Collisional effects on trapped particle modes in tandem mirrors

The effects of collisions on trapped particle modes in tandem mirrors are analyzed. Two regimes are considered, a low collisionality regime ω∼ωi*>νe and a high collisionality regime νi<ω<νe. The magnetic geometry of the equilibrium is left arbitrary and a pitch angle scattering operator is used to model the effects of collisions. For ω>νe electron collisions are found to destabilize an otherwise stable negative energy wave. Because of a boundary layer phenomenon, the growth rate scales as (νe‖ω*‖)1/2. For νi<ω<νe two modes are obtained: (a) a flute mode whose stability is determined by the flux tube integral of the beta-weighted curvature drive, and (b) a dissipative trapped ion mode driven unstable by the difference in collisionality between electrons and ions. To first order in (ω/νe), (νi/ω), (ωd/ω), and k2⊥ ρ2i the stability and eigenfrequency of the flute mode is independent of νi while the dissipative trapped ion mode is damped by increasing ion collisionality.