Alfvén-ion-cyclotron instability in mirror machines

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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.