Zonal-flow generation and saturation of electromagnetic ion-scale turbulence in tokamaks

Abstract

Local flux-tube gyrokinetic simulations of ion-scale turbulence in tokamak plasmas at finite plasma beta are conducted to investigate the generation of zonal flows via turbulent stresses. A parameter scan in the safety factor $q$ and electron beta $β_e$ reveals a transition from low- to high-transport states when $β_{\mathrm{eff}} \equiv q^2β_e$ exceeds a certain critical value $C_{\mathrm{nl}}$. While the linear stability limits for kinetic and ideal ballooning modes also scale as $β_e \propto 1/q^2$, they lie above the observed transition, indicating that the effect is not due to linear instabilities but to nonlinear dynamics. At low $β_{\mathrm{eff}}$, Reynolds stress dominates and drives zonal flows. At higher values, Maxwell stress becomes comparable, suppressing zonal-flow formation and leading to divergent transport. This nonlinear-transition boundary is determined for both the Cyclone Base Case and a spherical tokamak (ST40) configuration, suggesting that the relation $β_{\mathrm{eff}} = C_{\mathrm{nl}}$ may have broader applicability, though $C_{\mathrm{nl}}$ appears to be configuration-dependent. For the Cyclone Base Case, the ratio of energy transfer rates into zonal flows due to Maxwell and Reynolds stresses is observed empirically to scale as $β_e$ for $β_e$ below a critical value $β_{e,\mathrm{sb}}$ (scaling breakdown). The value of $β_{e,\mathrm{sb}}$ is found to increase with decreasing aspect ratio, suggesting that the linear scaling remains valid over a wider range of $β_e$ for more compact magnetic equilibria. This low-$β_e$ scaling provides the basis for a practical method to predict the nonlinear-transition threshold with minimal reliance on highly electromagnetic nonlinear simulations.