Weakly nonlinear aspects of a barotropically unstable polar vortex in a forced-dissipative system with a zonally asymmetric surface topography are investigated in order to obtain a deeper understanding of rather periodic variations of the winter circumpolar vortex in the Southern Hemisphere stratosphere that are characterized by the wave-wave interaction between the stationary planetary wave of zonal wavenumber 1 (denoted as Wave 1) and the eastward traveling Wave 2 as studied by Hio and Yoden in 2004. The authors use a spherical barotropic model with a forcing of zonally symmetric jet, dissipation, and sinusoidal surface topography. A parameter sweep experiment is performed by changing the amplitude of the surface topography, which forces the stationary Wave 1, and the width of the prescribed zonally symmetric jet, which controls the barotropic instability, to generate the traveling Wave 2. Several types of solutions from a time-independent solution to a nonperiodic irregular solution are obtained for the combination of these external parameters, but the predominant solution obtained in a wide parameter space is periodic.Details of the wave-wave interactions are described for the transition from a quasiperiodic vacillation to a periodic solution as the increase of the amplitude of topography. Phase relationships are locked at the transition, and variations of zonal-mean zonal flow and topographically forced Wave 1 synchronize with periodic progression of Wave 2 in the periodic solution. A diagnosis with a low-order "empirical mode expansion" of the vorticity equation gives a limited number of dominant nonlinear triad interactions among the zonal-mean, Wave-1 ,and Wave-2 components around the transition point.