Linear Instability Analysis for Toroidal Plasma Flow Equilibria

Abstract

The non-self-adjoint Frieman–Rotenberg equation for the linear ideal magnetohydrodynamic modes in flow equilibria is numerically solved in shaped finite-aspect ratio axisymmetric tokamak geometry. A quadratic form is derived from this equation, and, in particular, the self-adjoint force operator with finite toroidal rotation is cast into a manifestly self-adjoint form. The toroidal rotational velocities in the subsonic regime are considered. The quadratic form is discretized by a mixed finite-element procedure in the radial direction and by Fourier modes in the periodic directions. The mode frequency of the unstable mode is located by root searching, and the final root refinement is obtained by a rapid inverse iteration procedure for complex roots. The real part of then= 1 internal kink mode scales linearly with the plasma rotation, and the imaginary part of the unstable mode is at least an order of magnitude higher in the presence of high plasma rotation velocities. The kink mode is also found to be unstable at high rotation velocities, even when the axis safety factor is above unity. The instability characterized by these features is termed here as the “centrifugal” instability. The centrifugal kink instability would have finite real parts, as shown by the plasma rotation observed in plasma devices such as tokamaks. To explain the features of this mode, the plasma rotation should be taken into account. Therein lies the usefulness of the computational analysis presented here.