This is the first paper about fragmentation and mass outflow in molecular clouds by using three-dimensional magnetohydrodynamical (MHD) nested-grid simulations. The binary star formation process is studied, paying particular attention to the fragmentation of a rotating magnetized molecular cloud. We assume an isothermal rotating and magnetized cylindrical cloud in hydrostatic balance. Non-axisymmetric as well as axisymmetric perturbations are added to the initial state and the subsequent evolutions are studied. The evolution is characterized by three parameters: the amplitude of the non-axisymmetric perturbations, the rotation speed and the magnetic field strength. As a result, it is found that non-axisymmetry hardly evolves in the early phase, but begins to grow after the gas contracts and forms a thin disc. Disc formation is strongly promoted by the rotation speed and the magnetic field strength. There are two types of fragmentation: that from a ring and that from a bar. Thin adiabatic cores fragment if their thickness is less than 1/4 of the radius. For the fragments to survive, they should be formed in a heavily elongated barred core or a flat round disc. In the models showing fragmentation, outflows from respective fragments are found as well as those driven by the rotating bar or the disc.
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