The toroidal 1/1 internal kink mode

The m0/n=1/1 internal mode in toroidal plasma is shown to have an intrinsic dependence on a (global) epsilon beta-poloidal (εβp) parameter independent of the inverse aspect ratio ε, when the degree of compressibility is determined self-consistently. The εβp parameter introduces properties very different from those predicted from strict incompressibility, where the mode depends strongly on aspect ratio. An exact solution and scaling properties are presented numerically. A companion paper provides a supporting analysis based on small parameter solutions of the toroidal perturbation equations [L. E. Sugiyama, submitted to Phys. Plasmas (1998)]. The maximal ordering of the mode growth rate is γ=O(ε1/2) or (δq)1/2 at low beta, βp∼1. The compressible mode has two distinct regimes in εβp and ε=a/R. At moderate εβp approx-lt ε<1, the growth rate varies linearly with εβp-εβpc and is almost independent of ε when δq=1-q within q≤1 is small, δq∼ε. At very small ε<εβp, corresponding to large βp>1 and also to δq>ε, the growth rate varies as γ∝βp2. For the 1/1 mode, the small value of δq=1-q within the region q<1, satisfied by almost all tokamaks with q0<1, determines a number of basic properties of the toroidal mode. Small δq makes the m=1 component of the magnetic flux order δq, causing the m=2 component of the toroidal current density to be the same order of magnitude as m=1 on the q=1 surface and thereby giving the 1/1 mode a fundamental inboard/outboard asymmetry emdash on the outboard side of the torus the separatrix current sheet vanishes. The asymmetry strengthens with resistivity and in the nonlinear phase. copyright 1999 American Institute of Physics