eigenfrequencies of the rotating plasma kink modes have been experimentally observed and can be computationally
The non-self-adjoint Frieman-Rotenberg equation for the linear ideal magnetohydrodynamic modes in flow equilibria is numerically verified. Despite this, many existing numerical equilibrium solved in shaped finite-aspect ratio axisymmetric tokamak geomeand stability codes employ full geometric features but still try. A quadratic form is derived from this equation, and, in particular, do not consider plasma rotation. As a result, various linear the self-adjoint force operator with finite toroidal rotation is cast instability estimates used in the internal and external kink into a manifestly self-adjoint form. The toroidal rotational velocities in the subsonic regime are considered. The quadratic form is discre-calculations are approximate.
tized by a mixed finite-element procedure in the radial direction and
The situation with respect to the study of rotating plasma by Fourier modes in the periodic directions. The mode frequency of instabilities is changing, albeit slowly. Several linear rotathe unstable mode is located by root searching, and the final root tional instability calculations have been published recently refinement is obtained by a rapid inverse iteration procedure for in the astrophysical literature, as well as in the tokamak complex roots. The real part of the n ฯญ 1 internal kink mode scales linearly with the plasma rotation, and the imaginary part of the literature. Also, some results for the local stability of the unstable mode is at least an order of magnitude higher in the presrotating plasmas have been derived [3][4]. An analytical ence of high plasma rotation velocities. The kink mode is also found study of the ideal MHD, CGL, and Grad's GCP [5] equilibto be unstable at high rotation velocities, even when the axis safety ria with flows has been performed using a large aspectfactor is above unity. The instability characterized by these features is termed here as the ''centrifugal'' instability. The centrifugal kink ratio approximation. These results pertain mainly to high instability would have finite real parts, as shown by the plasma speed regimes. Also, considerable work has been done on rotation observed in plasma devices such as tokamaks. To explain the Kelvin-Helmholtz (KH) instability for specific equilibthe features of this mode, the plasma rotation should be taken into rium configurations. This instability is intensified at Alfven account. Therein lies the usefulness of the computational analysis velocities. The KH instability of velocity shear layers has presented here.