Tandem-mirror trapped-particle modes at arbitrary collisionality

Tandem-mirror trapped-particle modes are studied in a model system consisting of two connected square wells representing the solenoid and the end cells. Collisions are described by a Lorentz operator. A dispersion relation that is valid for arbitrary ν/ω (ω = wave frequency, ν = collision frequency) is derived. Four limits are investigated. When epsilonequivalentνR/sub am//ω <<1+p/sub i//p/sub e/, where R/sub am/ is the mirror ratio separating electrons trapped in the anchor from those passing to the solenoid and p/sub e/ and p/sub i/ are the fractions of passing electrons and ions, collisions destabilize a trapped-particle mode that is stable in the collisionless limit; the growth rate is proportional to epsilon1/sup //2 for epsilon<<1 and epsilon/ln epsilon for 1+p/sub i//p/sub e/ >>epsilon>>1. When epsilon >>1+p/sub i//p/sub e/, the trapped-particle mode becomes a weakly growing drift wave with growth rate proportional to epsilon-1 ln epsilon for ν/ω<<1 and ν-1 for ν/ω>>1; additionally identified are two flute modes, one of which is unstable for some parameters, and a strongly damped high-frequency mode