Stability regions of dissipative trapped-ion instability

Abstract

Conditions for the stability of the dissipative trapped ion mode in axisymmetric toroidal confinement systems are investigated, and the ranges of temperature and magnetic field for which the plasma is expected to be unstable are calculated numerically for several proposed Tokamaks. The collisional damping is studied as a boundary layer problem in the velocity space of the trapped ions. The Fokker-Planck equation is solved by means of a variational form with the ordering ν_iR/r ω 2π/τ_i, where R/r is the aspect ratio, ν_iR/r is the effective collision frequency, and τ_i is the bounce time. The relative damping rate is found to be proportional to (ν_iR/rω) ^1/2 [In (rω/ν_iR)^1/2]^−3/2 rather than ν_iR/rω. This damping term is compared with the electron driving term, which is proportional to rω/Rν_e, and with the Landau damping caused by resonant untrapped ions. The presence of a temperature gradient is found to be destabilizing. The Landau term provides relatively strong damping when dT/dr=0 but changes sign when d ln T/d In n > 2/3. In the latter case it is concluded that ion collisions are insufficient to stabilize the mode under typical operating conditions of the proposed machines. Finally, if d ln T/d In n > 1.75 the ion collisional damping itself changes to growth, and the mode cannot be stabilized.