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.
Topological study of active region 11158