W7-X EM turbulence

Motivation

• Status: most of stellarator turbulence studies are in a flux tube
• Problem: flux-tube modeling is more questionable in stellarators than it is in tokamaks

1. Flux tubes are not equivalent (in contrast to tokamaks): stellarators are intrinsically poloidally-global.
2. Radial bounce-averaged drifts do no vanish (in contrast to tokamaks): stellarators are intrinsically radially-global.
3. No good way to incorporate ambient $E_r$ in a flux tube.

• Status: only a handfull of global turbulence simulations (EUTERPE) is available for stellarators; mostly electrostatic
• Challenge: global electromagnetic turbulence in stellarator (W7-X) geometry

Geometry and profiles

Larger temperature implies larger $\rho_*$ (relevant values $T_e \sim 5 [keV]$)
Larger $\rho_*$ need less Fourier harmonics to resolve the same $k_{\perp}\rho_i \sim 1$

Time evolution ($\beta = 1.12\%$)

Heat and particle fluxes in gyro-Bohm units; profile evolution is not strong

Initialization with noise; instability starts to develop

Linear instability; ZF is excited; linear mode is weaken locally by ZF

The turbulence is pushed by ZF into regions of small gradient; saturation

Beta scan: electrostatic potential

on the left: zonal flow included. On the right: zonal flow excluded

• $\beta_{EU} = \frac{\mu_0 n_0 T_e}{B^2} = 0.00035$; $\beta = \frac{2 \mu_0 n_0 (T_i + T_e)}{B^2} = 0.14\%$

• $\beta_{EU} = \frac{\mu_0 n_0 T_e}{B^2} = 0.0007$; $\beta = \frac{2 \mu_0 n_0 (T_i + T_e)}{B^2} = 0.28\%$

• $\beta_{EU} = \frac{\mu_0 n_0 T_e}{B^2} = 0.0028$; $\beta = \frac{2 \mu_0 n_0 (T_i + T_e)}{B^2} = 1.12\%$

• $\beta_{EU} = \frac{\mu_0 n_0 T_e}{B^2} = 0.0042$; $\beta = \frac{2 \mu_0 n_0 (T_i + T_e)}{B^2} = 1.68\%$

Summary: zonal flow excited at larger $\beta$ is weaker

Evolution of electrostatic potential ($\beta=1.12\%$) plotted as a function of radius and toroidal mode number; including zonal flow and temperature evolution

Evolution of electrostatic potential ($\beta=1.12\%$) plotted as a function of radius and toroidal mode number; excluding zonal flow
Evolution of electrostatic potential ($\beta=1.12\%$) plotted in a poloidal cross-section; all modes included