Stability of the Resistive Sheet Pinch

Necessary and sufficient conditions for exponential stability of equilibria are obtained for the resistive, viscous, incompressible magnetohydrodynamic sheet pinch in a gravitational field. Insulating as well as conducting boundaries are considered. The unperturbed magnetic field B0 (horizontal), resistivity η0, viscosity ν0, and mass density ρ0 are permitted to be arbitrary (neither necessarily constant nor small) functions of the vertical coordinate z consistent with the equilibrium equations. In particular, it is shown that: (1) previous results claiming the existence of an exponentially growing “tearing” mode for a constant η0 equilibrium with zero gravity are incorrect, (2) a nonconstant η0 equilibrium with insulating boundaries cannot be gravitationally stabilized, and (3) a nonconstant η0 equilibrium with g(dρ0/dz) ≡ 0 and B0 nonunidirectional is exponentially unstable to all perturbations (i.e., waves of arbitrary wavelength) with certain directions of propagation.