Available energy of trapped electrons in Miller tokamak equilibria

Abstract

Available energy (Æ), which quantifies the maximum amount of thermal energy that may be liberated and converted into instabilities and turbulence, has shown to be a useful metric for predicting saturated energy fluxes in trapped-electron-mode-driven turbulence. Here, we calculate and investigate the Æ in the analytical tokamak equilibria introduced by Milleret al.(Phys. Plasmas, vol. 5, issue, 4, 1998, pp. 973–978). The Æ of trapped electrons reproduces various trends also observed in experiments; negative shear, increasing Shafranov shift, vertical elongation and negative triangularity can all be stabilising, as indicated by a reduction in Æ, although it is strongly dependent on the chosen equilibrium. Comparing Æ with saturated energy flux estimates from the TGLF (trapped gyro-Landau fluid) model, we find fairly good correspondence, showcasing that Æ can be useful to predict trends. We go on to investigate Æ and find that negative triangularity is especially beneficial in vertically elongated configurations with positive shear or low gradients. Furthermore, we extract a gradient-threshold-like quantity from Æ and find that it behaves similarly to gyrokinetic gradient thresholds: it tends to increase linearly with magnetic shear, and negative triangularity leads to an especially high threshold. We next optimise the device geometry for minimal Æ and find that the optimum is strongly dependent on equilibrium parameters, for example, magnetic shear or pressure gradient. Investigating the competing effects of increasing the density gradient, the pressure gradient, and decreasing the shear, we find regimes that have steep gradients yet low Æ, and that such a regime is inaccessible in negative-triangularity tokamaks.

Reducing transport via extreme flux-surface triangularity

“…However, this analysis does not take into account possible changes in turbulent correlations lengths that can strongly affect the efficacy of E A . It is noted here that correlation lengths, as calculated from nonlinear simulations, increase substantially from strongly negative to strongly PT, and that a nonlinear correction to E A scales as the squared correlation length [43], implying that the trend in E A can be reversed and matched qualitatively with the growth rates. However, such an analysis is beyond the scope of the present effort, and the results presented here primarily highlight the fact that an available-energy calculation not accounting for nonlinear changes can fail to predict linear growth rates.…”

Loss-cone stabilization in rotating mirrors: thresholds and thermodynamics

“…Recent evidence suggests that adding suitable adiabatic invariants to the Gardner free energy leads to a metric that can predict the saturation levels of some types of turbulence (Mackenbach, Proll & Helander 2022; Mackenbach et al. 2023 a , b ).…”