On the influence of the Hall effect on the spectrum of the ideal magnetohydrodynamic cylindrical pinch

Abstract

The influence of the Hall effect on the spectrum of ideal magnetohydrodynamics (MHD) is investigated for axisymmetric perturbations of an infinitely extended circular cylindricalZpinch with a diffuse current density. The pinch is bounded by an ideal conducting wall. The normal mode analysis expi(kz—ωt) of the linearized perturbations gives a non-selfadjoint eigenvalue problem. The complex eigenvaluesωare considered which depend on the equilibrium function, the axial mode numberk, and the dimensionless Hall effect parameterαwhich is inversely proportional to the square root of the mass density. A stabilizing influence of the Hall effect is shown by a sufficient stability criterion for equilibria without reversed magnetic fields which is more optimistic than the corresponding criterion of ideal MHD. If the Hall effect is retained in Ohm's law then there are, in contrast to ideal MHD, new simple points of accumulation of discrete eigenvalues when the radial mode numbernof the eigenfunctions tends to infinity. The accumulation points are real, they are given by the extrema of the equilibrium function and they depend on the axial mode numberk. The direction of accumulation is always along the real axis of the complex plane for equilibria without magnetic field reversal. From this it follows that accumulation points are no longer connected with stability criteria as is the case in ideal MHD. If the Hall effect parameterαtends to zero then the accumulation points coincide with the twofoldk−independent accumulation point of ideal MHD. If the Hall effect is not neglected then the pure growth rates of ideal MHD are replaced by complex frequencies with smaller growth rates; and ifαis greater than some critical value then the perturbation is stable. If thek0mode is stable then all modes withk≥k0are stable. For some continuous profiles of the current density the eigenvalue problem is solved numerically. The dependence of eigenvalues on the radial mode number and the dependence of the radial electron velocity on the Hall effect is shown. For equilibria with a reversal of the magnetic field, the unstable behaviour predicted by MHD theory cannot be excluded by consideration of the Hall effect.