Stability of force-free magnetic fields versus magnetic pitch

Abstract

Starting from the one-dimensional energy integral and related stability theorems given by Newcomb [Ann. Phys (NY)10, 232 (1960)] for a linear pinch system, this paper analyses the stability of one-dimensional force-free magnetic fields in cylindrical coordinates (r, θ, z). It is found that the stability of the force-free field is closely related to the radial distribution of the pitch of the field lines: h(r) = 2πrBz/Bθ. The following three types of force-free fields are proved to be unstable: (i) force-free fields with a uniform pitch; (ii) force-free fields with a pitch that increases in magnitude with r in the neighbourhood of r = 0(d[mid ]h[mid ]/dr > 0); and (iii) force-free fields for which (dh/dr)r=0 = 0, Bθ α rm in the neighbourhood of r = 0, and (h d2h/dr2)r=0 > −128π2/(2m+4)2. On the other hand, the stability does not have a definite relation to the maximum of the force-free factor α defined by [dtri ]×B = αB. Examples will be given to illustrate that force-free fields with an infinite force-free factor at the boundary are stable, whereas those with a force-free factor that is finite and smaller than the lowest eigenvalue of linear force-free field solutions in the domain of interest are unstable. The latter disproves the sufficient criterion for stability of nonlinear force-free magnetic fields given by Krüger [J. Plasma Phys.15, 15 (1976)] that a nonlinear force-free field is stable if the maximum absolute value of the force-free factor is smaller than the lowest eigenvalue of linear force-free field solutions in the domain of interest.