Approximate equations for plasmas in mirror machines

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Abstract
By expanding in terms of the parameter ϵ=ion Larmor radius/plasma radius (ϵ ≪ 1), sets of approximate equations are obtained from Vlasov's equation for the low-β inhomogeneous plasma equilibria which occur in the inhomogeneous magnetic fields of the mirror machines. The different sets correspond to different physical regimes which require different ordering of the parameters. Thus at very low densities (b2 = 4πnMc2/B2 ≪ 1), it is essential to order various terms in Vlasov's equation such that the precessional drift velocity is of the same order as the frequency of the motion. This implies ω≈ϵ4Ωi for a “long-thin” machine (r∼ϵ2 ρ), and ω≈ϵ2 Ωi for “short-fat” machine (r∼ρ). In either case the “closed” equation for the phase-averaged zero-order distribution function is found to be identical to the Northrop-Teller “Liouville equation” for the guiding centre with the μ∇B and the curvature drifts appearing in the same order as the frequency of the motion. For a high density plasma (b2 = 4πnMc2/B2 ≫ 1) the terms are ordered such that ω≈ϵ2Ωi for a “long-thin” machine (the finite Larmor radius case). The set of approximate equations for this case and the resulting eigenvalue equation are found to be completely analogous to those for the cylindrical geometry as found by Rosenbluth and Simon. In particular, for the “special case” of Rosenbluth and Simon the m = 1 mode is found to be neutral stable, in the absence of gravity as in the hydromagnetic limit.

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