Phase synchronization versus modulational instability for zonal flow generation and pattern formation

From global gyrokinetic simulations of toroidal ion temperature gradient-driven (ITG) turbulence, we identify two distinguished regimes where zonal flow generation and its radial pattern formation is governed by either phases or amplitudes of the turbulence spectrum. When the unstable region is wider than the correlation length of ITG modes, the zonal flow structure changes in a turbulence time scale. We newly uncover that the radial phases of ITG modes determine the evolution of the zonal flow structure. Synchronization of mode phases induced by the global zonal flow drives a fine-scale zonal flow pattern. With a narrow unstable region comparable to the mode correlation length, the phase effect almost vanishes. In this regime, we recover that the modulational instability promptly amplifies a coherent zonal flow, leading to a lower turbulence saturation level. This finding explains the reduction of turbulent transport at a narrow width of the strong gradient region, which has been attributed to the system size effects in ion-scale gyrokinetic turbulence.