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

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