A numerical study of flaring loop dynamics during magnetic reconnection

Adv. Space Res. Vol. 29, No. 10, pp. 1445-1449,2002 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0273-l 177/02 $22.00 + 0.00

Pergamon www.elsevier.com/locatelasr

PII: SO273-

A NUMERICAL DYNAMICS

1177(02)00201-6

STUDY

DURING

OF FLARING

MAGNETIC

LOOP

RECONNECTION

C. Fang, P. F. Chen, Y. H. Tang, and X. H. Di Department

of Astronomy,

Nanjing

University,

Nanjing 210093,

China

ABSTRACT 2.5-dimensional magnetic reconnection is numerically simulated for two cases, one with a high altitude of the reconnection point, the other with a low altitude. In the former case, bright loops appear to rise for a long time, with footpoints separating and the field lines below the bright loops shrinking. In the latter case, the bright loops cease to rise after a short period of reconnection and become rather stable. The results imply that the two types of solar flares, i.e., two-ribbon flares and compact flares, might be unified under a single magnetic reconnection model, where the height of the reconnection point leads to the bifurcation. 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved.

INTRODUCTION Solar flares are usually classified into two types, i.e., compact and two-ribbon flares (e.g., Sturrock 1980). For the latter, the rise of the flare-associated SXR (soft X-ray) loop system and the corresponding separation of the Ha ribbons can be easily explained by the so-called CSHKP model (e.g., Kopp and Pneuman 1976). Such apparent motions are due to the progressive appearance of higher bright loops, rather than mass motions (e.g., Martin 1979). The actual plasma flows in the loops are quite different and are due to chromospheric evaporation (Hirayama 1974), the gravitational draining of the cold plasma, and the shrinkage of the cold loops (Svestka et al. 1987; Forbes and Acton 1996). Compact flares aze usually attributed to the emerging flux reconnection model (Heyvaerts, Priest and Rust 1977). The discovery of a hard X-ray source above SXR loop by Yohkoh suggests that the reconnection process in compact flares should be the same as that in two-ribbon flares (Masuda et al. 1994). Such an idea is supported by more and more observational evidence, and it has been proposed that there is a need to unify the two types of flares (cf. Shibata 1997, and references therein). Based on numerical simulations, we have proposed a unified model for the two types of flares (Chen et al. 1999). This paper reviews our numerical research on the motions of the flaring loop, which are comparable with observations, and the possibility of unifying the two types of flares. NUMERICAL PROCEDURE The 2.5-D compressible resistive %

z

+

v . (pv)

MHD equations

are cast in the dimensionless

= 0

form

(1)

g+(v.V)v+iVP-:jxB-g=O

(2)

dB --Vx(vxB)+Vx(r]VxB)=O at

(3)

g+V.VT+(y-1)TV

~v_w-lb?. ppo

. J’J-

c 7”.

T: (B . VT)

B2

(4)

C. Fang et al.

1446

where V = &i& + $$. The dimensionless P = pRT, where R is the gas constant;lthe (r-l)noT at the surface of the Sun; C = 4

quantities p, v, B, T have their usual meaning; gas pressure ratio of specific heats y = 513; g is the acceleration of gravity is the heat conduction

coefficient.

The magnetic

field is given by

B = V x (I/J&,) + B,6,, where $ is the magnetic flux function. Eq.(3) is dissociated into two equations with The characteristic values of density, respect to $J and B,. j is the current density, and Q the resistivity. temperature, ratio of gas to magnetic pressures, and length are assumed to be pc = 3.34 x lo-l1 kg rnF3, TO = lo6 K, ,& = 0.1, LO = 2 x lo4 km, respectively. Then the isothermal sound speed 2ra = 128 km s-l, and the AlfvQn speed 0,~ = 575 km s-l. The time unit rA is defined as rA = LO/VA = 35 s. Since one purpose of this paper is to study the effect of the height (he) of the reconnection point (here meaning ‘X’ point), two models (denoted as Models A and B, respectively) are considered. Model A corresponds to the case with ho = 4, and Model B with ho = 1. The domain of simulation is -1 5 z < 1, The initial magnetic configuration is a 0 5 y 5 2ho, where x-axis is horizontal, and y-axis is upward. force-free field (current sheet) surrounded by a potential field, which is written as ~“0”‘~) 1c,=

-lx1 - +

- 1,

(Ix] < W),

+ w,

(Izl 2 w),

(5)

where the half width of the current sheet w = 0.1. The plasma is isothermal and in hydrostatic equilibrium b e fore anomalous resistivity 77is introduced into a local region: 1x1 5 0.1, (v=O, T = 1, and p = exp[(4-y)g]) resistivity to initiate the ]y - hoI 5 0.2, wh ere q = 0.02 cos( E) cos(2.5(y - ho)r). Th e use of a non-uniform reconnection was introduced by Ugai and Tsuda (1977). Due to the symmetry, calculation is made only in the right half region. The bottom (y=O) is a line-tying side (x = 1) are treated as open boundaries. Symmetry boundary. The top (y = 2ho) and the right-hand conditions are used for the left-hand side (x = 0). The numerical mesh consists of 41 nonuniformly spaced grid points along the x-direction and 181 uniformly spaced grid points along the y-direction. The nonlinear MHD equations are then numerically solved by a multistep implicit scheme (Hu, 1989). NUMERICAL RESULTS After localized resistivity is introduced in Model A, the oppositely-directed field lines beside the neutral line begin to reconnect. The magnetic tension of the reconnected field lines rapidly drives two opposite outflows (jets) to high speed, and these are accompanied by two symmetrical inflows. At the separatrix between the inflow and the outflow, a pair of MHD slow-mode shocks (which are switch-off shocks) develops as predicted by Petschek (1964). The included angle of the shock pair is about 3”. The upward jet, along with the frozen-in field lines, is ejected out of the top approximately with the Alfvdn speed VA; the downward jet collides with the closed field lines which are line-tied with the base to form a termination shock (Forbes and Priest, 1983). Along the magnetic loop which links the termination shock with the base, both the temperature and the density increase to form an SXR loop. As the reconnection goes on, the SXR loop along with the loop top (the termination shock) rises, and the two footpoints separate as shown in Figure 1. Note that the patterns of the upper half part and the lower half part show strong nonsymmetry, which mainly results from the different boundary conditions, i.e., the top is free, but the base is a line-tying boundary. The rise speed of the loop is about 30 km s-l, which is comparable with observations (e.g., Tsuneta 1996). The loop motions described above are due to the progressive appearance of newly-heated loops rather than mass motions, which is in accordance with the observations (e.g., Martin 1979). The plasma along the closed magnetic loop, in fact, falls down. Since the chromosphere is not included in our simulation region, chromospheric evaporation does not occur. Besides the apparent rise motion, the magnetic loop shrink slightly. Such an effect was discovered by observations (Svestka et al. 1987; Forbes and Acton 1996). Figure 2 shows the spatial change of the field line with $J = -1.2 from t = 20 ‘J-Ato t = 30 rA, where the amount of the shrinkage is about 5%. Figure 3 presents the y-plots of the z-component of the current density j, and the y-component of the magnetic force (~xB)~ of Model A along the y-axis at t = 30 7~. Compared with Figure 1, the loop top

Study

Numerical

of Flaring Loop Dynamics

1447

loop top: SXR

loop.

-1.0

-0.5

0.0

1.0

0.5

x

Fig.

1. Local distributions

of temperature

field in Model A at three separate

Fig.

and magnetic

2. Local distributions

magnetic

times.

field in Model

of temperature A at three

and

separate

times.

200

1,.

,

1

I.

,

I.

I,

I.

40

30

150

100

_,oo

:: :: !

t

I

1

0

I

20

i

,

I

12

.

3

1

I

I

4

5

6

,

I

l_2o

7

6

Y Fig.

3. y-plots

of z-component

solid line corresponds

corresponds potential

to j,,

to the sharp peaks of both j,

except near the base.

significantly

of the current

and the dotted

just

Besides,

near the reconnection

density j,

and y-component

of the magnetic

force (jxB),.

The

line to (jxB),.

and (jxB),,

below which the magnetic

it can be seen that the outflows are accelerated point

(B=O),

where two extreme

field is approximately by the magnetic

values of (j xB),

force

are coincident

with those of j,. In Model B, where the reconnection

point is low, the SXR

loop also appears.

it differs from Model A. One is that the slow shocks, which are characterized

However, in three aspects

by B, - 0 in the downstream,

are not so clear to be seen in Model A. The second is that the loop top (termination before t = 8 7~. Moreover, 5 km s-l

the rise speed of the SXR

near t = 30 7A, and even to 1 km s-l

a short period of rapid reconnection, temperature

and magnetic

shock)

loop drops down from 16 km s-l

near t = 40 7-A. The SXR

loop remains

rather

as seen in Figure 4, which shows the two-dimensional

field. Correspondingly,

smaller and smaller after an impulsive increase.

the magnetic

reconnection

is evident only

near t = 10 TA, to stable after

distributions

of

rate (R) of Model B becomes

The time profile of the R of Model B is compared

with that

of Model A in Figure 5. Here, reconnection rate R is defined by R = d&/dt, where q& is the magnetic flux function at the neutral point. The significant decrease of the R in Model B can be explained as follows: as illustrated

in Figure

3, the magnetic

loop top is close to the resistivity

field below the loop top is approximately region,

magnetic

reconnection

decelerates

potential. naturally

When

the rising

unless anomalous

C. Fang et al.

1448

Model E

i=8

Model

8

t=zo

Fig. 4. Distributions of temperature in Model B at three separate times.

resistivity resistivity

is re-introduced region.

Model B

t=30

and magnetic

at a higher region.

field

Fig. 5. Time profiles of magnetic reconnection rate (R) in Model A (solid line) and Model B (dotted line). R = d&/dt, where I,!J~is dimensionless, and the unit of t is 7~.

In other words, the top of the closed field can not exceed the

DISCUSSIONS Based on the fact that the usually classified two types of solar flares share many common features, e.g., the ejection of hot plasma, loop rise (although for the compact flares, the rise speed is much smaller), X-type or Y-type morphology suggesting the presence of current sheets or neutral points, change of field configuration, and so on, Shibata (1997) proposed to unify the two types of flares. The numerical results of this paper evoke us to consider such a proposal. In Model A, where the reconnection point is high, a bright loop with high temperature and density is formed, and appears as an SXR loop. The SXR loop keeps rising for a long time with two footpoints separating, which are the typical features of two-ribbon flares. Note that the old SXR loops do not cool down to form Ho loops since radiation loss is not considered at present. On the contrary, in Model B, where the reconnection point is low, the loop top and the loop motion last only for a short time, and the rise speed of the SXR loop decreases rapidly so that the SXR loop seems rather stable, which is similar to compact flares. It should be noted that the CSHKP model for two-ribbon flares and the emerging flux model for compact flares can both be reproduced in our simulations. In the former case the current sheet is formed by the filament eruption, and is vertically long, therefore, the reconnection point can be high; the current sheet in the latter case, however, is formed by the interaction of newly-emerging magnetic flux from the photosphere with the pre-existing magnetic field in the corona. The corresponding reconnection point must be low. Here, we might as well suppose that the anomalous resistivity is caused by ion-acoustic current instability with the critical condition: ud - B/(n&e) 2 (kT,/n~i)‘/~, where vd is the electron drift velocity, B the magnetic field, n, the electron density, Lo the characteristic length of the current sheet, e the electron charge, k the Boltzmann constant, T, the electron temperature, and rni the ion mass. From the formula, it can be deduced that the lower corona (with larger n,) requires stronger magnetic field, and vice versa. Our model then means that the reconnecting magnetic field is stronger for the compact flares, and weaker for the two-ribbon flares. Such a result is in agreement with Table I of Shibata (1997). Because a larger magnetic field may induce a larger electric field, so as to accelerate electrons to higher energy and form a more intensive hard X-ray source, it is inferred that compact flares should be stronger in hard X-rays than two-ribbon flares.

Numerical

Study of Flaring Loop Dynamics

1449

CONCLUSIONS 2.5-dirrlerlsioll;LI

magnrt,ic: reconnection

is numerically

current, sheet. s~~rrormtled by a potential different

field.

height,s t,o st,udy t,hc flaring loop dynamics.

1, Af%cr lo(:aliz(~tl ~mo11m1011sresistivity between separate.

is introduced,

As t,hc reconnection

The it(:t,ll;tl magnetic

reconnc>ction. 2.

The conclusions

the: inflow and the out,flow, with included

SXR. loop al)p~‘ars.

ant1 compare

with a force-free

with t,he reconnection

point at

are as follows:

a pair of slow MHD shocks appears at the separatrix angle of about

goes on, the SXR

loops shrink slightly.

favorably

solved for an initial configuration

Two models are considered

3”.

All of these features

with observations

Far below the neutral

loop rises apparently,

point,

are the natural

results of magnetic

of solar flares.

It is suggc>st,c,tl t)hat t,hc: t,wo types of solar flares can be unified under a single reconnection

which has t,hcLhc+ht

of t,hc recomlection

point as the parameter

the rcc:orlrlc~r,t,ioll l)oint, is high for the two-ribbon ACKNOWLEDGEMENTS The aut,hors t.llitnk two anonymous by NSFC

which determines

flares, and low for the compact

referees for the correction

and by a National

(No.4!.1990451)

an

and its two footpoints

Basic Research

of the manuscript.

Priorities

Project,

model,

which type occurs,

i.e.,

flares.

This work was supported

(G2000078402)

of China.

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