Unsteady magnetic reconnection in non-periodic multiple current sheets

LIU Yifan, WANG Xiaohu, ZHENG Huinan and WANG Shui Department of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China

Abstract The process of magnetic reconnection in non-periodic three-layer current sheets is studied numerically by using twedi- mensional magnetohydrodynamic simulation. The results show that unlike periodic current sheets, it is complex unsteady magnetic reconnection. It may be important for solar flare and corona heating.

Keywords: multiple current sheets, magnetic reconnection, numerical simulation.

MULTIPI~E current sheets are common in space plasma. For example, observations show that there are structures of multiple heliospheric current sheets in the solar wind'''. Many kinds of dynamic processes which occur in multiple current sheets may have significant influence on solar flares, the corona heating, and the coupling between the solar wind and the magnetosphere. Some authors have discussed the charac- teristics of tearing mode instability in two-layer or periodic current sheets by using analytical or numerical method^[^-^]. They found that the growth rate of asymmetric mode is much larger than that of syrnmet- ric mode. The purpose of this note is to study numerically the process of magnetic reconnection in non- periodic three-layer current sheets by using magnetohydrodynamic simulation. The results show that it is complex unsteady magnetic reconnection unlike periodic current sheets. Accompanying the development of magnetic reconnection, the initial reverse current sheet disappears gradually and a big magnetic island is formed in the corresponding region.

For two-dimensional ( a / a z = 0 ) and two-component ( B , = 0 , V, = 0 ) case, we consider the initial state of magnetic field B = B,( y ) i,, and

D / aB,tanh(f + D ) , - D < f G - -; 2

[B,tanh($ - D ) , D < f ,

where a and D are constants, which indicate the current intensity of the central reverse current sheet and the position of the current sheets on two sides, respectively. 6 is the characteristic half-thickness of cur- rent sheets. In order to study magnetic reconnection in non-periodic three-layer current sheets, we start from the following compressible magnetohydrodynamic equations[51 :

where + is the magnetic flux function ( B = - V x ( + i Z ) ) . The other parameters have ordinary mean- ings. In numerical calculation, eqs. (2)-(6) should be rewritten in dimensionless form. The length, magnetic field, magnetic flux function, density, pressure, velocity and time are normalized by 6 , B,,

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BULLETIN 6B,, p, , p , , V A , and T A respectively, where the parameters with subscript Go are the initial values as

1 y tends towards infinity. The A1fvi.n velocity VA, = B,/(4xp,)T , and the Alfven travel time r~ = 61 VAm .

Equations (2 ) - ( 6) can be numerically solved by a fourth-order Runge-Kutta scheme. The calcula- tion domain is - 106 < .r < 10 6 , - 106 < y < 106 . In order to compare it with the computation re- sultsof periodic multiple current sheets, we set the parameters in eq. ( 1 ) as a = 1, D = 3 . In x and y directions, the grid sizes are taken to be A x = 0 .25 6 and A y = 0.125 6 respectively. The time step is de- termined by the following condition:

where C, is the local sound speed. The boundary conditions at x = _f 106 can be periodic boundary condi- tions, but those at y = k 106 are free boundary conditions. At initial time t = 0, a small perturbation is applied to the whole calculated region with

B Z 1 = O , B Y 1 = e B , s i n - (:;a)* (8) where E is a small parameter taken to be 0.001. In the following computation, the other parameters are set as: the magnetic Lundquist number S = 100 , the ratio of plasma pressure to magnetic pressure Pm = 0 . 1 , and the ratio of specific heats y = 5 / 3 .

Figure 1 shows the evolution of magnetic field configuration with time. Due to the tearing mode in- stability, magnetic reconnection will take place in every current sheet before t reaches 2 0 ~ ~ . At t = 60r,, it can be seen clearly that thin and long magnetic islands are formed in three current sheets respec- tively. And these magnetic islands show the characteristics of antisymmetric mode. Accompanying the de-

Fig. 1 . Evolution of magnetic field configuration with time.

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velopment of magnetic reconnection process, the width of magnetic islands increases with time. At t = 140 r A , the half-width of central magnetic island reaches 1 . 4 6 , which is much larger than the half-width of initial central current sheet. In this period of time, the evolution of magnetic field configuration is basi-

cally identical to the numerical results of periodic multiple current sheetsL4]. However, the magnetic is- lands on two sides move gradually towards the region of central current sheet when its width increases with time. At t = 150 r , , the separation between the two magnetic islands at different sides reaches 2. 76 and new magnetic reconnection process will happen, i . e. the magnetic field lines on two sides recon- nect directly with that of central magnetic island. At the same time, the magnetic islands on up and down sides coalesce each other by the coalescence instability, and they evolve to two big closed field regions. The central magnetic island becomes smaller and smaller, and disappears finally. This configuration of magnetic field is also unstable. A new X line is formed at the origin ( x = 0 , y = 0 ) , and magnetic re- connection will be present between two closed field regions. The central region evolves into a wide current, sheet gradually. Then magnetic reconnection with two X lines happens and a big magnetic island is formed in the central region ( t = 300 r A ) . This magnetic field configuration will maintain stable. It can be con- cluded from the preceding results that magnetic reconnection in non-periodic three-layer current sheets is complex unsteady magnetic reconnection.

The profiles of current density along the axis x = 0 at different times are shown in fig. 2. There are three current sheets at t = 0 and the central positions are located at y = f 36 and y = 0 respectively. The intermediary current sheet at y = 0 is reverse. Accompanying the evolution of magnetic reconnection, the current density of three current sheets all decreases with time, while the width of current sheets increases gradually. The initial central reverse current sheet disappears after t reaches 260rA , and its current densi- ty becomes a small positive value. The maximum of current density on two sides is always located near y = + 36 , but drops to one fifth of the initial value at t = 500rA . These results show that a part of magnetic energy transfers continually to the thermal and kinetic energy of plasma in the process of mag- netic reconnection. Figs. 3 and 4 illustrate the distributions of plasma density (,o) , temperature ( T), current density ( j ) and velocity ( V ) at t = 160 r A and t = 300 r A respectively. As to the plasma respon-

Fig. 2 . Profiles of current density along the axis x = O .

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BULLETIN

Fig. 3 . Distributions of p , T, j and V at t = 1 6 0 r ~ .

Fig. 4 . Distributions of p , T, j and V at t = 3 0 0 r ~ . dence in the process of magnetic reconnection and its application in space plasma, we will discuss them in detail in another paper.

Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant No. 19573009)

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BULLETIN References

1 Crooker, N . U. , Siscoe, G. I<. , Shodhan, S . et a1 . , Multiple heliospheric current sheets and coronal streamer belt dynam- ics, JGR, 1993, 98(A6): 9371.

2 Pritchett, P. I-. , Lee, Y . C. , Drake, J . F. , Linear analysis of the double-tearing mode, Phys. Fluids, 1980, 23(7) : 1368.

3 Otto, A. , Birk, C;. T. , Resistive instability of periodic current sheets, Phys. Fluids, 1992, B4( 11 ) : 381 1 . 4 Yan, M. , Otto, A. , Muzzell, D. et a1 . , Tearing mode instability in a multiple current sheet system, JGR, 1994, 99(A5) :

8657. 5 Wang, S . , Zheng, H. N. , Magnetic reconnection in the earth's distant magnetotail ( I ) streaming tearing instability, Acta

Geophys. Sinicu, 1991, 34(6) : 663.

(Received May 15, 1997)

Experimental study of the metamorphic reaction involving melt under high temperature and high pressure

LIU ~ u l a i ' , SHEN ~ i h a n ' , GENG ~uansheng', XU xuechun2 and MA F?ui2 1. Institute of Geology, Chinese Academy of Geosciences, Beijing 100037, China; 2. Open Laboratory of Changchun Geological College, Changchun 130021, China

Abstract High P-T experiment with natural massive rock sample of garnet biotite plagioclase gneiss indicates that the metamorphic reaction involving melt (reaction between relic mineral phase and melt) is the most important reaction in granulite-facies metarnor- phism and accompanies anatexis process.

Keywords: metamorphic reaction involving melt, natural massive rock, garnet biotite plagioclase gneiss, high-temperature and high-pressure experiment.

THE P-T condition of peak metamorphism in most granulite-facies terrains overlaps that of dehydration- melting and partial melting of felsic minerals in many metamorphic rocks (especially in metapelites) . The modelling of dehydration-melting, metamorphic reaction and thermo-dynamics of mineral phases transfor- mation in metamorphic process is carried out mainly by experiment under high temperature and high pres-

sure. Most researchers, however, used powder rock from natural massive rock sample as starting material in their experiment. The destruction of original fabric in rocks must lead to change in characteristics of melting (including melting temperature), and also makes it impossible to observe real textural invasion in minerals and occurrence of melt. Moreover, the genetic features of transforming reac- tion between relic solid mineral and melt during dehydration-melting and partial melting of felsic minerals

could not be detected at The melting experiment using powder rock materials of garnet biotite plagioclase gneiss by Hoffer and rant"], for example, can only result in a simple melting reaction equa- tion: Bi + Gt + P1+ Qz= Opx + Crd + Melt + Oxide, because they cannot distinguish the influence of de- hydration-melting and partial melting on the transforming reaction of relic solid mineral-garnet in this case. To avoid the shortcoming of previous researches, this experiment was performed using natural mas- sive rock sample as starting material with the advantage of directly observing the genetic mechanism of metamorphic reaction involving melt under high temperature and high pressure.

1 Experimental sample, method and procedure

The sample for experiment is an Archean garnet biotite plagioclase gneiss (T310-1) from the Dapais- han district in northern Tianzhen of Shanxi Province. It is greyish rock of granoblastic texture (with a few garnet porphyroblasts) and gneissic structure without effect of magmatization. This gneiss consists of garnet ( 17.3 % ) , biotite ( 18.2 % ) , plagioclase (34.3 % ) , potash feldspar (7 .7 % ) , quartz (20.5 % ) and magnetite (2 .0 % ) . Biotite is the only hydrous mineral in this rock. The composition of this rock and its minerals is listed in table 1. The sample for experiment is a cylinder 4 cm in height and 2.5 cm in di- ameter. All cylinders were prepared from the same rock specimen showing homogeneous micro-texture

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