MHD Simulation of a Solar Flare Based on
Magnetic Reconnection Model

Yokoyama, T. & Shibata, K.
National Astronomical Observatoy of Japan

Abstract

Fig.1
Two-dimensional simulation of a solar flare is performed using a newly developed magnetohydrodynamic (MHD) code including nonlinear anisotropic heat conduction effect (Fig.1). We also derived a simple scaling law for the flare temperature described as T \propto B^{6/7} where T and B are the temperature at the flare loop apex and the coronal magnetic field strength.

Introduction

Fig.2 Fig.3
Fig.2 shows the soft X-ray image of a cusp-shaped flare observed with Yohkoh soft X-ray telescope (Tsuneta et al. 1992), and Fig. 3 shows the schematic picture of the magnetic reconnection model of a flare. In this model, the energy released by the reconnection is transported by heat conduction to the chromosphere, where the so-called ``chromospheric evaporation'' (Neupert 1968), i. e. an ablation of chromospheric plasma, occurs. In this study, we performed a two-dimensional MHD simulation of chromospheric evaporation associated with a solar flare based on this magnetic reconnection model.

Calculation Model

Fig.4
We solved two-dimensional MHD equations without gravitation. The nonlinear anisotropic heat conduction effect is taken into account. The initial condition (Fig.4) is in magnetohydrostatic equilibrium with antiparallel magnetic fields in the xz-plane, between which there is a current sheet. The initial gas consists of two layers: A hot corona with density of 10^{9} cm^{-3} and with temperature of 2 \times 10^{6} K. A low-temperature and dense region, whose density is 10^{5} times that of the other area, is located near the x-axis. The plasma beta, the ratio of the gas pressure to the magnetic pressure is taken to be \beta=0.2. So the magnetic field strength is 6 G and the Alfven speed is 400 km s^{-1}.

We assumed an anomalous resitivity model (e.g. Yokoyama & Shibata 1994). It increases with increasing drift velocity. The minimum magnetic Reynolds number defined as Rm= Cs \delta/\eta_{max} is 3, where Cs is the sound speed in the corona, \delta is the thickness of the current sheet, and \eta_{max} is the maximum value of the resistivity.

In these conditions, the Alfven time defined by the initial thickness of the current sheet divided by Alfven speed is 7 seconds and the heat conduction time is 2 seconds. In the initial 1 miniutes, we imposed a resistivity perturbation at this hatched region. Then magnetic reconnection starts at this point.

Results

Fig.5 Movie.1
Fig. 5 and Movie 1 (left; density, right; temperature) show the results. Because of the enhanced resistivity, magnetic reconnection starts. The reconnected field lines together with the frozen-in plasma are ejected from this X-point in both upward and downward directions due to the tension force of the reconnected field lines. At the boundary between the inflow and this outflow a slow-mode MHD shock is formed.

At the same time, a heat condcution front propagates from the hot region between the pair of shocks. The conduction of heat is only in the direction along the magnetic field line. The outer edge of the conduction front, therefore, traces the magnetic field lines extending from the X-point. This temperature distribution is very similar to the cusp-like structure of solar flare loops, which are observed by the soft X-ray telescope of Yohkoh satellite.

In the density distribution, a growing plasma mound can be seen. This is the direct consequence of the so-called chromospheric evaporation. The chromospheric plasma is heated up and expands suddenly due to the penetration of the heat conduction front. The induced pressure-gradient drives a back-flow toward the corona. This flow carries up the dense plasma into the corona.

Fig.6
On the other hand, the upward motion of the plasma driven by the tention force of the reconnected field lines will be observed as a plasmoid ejection (Fig. 6) The velocity of ejection is the Alfven speed, in this particular case 400 km/s.

`Observation' of the numerical flare

Fig.7
We derived soft X-ray (I-Be, I-thin Al) and radio (I free free) maps from the simulation results (Fig. 7). This X-ray distribution is very similar to the cusp-like structure of loops of a so-called long-duration-event (LDE) flare, which are observed by the soft X-ray telescope of Yohkoh satellite.

T - B Scaling Law

Fig.8
What determines the flare temperature and its distribution ? If we assume that the input of energy to a loop balances with the conduction cooling rate, the temperature at the loop apex is T \approx (2QL^{2}/\kappa_{0})^{2/7} where Q is the volumetric heating rate, L is the half-length of the loop (Fisher & Hawley 1990). In our simulations, the heating mechanism is magnetic reconnection so that the heating rate is described as Q=B^{2}/(4\pi) V_{in}/L 1/sin(\theta), where V_{in} is the inflow velocity (\approx 0.1 V_{A} from our result), and \theta is the angle between the slow-mode MHD shock and the loop and is approximately given by sin(\theta)\approx V_{in}/V_{A}. By manipulating these, we find

The simulation results in Fig. 8 show very good agreement with this scaling law. When \beta=0.01 that is the typical value in the real solar corona, this scaling law predicts the flare temperature T \approx 10 MK which is consistent with the observed flare temperature T \approx 10-20 MK (e.g. Tsuneta et al. 1992).

References

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Tsuneta, S. et al. (1992) PASJ, 44, L63
Yokoyama, T., & Shibata, K. (1994), ApJ, 436, L197
Yokoyama, T., & Shibata, K. (1997), ApJ, 474, L61
Yokoyama, T., & Shibata, K. (1998), ApJ, 494, L113
produced by T. Yokoyama in 1998.