1 MHD Simulations of 3D Reconnection Triggered by Finite Random Resistivity Perturbations T. Yokoyama Univ. Tokyo in collaboration with H. Isobe (Kyoto U.) Lab.-Hinode Reconnection Workshop Hayama

2 “Size-gap” problem in solar flares Spatial size necessary for the anomalous resistivity:  =  i  1 m  ; thickness of the current sheet  i  ; ion-gyro radius Spatial size of a flare – 10 4 –10 5 km Enormous gap with a ratio of 10 7 ! (c.f. Tajima Shibata 1997) For a rapid energy release by the fast reconnection, an anomalous resistivity is expected near the neutral point. It is hard to believe that a laminar flow structure is sustained in a steady state manner.

3 Fractal Current Sheet model ~1-10 m ~10 4 km >1 km Global current sheet Tajima & Shibata (1997) “Turbulent reconnection”: Matthaeus & Lamkin (1986), Strauss (1988), Lazarian & Vishniac (1999), Kim & Diamond (2001)...

4 Our final goal (but far away …) To investigate the physical nature (evolution, saturation, steadiness etc. ) of the 3-dimensional turbulent magnetic reconnection This study By using the resistive MHD simulations, the evolution is studied about a current sheet with initially imposed finite random perturbations in the resistivity. We focus on: –(1) Evolution of 3D structures –(2) Energy release rate

5 Model Periodic condition on all the boundaries 3D MHD eqs with the uniform resistivity plus finite resistivity perturbations spatially random; 50% max. amp. during t/(  /Cs)<4 128  10  32   x y z grid: 256 x 256 x 256

6 xy -plane ( z =0)  Jz Vx Vy Vz BxBy Bz x y Tiny structures evolve first. They are overcome by the larger wavelength mode, whose size are roughly consistent with the most unstable mode of the tearing instability. In later phase, the magnetic islands in the sheet coalesce with each other to form a few dominant X-points.

7 zy -plane, x =0 z y  Jz Vx Vy Vz BxBy Bz "Turbulent" structures

8 Evolution in the current sheet ( zy -plane, x =0)  Jz VxVyVz BxByBz z y Many tiny concentrations in the current are all X-points. They are associated with bipolar structure of the reconnected magnetic fields (Bx) and outflows (Vy).

9 z y BxVz Reconnection X-point x y JzP contour: Jz 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. y y

10 Power spectrum Bx(x=0) t/  s =1, 10, 50, 100, 200 Inverse cascade Evolution of “power-law”-like spectrum ky(Ly/2)ky(Ly/2) K IK time kz(Lz/2)kz(Lz/2)

11 MAMA t / (  /Cs) VxVyVxVy random perturbation Comparison with the Sweet-Parker reconnection t=90 random perturbation SP Sweet-Parker The energy release rate is slightly larger than the Sweet- Parker reconnection.

12 Discussion: divided-sheet vs. Sweet-Parker L single current sheet N -divided multi current sheet (e.g. Lazarian & Vishniac 1999) 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.

13 MAMA t / (  /Cs) VxVyVxVy 3D perturbation Comparison with the 2D turbulent case t=110 2D 2D perturbation ３D３D The energy release is weaker in the 3D case than 2D.

14 MAMA t / (  /Cs) 3D perturbation Comparison with the 2D turbulent case 2D 2D perturbation ３D３D The energy release is weaker in the 3D case than 2D. JzBxJzBx t=110

15 Discussion: 3D vs. 2D Why the energy release became less effective in the 3D case than in the 2D case ? In case of 3D, the distribution of the current density is not uniform with scattering concentrations here and there in the sheet. The filling factor is smaller than the 2D case. As a result, the efficiency of the energy release might be smaller. Current sheets Jz

16 Dependence on diffusivity  1.d-3 1.d-2 1.d-1 t / (  /Cs) The reconnection rate is: i.e., has a Sweet-Parker like dependence on R m.

17 Summary 3D structures in the z -direction evolve in time. A pair of Inflows toward the z -direction play role for this evolution. Slight increase (3%) of energy-release rate above the Sweet-Parker rate is observed. We interpreted it is due to the spatial division of the current sheets by the magnetic islands. Energy-release rate is reduced in 3D random case compared with 2D random case, presumably because of the reduction of the filling factor of the diffusion regions in the additional dimension. The dependence on the magnetic Reynolds number is consistent with that of the Sweet-Parker model.

18 uniform anomalous MAMA t / (  /Cs) VxVy VxVy anomalous resistivity uniform resistivity Case with anomalous resistivity t=140 In case with the anomalous resistivity, Petschek like structures (i.e. pairs of slow- mode shocks) appear, leading to the enhanced reconnection rate.

19 MAMA t / (  /Cs) VxVyVxVy Bz0/By0=0 Guide field (preliminary) t=150t=228 Bz0/By0=0.1 Bz0/By0=0

20 Bz0/By0= 0.1 xy -plane ( z =0) x y  Jz VxVyVz BxByBz

21 Bz0/By0 =0.1 zy- plane ( x =0) z y  Jz VxVyVz BxByBz

22 MAMA t / (  /Cs) VxVyVxVy Bz0/By0=1 Guide field t=110t=180 Bz0/By0=2 Bz0/By0=0.1 Bz0/By0=0

23 MAMA t / (  /Cs) VxVyVxVy random perturbation Comparison with the simple diffusion t=130 Random simple diffusion

24 Tanuma et al. (2001) 2D MHD simulations: Resistivity: uniform (Rm=150) + current-dependent Current-sheet thinning by the secondary tearing Impulsive nature of the inflow induced by the ejections of mag. islands

25 In the context of emerging flux study (Shimizu & Shibata 2006 private commu.)