Effect of a noncircular plasma cross section on the ideal (1, 1) mode in low-shear tokamaks

The effect of a noncircular shape of the plasma cross section on the ideal magnetohydrodynamic (MHD) stability of the (m,n)=(1,1) internal kink, or quasi-interchange (QI), mode in toroidal plasmas, where the safety factor q is close to unity in a wide area in the plasma core, is analyzed. The analysis is based on previously derived perturbation expansions of the ideal MHD equations for the (1, 1) mode, in the inverse aspect ratio, and in the ellipticity and the triangularity of the plasma cross section. For equilibria having a parabolic pressure profile and a constant central q=q0, with 0<q0-1<<1, a simple analytical formulation of the stability problem is obtained, and an expression for the eigenfunction in terms of a Bessel function is found. Furthermore, noncircularity of the cross section is found to induce a whole spectrum of (1, 1) ideal modes with eigenvalues approaching the continuum (ω/ωA)2=(q0-1)2/3 as the number of nodes in the radial direction goes to infinity. The QI mode is the most unstable mode in this class. If the pressure is low and/or the radius of the low-shear region is small, noncircularity has a small and, usually, destabilizing effect on the QI mode, whereas in plasmas with high pressure and a large radius of the low-shear region, the combined effects of vertical elongation and positive triangularity lead to a substantial improvement of the stability of this mode compared with the situation in plasmas with circular cross section