When a three-dimensional flow is simulated with vortex methods, the break-down of the vorticity occurs many times, in particular, in the case of the flow field with the velocity discontinuity. The purpose of this paper is to study the mechanism of the break-down of the vorticity in the vortex method. The present paper deals with the flow field in which the ring vortex with core is issued at each time step. This flow is similar to the ring jet, so that the flow changes from laminar flow to turbulent flow. In the present paper, the flow with initial disturbance is simulated with the discrete vortex method in which the PSE (Particle Strength Exchange) method is used for the viscous diffusion. Numerical results show that the flow is almost similar to the free jet and the break-down of the vorticity field occurs when the core radius is small.