We investigated the dynamic evolution of a 3-dimensional (3D) flux rope eruption and magnetic reconnection process in a solar flare, by simply extending 2-dimensional (2D) resistive magnetohydrodynamic simulation model of solar flares with low $beta$
plasma to 3D model. We succeeded in reproducing a current sheet and bi-directional reconnection outflows just below the flux rope during the eruption in our 3D simulations. We calculated four cases of a strongly twisted flux rope and a weakly twisted flux rope in 2D and 3D simulations. The time evolution of a weakly twisted flux rope in 3D simulation shows similar behaviors to 2D simulation, while a strongly twisted flux rope in 3D simulation shows clearly different time evolution from 2D simulation except for the initial phase evolution. The ejection speeds of both strongly and weakly twisted flux ropes in 3D simulations are larger than 2D simulations, and the reconnection rates in 3D cases are also larger than 2D cases. This indicates a positive feedback between the ejection speed of a flux rope and the reconnection rate even in the 3D simulation, and we conclude that the plasmoid-induced reconnection model can be applied to 3D. We also found that small scale plasmoids are formed inside a current sheet and make it turbulent. These small scale plasmoid ejections has role in locally increasing reconnection rate intermittently as observed in solar flares, coupled with a global eruption of a flux rope.
The plasmoid-induced-reconnection model explaining solar flares based on bursty reconnection produced by an ejecting plasmoid suggests a possible relation between the ejection velocity of a plasmoid and the rate of magnetic reconnection. In this stud
y, we focus on the quantitative description of this relation. We performed magnetohydrodynamic (MHD) simulations of solar flares by changing the values of resistivity and the plasmoid velocity. The plasmoid velocity has been changed by applying an additional force to the plasmoid to see how the plasmoid velocity affects the reconnection rate. An important result is that the reconnection rate has a positive correlation with the plasmoid velocity, which is consistent with the plasmoid-induced-reconnection model for solar flares. We also discuss an observational result supporting this positive correlation.
