Chaotic mixing and transport barriers around the wintertime stratospheric polar vortex are investigated using a nondivergent, barotropic model on a sphere. Barotropically unstable jets are given as the forcing in the forced-dissipative system. A flow with quasi-periodic time dependence is obtained after the initial transition. Several Lagrangian methods are used to examine the mixing processes in the flow. And an aperiodic flow of similar pattern is also investigated by comparing with the quasi-periodic flow.