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, ϕq. The nonlinear (second-order) sidebands are included in the perturbative expansion to derive the nonlinear equation for the evolution of ϕq. It is shown that effects of finite ϕq reduce the growth rate of zonal flow with a possibility of oscillatory regimes at a later stage.