Stability of long field-reversed configurations

Abstract

The stability of very elongated field-reversed configurations is solved by an expansion in the small parameter ε (inverse elongation). It is first shown that all possible unstable modes have small growth rates (γ ∼ε). The internal tilt mode is considered in detail. An explicit form for δW in leading order is derived, and leads to a quadratic form including Hall terms. A sufficient condition for stability is obtained by minimizing δW, leading to a field-line ordinary differential equation. Sufficient stability conditions are obtained from this formulation, and indicate stability for S*/E<2 (where S* is the ratio of separatrix radius to collisionless ion skin depth and E the elongation of the separatrix), if the local criterion is used. It is argued that a volume-averaged condition is more appropriate when finite ion orbit effects are included. This leads to S*/E<3.5–4 for stability, independent of separatrix shape or xs (separatrix radius to wall radius at the midplane). This condition for stability compares favorably with experimental observations.