Stabilisation of electrostatic electron-temperature-gradient instability in stellarators

We use the gyrokinetic code \texttt{stella} [Barnes, 2019] to study electron-temperature-gradient (ETG) instability [Dorland, 2000] in a W7-X discharge (20013) [Gonzalez, 2022] and a LHD discharge (113208). Linear electrostatic stability analysis reveals that microinstabilities, including ETG instability, in LHD are stabilized at k_yρe << 1, while W7-X is strongly unstable to ETG modes from k_yρ_i ~1 to k_yρ_e >> 1. Here, k_y is the binormal wavenumber, and ρ_e and ρ_i are the electron and ion gyroradii, respectively. We show that the stabilization of ETG instability in LHD is due to desirable properties of the magnetic geometry: specifically, in LHD (but not in W7-X) the electron magnetic drift frequency is fast compared with the ExB drift frequency, which results in weak toroidal ETG instability [Parisi, 2020]. Because stellarators typically have a large aspect ratio, the ratio of the major radius to the temperature gradient length scale, R/L_Te, is large. This large value of R/L_Te results in linear ETG modes with perpendicular wavenumbers that are much larger than their binormal wavenumber, and are challenging to resolve numerically. Such ETG modes are ubiquitous in the W7-X discharge we study. The relative stability of LHD to electron-gyroradius-scale instability suggests a path to stabilizing turbulence in stellarators with magnetic geometry.