On Phase Transition of ITG Turbulence in the Dimits shift
The transition between turbulent and zonal flow dominated states is investigated by varying the ion temperature gradient in nonlinear gyrokinetic simulations. Independent gradient scans reveal three distinct regimes: a turbulent regime at high gradients, a zonal flow dominated regime with strongly reduced heat transport at low gradients, and an intermediate regime characterized by intermittent switching between these two states. These different regimes can be classified using an order parameter, defined as the fraction of the zonal to total free energy in the system. To assess the memory effects, the temperature gradient is first lowered gradually from high values that result in fully developed turbulence, to lower values in the Dimits shift region that form strong zonal flows, and then is slowly increased back. Once the zonal flows form, and efficiently suppress turbulence, they can persist at higher gradients, leading to an asymmetric response implying a hysteresis loop. It is observed that, in the zonal flow dominated state, the free energy is mostly condensated in the largest radial scale, with a steep slope of the $k_x$ spectrum, while in the turbulent state, it exhibits a wider spectrum with two distinct slopes.