw J J ttgl J_ f B, r 3(rB«) _ dbrl _ By 9(sineB p )l 13^ 4ir\r[ 6V 36 \ rsin«9? J 739 4^\7^ a(r J B r ), 1 9( sin eft) dr r sin "7 9r J =<)i 0, (7) (8) (9) (10) 2. A MAGNETIC FIELD CONFIGURATION WITH X-TYPE NEUTRAL POINT IN THE SOLAR CORONA To discuss the dynamical processes in the solar corona, atfirst, we must construct a basic state of magnetic field which is agrees with the observations of die solar corona. By using a method in combination with analytical and numerical methods, a kind of stream-like magnetic configuration may be obtained (Wang and Wang, 1985a). Now we obtain a magnetic field configuration with a X-type neutral point by using the same method. Because the solar corona is constructed by the rare plasma, it can be used the following MHD equations where g = go(.r 0 fr) 2 gravitational acceleration at the solar surface. is the function of the radial distance r. r 0 is the radius of the sun, g a the The function A( r, 6) can be derived from Eq.(lO) 1 da r sin 9 dr Then, from Eqs.(7)-(9) and Eq.(ll), we can obtain (Wang and Wang, 1985) rr -L where B>{F) is an arbitrary function of F; R, F. G, P. are the following dimensiontess parameters (ID (i) and as at P r-(v x B) -0, AvR, (2) (3) (4) (5) Gtii.fl) -G[F(fl,«] - ^^B^(r,«), P(R,6) Po 1 (13) Ho where Ho,Ao,po,To being constants. The function F(R,6) satisfies the following nonlinear (6) elliptic differential equation where p, p, T and v are the pressure, density, temperature and velocity of plasma respectively, B is the magnetic induction (we take that the permeability is equal to unit), g the gravitational acceleration, R, the gas constant, i the ratio of specific heat, andwetakefss/sinthe following. TJ is the resistivity which is a constant. In order to discuss the large-scale structure of solar corona, we use the spherical coordinates (r, 9, p) and consider only the two-dimensional casc( ^ = 0). Then, for the basic state with 3 Z F 1 8R 1 + R* Observations show that the plasmatemperaturein the quiescent corona as slowly as considered approximately as isothermal, i.e. = I. In the case of the magnetic field being a zero component in the ^-direction, we discuss only the case in the meridional plane and put G(F) =0. (14)

3 As the choice of ft (F) is arbitrary, the solution of Eq.(14) is not unique. We may be chosen that the form of function ft(f) is a power series expansion of function F where C* are constants. In the following discussion, we take only die first two terms. Substituting Eq.(lS) into Eq.(14), then, Eq.{14) can be solved by the numerical method. (15) 3. NUMERICAL SIMULATION OF MAGNETIC RECONNECTION We shall further discuss the magnetic reconnection processes caused by the local heating in the closed magnetic field region over the equator in the solar corona. In the spherical coordinates(r,0,p) and the two-dimensional case [4- = 0 V the fundamental equations (l)-(4) may be written as dp 1 8 By using the central difference scheme, Eq.(14) can be written as a difference equation. Then, it is solved by the relax method. The initial form of function F( R, 6) in sthe process of iteration may be taken as m Oi + 02 iul(lo), ( lt)j R where a i and 02 are constants. As the physical quantities in the (olar corona, such as the density and pressure of plasma, the magnetic field, the gravitational field, and their derivation decrease rapidly with the increase of radial distance, we may take the difference mech in such away that the spacing in the 0-direction is uniform (A0 = ),butthespacinginther-directionisnonuniform(ajt =«RA6). The boundary conditions of equivalent extrapolate are used in our calculations. In the process of iteration, the tolerable error c is taken as 10 ~ 3 r *J LM t ~ 10" J. (17) In the calculation, we use the following initial values of the iteration: the coronal temperature is taken to be 2 x 10*K, the plasma density at the equator on the solar surface {9 = 90 ) 3 x 10~ w jcm~ 3 and 0 is equal to 1 in this position. The constants in Eq.(13) are taken to be H0 = ITQ.AQ = Borl&B. ft and Bo are the pressure of plasma and magnetic field at the equator on the solar surface respectively. The computation domain is taken to be 1 r 0 < r < 6r 0 ando <0<9O". Fig. 1 shows a kind of magnetic field configuration obtained. It can be shown that the magnetic field includes the closed-field region, the neutral line and the open-field region. The closed-field and the neutral line constitute a JC-type magnetic structure in the solar corona. The position of the neutral point is located at 2.6rg. The open-fields represent the coronal holes which are the sources of high-speed flow of the solar wind. The structure of magnetic field agrees with the observations and theoretical analyses of the large-scale magnetic field of the solar corona (Hundhausen, 1978). After solving the scalar function F( R, 9) by Eq.(14), then, the distribution of the magnetic field, the pressure and density of plasma in the magnetostatic equilibrium can be obtained by using Eqs.(ll), (12), (13), (15)andEq.(5). where v T, vg, v v and B T, B$, B v are the components of the plasma velocity ttand the magnetic field B, respectively. It can be shown that we have written the momentum equation (19)-(21) in nonconservational forms to avoid an error which an additional term as (V B)B is included, because Eq.(6) may not be strictly satisfied numerically. We use die full-implicit continuous Eulcrian scheme in the spherical coordinates to solve numerically Eqs. (18)-(25) and Eq.(5). A difference equation for the pressure of plasma pfj 1 at the (k+ \)th iteration step may be obtained, and it may be solved by the line-by-line method (Wang, (25)

4 Hu and Wu 1982). The iteration accuracy is til,. ftj.! < ^ 10- (26) The extensional mesh is chosen, and the following boundary conditions are used: (1) we use the usual symmetrical boundary conditions at the pole (.0 = 0) and the equator A part of the numerical results is shown in Fig.2, which is the configuration of the magnetic fields at t = 500a, 1000s, 1500a and 2000s, respectively. It shows the evolutions and reconnections of magnetic field in the expansion process of coronal plasma caused by a local heating at the solar surface. The following properties can be shown from Fig.2: (1) Starting the local heating in the closed magnetic region on the solar surface at t = 0, the magnetic field lines in the closed region rise up gradually and move slowly to both sides by the expansion of hot plasma in the local heating region. At the same time, the disturbance caused by the local heating propagates upward at a velocity about 500 tnu"' (corresponding to and Pi.l = W.2, Pi.tf = Pi,l =Pi,J Pi,U at 0, = 0, (27) the speed of the fast magneto-acoustic wave). At t = 360a, it propagates up to the vicinity of 2.6 r 0 which is the position of neutral point at the initial magnetostatic equilibrium, and it guides the first magnetic reconncction. The motion of magnetic field lines to both sides in the closed region may be also contributed for this magnetic reconnection process (Wang and Wang, 1987). Comparison of the configurations of magnetic field at t = 0«(Fig.1) and f = 500a (Fig.2a) indicates that the top of three magnetic field lines (the footpoints are located at 69.75", 74.25" and 78.75" on the solar surface, respectively) rise up to the vicinity of 2ro over the equator region at t = 500 s. at 0 = 90-0, (28) The magnetic line with the footpoint at on the solar surface has been evoluted to a closed magnetic line from an open line. It means that the first magnetic reconnection process has been finished. where the subscript M is the number of the mesh points in the ^-direction (in this case, we take M = 22). (2) The physical boundary conditions at bottom and the computation boundary conditions at the top in the computation domain (lr 0 ^ r < 6r 0, 0 < 8 <, 90 ) may be given with the projected characteristic method and the non-reflecting boundary conditions (Wang, Hu and Wu 1982; Hu and Wu 1984). If tv > 0, five quantities at the bottom are prescribed arbirarily (we prescribe p. p, 8 T, v# and v p ) and the other three («T, B» and B v ) are determined by the compatibility equations. For the boundaries of the top in the case of v t > 0, the five quantities may be given by the projected characteristic equations, and in addition, the other three arc given by non-reflecting boundary conditions. It should be must pointed out that the conductivity of plasma is finite, i.e. t) ^ 0, so the effects of finite conductivity on the boundary conditions are must considered. The results shown that the effects of finite conductivity in internal energy equation (25) is much small in our case. We have been simulated numerically the propagation characteristics of a coronal disturbance and accompanying physical processes in the solar corona with magnetostatic equilibrium caused by a radial ejection of cold mass in the neighbourhood of the equator at the solar surface (Wang and Wang, 1985). Now we discuss the effects of a local heating in the closed magnetic region over the equator on the magnetic configuration as Fig.l. Starting from t = 0, we introduce a local heating in the range lr 0 < r < 2r 0 8*178.75" < 8 < 90, and the temperature of plasma within the heating region is taken to be five times of the initial value. The plasma in the heating region is expanded and the magnetic field is deformed (2) As the continuous expansion of hot plasma by the local heating, the magnetic field lines in the closed region rise up continuously, and the neutral point rises also up gradually as well. Second magnetic reconnection may be driven by the upward plasma. It is similar to the magnetic reconnection which is driven by the rising prominence (Steele and Priest, 1989). It can be shown from Fig.2(b) that the magnetic field lines with the footpoints at 65.25" and 69.75" on the solar surface have been opened at t * 1000 s, and the top positions of magnetic lines with the footpoints at s, * and * have been located at 2.5 r<>, 2 3r 0 and 1.4 r 0, respectively. At the same time, the open magnetic lines on both sides beside the neutral point over the equator move slowly to the high-latitude regions. (3) Because of the effect of solar gravitational field, a part of the ejective mass falls. Accumulated by the ejective and falling mass, a high-density region keeps at a low position over the equator region (Wang, Hu and Wu, 1982; Wang and Wang, 1985). The magnetic field lines over the equator near the solar surface are deformed due to the effect of the accumulated plasma. As the continuous accumulation of mass, the magnetic field tines extend first toward both sides, then they gradually become flat and approach to the neighbourhood of the solar surface or bend as a concave (Wang and Wang, 1985; Wang and Wang 1985b). A new X -type neutral point has been formed at the vicinity of 1.2ro (Fig.2c). Finally, third magnetic reconnection appears at the new neutral point due to the compression of the accumulated plasma, and a magnetic island has been formed in the region 1.4 TO 2.6 ro over the equator (Fig.2d). (4) On the other hand, the results also show that the velocity of plasma is very small in 8

5 the magnetic rcconnection processes. The maximum of radial velocity above the neutral point is about 20kms~'. Obviously, the gravitational field of the sun resist the plasma motion towards the interplanetary space. It may be shown that the magnetic reconnection caused by the pure resistivity tearing mode may be unimportant for the coronal mass ejection. Acknowledgments The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. 4. SUMMARY Because the solar atmosphere is constructed by the rare plasma with high temperature. the magnetic field plays an important role in all solar physical phenomena. Magnetic reconnection which results in the rapid release to the plasma of energy stored in large-scale magnetic configurations, is a process of considerable importance. It may be associated with many dynamical processes (for example, solar flare, coronal mass ejection etc.). In this paper, a kind of magnetic reconnection processes caused by the local heating in the closed magnetic field region over the equator in the solar corona is discussed by using a numerical method. The result obtained in this paper is preliminary. However, we suggest two ideas about the magnetic reconnection in the solar corona. (1) The magnetic reconnection in the solar corona is a very complex process. Our results show that the magnetic reconnection may happen repeatedly at different regions in the solar corona. They are caused by different physical mechanisms. The origins of first reconnection are the disturbance caused by the local heating and the motion of magnetic field lines to both sides in the closed region. It may be a pun tearing mode reconnection process. Second reconnection is driven by the upward plasma, and third reconnection appears at the new neutral point which is formed by the continuous accumulation of th ejective and falling mass. These reconnection processes may be caused different physical phenomena. (2) Coronal mass ejection is also an important phenomenon which is assoicated with prominence and sometimes also with solar flares (Hundhausen, MacQueen and Sime, 1984; Pneuman 1980). Anzcr and Pneuman (1982) considered a magnetically coupled coronal mass ejection and prominence. Reconnection takes place under the priminence and this helps to drive the system upward. However, the importance of gravitational field of the sun on the coronal mass ejection is often neglected. For the plasma separate oneself from the grabvitational field of the sun, the radial velocity is larger than the escape velocity which is abour ons-' in the solar corona. The results of numerical simulation for magnetic reconnection caused by the pure resistivity tearing mode show that the radial velocity of plasma is smaller than die escape velocity when the effect of gravitational field is not considered (Forbes and Priest 1982,1983). Our results show the maximum of the radial velocity of plasma above the neutral point is only 20kms~' when the gravitational field is considered. So we suggest lhat the magnetic reconnection process caused by the resistivity tearing mode may be unimportant for the coronal mass ejection. 10 r T

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7 Figure Captions Fig. 1 A magnetic field configuration with Jf-type neutral point in the magnctostatic equilibhum. Fig.2 The evolution of the magnetic field configuration, (a) t = 500 a. (b)t = 1000 s; (c)t= 1500a. (d)t = 2000s Fig.l 13 r T

8 (ft) (c) (b) (d) Fig.2 IS Fig. 2 16