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 ${\beta }_e$ reveals a transition from low- to high-transport states when ${\beta }_{\mathrm{eff}}\equiv q^2{\beta }_e$ exceeds a certain critical value $C_{\mathrm{nl}}$. While the linear stability limits for kinetic and ideal ballooning modes also scale as ${\beta }_e\propto 1/q^2$, they lie above the observed transition, indicating that the effect is not due to linear instabilities but to nonlinear dynamics. At low ${\beta }_{\mathrm{eff}}$, 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 ${\beta }_{\mathrm{eff}}=C_{\mathrm{nl}}$ 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 ${\beta }_e$ for ${\beta }_e$ below a critical value ${\beta }_{e,\mathrm{sb}}$ (scaling breakdown). The value of ${\beta }_{e,\mathrm{sb}}$ is found to increase with decreasing aspect ratio, suggesting that the linear scaling remains valid over a wider range of ${\beta }_e$ for more compact magnetic equilibria. This low-${\beta }_e$ scaling provides the basis for a practical method to predict the nonlinear-transition threshold with minimal reliance on highly electromagnetic nonlinear simulations.