Nonlinear damping of zonal flows

The modulatonal instability theory for the generation of large-scale (zonal) modes by drift modes has been extended to the second order including the effects of finite amplitude zonal flows, ϕ_q. The nonlinear (second-order) sidebands are included in the perturbative expansion to derive the nonlinear equation for the evolution of ϕ_q. It is shown that effects of finite ϕ_q reduce the growth rate of zonal flow with a possibility of oscillatory regimes at a later stage.