Simulation Study of Three-Dimensional and Nonlinear Dynamics of Flux Rope in the Solar Corona

1. Simulation Study of Three-Dimensional and Nonlinear Dynamics of Flux Rope in the Solar Corona INOUE Satoshi(Hiroshima University/Earth Simulator Center), KUSANO Kanya(Earth Simulator Center) E-mail : inosato@hiroshima-u.ac.jp Abstract We numerically investigated the three-dimensional(3D) stability and the nonlinear dynamics of flux rope embedded in magnetic arcade. As a results, we found that the flux rope is unstable to the kink mode instability, as the system approach to the loss- of-equilibrium state. The 3D simulation shows that when the flux rope is long enough, it can escape from the arcade with almost constant speed after the accelerated launching due to the kink instability. This constant ascending is driven by magnetic reconnection on the current sheet, which is formed above the magnetic neutral line as a consequence of the instability. However, when the flux rope is short enough , the current sheet can not be maintained, so that the ascending is failed at some height. These results imply that the nonlinear effect mainly influenced by magnetic reconnection may determine whether the flux rope is ejected or not. 2Dsimulation Lx=0.6 Priest , Forbes(1990) proposed Loss-of-equilibrium model to explain filament eruption phenomena. This model is that the changing boundary condition causes filament eruption due to brake the equilibrium condition. Because filament is stability in two-dimensional space, it was able to arrive at loss-of-equilibrium point, so this model was not considered three-dimensional insta -bility. Therefore, we investigated the stability around loss-of- equilibrium point in three-dimensional space. Lx=1.5 Loss of Equilibrium point Fig3(a) Fig3(b) the flux rope, and the red surface is an isosurface of the current intensity |J|. We can see that firstly the center of the flux rope is lifted up due to the growth of kink mode, and secondary the field lines evolve nonlinearly. Fig(b) is the time profile of height and velocity of flux tube in the both cases. We showed that the ascending of short flux tube(Lx=0.6) is failed at certain height, whereas the long flux tube continuously ascends after the growth of the kink mode instability. In the final state of Fig(a), the current sheet is sustained in long flux tube, although it disappears in short one. It may be important the current sheet formation in the late phase that the filament continuously ascends. Fig1(a) Fig2(a) Fig1(a) is equilibrium line for Flux Rope embedded in coronal loop. Fig2(b) is the flux rope dynamics in two-dimensional space as M=1.5. Linear StabilityAnalysis The equilibriums from Priest, Forbes become unstable to kink mode instability. The equilibriums near the loss of equilibrium are more unstable than other equilibriums. Therefore it is pos- sible that filament eruption is occurred due to instability before reach the loss-of-equilibrium point. Discussion We carried out a hypothetical simulation to confirm the importanceof current sheet formation, i.e, magnetic reconnection on it. In the case of failed eruption(Lx=0.6, M=2.0875), we imposed an external force on the center of the flux rope from t=22.49 to 27.63. By this force, the stretching of the field line corresponding to feet of the arcade may further help the formation of current sheet, so that reconnection could be self-sustained. As a result, flux tube is enlarged and, by external force, forms current sheet in the lower part. Furthermore, the flux tube ascending because reconnection is self-sustained in the lower part after finishing external force without failing. Fig4(a) is represented the 3D flux rope structure acted external force, Fig4(b) is the current sheet distributions, Fig4(c) is time profile of flux rope height, red line is no external force, and blue line is acted external force. γ Loss of Equilibrium Point Fig2(a) Fig2(b) Fig2(c) Fig2(a) is equilibrium line for Flux Rope embedded in coronal loop. Fig2(b) is represented the linear growth rate vs. h/d. Fig2(c) is eigen-functions. Parameters, m : Dipole moment, d : Dipole depth from solar surface h : Filament height from solar surface I0 : Current in Flux Tube 3Dsimulation Fig4(a) Fig4(c) We show the results of the 3D MHD simulation. The different equilibria for M=2 and M=2.08 were used as the initial conditions, respectively. The Two simulations were carried out respectively for Lx=0.6 and 1.5 , in which the system length Lx=0.6 corresponds to the wave length giving the maximum growth rate. The eigen-functions obtained from the linear growth calculation are added to the equilibrium as small and initial perturbation. We represented the flux tube dynamics in the three-dimen -sional space. The upper panel is case for Lx = 0.6, the lower panel is for Lx = 1.5 at M=2. In these figures, each string represents magnetic field line, green surface is an isosurface of the strength Bx, which is along Summary Fig4(b) (1) It is important the formation of current sheet in the lower part that flux rope continuously ascends without failing at certain height. (2) There could be a critical height beyond which filament must exceed to escape to the infinity.