Trapped electron mode stability in regions of steep gradients

1 Introduction The trapped electron mode (TEM) is generally unstable in tokamaks for short poloidal wavelengths, k θ ρ i ~ 1, where k θ is the poloidal wavenumber and ρ i is the ion Larmor radius. Trapped electrons exist at lower collisionalities, ν* = ν eff /ω be < 1, where ν eff = ν e /ε with ν e the electron collision frequency and ε = r/R is the inverse aspect ratio, where r and R are the minor and major radii of the tokamak, respectively; ω be = ε 1/2 v the /Rq with v thj the thermal speed of species j and q the safety factor, is the bounce frequency of trapped electrons. However the nature of the TEM varies depending on the collisionality. In the dissipative regime, ν eff > ω, where ω is the mode frequency, there is a robust instability (DTEM) with growth rate γ/ω *e ~ ε 1/2 ω *e /ν eff , where ω *e = k θ ρ i v thi τ/L n , with L n the density profile scalelength, is the electron diamagnetic frequency and τ = T e /T i. On the other hand, in the absolutely collisionless limit there is an instability, driven by a drift Landau resonance with the magnetic curvature and grad-B drifts of trapped electrons, with γ/ω *e ~ ε 1/2 (R/L n) 5/2 exp(-R/L n). In a steep density profile, the collisionless γ is exponentially small and the question is: what dissipative mode exists in the limit ν eff < ω? Using a simple Krook collision operator there is an unstable mode with γ/ω *e ~ ν eff /ω due to finite Larmor radius effects (i.e. finite b = (k θ ρ i) 2), provided η e = L ne /L Te with L Te the electron temperature scalelength, is small enough, < O(b) – i.e. instability occurs for longer wavelengths. However this theory predicts a discontinuity in pitch angle, λ = (v ⊥ /v)