Spacecraft observations suggest that flux transfer events and interplanetary magnetic clouds may be associated with magnetic flux ropes which are magnetic flux tubes containing helical magnetic field lines. In the magnetic flux ropes, the azimuthal magnetic field (B&thgr;) is superposed on the axial field (Bz). In this paper the time evolution of a localized magnetic flux rope is studied. A two-dimensional compressible magnetohydrodynamic simulation code with a cylindrical symmetry is developed to study the wave modes associated with the evolution of flux ropes. It is found that in the initial phase both the fast magnetosonic wave and the Alfv¿n wave are developed in the flux rope. After this initial phase, the Alfv¿n wave becomes the dominant wave mode for the evolution of the magnetic flux rope and the radial expansion velocity of the flux rope is found to be negligible. Numerical results further show that even for a large initial azimuthal component of the magnetic field (B&thgr;≂1--4 Bz) the propagation velocity along the axial direction of the flux rope remains to be the Alfv¿n velocity. Diagnoses show that after the initial phase the transverse kinetic energy equals the transverse magnetic energy, which is characteristic of the Alfv¿n mode. It is also found that the localized magnetic flux rope tends to evolve into two separate magnetic ropes propagating in opposite directions. The simulation results are used to study the evolution of magnetic flux ropes associated with flux transfer events observed at the Earth's dayside magnetopause and magnetic clouds in the interplanetary space. ¿1991 American Geophysical Union