  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.
