Limit-cycle oscillations in turbulence–zonal flow dynamics: Observations in toroidal gyrokinetic simulations and sparse regression modeling

We investigate the dynamics of turbulence–zonal-flow interaction in toroidal ion-temperature-gradient (ITG) turbulence using global particle-in-cell gyrokinetic (GK) simulations. Under the conditions considered, we observe long-lasting, regular limit-cycle oscillations (LCOs) for different temperature gradients. Two factors are crucial for obtaining long-lasting and apparent oscillations: the narrow radial width of the turbulent region and the application of aggressive noise control. The former enables zonal flows to grow via modulational instability and interact locally with ITG modes, while the latter suppresses irregularity in the LCOs by reducing numerical particle noise. To assess whether these dynamics conform to the self-regulation framework, we adopt predator–prey-type equations for turbulence, zonal flow, and temperature gradient. The coefficients are identified from GK data using a regression method. The reduced model successfully reproduces the LCOs near the nonlinear threshold, thereby verifying the paradigm of zonal-flow regulation of turbulence in first-principles simulations. As the temperature gradient increases, however, we find the discrepancies between the model and GK simulations, implying modifications (or limitations) of the model equations in the strong-turbulence regime.