Low Beta Rigid Mode Stability Criterion for an Arbitrary Larmor Radius Plasma

Abstract

The low beta flute axisymmetric dispersion relation for rigid displacement perturbation of plasma equilibria with arbitrary Larmor radius particles and field line radius of curvature large compared to the plasma radius is derived. The equilibrium particle orbits are characterized by two constants of motion, energy and angular momentum, and a third adiabatic invariant derived from the rapid radial motion. The Vlasov equation is integrated, assuming that the mode frequency, axial “bounce” frequency, and particle drift frequency are small compared to the cyclotron frequency, and it is demonstrated that the plasma response to a rigid perturbation has a universal character independent of Larmor radius. As a result the interchange instability is the same as that predicted from conventional MHD theory. However, a new prediction, more optimistic than earlier work, is found for the low density threshold of systems like Migma, which are disc-shaped, that is, the axial extent Δz is less than the radial extent r0. The stability criterion for negative field line curvature χ where ωpi is the mean ion plasma frequency, Ωi, the ion cyclotron frequency, δh the hot particle to total ion particle ratio, χ/r is the ratio o f the field line curvature and the midplane radius, which in our model is treated as a constant over the entire plasma, χh is the Larmor radius of the energetic species, Z is approximately given by an interpolation formula which goes over to the correct limits if either . For Δz/r0 ≪ 1 the stability criterion is determined by the total particle number. Whereas the older theory (Δz/r0 ≫ 1) predicted instability at about the densities achieved in actual Migma experiments, the present theory (Δz/r0 ≪ 1) indicates that the experimental results are for plasmas with particle number below the interchange threshold.