Relaxed plasma-vacuum states in cylinders

Abstract

The variational principle for relaxed toroidal plasma-vacuum systems with pressure is applied to axially periodic circular cylinders. More precisely, equilibria with cylindrical symmetry are investigated for their potential to be a relaxed state. Such equilibria are characterized by their pinch ratio µ, the jump δ of the pitch angle of the magnetic field across the plasma-vacuum interface, a constant plasma pressure β, and the ratio l of wall radius over interface radius. In the limit of an infinitely long cylinder the necessary and sufficient condition for the equilibrium to be a relaxed state defines one or two intervals of allowed values of the pinch ratio µ, which depend still on the other parameters. These intervals are contained in the interval known from Taylor's theory, but are generally smaller. They are shrinking with increasing plasma pressure β or increasing radius ratio l. In particular, in the field-free limit case β = 1, for l exceeding the critical value l c ≈ 4.983, or for a vanishing δ, these intervals are zero.

Magnetohydrodynamic stability of plasmas with ideal and relaxed regions

“…Hole et al used a variational treatment to construct equilibrium solutions and analyse their stability in cylindrical geometry [7]. Using this analysis, they reproduced the earlier conclusion of Kaiser and Uecker [10], that stable plasmas exist in the case of a single-interface configuration providing that there is a jump in the rotational transform, ι -, at the plasma-vacuum interface. However, their work also raised questions.…”

“…A larger motivation for our study is to remove the singularity in the current density, thereby illuminating some limitations in the stability conclusions of Kaiser and Uecker [ [10]]. We will show that if the interface is resolved as an ideal region of nonzero width, the rotational transform profile, ι -(r), may pass through ideal resonances.…”

“…The factor of 2 arises from the normalisation, equation (22), since the two-interface configuration contains twice the number of interfaces. The expression for the single-interface eigenvalue, l, found by rearranging equation (40) using equation (10), is…”

Stepped pressure profile equilibria in cylindrical plasmas via partial Taylor relaxation

“…In this work, we generalize the analysis of Kaiser and Uecker [7] to an arbitrary number N of Taylor relaxed states, each separated by an ideal MHD barrier. The system is enclosed by a vacuum, and encased in a perfectly conducting wall.…”

“…Later analysis by Spies [6] extended the plasma model to include finite pressure. More recently, Kaiser and Uecker [7] analyzed the finite pressure model in cylindrical geometry.…”