Stability of force-free magnetic fields versus magnetic pitch

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, [theta], 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) = 2[pi]rBz/B[theta]. 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 R:]h[mid R:]/dr > 0); and (iii) force-free fields for which (dh/dr)r</em>=0 = 0, B</em>[theta] [alpha] r</em>m</em> in the neighbourhood of r</em> = 0, and (h d</em>2 h</em>/dr</em>2)r=0</em> > [minus sign]128[pi]2/(2m</em>+4)2. On the other hand, the stability does not have a definite relation to the maximum of the force-free factor [alpha] defined by [down triangle, open]×B = [alpha]B. 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.</em> 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.