1. 有限振幅拡散擾乱より発展する 磁気リコネクションの 3 次元シミュレーション 横山 央明（東京大学 ) 、磯部 洋明（京都大学） 理論懇シンポジウム 2008. 12. 15 - 17. 国立天文台 Temporal evolution of a current sheet with initial perturbations is studied by using the three-dimensional resistive magnetohydrodynamic (MHD) simulations. The magnetic reconnection is considered to be the main engine of the energy release in solar flares. The structure of the diffusion region is, however, not still understood under the circumstances with enormously large magnetic Reynolds number as the solar corona. In particular, the relationship between the flare's macroscopic physics and the microscopic ones are unclear. It is generally believed that the MHD turbulence should play a role in the intermediate scale. Our results are as follows: (1) In the early phase of the evolution, high wavenumber modes in the electric-current direction are excited and grow. The energy spectrum in the later stage approaches a power-law function. It indicates a turbulent nature of the system. (2) Many "X"-type neutral points (lines) are generated along the magnetic neutral line (plane) in the current sheet. When they evolve into the non-linear phase, three-dimensional structures also evolve. The spatial scale seems to be almost comparable with that in the xy-plane. (3) The energy release rate is reduced in case of 3D simulations compared with 2D ones probably because of the reduction of the inflow cross sections by the formation of patchy structures in the current sheet. Evolution of a current sheet with initial resistivity perturbations is studied by using the 3D resistive MHD simulations. Three-dimensional structures in the z -direction evolve in time. The spatial scale in the z -direction seems to be almost comparable with that in the xy - plane. Slight increase in the energy-release growth rate is observed due to the spatial division of the current sheets by the magnetic islands. Energy-release rate is reduced in 3D case compared with 2D case, presumably because of the reduction of the filling factor of the diffusion regions in the additional dimension. Summary & Discussion Tajima & Shibata, 1997, “Plasma Astrophysics”, Sec 3.3.2.8 Lazarian & Vishniac, 1999, ApJ, 517, 700 References Introduction ~10 m ~10 4 km ~1 km Tajima & Shibata (1997) The magnetic Reynolds number in the solar corona is enormously large when it is defined by using the Spitzer-type resistivity. This large number is approximately the ratio between the flare loop size and the diffusion region, which is less than centimeter accordingly. Even when the non-Spitzer-type resistivity caused by the plasma micro processes are taken into account, the scale size would be still the order of the ion Larmor radius, i.e. 10 meters in the corona. It is unreasonable to consider that such a small diffusion region controls the large- scale energy release in the flare in a steady manner. It is therefore claimed by some researchers that there must be a meso-size scale where the MHD turbulence plays a roll (e.g. Tajima & Shibata 1997). They suggest that there is a global current sheet in the diffusion region of a flare. The size of the global current sheet is less than the flare scale, but is not extremely small, say 1 km. The global current sheet contains many small magnetic islands, between which there are many small thin current sheets. These thin sheets have a fractal nature. In the smallest scale, the thickness is comparable to the Larmor radius. In this scale, the reconnection may proceed in a rapid time, then the size of islands grows and grows. Finally the reconnection process goes rapidly even in the large scale. Lazarian & Vishniac (1999) estimated the reconnection rate based on the turbulent reconnection and obtained the Alfven number of the reconnection inflow to be. Their estimation is based on an order-of-magnitude calculation and still needs to be proved by solving directly the full set of MHD equations. In order to study this, we are performing a series of MHD simulations of magnetic reconnection with finite amplitude fluctuations. Numerical Model Periodic condition on all the boundaries 16  128  10   x y z A current sheet with the anti-parallel magnetic fields ( ) is considered as an initial condition. We solve 3D MHD equations with the uniform resistivity whose magnetic Reynolds number is. Additional resistivity perturbations are imposed with random amplitudes (50% at the maximum with respect to the uniform background  ) only at the early stage ( ) in each simulation run. In the xy -plane, tiny structures evolve first. They are overcome by the larger wavelength mode. The size are roughly consistent with the most unstable mode of the tearing instability. In the zy -plane, by the imposed random perturbations, many tiny concentrations in the current are evolved in the initial phase. They are all X-points since we found bipolar structure of the reconnected x -component of the magnetic fields and corresponding pairs of y - directional outflows (see the figures in the r.h.s. which show the close-up plots of a selected X-point). A pair of z -directional flows into the X-point evolve for each of the current concentrations. It controls the evolution of 3D structure in the z -direction. Typical Case The plots in the r.h.s. is the power spectrum derived from the Fourier transform of the Bx distribution in the yz -plane ( x =0). It shows: Inverse cascade Evolution of a power-law spectrum z y BxVz x y JzP ky(Ly/2)ky(Ly/2) K IK time kz(Lz/2)kz(Lz/2) x y  Jz VxVyVz BxByBz z y  Jz VxVyVz BxByBz Comparison with the Sweet-Parker Reconnection The energy release rate is slightly larger than the Sweet- Parker reconnection. When the current sheet is divided into N parts, the length of each part becomes shorter. Then, keeping the aspect ratio of the diffusion region, the total amount of the released energy increases like this proportional to the square-root of the dividing number N. VxVy t =90 MAMA t / (  /Cs) L Sweet Parker sheet N -divided sheets (e.g. Lazarian & Vishniac 1999) SP Typical case The energy release is weaker in the 3D case than 2D. In case of 3D, the distribution of the current density is not uniform with scattering concentrations here and there in the sheet. As a result, the effective length in the z -direction might be smaller than 2D case and the efficiency of the energy release might be smaller also. JzBx t=110 diffusion regions Comparison with the 2D "turbulence" case MAMA t / (  /Cs) VxVy t=110 2D 3D The reconnection rate is: i.e., has a Sweet-Parker like dependence on R m. Dependence on R m  1.d-3 1.d-2 1.d-1 t / (  /Cs)  / A M In case with the anomalous resistivity, Petschek like structures (i.e. pairs of slow-mode shocks) appear, leading to the enhanced reconnection rate. VxVy t=140 Case with anomalous resistivity anomalous resistivity uniform resistivity where MAMA t / (  /Cs)