Low-Frequency Stability of Axisymmetric Toruses

The low-frequency stability of low-β collisionless plasma in axisymmetric toroidal configurations such as multipoles with and without applied toroidal fields is studied in some detail. Defining the ''bounce'' frequency ωb = (∫ dℓ/v||)-1 modes are investigated with ω> ωbe3, ωbi<ω<ωbe3 and ω<ωbi. Three classes are found: (1) flute modes, (2) ballooning and drift modes, and (3) trapped particle modes. Stability of the flutes requires, without a toroidal field, favourable ∫dℓ/B, and with a toroidal field a new generalization of the V'' criterion. Even with these criteria satisfied residual flute instabilities occur depending on resonant particle effects. Two modes are found: an electron flute with ω/ω⋇e > 0 destabilized if ωbi < ω < ωbe by electron resonances provided Ti /Te 0 destabilized if ω ≲ ωbi by ion resonances requiring finite Debye length and k⊥2λDi2 > (1 + Te/Ti)r/R, though having an exponentially small growth rate if (1 + Te / Ti ) 4πpc2/B2 > rR/L2. Here ω⋇i,e are the diamagnetic drift frequencies, r the density scale length, R the average radius of curvature, and L the connection length. For systems unstable to the generalized V'' criterion, the residual ion instability occurs at all densities and is insensitive to shear. Slight anisotropy or temperature gradient also affects the stability of the residual ion flute, but quasi-linear calculations indicate that the non-linear formation of a sufficient stabilizing anisotropy is slow. The drift ballooning mode local to a bad curvature region and the drift universal mode are unstable for ωbi < ω < ωbe provided Ti/Te < L2/rR and Ti, /Te < L/r, respectively. An eikonal procedure provides the effect of shear on the ballooning mode including geodesic curvature. Trapped particle models have ω < ωbi. Curvature-driven modes, insensitive to shear but stable in maximum-J configurations, are also stabilized if r/R is not too small, and criteria are given but can be very restrictive. For large k⊥ a new overstable mode relatively insensitive to shear, independent of curvature, and not involving resonant particles is found requiring terms of order (ω/ωbi)2 and k⊥2ai2, where ai is the ion Larmor radius: instability occurs if k⊥2ai2 > (1 + Ti/Te)r/L. (author)