Applying the nonlinear barotropic vorticity equation to the flow of moving waves, an ordinary differential equation with single degree offreedom of the autonomous dynamic system is derived. Based on the qual-itative and geometric analyses of the differential equation, the geometrictopological structure of the integral curve is investigated in the neighbor-hood of the balance points on the phase-plane. Barotropic instability criterion is established for two balance points respectively. Overall char-acteristics of the motion on the phase-plane and the relation of balancestates to the parameters are analysed and criteria similar to the above-stated are thus found in a more intuitive way.