Alfvén-ion-cyclotron instability in mirror machines

Abstract

The Alfvén‐ion‐cyclotron instability is studied for finite mirror‐confined plasmas with high beta without field reversal. Variation perpendicular to field lines is modeled by an effective k_⊥. Variation along a representative field line is treated using the Wentzel‐Kramer‐Brillouin approximation in two ways. First, the local dispersion relation is expanded about a wavenumber and frequency corresponding to absolute instability at the machine midplane. This yields a parabolic k_∥(s) and a frequency correction. Second, the local dispersion relation is evaluated exactly as a function of position, and the appropriate phase‐integral condition is used to fix the frequency. This condition is chosen using a generalized WKB formulation which is outlined. The two ways of obtaining the mode frequency agree closely. Stability boundaries are drawn in β_⊥−β_∥ space for two representative finite plasmas. The long thin approximation is used to model finite‐beta well deepening. For ease of computation, the bi‐Maxwellian ion velocity distribution is used. At high β, the stability boundaries are affected by the appearance of an additional root, with a larger parallel wavenumber and a lower frequency.