Rotational stability of a long, high-beta, field-reversed column

Rotationally driven modes are observed to be important in determining FRC stability and confinement. As a first model, we consider a long, rotating column. The $B$ is in the axial direction and the pressure gradient and centripetal acceleration are balanced by a strong radial gradient of $B$, which may pass through zero and reverse on axis, as in an FRC. The non-reversed version of this problem was considered earlier\footnote{J. P. Freidberg and L. D. Pearlstein, \textit{Phys. Fluids} \textbf{21}, 1207 (1978).} using an approximate solution of the ion kinetic equation. We simplify the present analysis by assuming incompressible motion of the plasma and including the gyro-viscous (GV) stress to construct an eigenvalue problem. This leads to a formally symmetric o.d.e. which contains the eigenfrequency in a complicated manner. Finite axial wavenlength is included to leading order. Regular solutions, which satisfy the outer boundary condition at a conducting wall, give normal modes, and are found by the shooting method. We show that this procedure reproduces the results of Ref. 1 and generalizes to more complicated equilibria having rotational shear and field reversal. In the case of field reversal, it is necessary to employ a GV form that is appropriate for small $B$. For this we use either the collisional form\footnote{N. Iwasawa, A. Ishida, L. C. Steinhauer, \textit{Phys. Fluids} \textbf{B8}, 1240 (2001).} or a new low-collisionality form which accounts for the de-magnetization of the ion orbits at small $B$. Results for FRC's with and without strong shear are presented.