Nonlinear dynamics of a zonal flow near marginal stability

The predator-prey model of the coupled spectral dynamics of drift-wave-zonal-flow turbulence is extended to the regime near the instability threshold. The problem reduces to a nonlinear integral equation, which is similar to that for the mode-particle system [H. L. Berk et al., Phys. Rev. Lett. 76, 1256 (1996)], but contains two additional terms due to drift-wave self-nonlinearity. The possibility of pulsating solutions of this equation is investigated. The critical parameter for the Hopf bifurcation and the pulsation period are in reasonable agreement with the bursting behavior recently observed in the global gyrokinetic simulations.