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Plasmoid formation in eruptive flares
Ah? Space Res. Vol. 19, No. 12, pp. 190~1906, 1997 0 1997 COSPAR. Publishedby ElsevierScience l&L All rightsmervd printedin Chat Britain M73-1177/97 $17.00 + 0.00 Pm so2734 177(97)ooo98-7
PLASMOID FORMATION IN ERUPTIVE FLARES T. Magara* and K. Shibata** * Department of Astronomy, Faculty of Science, Kyoto University, Sakyo-ku. Kyoto 606-01, Japan ** Nationul Astronomical Observator): Mitaka, Tokyo 181, Japan
ABSTRACT One phenomena Yohkoh has observed is plasmoid eruption in flares. Thus this is a key factor that must be explained in any flare mechanism. In order to understand the dynamics of a plasmoid, we performed a numerical MHD simulation and investigated the evolution of the coronal magnetic field, which is initially a force-free configuration. The main results are as follows. At first, small amount of dissipation, induced by the initial perturbation, occurs in the current sheet where the plasmoid forms. This plasmoid is slowly going upward by magnetic tension force of the reconnected magnetic fields produced by initial dissipation. The crucial point comes when the perpendicular magnetic fields are washed away from the reconnection point, after that the reconnection proceeds effectively so that the magnetic tension force of the reconnected fields becomes strong, which make the plasmoid be rapidly erupted upward. These are consistent with the observational results, which say that before the main energy release the plasmoid slowly rises and when the flare sets in it is rapidly accelerated upward. In this paper, we emphasize. on the role that the perpendicular magnetic fields play in the evolution of flare. CC) 1997 COSPAR. Published by ElsevierScience Ltd. MATHEMATICAL FORMULA Basic Equations We consider magnetized gas layers composing of ideal gas. The effects of gravity are neglected for simplicity. Using the Cartesian coordinates, the basic equations are:
(1)
p[~+(v.v)v]~-vP+~(vxB)xB,
(2)
~[~(~)+(v.v)(~)]=~lvxB12~
(3)
~$=VX(VXB)-VX(~VXB),
(4)
PAT.
(5)
Additionally, we use V - B =0 as an initial condition for Eq. (4). Here, all the symbols, such as P, p, T,
V,
1904
T. Magara aad K. Shibata
and B have their usual meanings, y is the adiabatic index, R is the gas constant, ~ is the mean molecular weight, and rl is the magnetic diffusivity. All of the physical values are dependent on both x and z coordinates, but constant along the y coordinate. Initial Configuration and Boundary Conditions Initially we assume a force-free configuration described by
2L Bo cos (,2_~x) e-/7, e Bx =- n-"Fl
By:-41
[2 L'~2 Bo cos (,ff--Ex) e-~,
B z = B o sin
40
(6) (7) (8)
e-//,
30
( - 2 L
20
10
h=6
Initial Perturbation For 0 < t < 2, the diffusivity coefficient is set to r I = 0 everywhere, I
except in the small region
z
[x2+(z-h)Z] 2 ,:0.7, ( h = 3, 6, 10, 20)
0 -2
-1
0
1
2
X
where r I = 1/15. Because of a finite value of resistivity in this small region can magnetic fields dissipate initially which arises the inflow toward this region. This flow make magnetic fields convex to the symmetric axis (x = 0), creating a neutral point on this axis (see Figure 3). Current densities will then increase in the neutral layer, eventually turning on anomalous resistivity (see below). Figure I shows both initial configuration and the site of initial perturbation.
Fig. I. Initial configuration (L = 1.0, H = 57.5, and h = 6) showing contours of magnetic fields. Hatched area represents the region where the artificial resistivity is turned in the initial perturbation. Anomalous Resistivity. We may safely assume that a fast magnetic Note: Figure is not to scale but reconnection is responsible for a violent energy release in flare. Among stretched horizontally. several physical processes capable of causing fast magnetic reconnections, anomalous resistivity will probably be one of the most important factors. Anomalous resistivity is known to have a close relationship with plasma micro-turbulence (cf. Parker, 1979). Suppose that the strength of magnetic fields changes so rapidly in space that the electron drift velocity can exceed its thermal velocity, excite plasma micro-turbulence, which provides a seed for anomalous resistivity. It is important to note that anomalous resistivity can localize the diffusion region into a fairly small area, thereby yielding the Petschek type configuration, which is actually favored in terms of causing a fast magnetic reconnection (Yokoyama and Shibata, 1994). We assume the following form for the anomalous resistivity in the present study except that the Vl is over 300, when rl is constantly set to 300.
Plasmoid Formation in Eruptive Flares
1905
orlv.l'l l>lvcl /0
(9)
otherwise,
where, v a is a relative ion-electron drift velocity, v c is a threshold velocity and rl c is a constant representing a magnitude of resistivity (see Ugai, 1986). In the present study, we assign v c = 13, 50, 70 and ~lc = 1/150. RESULT
1.0
3o .eco°. tio , j" z = -
7.32
+
.
In this paper, we present some results of one 0.8 0 typical case investigated by numerical simulation. 0.6 This case has H = 57.5, h = 6, and ~ = 0.2. Figure 0 2 shows the temporal variation of the position of 0.4 the plasmoid and the reconnection rate in this case. Here we define that the position of the plasmoid is 10 0.2 "O"-point on z-axis, where horizontal magnetic field vanishes and we derive the reconnection rate 0.0 from the electric field at the neutral point (see 0 5 10 15 20 25 Forbes and Priest, 1982). In this figure we show 0 two line-fittings for the temporal variation of the time position of the plasmoid, one of which is in the range of 1 < t < 8, and the other is 17 < t < 24. Fig. 2. Temporal variation of the position of Since this figure indicates that the plasmoid is plasmoid (+) and reconnection rate (solid line). Time rapidly accelerated in a short time range (10 < t < is normalized by sound crossing time. = 1.0 VA@ 15), we divide the evolution of the dynamics of the lime = 3.00 = 1.0 MAC time = 6 . 0 0 ~ plasmoid into three different phases. The first phase extends to t , o = 8, during that a plasmoid forms and starts to rise slowly by magnetic tension force of the reconnected fields produced by 3 0 initial dissipation. This is followed by the relatively short phase (10 < t < 15), in which the reconnection rate reaches its maximum and the plasmoid is violently erupted upward by the 1 well-strengthened tension force. (When the reconnection rate is 1 0 large, a lot of reconnected fields are produced near the neutral point, which exert a violent tension force to erupt plasmoid.) In 0- 2 - 1 0 1 2 -2 -1 0 2 the final phase (17 < t) the reconnection rate decreases and the acceleration of plasmoid becomes weak. In Figure 3 the temporal t i m e = 9 . 0 0 ~ = 1 , 0 V~0 time = 12.0 L = 1.0 V~ variation of perpendicular magnetic fields is displayed. By comparison with Figure 2 we can find that the change into dynamical phase is caused by the drain of the perpendicular 30 30 magnetic fields from the reconnection point.
Z2°z--6.48+ •
20
20
10
10
DISCUSSION From the results presented above, we can clearly understand the role that the perpendicular magnetic fields play in the dynamics of plasmoid. These fields bring about the transitions of the phases of evolution. When the perpendicular magnetic fields are around the reconnection points, the magnetic reconnection cannot proceed effectively so that the magnetic tension force of the reconnected fields is small and the upward motion of plasmoid is slow. This phase corresponds to the so-called pre-flare phase, in which the plasmoid or filament is observed to rise slowly.
ii ii ~)
I
0 2 -1
0
1
2
0 -2
-1
0
1
i 2
Fig. 3. Time evolution of the perpendicular magnetic fields. Black area means the region where the perpendicular fields dominate. Arrows mean fluid velocity normalized by VA0 (the Alfv6n velocity at the base, z = 0).
1906
T.
Magaraand K. Shibata
Meanwhile the reconnection jets and the X-rl~v Plmana /~ 1 magnetic tension force of the reconnected DbrpLo~ement /" i Tup /' I~ fields flow away the perpendicular fields from Full BeL (o) ~/ 1/ l:l[L1JrP~o. (~,) , .; / 1 the neutral point, and once these fields are expelled away from there, rapid magnetic reconnection starts, causing the plasmoid to be strongly accelerated upward by the well' Core strengthened tension force. Observationally, gradual rise of plasmoid in the pre-flare phase -I is followed by violent eruption of it in the impulsive phase (See Figure 4, Courtesy Ohyama et al.), which are theoretically considered by many workers (Mikic, et al., 1988, Priest, 1988, van Ballegooijen and Martens, 1989, Biskamp and Welter, 1989, Finn et al., 1990, Priest and Forbes, 1990, Forbes, 1990, and Choe, 1995). This paper 1093 Nov. 11 demonstrate one reliable idea that this phenomenon can be explained by the dynamics Fig. 4. Soft X-ray plasma displacement and the variation of the perpendicular magnetic fields. of soft X-ray emission with time. "Top" and "Core" show the displacement of the top and the center of the plasmoid, REFERENCES respectively. Courtesy Ohyama LD
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L1:18
11:20
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Biskamp, D., and Welter, H., Magnetic Arcade evolution and instability, in Solar Physics, 120, 49 (1989) Choe, G. S., Formation of Solar Prominences and Eruption of Solar Magnetic Arcade System, in Ph.D. Thesis (1995) Finn, J. M., and Chen, J., Equilibrium of Solar Coronal Arcades, in ApJ, 349, 345 (1990) Forbes, T. G., and Priest, E. R., Numerical Study of Line-tied Magnetic Reconnection, in Solar Physics, 81,303 (1982) Forbes, T. G., Numerical Simulation of a Catastrophe Model for Coronal Mass Ejections, in J.G.R., 95, 11919 (1990) Magara, T., Mineshege, S., Yokoyama, T., and Shibata, K., Numerical Simulation of Magnetic Reconnection in Eruptive Flares, in ApJ., 466, 1054 (1996) Mikic, Z., Barnes, D. C., Dynamical Evolution of a Solar Coronal Magnetic Field Arcade, in ApJ, 328, 830 (1998) Ohyama, M., and Shibata, K., X-ray Plasmoid Ejection in Eruptive Flares, in Magnetodynamic Phenomena in the Solar Atmosphere, ed. Y. Uchida, IAU Colloq., 153, in press Parker, E. N., Cosmical Magnetic Fields', Oxford Univ. Press, pp. 392-439, Oxford (1979) Priest, E. R., The Initiation of Solar Coronal Mass Ejection by Magnetic Nonequilibrium, in ApJ, 328, 848 (1988) Priest, E. R., and Forbes, T. G., Magnetic Field Evolution during Prominence Eruptions and Two-ribbon Flares, in Solar Physics, 126, 319 (1990) Ugai, M., Global Dynamics and Rapid Collapse of an Isolated Current-Sheet System Enclosed by Free Boundaries, Phys. Fluids, 2 9, 3659 (1986) van Ballegooijen, A. A., and Martens, P. C. H., Formation and Eruption of Solar Prominences, in ApJ, 343, 971 (1989) Yokoyama, T., and Shibata, K., What is the Condition for Fast Magnetic Reconnection, ApJ., 436, L197 (1994)
PLASMOID FORMATION IN ERUPTIVE FLARES T. Magara* and K. Shibata** * Department of Astronomy, Faculty of Science, Kyoto University, Sakyo-ku. Kyoto 606-01, Japan ** Nationul Astronomical Observator): Mitaka, Tokyo 181, Japan
ABSTRACT One phenomena Yohkoh has observed is plasmoid eruption in flares. Thus this is a key factor that must be explained in any flare mechanism. In order to understand the dynamics of a plasmoid, we performed a numerical MHD simulation and investigated the evolution of the coronal magnetic field, which is initially a force-free configuration. The main results are as follows. At first, small amount of dissipation, induced by the initial perturbation, occurs in the current sheet where the plasmoid forms. This plasmoid is slowly going upward by magnetic tension force of the reconnected magnetic fields produced by initial dissipation. The crucial point comes when the perpendicular magnetic fields are washed away from the reconnection point, after that the reconnection proceeds effectively so that the magnetic tension force of the reconnected fields becomes strong, which make the plasmoid be rapidly erupted upward. These are consistent with the observational results, which say that before the main energy release the plasmoid slowly rises and when the flare sets in it is rapidly accelerated upward. In this paper, we emphasize. on the role that the perpendicular magnetic fields play in the evolution of flare. CC) 1997 COSPAR. Published by ElsevierScience Ltd. MATHEMATICAL FORMULA Basic Equations We consider magnetized gas layers composing of ideal gas. The effects of gravity are neglected for simplicity. Using the Cartesian coordinates, the basic equations are:
(1)
p[~+(v.v)v]~-vP+~(vxB)xB,
(2)
~[~(~)+(v.v)(~)]=~lvxB12~
(3)
~$=VX(VXB)-VX(~VXB),
(4)
PAT.
(5)
Additionally, we use V - B =0 as an initial condition for Eq. (4). Here, all the symbols, such as P, p, T,
V,
1904
T. Magara aad K. Shibata
and B have their usual meanings, y is the adiabatic index, R is the gas constant, ~ is the mean molecular weight, and rl is the magnetic diffusivity. All of the physical values are dependent on both x and z coordinates, but constant along the y coordinate. Initial Configuration and Boundary Conditions Initially we assume a force-free configuration described by
2L Bo cos (,2_~x) e-/7, e Bx =- n-"Fl
By:-41
[2 L'~2 Bo cos (,ff--Ex) e-~,
B z = B o sin
40
(6) (7) (8)
e-//,
30
( - 2 L
20
10
h=6
Initial Perturbation For 0 < t < 2, the diffusivity coefficient is set to r I = 0 everywhere, I
except in the small region
z
[x2+(z-h)Z] 2 ,:0.7, ( h = 3, 6, 10, 20)
0 -2
-1
0
1
2
X
where r I = 1/15. Because of a finite value of resistivity in this small region can magnetic fields dissipate initially which arises the inflow toward this region. This flow make magnetic fields convex to the symmetric axis (x = 0), creating a neutral point on this axis (see Figure 3). Current densities will then increase in the neutral layer, eventually turning on anomalous resistivity (see below). Figure I shows both initial configuration and the site of initial perturbation.
Fig. I. Initial configuration (L = 1.0, H = 57.5, and h = 6) showing contours of magnetic fields. Hatched area represents the region where the artificial resistivity is turned in the initial perturbation. Anomalous Resistivity. We may safely assume that a fast magnetic Note: Figure is not to scale but reconnection is responsible for a violent energy release in flare. Among stretched horizontally. several physical processes capable of causing fast magnetic reconnections, anomalous resistivity will probably be one of the most important factors. Anomalous resistivity is known to have a close relationship with plasma micro-turbulence (cf. Parker, 1979). Suppose that the strength of magnetic fields changes so rapidly in space that the electron drift velocity can exceed its thermal velocity, excite plasma micro-turbulence, which provides a seed for anomalous resistivity. It is important to note that anomalous resistivity can localize the diffusion region into a fairly small area, thereby yielding the Petschek type configuration, which is actually favored in terms of causing a fast magnetic reconnection (Yokoyama and Shibata, 1994). We assume the following form for the anomalous resistivity in the present study except that the Vl is over 300, when rl is constantly set to 300.
Plasmoid Formation in Eruptive Flares
1905
orlv.l'l l>lvcl /0
(9)
otherwise,
where, v a is a relative ion-electron drift velocity, v c is a threshold velocity and rl c is a constant representing a magnitude of resistivity (see Ugai, 1986). In the present study, we assign v c = 13, 50, 70 and ~lc = 1/150. RESULT
1.0
3o .eco°. tio , j" z = -
7.32
+
.
In this paper, we present some results of one 0.8 0 typical case investigated by numerical simulation. 0.6 This case has H = 57.5, h = 6, and ~ = 0.2. Figure 0 2 shows the temporal variation of the position of 0.4 the plasmoid and the reconnection rate in this case. Here we define that the position of the plasmoid is 10 0.2 "O"-point on z-axis, where horizontal magnetic field vanishes and we derive the reconnection rate 0.0 from the electric field at the neutral point (see 0 5 10 15 20 25 Forbes and Priest, 1982). In this figure we show 0 two line-fittings for the temporal variation of the time position of the plasmoid, one of which is in the range of 1 < t < 8, and the other is 17 < t < 24. Fig. 2. Temporal variation of the position of Since this figure indicates that the plasmoid is plasmoid (+) and reconnection rate (solid line). Time rapidly accelerated in a short time range (10 < t < is normalized by sound crossing time. = 1.0 VA@ 15), we divide the evolution of the dynamics of the lime = 3.00 = 1.0 MAC time = 6 . 0 0 ~ plasmoid into three different phases. The first phase extends to t , o = 8, during that a plasmoid forms and starts to rise slowly by magnetic tension force of the reconnected fields produced by 3 0 initial dissipation. This is followed by the relatively short phase (10 < t < 15), in which the reconnection rate reaches its maximum and the plasmoid is violently erupted upward by the 1 well-strengthened tension force. (When the reconnection rate is 1 0 large, a lot of reconnected fields are produced near the neutral point, which exert a violent tension force to erupt plasmoid.) In 0- 2 - 1 0 1 2 -2 -1 0 2 the final phase (17 < t) the reconnection rate decreases and the acceleration of plasmoid becomes weak. In Figure 3 the temporal t i m e = 9 . 0 0 ~ = 1 , 0 V~0 time = 12.0 L = 1.0 V~ variation of perpendicular magnetic fields is displayed. By comparison with Figure 2 we can find that the change into dynamical phase is caused by the drain of the perpendicular 30 30 magnetic fields from the reconnection point.
Z2°z--6.48+ •
20
20
10
10
DISCUSSION From the results presented above, we can clearly understand the role that the perpendicular magnetic fields play in the dynamics of plasmoid. These fields bring about the transitions of the phases of evolution. When the perpendicular magnetic fields are around the reconnection points, the magnetic reconnection cannot proceed effectively so that the magnetic tension force of the reconnected fields is small and the upward motion of plasmoid is slow. This phase corresponds to the so-called pre-flare phase, in which the plasmoid or filament is observed to rise slowly.
ii ii ~)
I
0 2 -1
0
1
2
0 -2
-1
0
1
i 2
Fig. 3. Time evolution of the perpendicular magnetic fields. Black area means the region where the perpendicular fields dominate. Arrows mean fluid velocity normalized by VA0 (the Alfv6n velocity at the base, z = 0).
1906
T.
Magaraand K. Shibata
Meanwhile the reconnection jets and the X-rl~v Plmana /~ 1 magnetic tension force of the reconnected DbrpLo~ement /" i Tup /' I~ fields flow away the perpendicular fields from Full BeL (o) ~/ 1/ l:l[L1JrP~o. (~,) , .; / 1 the neutral point, and once these fields are expelled away from there, rapid magnetic reconnection starts, causing the plasmoid to be strongly accelerated upward by the well' Core strengthened tension force. Observationally, gradual rise of plasmoid in the pre-flare phase -I is followed by violent eruption of it in the impulsive phase (See Figure 4, Courtesy Ohyama et al.), which are theoretically considered by many workers (Mikic, et al., 1988, Priest, 1988, van Ballegooijen and Martens, 1989, Biskamp and Welter, 1989, Finn et al., 1990, Priest and Forbes, 1990, Forbes, 1990, and Choe, 1995). This paper 1093 Nov. 11 demonstrate one reliable idea that this phenomenon can be explained by the dynamics Fig. 4. Soft X-ray plasma displacement and the variation of the perpendicular magnetic fields. of soft X-ray emission with time. "Top" and "Core" show the displacement of the top and the center of the plasmoid, REFERENCES respectively. Courtesy Ohyama LD
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I
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I
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/'"
,
ID
IG
|Boiw
11.-04
Sm-fam~
l.l:l~
i
)
II:lt3
D
L1:18
11:20
!1:24
Biskamp, D., and Welter, H., Magnetic Arcade evolution and instability, in Solar Physics, 120, 49 (1989) Choe, G. S., Formation of Solar Prominences and Eruption of Solar Magnetic Arcade System, in Ph.D. Thesis (1995) Finn, J. M., and Chen, J., Equilibrium of Solar Coronal Arcades, in ApJ, 349, 345 (1990) Forbes, T. G., and Priest, E. R., Numerical Study of Line-tied Magnetic Reconnection, in Solar Physics, 81,303 (1982) Forbes, T. G., Numerical Simulation of a Catastrophe Model for Coronal Mass Ejections, in J.G.R., 95, 11919 (1990) Magara, T., Mineshege, S., Yokoyama, T., and Shibata, K., Numerical Simulation of Magnetic Reconnection in Eruptive Flares, in ApJ., 466, 1054 (1996) Mikic, Z., Barnes, D. C., Dynamical Evolution of a Solar Coronal Magnetic Field Arcade, in ApJ, 328, 830 (1998) Ohyama, M., and Shibata, K., X-ray Plasmoid Ejection in Eruptive Flares, in Magnetodynamic Phenomena in the Solar Atmosphere, ed. Y. Uchida, IAU Colloq., 153, in press Parker, E. N., Cosmical Magnetic Fields', Oxford Univ. Press, pp. 392-439, Oxford (1979) Priest, E. R., The Initiation of Solar Coronal Mass Ejection by Magnetic Nonequilibrium, in ApJ, 328, 848 (1988) Priest, E. R., and Forbes, T. G., Magnetic Field Evolution during Prominence Eruptions and Two-ribbon Flares, in Solar Physics, 126, 319 (1990) Ugai, M., Global Dynamics and Rapid Collapse of an Isolated Current-Sheet System Enclosed by Free Boundaries, Phys. Fluids, 2 9, 3659 (1986) van Ballegooijen, A. A., and Martens, P. C. H., Formation and Eruption of Solar Prominences, in ApJ, 343, 971 (1989) Yokoyama, T., and Shibata, K., What is the Condition for Fast Magnetic Reconnection, ApJ., 436, L197 (1994)