Loren Steinhauer, 11/23/98

Many analytical and numerical treatments have addressed the tilting mode. Early ideal-MHD theories (before 1985) for non-flowing FRCs consistently predicted instability. Later studies hinted that equilibria with more blunt separatrix shape and hollow current profile may be more stable.¹ However, the most complete stability analysis has just been completed: there the actual eigenvalues and eigenvector fields were found for both tilting and its related ballooning modes; all were found to be unstable in ideal-MHD theory for a wide range of equilibria.² The most successful tilting theories have included finite Larmor radius (FLR) effects, using either kinetic ions³ or a gyroviscous fluid.⁴ The latter led to the prediction of a marginal stability scaling law consistent with observations.

The FLR stability explanation, however, seems inadequate to explain several observations of robust stability: (1) observed global modes producing a restructuring rather than a disruption.⁵ (2) quiescent FRCs exhibit profile consistency.⁶ and (3) the relaxation of a spheromak (formed by merging) to an FRC, in which the toroidal field (and magnetic helicity) decays.⁷ These may be telltales of a rapid relaxation leading to an FRC stabilized by a combination of sheared flow and finite ion orbit effects.

Flow, an important stabilizing influence,⁸ has largely been overlooked in stability analyses. Recent analyses of the effect of flow on stability has shown that the kink mode in z-pinches can be stabilized with a sufficiently sheared axial flow.⁹ Sheared flow may also stabilize the sausage mode. Flow shear may provide stabilization by behaving like the magnetic shear arising from the axial field in a z-pinch, i.e. the conventional (Kruskal-Shafranov) method of stabilization. These results have implications not only for z-pinches, but for related configurations as well, including tokamaks and FRCs. Sheared flow may cause both improved global stability and reduced transport.

The possibility that the FRC may be a robust minimum energy state has been proposed.¹⁰ The modern relaxation theory is based on a two-fluid model; it generalizes the familiar one-fluid theory which has invariant magnetic helicity. The two-fluid theory is based on the invariance of both the ion and electron helicities: these generalize the magnetic helicity include the effect of mechanical momenta. The ruggedness of these "self helicities" is supported by the arguments of inverse cascade, selective decay, and stability to tearing modes.¹¹ Minimum energy equilibria in simplified geometries display qualitative features of, depending on the parameters, laboratory FRCs and reversed-shear tokamaks. The essential ingredient in these minimum energy states is sheared flow. The finite-beta theory of minimum energy states offers the basis for a new stability paradigm.

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2. N. Iwasawa and A. Ishida, in US-Japan Workshop on High-Beta Fusion Plasmas, Toki, 14-15 December 1998; to be submitted for publication in Phys. Plasmas.

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